Out of equilibrium dynamics in classical and

quantum systems

Beg Rohu Summer School

Leticia F. Cugliandolo

Universite Pierre et Marie Curie - Paris VI

Laboratoire de Physique Theorique et Hautes Energies

Institut Universitaire de France

June 28, 2009

June 2009

Contents

1 Introduction 41.1 Falling out of equilibrium . . . . . . . . . . . . . . . . . . . . . . 41.2 Classical glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Driven dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 Out of equilibrium environments . . . . . . . . . . . . . . 8

1.4 Quantum glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 The electron glass . . . . . . . . . . . . . . . . . . . . . . 81.4.2 Charged density waves . . . . . . . . . . . . . . . . . . . . 9

2 The Langevin equation 102.1 Derivation of the Langevin equation . . . . . . . . . . . . . . . . 102.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Irreversibility and dissipation. . . . . . . . . . . . . . . . . 142.2.2 Discretization of stochastic differential equations . . . . . 152.2.3 Generation of memory . . . . . . . . . . . . . . . . . . . . 152.2.4 Smoluchowski (overdamped) limit . . . . . . . . . . . . . 15

2.3 The basic processes . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.1 A constant force . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Relaxation in a quadratic potential . . . . . . . . . . . . . 192.3.3 Thermally activated processes . . . . . . . . . . . . . . . . 21

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3 Dynamics through a phase transition 243.1 Time-dependent Ginzburg-Landau . . . . . . . . . . . . . . . . . 253.2 Relaxation and equilibration time . . . . . . . . . . . . . . . . . . 283.3 Relaxation in equilibrium . . . . . . . . . . . . . . . . . . . . . . 293.4 Critical and subcritical coarsening . . . . . . . . . . . . . . . . . 293.5 Dynamic scaling hypothesis for sub-critical coarsening . . . . . . 303.6 The averaged domain length in subcritical coarsening . . . . . . . 34

3.6.1 Clean one dimensional cases with non-conserved orderparameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6.2 Non-conserved curvature driven dynamics (d > 2) . . . . 343.6.3 Conserved order parameter: bulk diffusion . . . . . . . . 363.6.4 Role of disorder: thermal activation . . . . . . . . . . . . 363.6.5 Temperature-dependent effective exponents . . . . . . . . 38

3.7 Scaling functions for subcritical coarsening . . . . . . . . . . . . . 393.7.1 Breakdown of dynamic scaling . . . . . . . . . . . . . . . 39

3.8 Critical coarsening . . . . . . . . . . . . . . . . . . . . . . . . . . 403.8.1 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.8.2 Crossover between critical and sub-critical coarsening: λ(T ) 40

3.9 Annealing and the Kibble-Zurek mechanism . . . . . . . . . . . . 413.10 An instructive case: the large N approximation . . . . . . . . . . 413.11 Nucleation and growth . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Classical formalism and glassy dynamics 474.1 Classical statics: the reduced partition function . . . . . . . . . . 474.2 Classical dynamics: generating functional . . . . . . . . . . . . . 48

4.2.1 Generic correlation and response. . . . . . . . . . . . . . . 494.2.2 FDT as a consequence of a symmetry . . . . . . . . . . . 514.2.3 Equations on correlations and linear responses . . . . . . 52

4.3 The random manifold . . . . . . . . . . . . . . . . . . . . . . . . 534.3.1 The dynamic equations . . . . . . . . . . . . . . . . . . . 554.3.2 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4 Finite N case: simulations . . . . . . . . . . . . . . . . . . . . . . 634.5 Interacting systems with no quenched disorder . . . . . . . . . . 63

5 Quantum formalism and glassy dynamics 645.0.1 Feynman path integral for quantum time-ordered averages 645.0.2 The Matsubara imaginary-time formalism . . . . . . . . . 655.0.3 The equilibrium reduced density matrix . . . . . . . . . . 65

5.1 Quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 675.1.1 Schwinger-Keldysh path integral . . . . . . . . . . . . . . 675.1.2 Green functions . . . . . . . . . . . . . . . . . . . . . . . . 685.1.3 Generic correlations . . . . . . . . . . . . . . . . . . . . . 705.1.4 Linear response and Kubo relation . . . . . . . . . . . . . 715.1.5 Quantum FDT . . . . . . . . . . . . . . . . . . . . . . . . 71

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5.1.6 The influence functional . . . . . . . . . . . . . . . . . . . 745.1.7 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . 755.1.8 Transformation to ‘MSR-like fields’ . . . . . . . . . . . . . 795.1.9 Classical limit . . . . . . . . . . . . . . . . . . . . . . . . . 805.1.10 A particle or manifold coupled to a fermionic reservoir . 805.1.11 Other baths . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Quantum driven coarsening . . . . . . . . . . . . . . . . . . . . . 845.3 Quantum aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6 Summary 87

A Conventions 89A.1 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 89A.2 Commutation relations . . . . . . . . . . . . . . . . . . . . . . . . 89A.3 Time ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

B The instanton calculation 90

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1 Introduction

1.1 Falling out of equilibrium

In standard condensed matter or statistical physics focus is set on equilib-rium systems. Microcanonical, canonical or grand canonical ensembles are useddepending on the conditions one is interested in. The relaxation of a tiny per-turbation away from equilibrium is also sometimes described in textbooks andundergraduate courses.

More recently, attention has turned to the study of the evolution of similarmacroscopic systems in far from equilibrium conditions. These can be achievedby changing the properties of the environment (e.g. the temperature) in acanonical setting or by changing a parameter in the system’s Hamiltonian in amicrocanonical one. The procedure of rapidly (ideally instantaneously) changinga parameter is called a quench. Right after both types of quenches the initialconfiguration is not one of equilibrium at the new conditions and the systemssubsequently evolve in an out of equilibrium fashion. The relaxation towards thenew equilibrium (if possible) could be fast (and not interesting for our purposes)or it could be very slow (and thus the object of our study). There are plenty ofexamples of the latter including systems quenched through a phase transitionand later undergoing domain growth, and problems with competing interactionsthat behave as glasses.

Out of equilibrium situations can also be established by driving a system,that otherwise would reach equilibrium in observable time-scales, with an exter-nal pertubation. In the context of macroscopic systems an interesting exampleis the one of sheared complex liquids. Yet another interesting case is the oneof powders that stay in static metastable states unless externally perturbed bytapping, vibration or shear that drives them out of equilibrium and makes themslowly evolve towards more compact configurations. But also small systemscan be driven out of equilibrium with external perturbations and the quantumcounterparts of these cases are of special interest at present.

In the study of macroscopic out of equilibrium systems a number of questionsone would like to give an answer to naturally arise. Among these are:

– Is the (instantaneous) structure out of equilibrium similar to the one inequilibrium?

– What kind of relaxation takes place after the quench? Does the systemquickly settle into a stationary state?

– Can one describe the states of the system after the quench with some kindof effective equilibrium-like measure?

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– Are there thermodynamic concepts, such as temperature, entropy, free-energy, playing a role in the non-equilibrium relaxation?

In the last 20 years or so a rather complete theory of the dynamics of classi-cal macroscopic systems evolving slowly out of equilibrium, in a small entropyproduction limit (asymptotic regime after a quench, small drives), that encom-passes the situations described above has been developed. This is a mean-fieldtheory type in the sense that it applies strictly to models with long-range inter-actions or in the infinite dimensional limit. It is, however, proposed that manyaspects of it also apply to systems with short-range interactions although withsome caveats.

In several cases of practical interest, quantum effects play an important role.For instance, glassy phases at very low temperatures have been identified in alarge variety of materials (spin-glass like systems, interacting electrons with dis-order, materials undergoing super-conductor transitions, metallic glasses, etc.).Clearly, the driven case is also very important in systems with quantum fluctu-ations. Take for instance a molecule or an interacting electronic system drivenby an external current applied via the coupling to leads at different chemicalpotential. It is then necessary to settle whether the approach developed and theresults obtained for the classical dynamics in a limit of small entropy productioncarries through when quantum fluctuations are included.

In these notes we start by exposing some examples of the phenomenologyof out of equilibrium dynamics we are interested in. We first describe classicalproblems and their precise setting and we next present some quantum out ofequilibrium situations. Next we go into the formalism used to deal with theseproblems. The basic techniques used to study classical glassy models with orwithout disorder are well documented in the literature (the replica trick, scalingarguments and droplet theories, the dynamic functional method used to de-rive macroscopic equations from the microscopic Langevin dynamics, functionalrenormalization, Montecarlo and molecular dynamic numerical methods). Onthe contrary, the techniques needed to deal with the statics and dynamics ofquantum macroscopic systems are much less known in general. I shall brieflydiscuss the role played by the environment in a quantum system and introduceand compare the equilibrium and dynamic approaches.

Concretely, we recall some features of the Langevin formalism and its gen-erating function that bears a strong resemblance with the Schwinger-Keldyshpath-integral formulation of the quantum real-time dynamics. We dwell initiallywith some emblematic aspects of classical macroscopic systems slowly evolvingout of equilibrium that will be useful to understand the quantum cases.

Concerning models, we focus on two, that are intimately related: the O(N)model in the large N limit that is used to describe coarsening phenomena, andthe random manifold, that finds applications to many physical problems likecharge density waves, high-Tc superconductors, etc. Both problems are of field-theoretical type and can be treated both classically and quantum mechanically.These two models are ideal for the purpose of introducing and discussing for-

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malism and some basic ideas we would wish to convey in these lectures.

1.2 Classical glasses

Without entering into the rich details of the glass phenomenology we justmention here that their key feature is that below some characteristic temper-ature Tg, or above a critical density ρg, the relaxation time goes beyond theexperimentally accessible time-scales and the system is next bound to evolveout of equilibrium. Although the mechanism leading to such a slow relaxationis unknown – and might be different in different cases – the out of equilibriumrelaxation presents very similar properties. The left panel in Fig. 1 shows oneaspect of glassy dynamics, aging, as shown by the two-time relation of the self-correlation of a colloidal suspension [28], that is remarkably similar to the decayof the magnetic correlation in the Ising model shown in the right panel and inFig. 7.

There are many types of glasses and they occur over an astounding range ofscales from macroscopic to microscopic. Macroscopic examples include granularmedia like sand and powders. Unless fluidized by shaking or during flow thesequickly settle into jammed, amorphous configurations. Jamming can also becaused by applying stress, in response to which the material may effectivelyconvert from a fluid to a solid, refusing further flow. Temperature (and ofcourse quantum fluctuations as well) is totally irrelevant for these systems sincethe grains are typically big, say, of 1mm radius. Colloidal suspensions containsmaller (typically micrometre-sized) particles suspended in a liquid and formthe basis of many paints and coatings. Again, at high density such materialstend to become glassy unless crystallization is specifically encouraged (and caneven form arrested gels at low densities if attractive forces are also present).On smaller scales still, there are atomic and molecular glasses: window glass isformed by quick cooling of a silica melt, and of obvious everyday importance.The plastics in drink bottles and the like are also glasses produced by cooling,the constituent particles being long polymer molecules. Critical temperaturesare of the order of 80C for, say, PVC and these systems are glassy at roomtemperature. Finally, on the nanoscale, glasses are also formed by vortex linesin type-II superconductors. Atomic glasses with very low critical temperature,of the order of 10mK, have also been studied in great detail.

1.3 Driven dynamics

An out of equilibrium situation can be externally maintained by applyingforces and thus injecting energy into the system and driving it. There are severalways to do this and we explain below two quite typical ones that serve also astheoretical tradition examples.

6

0.14

0.12

0.10

0.08

0.06

0.04

0.02

|g1(

t w,t)

|2

0.01 0.1 1 10 100 1000t (sec)

twVarious shear histories

a)

b)

0.1

1

1 10 100 1000

C(t

,t w)

t-tw

tw=248

163264

128256512

Figure 1: Left: two-time evolution of the self-correlation in a colloidal suspensioninitialized by applying a shearing rate [28]. The longer the waiting time theslower the decay. Right: two-time evolution in the bi-dimensional Ising modelquenched below its phase transition at Tc. A two-scale relaxation with a clearplateau at a special value of the correlation is seen in the double logarithmicscale. We shall discuss this feature at length in the lectures.

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1.3.1 Rheology

Rheological measurements are common in soft condensed matter; they con-sist in driving the systems out of equilibrium by applying an external force thatdoes not derive from a potential (e.g. shear, shaking, etc.). The dynamics ofthe system under the effect of such a strong perturbation is then monitored.

The effect of shear on domain growth is one of great technological and the-oretical importance. The growth of domains is anisotropic and the there mightbe different growing lengths in different directions. Moroever, it is not clearwhether shear might interrupt growth altogether giving rise to stationary stateor whether coarsening might continue for ever.

1.3.2 Out of equilibrium environments

Another setting is to couple the system to different external reservoirs allin equilibrium but at different temperature or chemical potential thus inducinga heat or a particle current through the system. This set-up is relevant toquantum situations in which one can couple a system to, say, a number of leadsat different chemical potential.

1.4 Quantum glasses

A number of interacting systems at very low temperature in which quantumfluctuations cannot be neglected are currently being studied experimentally.Two examples of different nature are discussed below. In the first model, quan-tum effects are at the origin of a peculiar effective classical stochastic modelwith quenched disorder. In the second case, an elastic field theory is developedto describe charge density waves and Wigner crystals and this theory can betreated both classically and quantum mechanically.

1.4.1 The electron glass

Recent experimental studies of hopping conductance in Anderson insulatorsshowed striking non-equilibrium effects that persist for long times at low temper-ature. These results support early theoretical predictions of a low-temperatureglassy phase in interacting disordered electron systems in the strongly localizedlimit. In this regime the essential physics is well captured by the Coulombglass model that describes a system of interacting electrons hopping betweenrandomly distributed sites that correspond to the localization centers of thesingle-electron wavefunctions. The statics of Coulomb glass models has beenextensively studied during the last thirty years and more recently attention hasbeen payed to their out of equilibrium relaxation.

The low-T hopping conductance is dominated by the diffusion of carriers ona set of conducting percolation paths that is altered by the hops of individualcarriers that change the random potential felt by the rest. Within any chosen

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time-interval electrons separate, basically, in two groups, the ones that diffuseaway from their initial position and the ones that remain confined in relativelysmall regions in space. After a quench the excess of eletrons with metallichopping decreases logarithmically.

The Hamiltonian of the classical Coulomb glass is

H =∑

i

niϕi +e2

2κ

∑

i6=j

(ni − κ)(nj − κ)

|~Ri − ~Rj |(1)

where ~Ri is the center of localization of a single-particle localized electronicstate, ϕi is the energy of the state, e and κ are the electron charge and themedium’s dielectric constant, ni = 0, 1 are the occupancy of the electronicstates. NK =

∑Ni ni ensures charge neutrality and one usually takes K = 1/2

the particle-hole symmetric case. ~Ri and ϕi are random variables. One canstudy a lattice model in which only ϕi are random or a random site model inwhich ϕi = 0 and the positions are random.

The transition rates that model phonon assisted processes are

Γa→b = τ−10 e−2Rab/ξ min[1, e−∆EAb/(kBT )] (2)

with τ0 a microscopic time, ξ the spatial extension of the localized wave-function,Rab = |~Ra − ~Rb| and ∆Eab the total energy difference. The first term reflectsthe exponential decay of electron-phonon matrix element between two electronicwave functions and it contributes to slow down the dynamics since elctron hopsthat decrease the energy are accepted only when they take place between sitesthat are closer than the localization length ξ.

The T -dependence of the relaxation time in the model defined above is shownin Fig. 3.

1.4.2 Charged density waves

See [7] and references therein.

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2 The Langevin equation

Examples of experimental and theoretical interest in condensed matter andbiophysics in which quantum fluctuation can be totally neglected are manifold.In this context one usually concentrates on systems in contact with an environ-ment: one selects some relevant degrees of freedom and treats the rest as a bath.It is a canonical view. Among these instances are colloidal suspensions which areparticles suspended in a liquid, typically salted water, a ‘soft condensed matter’example; spins in ferromagnets coupled to lattice phonons, a ‘hard condensedmatter’ case; and proteins in the cell a ‘biophysics’ instance. These problems aremodelled as stochastic processes with Langevin equations, the Kramers-Fokker-Planck formalism or master equations depending on the continuous or discretecharacter of the relevant variables and analytic convenience.

The Langevin equation is a stochastic differential equation that describesphenomenologically a large variety of problems. It models the time evolutionof a set of slow variables coupled to a much larger set of fast variables that areusually (but not necessarily) assumed to be in thermal equilibrium at a giventemperature. We first introduce it in the context of Brownian motion and wederive it in more generality in Sect. .

The Langevin equation1 for a particle moving in one dimension in contactwith a white-noise bath reads

mv + γ0v = F + ξ , v = x , (3)

with x and v the particle’s position and velocity. ξ is a Gaussian white noisewith zero mean and correlation 〈ξ(t)ξ(t′)〉 = 2γ0kBTδ(t− t′) that mimics ther-mal agitation. γ0v is a friction force that opposes the motion of the particle.The force F designates all external deterministic forces and depends, in general,on the position of the particle x only. In cases in which the force derives from apotential, F = −dV/dx. The generalization to higher dimensions is straightfor-ward. Note that γ0 is the parameter that controls the strength of the couplingto the bath (it appears in the friction term as well as in the noise term). In thecase γ0 = 0 one recovers Newton equation of motion. The relation between thefriction term and thermal correlation is non-trivial, as we shall discuss below.

2.1 Derivation of the Langevin equation

Let us take a system in contact with an environment. The interacting sys-tem+environment ensemble is ‘closed’ while the system is ‘open’. The nature

1P. Langevin, Sur la theorie du mouvement brownien, Comptes-Rendus de l’Academie desSciences 146 (1908), 530-532.

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of the environment, e.g. whether it can be modeled by a classical or a quantumformalism, depends on the problem under study. A derivation of a generalizedLangevin equation with memory is very simple starting from Newton dynamicsof the full system [38, 30]. We shall then study the coupled system

Htot = Hsyst +Henv +Hint +Hcounter . (4)

For simplicity we focus on a single particle moving in d = 1: Hsyst is theHamiltonian of the isolated particle, Hsyst = p2/(2M) + V (x), with p and x itsmomentum and position. Henv is the Hamiltonian of a thermal bath that, forsimplicity, we take to be an ensemble of N independent Harmonic oscillatorswith masses ma and frequencies ωa, a = 1, . . . , N

Henv =

N∑

a=1

π2a

2ma+maω

2a

2q2a (5)

with πa and qa their momenta and positions. This is indeed a very usual choicesince it may represent phonons. Hint is the coupling between system and envi-ronment. We shall restrict the following discussion to a linear interaction in theoscillator coordinates, qa, and in the particle coordinate, Hint = x

∑Nba=1 caqa,

with ca the coupling constants. The calculations can be easily generalized toan interaction with a more complicated dependence on the system’s coordinate,V(x), that may be dictated by the symmetries of the system. We shall discussthe last term, Hcounter, below. The generalization to more complex systemsand/or to more complicated baths and higher dimensions is straightforward.

Hamilton’s equations for the particle are

x(t) =p(t)

m, p(t) = −V ′[x(t)] −

N∑

a=1

c2amaω2

a

x(t) −N∑

a=1

caqa(t) , (6)

while the dynamic equations for each member of the environment read

qa(t) =πa(t)

ma, πa(t) = −maω

2aqa(t) − cax(t) , (7)

showing that they are all massive harmonic oscillators forced by the chosenparticle. These equations are readily solved yielding

qa(t) = qa(0) cos(ωat) +πa(0)

maωasin(ωat) −

camaωa

∫ t

0

dt′ sin[ωa(t− t′)]x(t′) (8)

with qa(0) and πa(0) the initial coordinate and position at time t = 0 when theparticle is set in contact with the bath. It is convenient to integrate by partsthe last term. The replacement of the resulting expression in the last term inthe rhs of eq. (6) implies

p(t) = −V ′[x(t)] + ξ(t) −∫ t

0 dt′ Γ(t− t′)x(t′) , (9)

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with the symmetric and stationary kernel Γ given by

Γ(t− t′) =∑N

a=1c2a

maω2a

cos[ωa(t− t′)] , (10)

Γ(t− t′) = Γ(t′ − t), and the time-dependent force ξ given by

ξ(t) = −∑Na=1 ca

[

πa(0)maωa

sin(ωat) +(

qa(0) + cax(0)maω2

a

)

cos(ωat)]

. (11)

This is the equation of motion of the reduced system.For a non-linear coupling the Langevin equation reads

p(t) = −V ′[x(t)] + ξ(t)V ′[x(t)] −∫ t

0

dt′ Γ(t− t′)V ′[x(t′)]x(t′) (12)

with the same Γ as in eq. (10) and ξ(t) given by eq. (11) with x(0) → V [x(0)].The noise appears now multiplying a function of the particles’ coordinate.

The third term on the rhs of eq. (9) represents a rather complicated frictionforce. Its value at time t depends explicitly on the history of the particle attimes 0 ≤ t′ ≤ t and makes the equation non-Markovian. One can rewrite it asan integral running up to a total time T > max(t, t′) introducing the retardedfriction:

γ(t− t′) = Γ(t− t′)θ(t− t′)] . (13)

Until this point the dynamics of the system remain deterministic and arecompletely determined by its initial conditions as well as those of the reservoirvariables. The statistical element comes into play when one realizes that it isimpossible to know the initial configuration of the large number of oscillatorswith great precision and one proposes that the initial coordinates and momentaof the oscillators have a canonical distribution at an inverse temperature β.Then, πa(0), qa(0) are initially distributed according to

P (πa(0), qa(0)) = Z−1env e

−βHenv[πa(0),qa(0)] (14)

with Henv = Henv +Hcounter +Hint, that can be rewritten as

Henv =N∑

a=1

[

maω2a

2

(

qa(0) +ca

maω2a

x(0)

)2

+π2a(0)

2ma

]

. (15)

The randomness in the initial conditions gives rise to a random force acting onthe reduced system. Indeed, ξ is now a Gaussian random variable, that is tosay a noise, with

〈ξ(t)〉 = 0 , 〈ξ(t)ξ(t′)〉 = kBT Γ(t− t′) . (16)

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One can easily check that higher-order correlations vanish for an odd number ofξ factors and factorize as products of two time correlations for an even numberof ξ factors. In consequence ξ has Gaussian statistics. Defining the inverse ofΓ over the interval [0, t],

∫ t

0dt′′ Γ(t − t′′)Γ−1(t′′ − t′) = δ(t − t′), one has the

Gaussian pdf:

P [ξ] = Z−1e− 1

2kBT

∫

t

0dt∫

t

0dt′ ξ(t)Γ−1(t−t′)ξ(t′)

. (17)

Z is the normalization. A random force with non-vanishing correlations on afinite support is usually called a coloured noise. Equation (9) is now a gen-uine Langevin equation. A multiplicative retarded noise arises from a model inwhich one couples the coordinates of the oscillators to a generic function of thecoordinates of the system, see eq. (12).

The use of an equilibrium measure for the oscillators implies the invarianceunder time translation of the friction kernel, Γ(t − t′), and its relation to thenoise-noise correlation, which are proportional, with a constant of proportion-ality of value kBT . This is a generalized form of the fluctuation-dissipationrelation, and it applies to the environment.

Different choices of the environment are possible by selecting different en-sembles of harmonic oscillators. The simplest one, that leads to an approximateMarkovian equation, is to consider that the oscillators are coupled to the par-ticle via coupling constants ca = ca/

√N with ca of order one. In the limit

N → ∞ one defines

S(ω) ≡ 1N

∑Na=1

c2amaωa

δ(ω − ωa) (18)

and rewrites the kernel Γ as

Γ(t− t′) =∫∞

0dω S(ω)

ω cos[ω(t− t′)] . (19)

A common choice is

S(ω)ω = 2γ0

(

|ω|ω

)α−1

fc

(

|ω|Λ

)

. (20)

The function fc(x) is a high-frequency cut-off of typical width Λ and is usuallychosen to be an exponential. The frequency ω ≪ Λ is a reference frequency thatallows one to have a coupling strength γ0 with the dimensions of viscosity. Ifα = 1, the friction is said to be Ohmic, S(ω)/ω is constant when |ω| ≪ Λ as fora white noise. When α > 1 (α < 1) the bath is superOhmic (subOhmic). Theexponent α is taken to vary in the interval (0, 2) to avoid divergencies. For theexponential cut-off the integral over ω yields

Γ(t) = 2γ0ω−α+1 cos[α arctan(Λt)]

[1 + (Λt)2]α/2Γa(α) Λα (21)

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with Γa(x) the Gamma-function, that in the Ohmic case α = 1 reads

Γ(t) = 2γ0Λ

[1 + (Λt)2], (22)

and in the Λ → ∞ limit becomes a delta-function, Γ(t) → 2γ0δ(t). At longtimes, for any α ∈ (0, 2), one has

limΛt→∞

Γ(t) = 2γ0ω−α+1 cos(απ/2)Γa(α) Λ−1 t−α−1 (23)

a power law decay.Time-dependent, f(t), and constant non-potential forces, fnp, as the ones

applied to granular matter and in rheological measurements, respectively, aresimply included in the right-hand-side (rhs) as part of the deterministic force.When the force derives from a potential, F (x, t) = −dV/dx.

In so far we have discussed systems with position and momentum degreesof freedom. Other variables might be of interest to describe the dynamics ofdifferent kind of systems. In particular, a continuous Langevin equation forclassical spins can also be used if one replaces the hard Ising constraint, si = ±1,by a soft one implemented with a potential term of the form V (si) = u(s2i − 1)2

with u a coupling strength (that one eventually takes to infinity to recovera hard constraint). The soft spins are continuous unbounded variables, si ∈(−∞,∞), but the potential energy favors the configurations with si close to±1. Even simpler models are constructed with spherical spins, that are alsocontinuous unbounded variables globally constrained to satisfy

∑Ni=1 s

2i = N .

The extension to fields is straightforward and we shall discuss one when dealingwith the O(N) model.

Another derivation of the Langevin equation uses collision theory and admitsa generalization to relativistic cases [39].

2.2 Properties

We briefly discuss here some important properties of the Langevin equation.

2.2.1 Irreversibility and dissipation.

The friction force −γ0v in eq. (3) – or its retarded extension in the non-Markovian case – explicitly breaks time-reversal (t → −t) invariance, a propertythat has to be respected by any set of microscopic dynamic equations. Newtonequation describing the whole system, the particle and all the molecules of thefluid, are time reversal invariant. However, time-reversal can be broken in thereduced equation in which the fluid is treated in an effective form and the factthat it is in equilibrium is assumed from the start.

Even in the case in which all forces derive from a potential, F = −dV/dx,the energy of the particle, mv2/2+V , is not conserved and, in general, flows to

14

the bath leading to dissipation. At very long times, however, the particle mayreach a stationary regime in which the particle gives and receives energy fromthe bath at equal rate, on average.

2.2.2 Discretization of stochastic differential equations

The way in which a stochastic differential equation with white noise is to bediscretized is a subtle matter that we shall not discuss in these lectures, unlesswhere it will be absolutely necessary. There are basically two schemes, calledthe Ito and Stratonovich calculus, that are well documented in the literature.

In short, we shall use the Ito prescription in which the pair velocity-positionof the particle at time t + δ, with δ an infinitesimal time-step, depends on thepair velocity-position at time t and the value of the noise at time t. Thus, thefull set of equations defines a Markov process, that is a stochastic process thatdepends on its history only through its very last step.

2.2.3 Generation of memory

The Langevin equation (3) is actually a set of two first order differentialequations. Notice, however, that the pair of first-order differential equationscould also be described by a single second-order differential equation:

mx+ γ0x = F + ξ . (24)

Having replaced the velocity by its definition in terms of the position the Markovcharacter of the process is lost. This is a very general feature: by integratingaway some degrees of freedom (the velocity in this case) one generates memoryin the evolution. Generalizations of the Langevin equation, such as the one thatwe have just presented, and the ones that will be generated to describe the slowevolution of super-cooled liquids and glasses in terms of correlations and linearresponses, do have memory.

2.2.4 Smoluchowski (overdamped) limit

In many situations in which friction is very large, the characteristic time forthe relaxation of the velocity degrees of freedom to their Maxwellian distribu-tion, tvr , is very short. In consequence, observation times are very soon longerthan this time-scale, the inertia term mv can be dropped, and the Langevinequation becomes

γ0x = F + ξ . (25)

Indeed, this overdamped limit is acceptable whenever the observation times aremuch longer than the characteristic time for the velocity relaxation. Inversely,the cases in which the friction coefficient γ0 is small are called underdamped.

15

In the overdamped limit with white-noise the friction coefficient γ0 can beabsorbed in a rescaling of time. One defines the new time τ

t = γ0τ (26)

the new position, x(τ) = x(γ0τ), and the new noise η(τ) = ξ(γ0τ). In the newvariables the Langevin equation reads ˙x(τ) = F (x, τ) + η(τ) with 〈η(τ)η(τ ′)〉 =2kBTδ(τ − τ ′).

2.3 The basic processes

We shall discuss the motion of the particle in some 1d representative poten-tials: under a constant force, in a harmonic potential, in the flat limit of thesetwo (Fig. 2) and the escape from a metastable state and the motion in a doublewell potential (Fig. 3).

x

V

x

V

x

V

Figure 2: Sketches of three representative one-dimensional potentials.

2.3.1 A constant force

Let us first consider the case of a constant force, F , and study the evolutionof its velocity and position to show that these variables behave very differently.

The velocity

The time-dependent velocity follows from the integration of eq. (3) over time

v(t) = v0 e−γ0mt +

1

m

∫ t

0

dt′ e−γ0m

(t−t′) [F + ξ(t′) ] , v0 ≡ v(t = 0) .

The velocity is a Gaussian variable that inherits its average and correlationsfrom the ones of ξ. Using the fact that the noise has zero average

〈v(t)〉 = v0 e−γ0mt +

F

γ0

(

1 − e−γ0mt)

→ F

γ0when t≫ tvr ≡ m

γ0.

16

For all initial conditions v0 the averaged velocity decays exponentially with arelaxation time tvr = m/γ0.

The velocity mean-square displacement is

〈(v(t) − 〈v(t)〉)2〉 = kBTm

(

1 − e−2γ0mt)

→ kBTm (Equipartition) (27)

independently of F . This asymptotic limit is the result expected from equipar-tition of the kinetic energy, 〈(v(t) − 〈v(t)〉)2〉 → 〈(v − 〈v〉2eq〉eq that implies forthe kinetic energy 〈K〉eq = kBT/2. In the heuristic derivation of the Langevinequation the amplitude of the noise-noise correlation, say A, is not fixed. Thesimplest way to determine this parameter is to require that equipartition forthe kinetic energy holds A/(γ0m) = T/m and hence A = γ0T . This relation isknown under the name of fluctuation–dissipation theorem (fdt) of the secondkind in Kubo’s nomenclature. It is important to note that this fdt character-izes the surrounding fluid and not the particle, since it relates the noise-noisecorrelation to the friction coefficient. In the case of the Brownian particle thisrelation ensures that after a transient of the order of tvr , the bath maintains themean kinetic energy of the particle constant and equal to its equilibrium value.The Gaussian distribution of the noise and the linear relation linking it to thevelocity imply that the velocity of the particle is indeed distributed accordingto Maxwell’s distribution (actually an extension of it that takes care of the factthat 〈v(t)〉 → F/γ0), with the form P (v, t) → exp[−β(v − γ−1

0 F )2/(2m)]. Weshall see later that even when the environment satisfies (fdt) the system incontact with it does not necessarily follow and satisfy it itself.

The velocity two-time connected correlation reads

〈[v(t)−〈v(t)〉][v(t′)−〈v(t′)〉〉 =kBT

m

[

e−γ0m

|t−t′| − e−γ0m

(t+t′)]

→ kBT

me−

γ0m

|t−t′| .

This is sometimes called the Dirichlet correlator. This and all other higher-ordervelocity correlation functions approach a stationary limit when the shortest timeinvolved is longer than tvr . In particular, at t = t′ on recovers the mean-squaredisplacement computed above.

The position

The particle’s position has a very different behaviour. A simple calculationyields

x(t) = x0 + v0 tvr +

F

γ0(t− tvr) +

1

m

∫ t

0

dt′ e−γ0m

(t−t′) ξ(t′) (28)

and then

〈x(t)〉 → x0 + v0 tvr +

F

γ0(t− tvr) (29)

for t≫ tvr (with exponentially decaying corrections at shorter times). Note thereduction with respect to ballistic motion (x ∝ Ft2) due to the friction drag.

17

The position mean-square displacement approaches

〈(x(t) − 〈x(t)〉)2〉 → 2 kBTγ0

t (Normal diffusion) (30)

in the usual t ≫ tvr limit, that is to say normal diffusion with the diffusionconstant Dx = kBT/γ0. This expression can be computed using x(t)−〈x(t)〉 asobtained from the v(t)−〈v(t)〉 above (and it is quite a messy calculation) or onecan simply go to the Smoluchowski limit, taking advantage of the knowledge ofwhat we have just discussed on the behaviour of velocities, and obtain diffusionin two lines. Similarly, the particle displacement between two different times tand t′ is

∆(t, t′) ≡ 〈[x(t) − 〈x(t)〉][x(t′) − 〈x(t′)〉]〉 → 2kBT

γ0|t− t′| . (31)

It is interesting to note that the force dictates the mean position but it does notmodify the fluctuations about it.

The averaged potential energy diverges in the long-time limit since the po-tential is unbounded in the x→ ∞ limit: 〈H〉 = kBT/2−F 〈x(t)〉 → −∞ whent→ ∞.

Two kinds of variables

This example shows that even in this very simple problem the velocity andposition variables have distinct behaviour: the former is in a sense trivial, afterthe transient tvr and for longer times, all one-time functions of v−F/γ0 saturateto their equilibrium values, the correlations are stationary and so on; instead,the latter remains non-trivial and evolving out of equilibrium. One can looselyascribe the different behaviour to the fact that the velocity feels a confiningpotential K = mv2/2 while the position feels an unbounded potential V = −Fxin the case in which a force is applied, or a flat potential V = 0 if F is switchedoff. In none of these cases the potential is able to take the particle to equilibriumwith the bath. The particle slides on the lope and its excursions forward andbackward from the mean get larger and larger as time increases.

Quite generally, the classical problems we are interested in are such that thefriction coefficient γ0 is large and the inertia term can be neglected. We shalldo it in the rest of the lectures.

Ergodicity

The ergodic hypothesis states that, in equilibrium, one can exchange ensem-ble averages by time averages and obtain the same results. Out of equilibriumthis hypothesis is not expected to hold and one can already see how dangerousit is to take time-averages in these cases by focusing on the simple velocity vari-able. Ensemble and time averages coincide if the time-averaging is done overa time-window that lies after tvr but it does not if the integration time-intervalgoes below tvr .

18

2.3.2 Relaxation in a quadratic potential

Another relevant example is the relaxation of a particle in a harmonic po-tential, with its minimum at x∗ 6= 0:

V (x) =k

2(x− x∗)2 . (32)

This problem can be solved exactly for all values of γ0 but the calculation isslightly tedious. The behaviour of the particle velocity has already been clarifiedin the constant force case. We now focus on the overdamped limit,

γ0x = −k(x− x∗) + ξ , (33)

with k the spring constant of the harmonic well, that can be readily solved,

x(t) = x0 e− kγ0t+ γ−1

0

∫ t

0

dt′ e− kγ0

(t−t′)[ξ(t′) + kx∗] , x0 = x(0) . (34)

This problem become formally identical to the velocity dependence in the pre-vious example.

Convergence of one-time quantities

The averaged position is

〈x(t)〉 = x0e− kγ0t → x∗ txr ≫ γ0/k (Convergence) (35)

Of course, one-time quantities should approach a constant asymptotically if thesystem equilibrates with its environment.

Two-time quantities

The two-time connected correlation (where one extracts, basically, the asymp-totic position x∗) reads

〈δx(t)δx(t′)〉 = x20 e

− kγ0

(t+t′) + kBT k−1 e−kγ0

(t+t′)[

e2kγ0

min(t,t′) − 1]

. (36)

Again, the Dirichlet correlator. For at least one of the two times going well be-yond the position relaxation time txr = γ0/k the memory of the initial conditionis lost and the connected correlation becomes stationary:

Cc(t, t′) = 〈δx(t)δx(t′)〉 → kBT k−1 e−

kγ0

|t−t′| . (37)

Note that when the time-difference t − t′ diverges the average of the productfactorizes, in particular, for the correlation one gets

〈x(t)x(t′)〉 → 〈x(t)〉〈x(t′)〉 → x∗〈x(t′)〉 (38)

19

for any t′, even finite. We shall see this factorization property at work later inmore complicated cases.

More generally one can show that for times t1 ≥ t2 ≥ . . . ≥ tn ≥ txr :

〈x(t1 + ∆) . . . x(tn + ∆)〉 = 〈x(t1) . . . x(tn)〉 (TTI) (39)

for all delays ∆. Time-translation invariance (TTI) or stationarity is one genericproperty of equilibrium dynamics. Another way of stating (39) is

〈x(t1) . . . x(tn)〉 = f(t1 − t2, . . . , tn−1 − tn) . (40)

Fluctuation-dissipation theorem (FDT)

One can also compute the linear response to an infinitesimal perturbationthat couples linearly to the position changing the energy of the system as H →H − fx at a given time t′:

R(t, t′) =δ〈x(t)〉fδf(t′)

∣

∣

∣

∣

f=0

. (41)

The explicit calculation yields

R(t, t′) = γ−10 e−kγ

−1

0(t−t′) θ(t− t′)

R(t, t′) = 1kBT

∂Cc(t,t′)

∂t′ θ(t− t′) (FDT) (42)

The last equality holds for times that are longer than txr . It expresses thefluctuation-dissipation theorem (fdt), a model-independent relation betweenthe two-time linear response and correlation function. Similar - though morecomplicated - relations for higher-order responses and correlations also exist inequilibrium. There are many ways to prove the fdt for stochastic processes. Weshall discuss one of them in Sect. that is especially interesting since it applieseasily to problems with correlated noise.

It is instructive to examine the relation between the linear response and thecorrelation function in the limit of a flat potential (k → 0). The linear responseis just γ−1

0 θ(t− t′). The Dirichlet correlator approaches the diffusive limit:

〈δx(t)δx(t′)〉 = 〈x(t)x(t′)〉 → x20 + 2γ−1

0 kBT min(t, t′) for k → 0 (43)

and its derivative reads ∂t′〈δx(t)δx(t′)〉 = 2γ−10 kBT θ(t− t′). Thus,

R(t, t′) =1

2kBT∂t′〈δx(t)δx(t′)〉 θ(t− t′)

R(t, t′) = 12kBT

∂t′Cc(t, t′) θ(t− t′) (FDR for diffusion) (44)

20

A factor 1/2 is now present in the relation between R and Cc. It is anothersignal of the fact that the coordinate is not in equilibrium with the environment.

Reciprocity or Onsager relations

Let us compare the two correlations 〈x3(t)x(t′)〉 and 〈x3(t′)x(t)〉 within theharmonic example. One finds 〈x3(t)x(t′)〉 = 3〈x2(t)〉〈x(t)x(t′)〉 and 〈x3(t′)x(t)〉 =3〈x2(t′)〉〈x(t′)x(t)〉. Given that 〈x2(t)〉 = 〈x2(t′)〉 → 〈x2〉eq and the fact thatthe two-time self-correlation is symmetric,

〈x3(t)x(t′)〉 = 〈x3(t′)x(t)〉 . (45)

With a similar argument one shows that for any functions A and B of x:

〈A(t)B(t′)〉 = 〈A(t′)B(t)〉CAB(t, t′) = CAB(t′, t) (Reciprocity) (46)

This equation is known as Onsager relation and applies to A and B that areeven under time-reversal (e.g. they depend on the coordinates but not on thevelocities or they have an even number of verlocities).

All these results remain unaltered if one adds a linear potential −Fx andworks with connected correlation functions.

2.3.3 Thermally activated processes

The phenomenological Arrhenius law 2 yields the typical time needed toescape from a potential well as an exponential of the ratio between the heightof the barrier and the thermal energy scale kBT , (with prefactors that can becalculated explicitly, see below). This exponential is of crucial importance forunderstanding slow (glassy) phenomena, since a mere barrier of 30kBT is enoughto transform a microscopic time of 10−12s into a macroscopic time scale. SeeFig. 3-right for a numerical study of the Coulomb glass that demonstrates theexistence of an Arrhenius time-scale in this problem. In the glassy literaturesuch systems are called strong glass formers as opposed to weak ones in whichthe characteristic time-scale depends on temperature in a different way.

In 1940 Kramers estimated the escape rate from a potential well as the oneshown in Fig. 3-center due to thermal fluctuations that give sufficient energyto the particle to allow it to surpass the barrier. After this seminal paper thisproblem has been studied in great detail [39] given that it is of paramountimportance in many areas of physics and chemistry. An example is the problemof the dissociation of a molecule where x represents an effective one-dimensionalreaction coordinate and the potential energy barrier is, actually, a free-energybarrier.

2S. A. Arrhenius, On the reaction velocity of the inversion of cane sugar by acids,Zeitschrift fr Physikalische Chemie 4, 226 (1889).

21

x

V

x

VFigure 3: Left: sketch of a double-well potential. Center: sketch of a potentialwith a local minimum. Right: correlation function decay in a classical model ofthe 3d Coulomb glass at nine temperatures ranging from T = 0.1 to T = 0.05in steps of 0.05 and all above Tg. In the inset the scaling plot C(t) ∼ f(t/tA)with a characteristic time-scale, tA, that follows the Arrhenius activated law,tA ≃ 0.45/T . Figure taken from [22].

Kramers assumed that the reaction coordinate is coupled to an equilibratedenvironment with no memory and used the probability formalism in which theparticle motion is described in terms of the time-dependent probability den-sity P (x, v, t) (that for such a stochastic process follows the Kramers partialdifferential equation).

If the thermal energy is at least of the order of the barrier height, kBT ∼ ∆V ,the reaction coordinate, x, moves freely from the vicinity of one well to thevicinity of the other.

The treatment we discuss applies to the opposite weak noise limit in whichthe thermal energy is much smaller than the barrier height, kBT ≪ ∆V , therandom force acts as a small perturbation, and the particle current over the topof the barrier is very small. Most of the time x relaxes towards the minimum ofthe potential well where it is located. Eventually, the random force drives it overthe barrier and it escapes to infinity if the potential has the form in Fig. 3-center,or it remains in the neighbourhood of the second well, see Fig. 3-left.

The treatment is simplified if a constant current can be imposed by injectingparticles within the metastable well and removing them somewhere to the rightof it. In these conditions Kramers proposed a very crude approximation wherebyP takes the stationary canonical form

Pst(x, v) = N e−βv2

2−βV (x) . (47)

If there is a sink to the right of the maximum, the normalization constant N isfixed by further assuming that Pst(x, v) ∼ 0 for x ≥ x > xmax. The resulting

22

integral over the coordinate can be computed with a saddle-point approximationjustified in the large β limit. After expanding the potential about the minimumand keeping the quadratic fluctuations one finds

N−1 =2π

β√

V ′′(xmin)e−βV (xmin) .

The escape rate, r, over the top of the barrier can now be readily computed bycalculating the outward flow across the top of the barrier:

r ≡ 1

tA≡∫ ∞

0

dv vP (xmax, v) =

√

V ′′(xmin)

2πe−β(V (xmax)−V (xmin)) . (48)

Note that we here assumed that no particle comes back from the right of thebarrier. This assumption is justified if the potential quickly decreases on theright side of the barrier.

The crudeness of the approximation (47) can be grasped by noting thatthe equilibrium form is justified only near the bottom of the well. Kramersestimated an improved Pst(x, v) that leads to

r =

(

γ2

4 + V ′′(xmax))1/2

− γ2

√

V ′′(xmax)

√

V ′′(xmin)

2πe−β(V (xmax)−V (xmin)) . (49)

This expression approaches (48) when γ ≪ V ′′(xmax), i.e. close to the under-damped limit, and

r =

√

V ′′(xmax)V ′′(xmin)

2πγe−β(V (xmax)−V (xmin)) (50)

when γ ≫ V ′′(xmax), i.e. in the overdamped limit (see Sect. for the definitionof these limits).

The inverse of (49), tA, is called the Arrhenius time needed for thermal acti-vation over a barrier ∆V ≡ V (xmax)−V (xmin). The prefactor that characterisesthe well and barrier in the harmonic approximation is the attempt frequencywith which the particles tend to jump over the barrier. In short,

tA ≃ τ eβ|∆V | (Arrhenius time) (51)

The one-dimensional reaction coordinate can be more or less easily identifiedin problems such as the dissociation of a molecule. In contrast, such a singlevariable is much harder to visualize in an interacting problem with many degreesof freedom. The Kramers problem in higher dimensions is highly non-trivial and,in the infinite-dimensional phase-space, is completely out of reach.

23

3 Dynamics through a phase transition

Take a piece of material in contact with an external reservoir. The materialwill be characterized by certain observables, energy density, magnetization den-sity, etc. The external environment will be characterized by some parameters,like the temperature, magnetic field, pressure, etc. In principle, one is able totune the latter and study the variation of the former. Note that we are using acanonical setting in the sense that the system under study is not isolated butopen.

Sharp changes in the behavior of macroscopic systems at critical points (orlines) in parameter space have been observed experimentally. These correspondto equilibrium phase transitions, a non-trivial collective phenomenon appearingin the thermodynamic limit. We shall assume that the main features of, andanalytic approaches used to study, such phase transitions are known.

Imagine now that one changes an external parameter instantaneously orwith a finite rate going from one phase to another in the (equilibrium) phasediagram. The kind of internal system interactions are not changed. In thestatistical physics language the first kind of procedure is called a quench and thesecond one an annealing and these terms belong to the metalurgy terminology.We shall investigate how the system evolves by trying to accomodate to thenew conditions and equilibrate with its environment. We shall first focus on thedynamics going through phase transitions between well-known phases (in thesense that one knows the order parameter, the structure, and all thermodynamicproperties on both sides of the transition). Later we shall comment on cases inwhich one does not know all characteristics of one of the phases and sometimesone does not even know whether there is a phase transition.

The evolution of the free-energy landscape (as a function of an order pa-rameter) with the control parameter driving a phase transition is a guideline tograsp the dynamics following a quench or annealing from, typically, a disorderedphase into the ordered one. See Fig. 4 for a sketch. Basically, we shall discusstwo instances: the initial state becomes unstable, that is to say a maximum, andthe phase transition is of second-order (see Fig. 4-left) or metastable, that is tosay a local minimum, and the phase transition is of first order (see Fig. 4-right)in the final externally imposed conditions.3 In the former case the orderingprocess occurs throughout the material, and not just at nucleation sites. Twotypical examples are spinodal decomposition, i.e. the method whereby a mix-ture of two or more materials can separate into distinct regions with differentmaterial concentrations, or magnetic domain growth in ferromagnetic materials.Instead, in the latter case, the stable phase conquers the system through the

3Strictly speaking metastable states with infinite life-time exist only in the mean-field limit.

24

lower criticalcritical

upper critical

φ

f

lower criticalcritical

upper critical

φ

f

Figure 4: Left: second-order phase transition. Right: first order phase transi-tion.

nucleation of a critical localized bubble via thermal activation and its furthergrowth.

Having described the dependence of the free-energy landscape on the exter-nal parameters we now need to choose the microscopic dynamics of the orderparameter. Typically, one distinguishes two classes: one in which the orderparameter is locally conserved and another one in which it is not. Conservedorder parameter dynamics are found for example in phase separation in mag-netic alloys or inmiscible liquids. Ferromagnetic domain growth is an exempleof the non-conserved case.

3.1 Time-dependent Ginzburg-Landau

The kinetics of systems undergoing an ordering process is an importantproblem for material science but also for our generic understanding of patternformation in nonequilibrium systems. The late stage dynamics is believed tobe governed by a few properties of the systems whereas material details shouldbe irrelevant. Among these relevant properties one may expect to find thenumber of degenerate gound states, the nature of the conservation laws and thehardness or softness of the domain walls. Thus, classes akin to the universalityones of critical phenomena have been identified. These systems constitute afirst example of a problem with slow dynamics. Whether all systems with slowdynamics, in particular structural and spin glasses, undergo some kind of simplethough slow domain growth is an open question.

Take a magnetic system, such as the ubiquitous Ising model with ferro-magnetic uniform interactions, and quench it into the low temperature phasestarting from a random initial condition. Classically, the spins do not have anintrinsic dynamics; it is defined via a stochastic rule of Glauber, Metropolis

25

or similar type with or without locally conserved magnetization. For the pur-pose of the following discussion it is sufficient to focus on non-conserved localmicroscopic dynamics. Time-dependent macroscopic observables are then ex-pressed in terms of the values of the spins at each time-step. For instace, themagnetization density and its two-time self correlation function are defined as

m(t) ≡ N−1N∑

i=1

〈 si(t) 〉 , C(t, t′) ≡ N−1N∑

i=1

〈 si(t)si(t′) 〉 , (52)

where the angular brackets indicate an average over many independent runs (i.e.random numbers) starting from identical initial conditions and/or averages overdifferent initial configurations.

The ferromagnetic interactions tend to align the neighbouring spins in par-allel direction and in the course of time domains of the two ordered phases formand grow, see Fig. 5. At any finite time the configuration is such that both typesof domains exist. If one examines the configurations in more detail one reckonsthat there are some spins reversed within the domains. These ‘errors’ are dueto thermal fluctuations and are responsible of the fact that the magnetizationof a given configuration within the domains is smaller than one and close to theequilibrium value at the working temperature (apart from fluctuations due tothe finite size of the domains). The total magnetization, computed over the fullsystem, is zero (up to fluctuating time-dependent corrections that scale withthe square root of the inverse system size). The thermal averaged spin, 〈si(t)〉vanishes for all i and all finite t, see below for a more detailed discussion of thetime-dependence. As time passes the typical size of the domains increases in away that we shall also discuss below.

In order to treat phase-transitions and the coarsening process analytically itis preferable to introduce a coarse-grained description in terms of a continuouscoarse-grained field,

φ(~x, t) ≡ 1

V

∑

i∈V~x

si(t) , (53)

the fluctuating magnetization density. The partition function can be expressedas Z =

∫

Dφ e−βF [φ] with the Landau-Ginzburg free-energy functional

F [φ] =

∫

ddx c

2[∇φ(~x, t)]2 + V [φ(~x)]

(54)

andV (φ) = (φ2

0 − φ2)2 . (55)

The first term in eq. (54) represents the energy cost to create a domain wallor the elasticity of an interface. The second term depends on temperature(it contains an entropic contribution) and below the critical point Tc it hasa double well structure with two minima, φ = ±φ0 = 〈φ〉eq , that correspond

26

Figure 5: Snapshot of the 2d Ising model at a number of Monte Carlo stepsafter a quench from infinite to a subcritical temperature. Left: the up anddown spins on the square lattice are represented with black and white sites.Right: the domain walls are shown in black.

to the equilibrium states at low temperatures. Equation (54) is exact for afully connected Ising model where V (φ) arises from the multiplicity of spinconfigurations that contribute to the same φ(~x) = m. The order-parameterdependent free-energy density reads f(m) = −Jm2−hm+kBT (1+m)/2 ln[(1+m)/2] + (1 − m)/2 ln[(1 − m)/2] that close to the critical point where m ≃ 0becomes f(m) ≃ (kBT−2J)/2m2−hm+kBT/12m4 demonstrating the passagefrom a harmonic form at kBT > kBTc = 2J a quartic well at T = Tc and to adouble-well structure at T < Tc.

When discussing dynamics one should write down the stochastic evolutionof the individual spins and compute time-dependent averaged quantities as theones in (52). This is the procedured used in numerical simulations. Analyticallyit is more convenient to work with a field-theory and an evolution equation ofLangevin-type. This is the motivation for the introduction of continuous fieldequations that regulate the time-evolution of the coarse-grained order parame-ter. Ideally these equations should be derived from the spin stochastic dynamicsbut in practice they are introduced phenomenologically. In the magnetic caseas well as in many cases of interest, the domain wall and interface dynamicscan be argued to be overdamped. The field evolution is then proposed to begiven by the phenomenological Allen-Cahn equation with random noise, thetime-dependent Ginzburg-Landau equation or model A in the classification ofHohenberg-Halperin:

∂φ

∂t= −δF [φ]

δφ+ ξ . (56)

27

This is the field-theoretical extension of the Langevin equation in which thepotential is replaced by the order-parameter-dependent free-energy in eq. (54)and one uses a double-well potential form. This equation does not conserve theorder parameter. ξ is a noise taken to be Gaussian distributed with zero meanand correlations

〈ξ(~x, t)ξ(~x′, t′)〉 = 2kBTδd(~x− ~x′)δ(t− t′) . (57)

The friction coefficient has been absorbed in a redefinition of time. Initial con-ditions are usually chosen to be random with short-range correlations

[φ(~x, 0)φ(~x′, 0) ] = ∆δ(~x − ~x′) (58)

thus mimicking the high-temperature configuration ([. . .] represent the averageover its probability distribution). A dynamic phase transition arises at a criticalTc; above Tc the system freely moves above the two minima and basically ignoresthe double well structure while below Tc this is important.

Cases with vectorial or even tensorial order parameters can be treated sim-ilarly and are also of experimental relevance.

3.2 Relaxation and equilibration time

Coarsening from an initial condition that is not correlated with the equi-librium state and with no bias field does not take the system to equilibriumin finite times with respect to a function of the system’s linear size, L. Moreexplicitly, if the growth law is a power law [see eq. (75)] one needs times of theorder of Lzd to grow a domain of the size of the system. This gives a roughidea of the time needed to take the system to one of the two equilibrium states.For any shorter time, domains of the two types exist and the system is out ofequilibrium.

We thus wish to distinguish the relaxation time, tr, defined as the timeneeded for a given initial condition to reach equilibrium with the environment,from the decorrelation time, td, defined as the time needed for a given config-uration to decorrelate from itself. At T < Tc the relaxation time of any initialcondition that is not correlated with the equilibrium state diverges with thelinear size of the system. The self-correlation of such an initial state evolving atT < Tc decays as a power law and although one cannot associate to it a decaytime as one does to an exponential, one can still define a characteristic timethat turns out to be related to the age of the system.

In contrast, the relaxation time of an equilibrium magnetized configurationat temperature T vanishes since the system is already equilibrated while thedecorrelation time is finite and given by td ∼ |T − Tc|−νzeq .

The lesson to learn from this comparison is that the relaxation time and thedecorrelation time not only depend on the working temperature but they alsodepend strongly upon the initial condition. Moreover, in all low-temperature

28

cases we shall study the relaxation time depends on (N,T ) while the decorrela-tion time depends on (T, tw). For a random initial condition one has

tφr ≃

finite T > Tc ,

|T − Tc|−νzeq T>∼ Tc ,

Lzd T < Tc .

3.3 Relaxation in equilibrium

For concreteness, let us use a magnetic system as an example. If one quenchesit to a T > Tc the relaxation time, tr, needed to reach configurations sampledby the Boltzmann measure does not diverge with the size of the sample. Once ashort transient overcome, the average of a local spin approaches the limit givenby the Boltzmann measure, 〈si(t)〉 → 〈si〉eq = m = 0, for all i.

The relaxation of the two-time self-correlation at T > Tc, when the time t′

is chosen to be longer than tr, decays exponentially

〈si(t)si(t′)〉 ≃ e−(t−t′)/td (59)

with a decorrelation time that increases with decreasing temperature and closeto (but still above) Tc diverges as a power law, td ∼ (T − Tc)

−νzeq , with ν thecritical exponent characterizing the divergence of the equilibrium correlationlength, ξ ∼ (T − Tc)

−ν , and zeq the equilibrium exponent that links times and

lengths, ξ ∼ t1/zeqd . The divergence of td is the manifestation of critical slowing

down. The asympotic value verifies

limt−t′≫t′

〈si(t)si(t′)〉 = 〈si(t)〉〈si(t′)〉 = 〈si〉eq〈si〉eq = m2 = 0 , (60)

cfr. eq. (38).The relaxation of the two-time self-correlation at T < Tc, when the time t′

is chosen to be longer than tr, that is to say, once the system has thermalizedin one of the two equilibrium states, also decays exponentially

〈si(t)si(t′)〉 ≃ e−(t−t′)/td (61)

with a decorrelation time that decreases with decreasing temperature and closeto Tc (but below it) also diverges as a power law, td ∼ (T − Tc)

−νzeq . Theasympotic value verifies

limt−t′≫t′

〈si(t)si(t′)〉 = 〈si(t)〉〈si(t′)〉 = 〈si〉eq〈si〉eq = m2 6= 0 , (62)

cfr. eqs. (38) and (60).

3.4 Critical and subcritical coarsening

29

The ordering process is characterized by the growth of a typical length,R(T, t). The growth regimes are summarized in the following equation and inFig. 6:

R(T, t) ∼

→ ξ(T ) T > Tc saturationRc(t) < ξ(T ) T = Tc critical coarseningR(T, t) > ξ(T ) T < Tc sub-critical coarsening

(63)

After a quench to the high temperature phase T > Tc the system first ‘or-ders’ until reaching the correlation length ξ and next relaxes in equilibrium asexplained in the previous section. The correlation length could be very shortand the transient non-equilibrium regime be quite irrelevant (T ≫ Tc). In thecritical region, instead, the correlation length grows and it becomes important.In a critical quench the system never orders sufficiently and R(t) < ξ for allfinite times. Finally, a quench into the subcritical region is characterized bytwo growth regimes: a first one in which the critical point dominates and thegrowth is as in a critical quench; a second one in which the proper sub-criticalordering is at work. The time-dependence of the growth law is different in thesetwo regimes as we shall see below.

3.5 Dynamic scaling hypothesis for sub-critical coarsening

Monte Carlo simulations of the Ising model and other systems quenchedbelow criticality and undergoing domain growth demonstrate that in the longwaiting-time limit t′ ≫ t0, the spin self-correlation 〈si(t)si(t′)〉 separates intotwo additive terms

C(t, t′) ∼ Cst(t− t′) + Cag(t, t′) Additive separation (64)

see Fig. 7, with the first one describing equilibrium thermal fluctuations withinthe domains,

Cst(t− t′) =

1 − 〈si〉2eq = 1 −m2 , t− t′ = 0 ,0 , t− t′ → ∞ ,

(65)

and the second one describing the motion of the domain walls

Cag(t, t′) = 〈si〉2eq fc

(

R(t)

R(t′)

)

=

〈si〉2eq , t′ → t− ,0 , t− t′ → ∞ .

(66)

To ease the notation we have not written the explicit T -dependence in R that,as we shall see below, is less relevant than t. Note that by adding the twocontributions one recovers C(t, t) = 1 as expected and C(t, t′) → 0 when t≫ t′.The first term is identical to the one of a system in equilibrium in one of the twoordered states, see eq. (60) for its asymptotic t− t′ ≫ t′ limit; the second one isinherent to the out of equilibrium situation and existence and motion of domain

30

R(T, t)

Tc

ξ

Figure 6: Sketch of the growth process in a second-order phase transition. Thethick line is the equilibrium correlation length ξ ≃ |T − Tc|−ν . The thin ar-rows indicate the growing length R in the critical coarsening and sub-criticalcoarsening regimes.

31

0.1

1

1 10 100 1000

C(t

,t w)

t-tw

tw=248

163264

128256512

0.1

1

1 10 100 1000

C(t

,t w)

t-tw

tw=248

163264

128256512

Figure 7: The two-time self-correlation in the 2dIM with non-conserved orderparameter dynamics at several waiting-times given in the key at temperatureT = 0.5 (left) and T = 2 (right). Data obtained with Monte Carlo simulations.Note that the plateau is located at a lower level in the figure on the rightconsistently with the fact that 〈φ〉eq decreases with increasing temperature.

walls. They vary in completely different two-time scales. The first one changeswhen the second one is fixed to 〈si〉2eq, at times such that R(t)/R(t′) ≃ 1. Thesecond one varies when the first one decayed to zero. The mere existence of thesecond term is the essence of the aging phenomenon with older systems (longert′) having a slower relaxation than younger ones (shorter t′). The scaling ofthe second term as the ratio between ‘two lengths’ is a first manifestation ofdynamic scaling.

A decorrelation time can also be defined in this case by expandind the ar-gument of the scaling function around t′ ≃ t. Indeed, calling ∆t ≡ t − t′

one has R(t)/R(t′) ≃ R(t′ + ∆t)/R(t′) ≃ [R(t′) + R′(t′)∆t]/R(t′) ≃ 1 +∆t/[d lnR(t′)/dt′]−1 and one identifies a t′-dependent decorrelation time

td ≃ [d lnR(t′)/dt′]−1 decorrelation time (67)

which is, in general, a growing function of t′.The dynamic scaling hypothesis states that at late times and in the scaling

limitr ≫ ξ(g) , R(g, t) ≫ ξ(g) , r/R(g, t) arbitrary , (68)

where r is the distance between two points in the sample, r ≡ |~x − ~x′|, andξ(g) is the equilibrium correlation length that depends on all parameters (T andpossibly others) collected in g, there exists a single characteristic length, R(g, t),such that the domain structure is, in statistical sense, independent of time whenlengths are scaled by R(g, t). Time, denoted by t, is typically measured from theinstant when the critical point is crossed. In the following we ease the notation

32

-0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120

C(r

,t)

r

t=48

163264

128256512

1024

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6

C(r

,t)

r / t1/2

t=48

163264

128256512

1024

Figure 8: The equal-time correlation as a function of distance for the 2dIM.Raw (left) and scaled (right) data.

and write only the time-dependence in R. This hypothesis has been provedanalytically in very simple models only, such as the one dimensional Ising chainwith Glauber dynamics or the Langevin dynamics of the d-dimensional O(N)model in the large N limit (see Sect. ).

We now explain what the scaling hypothesis means in practice. In the coarse-grained formulation the two-time and two-point dependent order-parameter cor-relation function is

C(~x, t; ~x′, t′) = C(~r; t, t′) = 〈φ(~x, t)φ(~x′, t′)〉 (69)

where we assumed that there is translational invariance in space. (Similarexpressions can be written in the spin language.) Since the averaged order-parameter vanishes at all times when the system undergoes coarsening this def-inition coincides with the one of connected correlations. Its Fourier transformis

S(~k; t, t′) = 〈φ(~k, t)φ(−~k, t′)〉 . (70)

In the late stages of the coarsening process the spherically averaged structurefactor S(k, t) at equal times t = t′ can be measured experimentally with small-angle scattering of neutrons, x-rays or light. In the limits given in (68), wherewe focus on the aging relaxation only, see below, it has been found to satisfyscaling:

S(k, t) ∼ 〈φ〉2eq Rd(t)G[kR(t)] . (71)

Correspondingly, the real-space correlation function is expected to behave as

C(r, t) ∼ 〈φ〉2eq F [r/R(t)] . (72)

G is the Fourier transform of F . See Fig. 8 for a test of the scaling hypothesisin the MC dynamics of the 2dIM.

33

Similarly, the two-times pair correlation function satisfies

C(r; t, t′) ∼ 〈φ〉2eq f(

r

R(t),R(t)

R(t′)

)

(73)

In order to fully characterise the correlation functions one then has to de-termine the typical growing length, R, and the scaling functions, fc, F , etc. Itturns out that the former can be determined with semi-analytic arguments andthe predictions are well verified numerically – at least for clean system. We shallsummarize the outcome in Sect. . The latter, instead, are harder to obtain. Weshall give a very brief state of the art report in Sect. . For a much more detaileddiscussion of these methods see the review articles in [23].

3.6 The averaged domain length in subcritical coarsening

The time-dependent typical domain length, R(t), is determined numericallyby using several indirect criteria or analytically within certain approximations.The most common ways of measuring R are with numerical simulations of lat-tice models or the numerical integration of the continuous partial differentialequation for the evolution of the order parameter. In both cases one

– Computes the ‘inverse perimeter density’ R(t) = −〈H〉eq/[〈H(t)〉− 〈H〉eq]with 〈H(t)〉 the time-dependent averaged energy and 〈H〉eq the equilibriumenergy both measured at the working temperature T .

– Puts the dynamic scaling hypothesis to the test and extracts R from theanalysis.

3.6.1 Clean one dimensional cases with non-conserved order pa-rameter

In one dimension, a space-time graph allows one to view coarsening as thediffusion and annhilitation upon collision of point-like particles that representthe domain walls. In the Glauber Ising chain with non-conserved dynamics onefinds that the typical domain length grows as t1/2 while in the continuous casethe growth is only logarithmic, ln t.

3.6.2 Non-conserved curvature driven dynamics (d > 2)

The time-dependent Ginzburg-Landau model allows us to gain some insighton the mechanism driving the domain growth and the direct computation ofthe averaged domain length. In clean systems temperature does not play avery important role in the domain-growth process, it just adds some thermalfluctuations within the domains, as long as it is smaller than Tc. In dirty casesinstead temperature triggers thermal activation.

We focus first on the clean cases at T = 0 and only later we discuss thermaleffects. Equation (56) for θ = 0 is just a gradient descent in the energy landscape

34

φ

g

xyt1t2

g

φo < 0 φo > 0 φo < 0 φo > 0

t1 t2

Figure 9: Left: domain wall profile. Right: view from the top. (g is n.)

F . Two terms contribute to F : the bulk-energy term that is minimized byφ = ±φ0 and the elastic energy (∇φ)2 which is minimized by flat walls if present.As a consequence the minimization process implies that regions of constant field,φ(~x, t) = ±φ0, grow and they separated by flatter and flatter walls.

Take a flat domain wall separating regions where the configuration is theone of the two equilibrium states, φ(~x, t) = ±φ0 + δφ(~x, t). Linearizing eq. (56)around ±φ0 and looking for static configurations, i.e. δφ(~x, t) = δφ(~x) =δφ(n) where n is the distance from the wall along the normal direction onefinds d2δφ(n)/dn2 = −V ′′(φ0)δφ(n). This equation has the solution δφ(n) ∼e−

√V ′′(φ0)n where n is the perpendicular distance to the wall . The order pa-

rameter approaches ±φ0 on both sides of the wall very rapidly. This meansthat the free-energy of a configuration with an interface (sum of the elasticand potential terms) is concentrated in a very narrow region close to it. Inconsequence, the domain-wall curvature is the driving force for domain growth.

Allen and Cahn showed that the local wall velocity is proportional to thelocal curvature working with the GL equation at θ = 0. The proof goes asfollows. Take the GL equation and trasform the derivatives to apply in thedirection normal to the wall:

∂φ(~x, t)

∂t= − ∂φ(~x, t)

∂n

∣

∣

∣

∣

t

∂n

∂t

∣

∣

∣

∣

φ

, ~∇φ(~x, t) =∂φ(~x, t)

∂n

∣

∣

∣

∣

t

n ,

∇2φ(~x, t) =∂2φ(~x, t)

∂n2

∣

∣

∣

∣

t

+∂φ(~x, t)

∂n

∣

∣

∣

∣

t

~∇ · n

where the subscripts mean that the derivatives are taken at t or φ fixed. Using

now ∂2φ(~x,t)∂n2 |t = V ′(φ) (note that the derivative is taken at fixed t) in the GL

equation one finds the Allen-Cahn result

v ≡ ∂tn|φ = −~∇ · n ≡ −κ (74)

35

valid in all d with κ the geodesic curvature.Eqaution (74) allows one to get an intuition about the typical growth law

in such processes. Take a spherical wall in any dimension. The local curvatureis constant and κ = (d − 1)/R where R is the radius of the pshere within thehull. Equation (74) is recast as dR/dt = −(d − 1)/R that implies R2(t) =R2(0) − 2(d− 1)t.

A closer look at the 2d equation allows one to go beyond and prove, inthis case, that all areas enclosed by domain walls irrespective of their beingother structures within (the so-called hull-enclosed areas) tend to diminish atconstant rate dA/dt = −λ. This, of course, does not mean that all domainsreduce their area since a domain can gain area from the disappearance of aninternal domain of the opposite sign, for instance. The proof is simple and justuses the Gauss-Bonnet theorem: dA

dt =∮

~v∧d~ℓ =∮

vdℓ. The local wall-velocity,~v, is proportional to the local geodesic curvature, κ, and the Gauss-Bonnettheorem implies

∮

κdℓ = 2π for a planar 2d manifold with no holes. Therefore,the hull-enclosed area decreases with constant velocity for any geometry.

There are a number of ways to find the growth law

R(t) = λ t1/zd (75)

with zd the dynamic exponent, in pure and isotropic systems (see [23]). Theeffects of temperature enter only in the parameter λ and, for clean systems,growth is slowed down by an increasing temperature since thermal fluctuationtend to roughen the interfaces thus opposing the curvature driven mechanism.We estimate the T dependence of λ in Sect. .

In curvature driven Ising or Potts cases with non-conserved order parameterthe domains are sharp and zd = 2 with λ a weakly T -dependent coefficient. Forsystems with continuous variables such as rotors or XY models and the sametype of dynamics, a number of computer simulations have shown that domainwalls are thicker and zd = 4.

3.6.3 Conserved order parameter: bulk diffusion

A different type of dynamics occurs in the case of phase separation (thewater and oil mixture ignoring hydrodynamic interactions or a binary allow).In this case, the material is locally conserved, i.e. water does not transform intooil but they just separate. The main mechanism for the evolution is diffusion ofmaterial through the bulk of the opposite phase. After some discussion, it wasestablished, as late as in the early 90s, that for scalar systems with conservedorder parameter zd = 3.

3.6.4 Role of disorder: thermal activation

36

Figure 10: Time evolution of a spin configuration; three snapshots of a 2dIM ona square lattice with locally conserved order parameter (Kawasaki dynamics).

The situation becomes much less clear when there is quenched disorder inthe form of non-magnetic impurities in a magnetic sample, lattice dislocations,residual stress, etc. Qualitatively, the dynamics are expected to be slower thanin the pure cases since disorder pins the interfaces. In general, based on anargument due to Larkin (and in different form to Imry-Ma) one expects that ind < 4 the late epochs and large scale evolution is no longer curvature driven butcontrolled by disorder. Indeed, within a phase space view disorder generatesmetastable states that trap the system and thus slow down the relaxation.

A hand-waving argument to estimate the growth law in dirty systems isthe following. Take a system in one equilibrium state with a domain of linearsize R of the opposite equilibrium state within it. This configuration could bethe one of an excited state with respect to the fully ordered one with absoluteminimum free-energy. Call ∆F (R) the free-energy barrier between the excitedand equilibrium states. The thermal activation argument (see Sect. ) yields theactivation time scale for the decay of the excited state (i.e. erasing the domainwall)

tA ∼ τ e∆F (R)/(kBT ) . (76)

For a barrier growing as a power of R, ∆V (R) ∼ Υ(T, J)Rψ (where J representsthe disorder) one inverts (76) to find the linear size of the domains still existingat time t, that is to say, the growth law

R(t) ∼(

kBTΥ(T ) ln t

τ

)1/ψ

. (77)

All smaller fluctuation would have disappeared at t while typically one wouldfind objects of this size. The exponent ψ is expected to depend on the dimen-

37

sionality of space but not on temperature. In ‘normal’ systems it is expectedto be just d − 1 – the surface of the domain – but in spin-glass problems, itmight be smaller than d − 1 due to the presumed fractal nature of the walls.The prefactor Υ is expected to be weakly temperature dependent.

One assumes that the same argument applies out of equilibrium to the re-conformations of a portion of any domain wall or interface where R is the ob-servation scale.

However, already for the (relatively easy) random ferromagnet there is noconsensus about the actual growth law. In these problems there is a competi-tion between the ‘pure’ part of the Hamiltonian, that tries to minimize the total(d−1) dimensional area of the domain wall, and the ‘impurity’ part that makesthe wall deviate from flatness and pass through the locations of lowest localenergy (think of Jij = J + δJij with J and δJij contributing to the pure andimpurity parts of the Hamiltonian, respectively). The activation argument ineq. (76) together with the power-law growth of barriers in ∆F (R) ∼ Υ(T, J)Rψ

imply a logarithmic growth of R(t). Simulations, instead, suggest a power lawwith a temperature dependent exponent. Whether the latter is a pre-asymptoticresult and the trully asymptotic one is hidden by the premature pinning of do-main walls or it is a genuine behaviour invalidating ∆F (R) ∼ Υ(T, J)Rψ oreven eq. (76) is still an open problem. See the discussion below for a plausi-ble explanation of the numerical data that does not invalidate the theoreticalexpectations.

In the 3d RFIM the curvature-driven growth mechanism that leads to (75)is impeded by the random field roughening of the domain walls. The depen-dence on the parameters T and h has been estimated. In the early stages ofgrowth, one expects the zero-field result to hold with a reduction in the ampli-tude R(t) ∼ (A−Bh2) t1/2. The time-window over which this law is observed nu-merically decreases with increasing field strength. In the late time regime, wherepinning is effective Villain deduced a logarithmic growth R(t) ∼ (T/h2) ln t/t0by estimating the maximum barrier height encountered by the domain wall andusing the Arrhenius law to derive the associated time-scale.

In the case of spin-glasses, if the mean-field picture with a large number ofequilibrium states is realized in finite dimensional models, the dynamics wouldbe one in which all these states grow in competition. If, instead, the phenomeno-logical droplet model applies, there would be two types of domains growing andR(t) ∼ (ln t)1/ψ with the exponent ψ satisfying 0 ≤ ψ ≤ d − 1. Some refinedarguments that we shall not discuss here indicate that the dimension of thebulk of these domains should be compact but their surface should be roughwith fractal dimension ds > d− 1.

3.6.5 Temperature-dependent effective exponents

The fact that numerical simulations of dirty systems tend to indicate that thegrowing length is a power law with a T -dependent exponent can be explained

38

as due to the effect of a T -dependent cross-over length LT . Indeed, if belowLT ∼ T φ the growth process is as in the clean limit while above LT quencheddisorder is felt and the dynamics is thermally activated:

R(t) ∼

t1/zd for R(t) ≪ LT ,(ln t)1/ψ for R(t) ≫ LT .

(78)

These growth-laws can be first inverted to get the time needed to grow a givenlength and then combined into a single expression that interpolates between thetwo regimes:

t(R) ∼ e(R/LT )ψRzd (79)

where the relevant T -dependent length-scale LT has been introduced. Now, bysimply setting t(R) ∼ Rz(T ) one finds z(T ) − zd ≃ z/LψT (valid for times such

that tψ/zd ln t is almost constant). Similarly, by equating t(R) ∼ exp(Rψ(T )/T )one finds that ψ(T ) is a decreasing function of T approaching ψ at high T .

3.7 Scaling functions for subcritical coarsening

Even though the qualitative behaviour of the solution to eq. (56) is easy tograsp, it is still too difficult to solve analytically and not much is known exactlyon the scaling functions. A number of approximations have been developed butnone of them is fully satisfactorily (see [23] for a critical review of this problem).

At very short distances (large wave-vector) the equal-time correlation func-tion (structure factor) verifies Porod’s law the derivation of which goes as fol-lows. The field product at two points ~x and ~x+ ~r with ξ ≪ r ≪ L is -1 if thereis wall in between or 1 otherwise. At long times the probability of finding morethan one wall is negligible. Then, C(r, t) ≃ (−1)r/L + (1 − r/L) = 1 − 2r/L,for r ≪ L, since r/L is the probability for a rod of length r to cut a wall.This expression is non-analytic at ~r/L = ~0. The structure factor scales asS(k, t) ∼ L−1k−(d+1), for kL≫ 1.

The super-universality hypothesis states that in cases in which temperatureand quenched disorder are ‘irrelevant’ in the sense that they do not modify thenature of the low-temperature phase (i.e. it remains ferromagnetic in the caseof ferromagnetic Ising models) the scaling functions are not modified. Only thegrowing length changes from the, say, curvature driven t1/2 law to a logarithmicone. This hypothesis has been verified in a number of two and three dimensionalmodels including the RBIM and the RFIM.

3.7.1 Breakdown of dynamic scaling

Some special cases in which dynamic scaling does not apply have also beenexhibited. Their common feature is the existence of two (or more) growinglengths associated to different ordering mechanisms. An example is given bythe Heisenberg model at T → 0 in which the two mechanisms are related to the

39

vectorial ordering within domains separated by couples of parallel spins thatannhilate in a way that is similar to domain-wall annihilation in the Ising chain.

3.8 Critical coarsening

3.8.1 Scaling

Imagine now that one quenches a system to the critical point of a second-order phase transition. The equilibrium order parameter vanishes. Still, theequilibrium correlation length diverges and, roughly speaking, an equilibriumconfiguration is made of structures of all sizes. The evolution of a high tem-perature initial condition undergoes critical coarsening with growth of structureof linear size R(t). However, the scaling of the correlations cannot then havea stationary regime of the form discussed above for sub-critical quenches. Thescaling form found for two-time correlations is [24]

C(t, tw) ∼ (t− tw + t0)−b f

(

R(t)

R(tw),t

t0

)

(80)

(t0 is a microscopic cut-off that cures a divergence at very short time-differences.)

3.8.2 Crossover between critical and sub-critical coarsening: λ(T )

In the case of non-conserved scalar order-parameter dynamics behaves as

R(t) ∼ t1/zeq with zeq ≃ 2.13 (81)

with zeq the equilibrium dynamics exponent (note that zeq is different fromzd). We shall not discuss critical dynamics in detail; this problem is treatedanalytically with dynamic renormalization group techniques and it is very welldiscussed in the literature [24].

Matching critical coarsening with sub-critical one allows one to find the T -dependent prefactor λ [6]. The argument goes as follows. The out of equilibriumgrowth at criticality and in the ordered phase are given by

R(t) ∼ t1/zeq at T = Tc ,

(λ(T )t)1/zd at T < Tc .(82)

zeq is the equilibrium dynamic critical exponent and zd the out of equilibriumgrowth exponent. Close but below criticality one also has

R(t) ∼ ξ−a t1/zd f

(

t

ξzeq

)

at T = Tc − ǫ (83)

with ξ the T -dependent equilibrium correlation length, ξ(T ) ∼ (Tc − T )−ν.The last factor tends to one when R(t) ≫ ξ that is to say when the argument

40

diverges, f(x→ ∞) → 1, but is non-trivial when R(t) ∼ ξ and the argument isfinite. In particular, we determine its behaviour at small argument by requiringthat eq. (83) matches the subcritical growing length which is achieved by (i)recovering the correct t dependence, (ii) cancelling the ξ factor. (i) implies

f(x) ∼ x−1/2−1/zeq for x→ 0 . (84)

Then eq. (83) becomes

R(t) ∼ ξ−a+zeq/2−1 t1/zeq (85)

and to eliminate ξ we needa = zeq/2 − 1 . (86)

Comparing now the subcritical growing length and (83) in the very long timeslimit such that R(t) ≫ ξ and f(x) → 1:

[λ(T )]1/zd ∼ ξ−a ∼ (Tc − T )ν(zeq−2)/2 . (87)

3.9 Annealing and the Kibble-Zurek mechanism

There has been recent interest on understanding how a finite rate coolingaffects the defect density found right after the quench. A scaling form involvingequilibrium critical exponents was proposed by Zurek following work by Kibble.The interest is triggered by the similarity with the problem of quantum quenchesin atomic gases, for instance. An interplay between critical coarsening (thedynamics that occurs close in the critical region) that is usually ignored (!)and sub-critical coarsening (once the critical region is left) is the mechanismdetermining the density of defects right after the end of the cooling procedure.

The growing length satisfies a scaling law

R(t, ǫ(t)) ∼ ǫ−ν(t) f [tǫzeqν(t)] ǫ(t) = |T (t) − Tc|

f(x) →

ct x≪ −1 Equilibrium at high T√x x≫ 1 Coarsening at low T

with t measured from the instant when the critical point is crossed and x ∈(−1, 1) is the critical region. A careful analysis of this problem can be found in[11].

3.10 An instructive case: the large N approximation

A very useful approximation is to upgrade the scalar field to a vectorial onewith N components

φ(~x, t) → ~φ(~x, t) = (φ1(~x, t), . . . , φN (~x, t)) , (88)

41

and modify the free-energy

F =

∫

ddx

[

1

2(∇~φ)2 +

N

4(φ2

0 −N−1φ2)2]

, (89)

with φ2 =∑Nα=1 φ

2α and φ0 finite. The T = 0 dynamic equation then becomes

∂tφα(~x, t) = ∇2φα(~x, t) − 4φα(~x, t) [φ20 −N−1φ2(~x, t)] (90)

and it is clearly isotropic in the N dimensional space implying

Cαβ(~x, t; ~x′, t′) = δαβC(~x, t; ~x′, t′) (91)

In the limit N → ∞ while keeping the dimension of real space fixed to d,the cubic term in the right-hand-side can be replaced by

−φα(~x, t)N−1φ2(~x, t) → −φα(~x, t)N−1[φ2(~x, t) ]ic ≡ −φα(~x, t) a(t) (92)

since N−1φ2(~x, t) does not fluctuate, it is equal to its average over the initialconditions and it is therefore not expected to depend on the spatial position if theinitial conditions are chosen from a distribution that is statistically translationalinvariant. For the scalar field theory the replacement (92) is just the Hartreeapproximation. The dynamic equation is now linear in the field φα(~x, t) thatwe rename φ(~x, t) (and it is now order 1):

∂tφ(~x, t) = [∇2 + a(t)]φ(~x, t) , (93)

where the time-dependent harmonic constant a(t) = φ20 − [φ2(~x, t)]ic = φ2

0 −[φ2(~0, t)]ic has to be determined self-consistently. Equation (93) can be Fouriertransformed

∂tφ(~k, t) = [−k2 + a(t)]φ(~k, t) , (94)

and it takes now the form of almost independent oscillators under differenttime-dependent harmonic potentials coupled only through the self-consistentcondition on a(t). The stability properties of the oscillators depend on the signof the prefactor in the rhs. The solution is

φ(~k, t) = φ(~k, 0) e−k2t+

∫

t

0dt′ a(t′)

(95)

and the equation on a(t) reads:

a(t) = φ20 − ∆ e

2∫

t

0dt′a(t′)

(

2π

4t

)d/2

, (96)

where one used a delta-correlated Gaussian distribution of initial conditionswith strength ∆. The self-consistency equation is not singular at t = 0 sincethere is an underlying cut-off in the integration over k corresponding to the

42

inverse of the lattice spacing, this implies that times should be translated ast→ +1/Λ2.

Without giving all the details of the calculation, eq. (96), generalized to thefinite T case, can be solved at all temperatures [12]. One finds that there existsa finite Tc(d) and

Upper-critical quench

a(t) → −ξ−2 (97)

with ξ the equilibrium correlation length, and the ‘mass’ saturates to a finitevalue: −k2 + a(t) → −k2 − ξ−2.

Critical quench

a(t) → −w/(2t) with w = 0 for d > 4 and w = (d− 4)/2 for d < 4 . (98)

The dynamics is trivial for d > 4 but there is critical coarsening in d < 4.

Sub-critical coarsening

In the long times limit in which the system tends to decrease its elastic andpotential energies [φ2(~x, t) ]ic must converge to φ2

0 6= 0 below criticality and this

imposes 2∫ t

0 dt′ a(t′) ≃ d

2 ln(t/t0) with t0 = π/2 (∆/φ20)

2/d at large times, i.e.

a(t) ≃ d

4tfor t≫ t0 (99)

and the time-dependent contribution to the spring constant vanishes asymptoti-cally. Knowing the long-time behaviour of a(t) implies that each mode [φ(~k, t)]icwith ~k 6= 0 vanishes asymptotically but the ~k = 0 mode grows as td/4. Thegrowth of the ~k = 0 reflects the domain growth process whereby all modulationtend to disappear and the configuration gets more and more uniform as timepasses.

We focus now on the two interesting cases: quenches to Tc and T < Tc. Theasymptotic behaviour of the space-time correlation function in the aging regimeis

[φ(~x, t)φ(~x′, t′) ]ic = φ20

[

4tt′

(t+ t′)2

]d/4

exp

[

− (~x− ~x′)2

4(t+ t′)

]

, (100)

for t ≥ t′ for a quench to T < Tc and

[φ(~x, t)φ(~x′, t′) ]ic = φ20 t

′1−d/2f(t/t′) exp

[

− (~x− ~x′)2

4(t+ t′)

]

, (101)

for a quench to Tc. We focus on d < 4. This expression captures the mainfeatures of the domain growth process:

• In Fourier space all k 6= 0 modes have an exponential decay while thek = 0 one is fully massless asympotically and diffuses.

43

• For any finite and fixed (~x − ~x′), in the long times limit the exponentialfactor approaches one and one obtaines a function of t′/t only.

• Due to the exponential factor, for fixed but very large time t and t′ thecorrelation falls off to zero over a distance |~x− ~x′| ∝

√t+ t′. This means

that, at time t, the typical size of the regions in the states ±φ0 is R(t) ∝t1/2. This holds for critical and sub-critical quenches as well and it is apeculiar property of the large N O(N) model that has zeq = zd.

• For fixed |~x−~x′|, the correlation always falls to zero over a time separationt − t′ which is larger than t′. This means that the time it takes to thesystem to decorrelate from its configuration at time t′ is of the order oft′ itself. The age of the system is the characteristic time-scale for thedynamical evolution: the older is the system, the slower is its dynamics.After a time of the order of the age of the system any point ~x will be sweptby different domain walls and the correlation will be lost.

• In a critical quench the correlation always decays to zero due to the pref-actor that goes as t(2−d)/2 and vanishes in all d > 2. The aging curveshave an envelope that approaches zero as a power law.

• In a sub-critical quench, for any finite and fixed (~x − ~x′), in the long t′

and t limit such that t′/t → 1 the time dependence disappears and thecorrelation between two points converges to φ2

0. This means that, typically,if one looks at a finite spatial region on a finite time-scale this region willbe in one of the two states ±φ0, i.e. within a domain.

Note that we have obtained the field and then computed correlations fromthe time-dependent configuration. We have not needed to compute the linearresponse. We shall see later that in other more complex glassy systems onecannot follow this simple route and needs to know how the linear responsebehave. We refer to the reviews in [26] for detailed accounts on the behaviourof the linear response in critical dynamics.

3.11 Nucleation and growth

In a first-order phase transition the equilibrium state of the system changesabruptly. Right at the transition the free-energies of the two states involvedare identical and the transition is driven by lowering the free-energy as the newphase forms, see Fig. 4. The original phase remains meta-stable close to thetransition. The nucleation of a sufficiently large bubble of the trully stable phaseinto the metastable one needs to be thermally activated to trigger the growthprocess [25]. The rate of the process can be very low or very fast dependingon the height of the free-energy barrier between the two states and the ambienttemperature.

44

Two types of nucleation are usually distinguished: homogeneous (occuryingat the bulk of the meta-stable phase) and heterogeneous (driven by impuritiesor at the surface). The more intuitive examples of the former, on which we focushere, are the condensation of liquid droplets from vapour and the crystallizationof a solid from the melt.

The classical theory of nucleation applies to cases in which the identificationof the nucleous is easy. It is based on a number of assumptions that we now list.First, one associates a number of particles to the nucleous (although in someinteresting cases this is not possible and a different approach is needed). Second,one assumes that there is no memory for the evolution of the number densitiesof clusters of a given size in time (concretely, a Markov master equation is used).Third, one assumes that clusters grow or shrink by attachment or loss of a singleparticle, that is to say, coallescence and fission of clusters are neglected. Thus,the length-scale over which the slow part of the dynamics takes place is the oneof the critical droplet size, the first one to nucleate. Fourth, the transition ratessatisfy detail balance and are independent of the droplet form. They just dependon the free-energy of the droplet with two terms: a contribution proportionalto the droplet volume and the chemical potential difference between the stableand the metastable states, ∆f , and a contribution proportional to the bubblesurface that is equal to the surface area times the surface tension, σ, that isassumed to be the one of coexisting states in equilibrium - that is to say theenergy of a flat domain wall induced by twisted boundary conditions. Fift, thebubble is taken to be spherical and thus dependent of a single parameter, theradius. Thus

∆F [R] = σ Ωd−1 Rd−1 − |∆f | Ωd R

d (102)

for d > 1. Ωd is the volume of the unit sphere in d dimensions. For small radiithe surface term dominates and it is preferable to make the droplet disappear.In contrast, for large radii the bulk term dominates and the growth of the bubbleis favoured by a decreasing free-energy. Thus the free-energy difference has amaximum at

R∗ =(d−1) Ωd−1 σd Ωd |∆f | ∝ σ|∆f |−1 (103)

and the system has to thermally surmount the barrier ∆F ∗ ≡ ∆F [R∗]. TheKramers escape theory, see Sect. , implies that the nucleation rate or the av-erage number of nucleations per unit of volume and time is suppressed by theArrhenius factor

rA = t−1A ∼ e−β∆F∗

with ∆F ∗ = (d−1)d−1

ddΩdd−1

Ωd−1

d

σd

|∆f |d−1 (104)

As expected, ∆F ∗ increases with increasing σ and/or |∆f |−1 and r−1 vanishesfor T → 0 when thermal agitation is switched off. The implicit assumption isthat the time to create randomly the critical droplet is much longer than the timeinvolved in the subsequent growth. The relaxation of the entire system is thus

45

expected to be given by the inverse probability of escape from the metastablewell. The determination of the prefactor [that is ignored in eq. (104)] is a hardtask.

46

4 Classical formalism and glassy dynamics

In this Section we discuss some static and dynamic aspects of classical sta-tistical systems in the canonical ensemble. The features we recall here are usefulto better grasp the quantum problems to be dealt with later. We also reviewsome aspects of the formalism used to treat quantum systems. We set the no-tation that we shall use throughout the lectures. In particular, we choose theconventions used in the definitions of the bosonic and fermionic Green functions.

In the rest of the lectures we shall heavily use the path integral formalismin its classical and quantum versions.

4.1 Classical statics: the reduced partition function

In order to analyze the statistical static properties of the classical coupledsystem, we study the partition function or Gibbs functional, Ztot that reads

Ztot[η] =∑

conf osc

conf syst

exp(−βHtot − βηx) (105)

where the sum represents an integration over the phase space of the full system,the particle’s and the oscillators’, and η is a source. Having chosen a quadraticbath and a linear coupling, the integration over the oscillators’ coordinates andmomenta can be easily performed. This yields the reduced Gibbs functional

Zred[η] ∝∑

conf syst

exp

[

−β(

Hsyst +Hcounter + ηx− 1

2

Nb∑

a=1

c2amaω2

a

x2

)]

.(106)

The ‘counterterm’ Hcounter is chosen to cancel the last term in the exponentialand it avoids the renormalization of the particle’s mass (the coefficient of thequadratic term in the potential) due to the coupling to the environment thatcould have even destabilize the potential taking negative values. An alternativeway of curing this problem would be to take a vanishingly small coupling to thebath in such a way that the last term must vanish by itself (say, all ca → 0).However, this might be problematic when dealing with the stochastic dynamicssince a very weak coupling to the bath implies also a very slow relaxation. It isthen conventional to include the counterterm to cancel the mass renormalization.One then finds

Zred[η] ∝∑

conf syst

exp [−β (Hsyst + ηx)] = Zsyst[η] . (107)

The interaction with the reservoir does not modify the statistical properties ofthe particle since Zred ∝ Zsyst. We shall see in Sect. that this does not happen

47

quantum mechanically. (For a non-linear coupling Hint =∑

α cαqαV(x) the

counterterm is Hcounter = 12

∑

αc2α

mαω2α[V(x)]2.)

4.2 Classical dynamics: generating functional

In Sect. we showed a proof of the Langevin equation based on the integrationover a large ensemble of harmonic oscillators that act as a bath with which thesystem is set in contact.

The dynamic generating functional is defined as [40]

Zd[η] ≡∫

DξdP (t0) e− 1

2kBT

∫

T

t0dt′∫

T

t0dt′′ ξ(t′)Γ−1(t′−t′′)ξ(t′′)

×e∫

T

t0dt′ η(t′)xξ(t

′).

xξ(t) is the solution to the Langevin equation with initial condition x(t0), x(t0)at the initial time t0. The factor dP (t0) is the measure of the initial condition,dP (t0) ≡ dx0dx0P [x(t0), x(t0)]. The Gaussian factor is proportional to P [ξ] the

functional probability measure of the noise. The measure is Dξ ≡ ∏Nk=0 dξ(tk)

with k = 0, . . . ,N , tk = t0 + k(t − t0)/N and N → ∞ while (t − t0) remains

finite. The inverse kernel Γ−1 is defined within the interval [t0, T ]:∫ T

t0dt′′Γ(t−

t′′)Γ−1(t′′ − t′) = δ(t− t′).A very useful expression for Zd[η], usually called the Martin-Siggia-Rose

generating functional (actually derived by Jenssen), is obtained by introducingan identity

Eq[x(t)] ≡ mx(t) +

∫ T

t0

dt′ γ(t− t′)x(t′) + V ′[x(t)] = ξ(t) (108)

with the delta function

1 =

∫

Dx δ [Eq[x(t)] − ξ(t)]

∣

∣

∣

∣

detδEq[x(t)]

δx(t′)

∣

∣

∣

∣

, (109)

with Dx ≡∏Nk=1 dx(tk). The factor |det . . . | is the determinant of the operator

δ(t − t′)m∂2t + V ′′[x(t)] + γ(t − t′)∂t′ and ensures that the integral equals

one.4 The delta function can be exponentiated with an auxiliary field (using theFourier representation of the δ-function). The determinant can be exponentiatedwith time-dependent anticommunting variables – opening the way to the use ofsuper-symmetry [20], a subject that we shall not touch in these notes. However,since it does not yield a relevant contribution to the kind of dynamics we are

4Its origin is in the change of variables. In the same way as in the one dimensional integral,∫

dx δ[g(x)] =∫

dz 1/|g′[g−1(z)]| δ(z) = 1/|g′[g−1(0)]|, to get 1 as a result one includes the

inverse of the Jacobian within the integral:∫

dx δ[g(x)] |g′(x)| = 1.

48

interested in, we forget it (one can be more precise and show that using the Itoconvention to discretize stochastic differential equations the determinant equalsone and it is not relevant in other cases either). Zd reads

Zd[η, η] ≡∫

DξDxDixdP (t0)

×e−∫

T

t0dt′ ix(t′)

[

mx(t′)+∫

T

t0dt′′ γ(t′−t′′)x(t′′)+V ′[x(t′)]−ξ(t′)

]

×e−1

2kBT

∫

T

t0dt′∫

T

t0dt′′ ξ(t′)Γ−1(t′−t′′)ξ(t′′)+

∫

T

t0dt′ [η(t′)x(t′)+η(t′)ix(t′)]

where we have introduced a new source η(t), coupled to the auxiliary field ix(t).The integration over the noise ξ(t) is Gaussian and it can be readily done; ityields

+kBT

2

∫ T

t0

dt′∫ T

t0

dt′′ ix(t′) Γ(t′ − t′′) ix(t′′) (110)

and, for a coloured bath, the environment generates a retarded interactionin the effective action. In the usual white noise case, this term reduces to,

kBTγ0

∫ T

t0dt′ [ix(t′)]2, a local expression. In the end, the generating function

and resulting Martin-Siggia-Rose-Jenssen-deDominicis (MSRJD) action read

Zd[η, η] ≡∫

DxDix dP (t0) eS[x,ix,η,η]

S[x, ix, η, η] = −∫

dt′ ix(t′)

mx(t′) +

∫ T

t0

dt′′ γ(t′ − t′′)x(t′′) + V ′[x(t′)]

+kBT

2

∫

dt′∫

dt′′ ix(t′)Γ(t′ − t′′)ix(t′′) + sources . (111)

All integrals runs over [t0, T ].Interestingly enough, the dynamic generating functional at zero sources is

identical to one for any model:

Zd[η = 0, η = 0] = 1 (112)

as can be concluded from its very first definition.

4.2.1 Generic correlation and response.

The mean value at time t of a generic observable A is

〈A(t)〉 =

∫

DxDix dP (t0) A[x(t)] eS[x,ix] . (113)

The self-correlation and linear response of x are given by

C(t, t′) = 〈x(t)x(t′)〉 =1

Zd[η, η]

δ2Zd[η, η]

δη(t)δη(t′)

∣

∣

∣

∣

η=η=0

=δ2Zd[η, η]

δη(t)δη(t′)

∣

∣

∣

∣

η=η=0

(114)

49

R(t, t′) =δ〈x(t)〉δh(t′)

∣

∣

∣

∣

h=0

= 〈x(t)δS[x, ix;h]

δh(t′)〉 = 〈x(t)ix(t′)〉

=1

Zd[η, η]

δ2Zd[η, η]

δη(t)δη(t′)

∣

∣

∣

∣

η=η=0

=δ2Zd[η, η]

δη(t)δη(t′)

∣

∣

∣

∣

η=η=0

(115)

with h(t′) a small field applied at time t′ that modifies the potential energy V →V − h(t′)x(t′). The ix auxiliary function is sometimes called the response fieldsince it allows one to compute the linear response by taking its correlations withx. Similarly, we define the two-time correlation between two generic observablesA and B,

CAB(t, t′) ≡∫

DxDixdP (t0)A[x(t)]B[x(t′)] eS[x,ix] = 〈A[x(t)]B[x(t′)]〉 (116)

and the linear response of A at time t to an infinitesimal perturbation linearlyapplied to B at time t′ < t,

RAB(t, t′) ≡ δ〈A[x(t)]〉fBδfB(t′)

∣

∣

∣

∣

fB=0

, (117)

with V (x) 7→ V (ψ) − fBB(x). The function B(x) depends only on x (or onan even number of time derivatives, that is to say, it is even with respect tot → −t). By plugging eq. (113) in this definition we get the classical Kuboformula for generic observables:

RAB(t, t′) = 〈A[x(t)]δS[x, ix; fB]

δfB(t′)〉∣

∣

∣

∣

fB=0

= 〈A[x(t)]ix(t′)B′[x(t′)]〉 (118)

with B′[x(t′)] = ∂xB[x(t′)]. This relation is also causal and hence proportionalto θ(t− t′); it is valid in and out of equilibrium.

If the system has quenched random exchanges or any kind of disorder, onemay be interested in calculating the averaged correlations and responses overdifferent realizations of disorder. Surprisingly, this average is very easy to per-form in a dynamic calculation [41]. The normalization factors 1/Zd[η, η] in(114) and (115) have to be evaluated at zero external sources in which they aretrivially independent of the random interactions. Hence, it is sufficient to aver-age Zd[η, η] over disorder and then take variations with respect to the sourcesto derive the thermal and disorder averaged two-point functions. This propertycontrasts with an equilibrium calculation where the expectation values are givenby 〈A〉 = 1/Z

∑

confA exp(−βH), with · denoting the disorder average. In this

case, the partition function Z depends explicitly on the random exchanges andone has to introduce replicas [21] to deal with the normalization factor and dothe averaging.

Having assumed the initial equilibration of the environment ensures thata normal system will eventually equilibrate with it. The interaction with the

50

bath allows the system to dissipate its energy and to relax until thermalization isreached. However, in some interesting cases, as glassy models, the time neededto equilibrate is a fast growing function of N , the number of dynamic degreesof freedom. Thus, the evolution of the system in the thermodynamic limitoccurs out of equilibrium. In real systems, a similar situation occurs whenthe equilibration time crosses over the observation time and falls out of theexperimental time-window.

A final interesting remark on the relevance of quenched disorder is the fol-lowing. When a system with quenched disorder evolves out of equilibrium atfinite temperature, the correlation function and the response function do notdepend on the realization of disorder if the size of the system is large enough(the realization of disorder has to be a typical one). These quantities are self-averaging. This statement is easily checked in a simulation. When times getsufficiently long as to start seeing the approach to equilibrium, dependencies onthe realizations of disorder appear.

4.2.2 FDT as a consequence of a symmetry

If the initial time t0 is set to t0 = −T and the system is in equilibrium atthis instant, P−T is given by the Boltzmann measure. One can then check thatthe action, S[x, ix], is fully invariant under the field transformation Tc definedas

Tc ≡

xu 7→ x−u ,ixu 7→ ix−u + βdux−u .

(Note that dux−u = −d−ux−u. This transformation does not involve the kernelΓ and it includes a time-reversal. The invariance is achieved independentlyby the deterministic and dissipative terms in the action. The former includesthe contribution from the initial condition, lnPt0 . Moreover, the path-integralmeasure is invariant since the Jacobian is also block triangular with ones on thediagonal.

This symmetry implies

〈xtixt′〉S[x,ix] = 〈TcxtTcixt′〉S[Tcx,Tcix]

= 〈x−tix−t′〉S[x,ix] + β〈x−tdt′x−t′〉S[x,ix] (119)

that, using time-translational invariance and τ ≡ t− t′, becomes

R(τ) −R(−τ) = −βdτC(−τ) = −βdτC(τ) . (120)

For generic observables one can similarly apply the Tc transformation to expres-sion (118) of the linear response

RAB(τ) −RArBr(−τ) = −βdτCAB(−τ) = −βdτCAB(τ) . (121)

where we defined Ar and Br as

Ar[x(t)] ≡ A[x(−t),−t] . (122)

51

Take for instance a function A[x(t), t] =∫

duf(x(u))δ(u− t)+∫

duf(x(u))δ(u−t)+∫

duf(x(u))δ(u−t)+. . . then Ar[x(t), t] = A[x(−t),−t] =∫

duf(x(−u))δ(u+t) +

∫

duf(x(−u))δ(u+ t) +∫

duf(x(−u))δ(u + t) + . . ..Relations between higher order correlation functions evaluated at different

times t1, t2, . . . tn are easy to obtain within this formalism.

4.2.3 Equations on correlations and linear responses

Take any Langevin process in the Martin-Siggia-Rose path-integral formal-ism. From the following four identities

⟨

δix(t)

δix(t′)

⟩

=

⟨

δx(t)

δx(t′)

⟩

= δ(t− t′) ,

⟨

δx(t)

δix(t′)

⟩

=

⟨

δix(t)

δx(t′)

⟩

= 0 , (123)

where the angular brackets indicate an average with the Martin-Siggia-Roseweight, after an integration by parts, one derives four equations

⟨

x(t)δS

δx(t′)

⟩

= −δ(t− t′) ,

⟨

ix(t)δS

δix(t′)

⟩

= −δ(t− t′) , (124)

⟨

x(t)δS

δix(t′)

⟩

= 0 ,

⟨

ix(t)δS

δx(t′)

⟩

= 0 . (125)

The second and third one read⟨

ix(t)

mx(t′) +

∫

dt′′ γ(t′ − t′′) x(t′′) + V ′[x(t′)]

⟩

+kBT

∫

dt′′ Γ(t′ − t′′) 〈ix(t)ix(t′′)〉 = δ(t− t′) ,

⟨

x(t)

mx(t′) +

∫

dt′′ γ(t′ − t′′) x(t′′) + V ′[x(t′)]

⟩

+kBT

∫

dt′′ Γ(t′ − t′′) 〈x(t)ix(t′′)〉 = 0 , (126)

while the other ones, once causality is used (basically 〈x(t′)ix(t)〉 = 0 for t > t′

and 〈ix(t)ix(t′)〉 = 0) do not yield further information. All terms are eas-ily identified with the four types of two-time correlations apart from the onesthat involve the potential and are not necessarily quadratic in the fields. Thelinear terms in two-time functions can be put together after identifying thefree-operator

G−10 (t′, t′′) = δ(t′ − t′′)m

d2

dt′′2+ γ(t′ − t′′)

∂

∂t′′(127)

The non-linear terms can be approximated in a number of ways: perturba-tion theory in a small parameter, Gaussian approximation of the MSR action,

52

self-consistent approximations, etc. The choice should be dictated by someknowledge on the system’s behaviour one wishes to reproduce. In short then

0 =

∫

dt′′G−10 (t′, t′′)C(t′′, t) + 〈x(t)V ′[x(t′)]〉 + kBT

∫

dt′′Γ(t′ − t′′)R(t, t′′) ,

δ(t− t′) =

∫

dt′′G−10 (t′, t′′)R(t′′, t) + 〈ix(t)V ′[x(t′)]〉 . (128)

4.3 The random manifold

The problem of an oriented manifold embedded in a space with randomimpurities is relevant for a large class of physical systems. The dimension ofthe manifold is called d while the dimension of the embedding space is calledN ; there are then N transverse dimensions to the manifold. Both d and N areimportant to determine the behaviour of the manifold. Different choices of dand N are associated with different physical problems. If N = 1 the modeldescribes the interface between two coexisting phases. For d = 1 the model isthat of a directed polymer which also describes the interaction of a flux line ina type II superconductor. This problem shares many of the interesting featuresof other disordered system such as spin-glasses.

Attention has been first paid to the study of the equilibrium properties of themanifold. Mainly two approaches have been applied. On the one hand, Mezardand Parisi [9] proposed a Gaussian Variational method complemented by thereplica analysis with an ensuing replica symmetry breaking solution. This studyallowed to obtain very interesting results such as non-trivial - Flory-like - ex-ponents for the displacement of the manifold, sample to sample susceptibilityfluctuations, etc. On the other hand, the functional renormalization group andthe relation between the two in the large N limit was discussed by Le Doussaland collaborators [4]. The extension of the large N treatment and the GaussianVariational Approach (GVA) to the out of equilibrium relaxation has been de-veloped and we discuss it below [10]. The dynamic functional renormalizationgroup to treat the out of equilibrium dynamics is not ready yet.

The model of a manifold of internal dimension d embedded in a randommedium is described, in terms of an N component displacement field, ~φ =(φ1, φ2, . . . , φN ), by the Hamiltonian

H =

∫

ddx

[

1

2(~∇~φ(~x))2 + V (~φ(~x), ~x) +

µ

2~φ2(~x)

]

. (129)

µ is a mass, which effectively constraints the manifold to fluctuate in a restrictedvolume of the embedding space. The elastic terms has to be read as

~∇~φ(~x) =

d∑

i=1

N∑

α=1

∂

∂xiφα(~x)

∂

∂xiφα(~x) . (130)

53

The Hamiltonian is invariant under rotations of the vector ~φ. (~φ, ~x) = ~φ(~x) is apoint in the N + d dimensional space. V is a Gaussian random potential withzero mean and correlations

V (~φ, ~x)V (~φ′, ~x′) = −Nδd(~x− ~x′) ∆

(

|~φ− ~φ′|2N

)

. (131)

The factors N are introduced to have a good N → ∞ limit. The overlinerepresents the average over the distribution of the random potential V . Welabel with greek indices the N -components of the field ~φ, φα, α = 1, . . . , N .The internal coordinate ~x has d components ~x = (x1, . . . , xd) and we label themwith latin indices, xi, i = 1, . . . , d. Note the similarity with the O(N) coarseningproblem. We study this problem in the infinite volume limit −∞ < φα < ∞and −∞ < xi <∞.

Equilibrium properties

The equilibrium properties of the manifold are often described in terms of thedisplacement of the manifold Dst(~x, ~x′) that is characterized by the roughnessexponent ζ:

Dst(~x, ~x′) ≡ 〈(

~φ(~x) − ~φ(~x′))2

〉 ∼ |~x− ~x′|2ζ . (132)

The angular brackets denote a thermal average. The random potential andthe thermal fluctuations make the manifold roughen and this impies that thefluctuations of the field ~φ diverge at large distances |~x− ~x′| ≫ 1.

In the absence of disorder the manifold is flat if the internal dimension islarger than two: for d > 2, ζ = 0. Instead, if the internal dimension is smallerthat two the manifold roughens: for d < 2, ζ = (2 − d)/2. The situationchanges in the presence of disorder where ζ depends on d and N non-trivially.For instance, for d > 4 the manifold remains flat; for 2 < d < 4 there is onlya disorder phase with a non-trivial exponent ζ, finally, for d < 2, and largeenough N there is a high-temperature phase and a randomness dominated low-temperature phase with ζ increasing with N and approaching 1/2 for N → ∞as suggested by numerical simulations for finite N and calculations for N → ∞.Whether there is a finite upper critical Nc such that for N > Nc ζ = 1/2 is notclear.

There are several models which can be studied for N → ∞ corresponding todifferent choices of the random potential correlation ∆(z). A common choice isthe ‘power law model’

∆(z) =(θ + z)1−γ

2(1 − γ). (133)

There are two physically distinct cases. If γ(1 − d/2) < 1 the correlations growwith the distance z and the potential is called long-range. If γ(1− d/2) > 1 the

54

tt′

φ

xx′

Figure 11: Sketch of the field configuration for N = d = 1. The arrows indicatethe local displacements that are correlated in the computation of B(~x, t; ~x′, t′).The usual roughness is computed, instead, on an instantaneous configuration.

correlations decay with the distance and the potential is called short-range. Thestatics of these models, studied with the Gaussian Variational method comple-mented by the replica method [9], is solved with a one step replica symmetricansatz in the short-range case and with a full replica symmetry breaking schemein the long-range one.

4.3.1 The dynamic equations

A dynamic Gaussian variational approximation becomes exact in the limitN → ∞. We consider the overdamped Langevin dynamics with white noise

∂~φ(~x, t)

∂t= − δH

δ~φ(~x, t)+ ~ξ(~x, t) , (134)

with 〈ξα(~x, t)〉 = 0 and 〈ξα(~x, t)ξβ(~x′, t′)〉 = 2kBT δαβ δd(~x − ~x′)δ(t − t′). This

is a time-dependent Ginzburg-Landau approach in which the Hamiltonian Hhas to be interpreted as being now a functional of ~φ(~x, t), including an integralover time.

55

In the dynamics the average over disorder can be done without introducingreplicas just because the dynamic generating functional in the absence of sourcesis equal to one. Thus, one can work with the averaged ZD directly:

ZD =

∫

DφDiφ dV P [V ] eS[φ,iφ;V ] (135)

with

P [V ] ∝ e12

∫

dNφdNφ′ddxddx′ V [~φ,~x] N−1δ(~x−~x′) ∆−1(|~φ−~φ′|2/N) V [~φ′,~x′] (136)

and the inverse is in the operational sense. The quenched potential appears inthe MSRJD action in the term

∑

α

∫

dtddx iφα(~x, t)∂

∂φα(~x, t)V [~φ(~x, t), ~x] (137)

that is local in time and space. Thus the Gaussian average over V gives rise to

−iφα(~x, t) N∂2 ∆(|(~φ(~x, t) − ~φ(~x, t′)|2/N)

∂φα(~x, t)∂φβ(~x, t′)iφβ(~x, t

′) (138)

and we should take∑

αβ

∫

dtdt′ddx of this expression. This term is still localin space but delayed in time. Moreover, it is a highly non-linear contributionto the action. Equations (124) and (125) then give rise to slighly more complexequations. These terms can now be treated in an approximation in which one

assumes that the fields ~φ and i~φ have Gaussian statistics. Using 〈ϕαF [~ϕ]〉 =

∑

β〈ϕαϕβ〉〈δF [~ϕ]/δϕβ〉 one has

δαβδ(t− t′)δd(~x − ~x′) =(

∂t −∇2 + µ)

Rαβ(~x, t; ~x′, t′)

−∑

γδǫ

∫

dt′′Rδγ(~x, t; ~x, t′′) [Rǫβ(~x, t; ~x

′, t′) −Rǫβ(~x, t′′; ~x′, t′)]

×N⟨

∆(4)αγδǫ

(

[~φ(~x, t) − ~φ(~x, t′)]2/N)⟩

2kBTRαβ(~x′, t′; ~x, t) =

(

∂t −∇2 + µ)

Cαβ(~x, t; ~x′, t′)

−∑

γ

∫

dt′′Rαγ(~x′, t′; ~x, t′′) N

⟨

∆′′βγ

(

[~φ(~x, t) − ~φ(~x, t′)]2/N)⟩

+∑

γδǫ

∫

dt′′Rδγ(~x, t; ~x, t′′) [Cǫα(~x, t; ~x, t′) − Cǫα(~x, t′′; ~x, t′)]

×N⟨

∆(4)βγδǫ

(

[~φ(~x, t) − ~φ(~x, t′)]2/N)⟩

(139)

The correlations and linear responses in these equations are averaged over thenoise and the quenched disorder

Cαβ(~x, t; ~x′, t′) = 〈φα(~x, t)φβ(~x′, t′)〉 Rαβ(~x, t; ~x

′, t′) = 〈φα(~x, t)iφβ(~x′, t′)〉 .

56

Assuming now the isotropy in the N -directions,

Rαβ = δαβR , Cαβ = δαβC (140)

the sums over components simplify; moreover, in the large N limit only thefollowing averages intervene:

⟨

∆′′αβ(ϕ

2/N)⟩

= δαβ∆′(〈ϕ2〉/N)

∑

γ

⟨

∆(4)γγαβ(ϕ

2/N)⟩

= 4δαβ∆′′(〈ϕ2〉/N) .

One finally has the dynamic equations for the correlations and linear re-sponses

∂R(~x, t; ~x′, t′)

∂t= (∇2 − µ)R(~x, t; ~x′, t′) + δd(~x− ~x′) δ(t− t′) (141)

+4

∫ t

0

ds V ′′(B(~x, t; ~x, s)) R(~x, t; ~x, s) (R(~x, t; ~x′, t′) −R(~x, s; ~x′, t′))

∂C(~x, t; ~x′, t′)

∂t= (∇2 − µ)C(~x, t; ~x′, t′) + 2kBT R(~x′, t′; ~x, t)

+2

∫ t′

0

ds V ′(B(~x, t; ~x, s)) R(~x′, t′; ~x, s)

+4

∫ t

0

ds V ′′(B(~x, t; ~x, s)) R(~x, t; ~x, s) (C(~x, t; ~x, t′) − C(~x, s; ~x′, t′)) .

with

B(~x, t; ~x′, t′) ≡ N−1〈[~φ(~x, t) − ~φ(~x, t′)] [~φ(~x′, t) − ~φ(~x′, t′)]〉= C(~x, t; ~x′, t) − C(~x, t; ~x′, t′) − C(~x, t′; ~x′, t) + C(~x, t′; ~x′, t′) .

We use the Ito convention limǫ→0R(~x, t; ~x′, t−ǫ) = δd(~x−~x′), limǫ→0R(~x, t; ~x, t+ǫ) = 0. Using the fact that R(~x′, t′; ~x, t) vanishes for t > t′ and all ~x, ~x′ one alsohas

1

2

dC(~x, t; ~x′, t)

dt= (∇2 − µ)C(~x, t; ~x′, t) + kBT

+2

∫ t

0

ds ∆′(B(~x, t; ~x, s)) R(~x, t; ~x, s) + 2

∫ t

0

ds ∆′′(B(~x, t; ~x, s)) R(~x, t; ~x, s)

× (C(~x, t; ~x′, t) − C(~x, s; ~x′, s) +B(~x, t; ~x′, s)) . (142)

In these equations there are two external parameters T and µ. In deriving theseequations a mean over random initial condition φ(~x, t = 0) [with a Gaussiandistribution of variance C(~x, 0; ~x′, 0)] is implicit. One could certainly obtain thedynamical equations for a particular initial condition φ(t = 0) = φo or for adifferent distribution of initial conditions and attempt to study their effects in

57

detail. We shall not do so here but we shall restrict our analysis to the studyof the above dynamical equations.

The dynamics is described by the set (141)-(142) of non-linear coupledintegro-differential equations that admit a unique solution for each ‘initial auto-correlation’ C(~x, 0; ~x′, 0). The set of equations is causal. One can in principleuse a numerical algorithm to iterate the equations and construct the space de-pendent solution step by step in time.

We shall choose a space homogeneous initial condition such that the corre-lations at all subsequent times are translational invariant in space, they dependonly upon space-differences

C(~x, 0; ~x′; 0) = C(~x − ~x′; 0, 0) ⇒ C(~x, t; ~x′; t′) = C(~x− ~x′; t, t′) (STI)

As a consequence, the dynamical equations for the Fourier transformed - withrespect to ~x − ~x′ - correlation and response functions are easier to treat. Theyread

∂R(~k; t, t′)

∂t= −(k2 + µ)R(~k, t; t′)

+4

∫ t

0

ds ∆′′(B(t, s)) R(t, s)(

R(~k; t, t′) −R(~k; s, t′))

,

∂C(~k; t, t′)

∂t= −(k2 + µ)C(~k; t, t′) + 2

∫ t′

0

ds ∆′(B(t, s))R(~k; t′, s)

+4

∫ t

0

ds ∆′′(B(t, s))R(t, s)[C(~k; t, t′) − C(~k; s, t′)] + 2kBTR(~k; t′, t) ,

1

2

dC(~k, t, t)

dt= −(k2 + µ)C(~k; t, t) + kBT + 2

∫ t

0

ds ∆′(B(t, s))R(~k; t, s)

+2

∫ t

0

ds ∆′′(B(t, s)) R(t, s)(

C(~k; t, t) − C(~k; s, s) +B(~k; t, s))

,

where B(t, t′) ≡∫

kB(~k; t, t′) = B(~x = 0; t, t′) and R(t, t′) ≡

∫

kR(~k; t, t′) =

R(~x = 0; t, t′).∫

k ≡∫

(ddk)/(2π)d represents the integration over the k-space.

There is no angle dependence in the ~k-space; thus

R(~k; t, t′) = R(k; t, t′) , C(~k; t, t′) = C(k; t, t′) . (143)

It is convenient to write down the equations in terms of B(k; t, t′):

∂B(k; t, t′)

∂t= −(k2 + µ) [C(k; t, t) − C(k; t′, t′) +B(k; t, t′)] + 2kBT

+ 4

∫ t

0

ds ∆′(B(t, s)) (R(k; t, s) −R(k; t′, s))

+ 4

∫ t

0

ds ∆′′(B(t, s))R(t, s) [B(k; t, s) +B(k; t, t′) −B(k; s, t′)] .

58

These equations can also be written in a more compact form

∫

dt′′ G−10 (k; t, t′′)R(k; t′′, t′) = δ(t− t′)

+

∫ t

0

dt′′ ΣR(t, t′′)R(k; t′′, t′) , (144)

∫

dt′′ G−10 (k; t, t′′)C(k; t′′, t′) =

∫ t′

0

dt′′ ΣK(t, t′′)R(k; t′, t′′)

+

∫ t′

0

dt′′ ΣR(t, t′′)C(k; t′′, t′) + 2kBTR(k; t′, t) , (145)

with

G−10 (k; t, t′) ≡ δ(t− t′)

[

∂

∂t′+ k2 − µ+ a(t)

]

(146)

a(t) ≡∫ t

0

dt′′ ΣR(t, t′′) (147)

ΣR(t, t′) ≡ −4∆′′(B(t, t′))R(t, t′)

ΣK(t, t′) ≡ 2∆′(B(t, t′)) (148)

Note that the non-interacting part of the equations has the same structure as inthe O(N) ferromagnetic model. The quenched disorder generated the interactionterms that involve ΣR and ΣK .

The problem is now reduced to solving this set of integro-differential equa-tions. Let us summarize the steps followed

• We derive integro-differential equations for macroscopic two-time spacedependent correlations and linear responses using a Gaussian decoupling.One can prove that this procedure is exact in the limit N → ∞.

• Due to isotropy in the N -dimensional space we reduced the α-coordinatedependence and focused in one (any) αα component.

• For initial conditions that are translational invariant in the d-dimensionalspace, the time-dependent evolution respects this property.

• Causality implies that we can set to zero two of the four correlators, thehat-hat one and the advanced response.

• The Fourier mode dependence is then relatively similar to the one found inthe O(N) coarsening model: the modes are coupled only through ‘global’quantities, B(t, t′) and R(t.t′) in this case.

• The equations for correlations and linear responses are coupled in a highlynon-trivial manner. One cannot disentagle this coupling without knowing

59

which is the relation between correlations and linear responses, that is tosay, knowing what happens with the fluctuation dissipation theorem outof equilibrium.

4.3.2 Relaxation

First one can solve the equations numerically and get an idea of what thesystem is doing.

The lessons learned from coarsening suggest that two-time correlations mighthave a separation of time-scales with a stationary regime for t− t′ ≪ t′ and anaging one for much longer time-differences. This is indeed what happens in therandom manifold problem too and what allows us to solve the dynamic equationsin the long times limit where the separation of time scales is sharp. The agingbehaviour depends strongly on the kind of quenched disorder, whether it isshort-range or long-range. For ease of discussion we shall focus on the short-range case only, which is the one that resembles most the simple coarseningproblems.

Phase transition

The study of the free-energy using the replica trick and a Gaussian varia-tional approach yields a phase transition at a finite temperature Tc. The natureof the transition depends on the range of the disorder correlation. In the fol-lowing we review the main features of the dynamics in the low-T phase and wefocus on models with short-ranged disorder correlators.

One-time quantities

The manifold is not expected to diffuse; thus one-time quantities includ-ing C(k; t, t) should approach a constant limit when times diverge. Thus, theequation for C(t, t) simplifies in this limit and, moreover, the evolution can bedescribed by a set of two equations involving R and B, only, eqs. (145) and(144).

The analysis of the these equations shows that below Td the energy densityapproaches an asymptotic value that is not the one computed in equilibrium(studying the Gibbs-Boltzmann measure). The reason for this is that belowTd there is a proliferation of metastable states that “trap” the system withoutletting it relax to equilibrium. The barriers between these states diverge withthe size of the system and are then unsurmountable in the thermodynamiclimit. The “basin of attraction” of the typical initial conditions is what hasbeen named the threshold, a region in the free-energy landscape that is madeof flat directions and does not stop the dynamics.

k-dependence

60

The k-dependence is specially simple and can be read from the dynamicequations. Indeed, one first writes the Schwinger-Dyson equations in a matricialform

∫

dt′′G−10 (k; t, t′′)Q(k; t′′, t) = δ(t− t′) + Σ(t, t′′)Qt(k; t′′, t′) (149)

with G−10 (k; t, t′′) = G−1

0 (k = 0; t, t′′) + k2δ(t − t′′) and Qt(k; t, t′) = Q(k; t′, t).One then inverts this equation noting that Σ(t, t′′) only depends on Q(t, t′′) butnot on Q(k; t, t′′). Then,

Q(k; t, t′) = [k2δ(t− t′) +Q−10 (t, t′)]−1 (150)

Thus, one needs to solve the equation for the mode k = 0 and then this ‘algebraicrelation’ yields the k dependence.

Separation of time scales

The relaxation of Bk and Rk show a neat separation of time scales betweena stationary regimes (at short time-differences) and an aging regime (at longtime-differences). The separation of time-scales is then described by

Bk(t, tw) ≃ bFk (τ) + bk(τ + tw, tw) , Rk(t, tw) ≃ rFk (τ) + rk(τ + tw, tw) ,

Taking first the long waiting-time limit one has

limtw→∞

Bk(t, tw) = bFk (τ) ,

limτ→∞

bFk (τ) = beak

limtw→∞

bk(τ + tw, tw) = 0 . (151)

For all tw and sufficiently large τ , Bk(τ + tw, tw) continues to grow beyond beakup to

b0k ≡ limτ→∞

Bk(τ + tw, tw) (152)

and

limτ→∞

bk(τ + tw, tw) = b0k − beak . (153)

beak is the mode-dependent Edwards-Anderson order parameter and b0k is themaximum distance allowed by the harmonic well controlled by µ. The functionsbFk and bk vary in completely different two-time regimes, the stationary andaging ones, respectively. A similar separation applies to Rk and its integral

χk(t, tw) =

∫ t

tw

dt′Rk(t, t′) . (154)

61

As regards the measurements of noise and susceptibility the stationary regimecorresponds to high frequencies while the aging one corresponds to low frequen-cies (scaling with the waiting time) such that noise and susceptibility dependon tw. In a domain-growth process the former corresponds to the fast ther-mal fluctuations of the spins around their mean magnetisation while the lattercorresponds to the actual growth of the domains.

Stationary regime

In this time regime the displacement is time-translation invariant (TTI) andthe response rFk (τ) = limtw→∞Rk(τ + tw, tw) satisfies FDT

rFk (τ) = 1/(2kBT ) ∂τ bFk (τ) θ(τ) (FDT) (155)

The analysis of the stationary equation on bFk (τ) yields

bea =4kBTcd(d− 2)

[

−4cd∆′′(bea)

](d−2)/(4−d)

beak =2kBT

k2 + a+ µ

a+ µ =[

−4cd∆′′(bea)

]2/(4−d)

(156)

for d < 4 and cd =∫

k(k2+1)−2. a, the anomaly, is a kind of effective mass. The

‘FDT-regime’ is very much like an equilibration relaxation in a harmonic well,the manifold looks pinned with an effective mass a+ µ. However, the approachto the asymptotic beak is achieved with a power law

bFk (τ) → beak − ctk τ−a(T ) (157)

with a(T ) a temperature-dependent exponent that increases from a(0) = 1/2 toa(Tc) = ac but does not depend on k. (While in the harmonic well the approach

is exponential 〈(x(t) − x(t′))2〉 = 2kBT [1 − e−(k2+µ)(t−t′)] for γ = 1.)Note that in the zero temperature limit the stationary part of B vanishes

(beak ∝ T ) but it has a non-trivial contribution to rFk and χFk .

Aging regime

The growth of Bk now depends on tw: the larger tw the slower the motionof the system, it ages. Thus pinning is gradual: the older the system the longerit is pinned but it is not pinned forever. The aging time-regime is defined (fortw → ∞) as the times τ such that Bk(τ + tw, tw) > beak .

The aging dynamics are summarized by the following two important features:

b(t, tw) ≃ fb

(

h(t)

h(tw)

)

χ(t, tw) ≃ fχ

(

h(t)

h(tw)

)

(158)

62

and the FDT relation takes a non-trivial form

χ(t, tw) = fχ[

f−1b (b)

]

(159)

Moreover, one finds

χ(t, tw) =1

2Teff

(b(t, tw)−bea)+ 1

2Tbea(t, tw) for times such thatb > bea .

(160)Teff is an effective temperature that replaces the bath temperature in the agingregime [2] A more careful analysis of the FDRs for different k’s yields

Teff[k;B(k)] = Teff(B) . (161)

The meaning of this relation is the following: taking a large time tw and alarger or equal time t one can measure the displacement B(k) = B(k; t, tw). Atthe same times, there is an associated averaged displacement averaged over themodes B(t, tw). If one measures then the FDT violating factors Teff(k,B(k))and Teff(B) the relation (161) tells us that the numerical values measured mustbe equal. The effective temperature does not depend on the wave vector.

The departure from the plateau is given by

b(t, tw) − bea ≃ lnb(T )

[

h(t)

h(tw)

]

≃(

τ

d lnh(tw)/dtw

)b(T )

(162)

and b(T ) is another non-trivial exponent that depends on T and d. The asymp-totic value is

b0k =Teff

T

[

1

k2 + µ+ a− 1

k2 + µ

]

. (163)

The effective temperature takes the value

T

Teff

= −M4

[

∆′(bea) − ∆′(b0)]

. (164)

4.4 Finite N case: simulations

Many simulations in finite d and finite N have been performed, especially inthe case d = N = 1. The results are similar to the previous ones but with, alsosome important differences. A summary of the situation is given in [1]

4.5 Interacting systems with no quenched disorder

It turns out that the dynamics of systems of particles in interaction (e.g.Lennard-Jones mixtures an the like) once treated with self-consistent approxi-mations, mode-coupling and similar ones, yield coupled integro-differential equa-tion with very similar structure to the ones derived above. The results obtainedfor the random manifold (although in not all their details) apply then to glassysystems of more traditional nature, as the ones shown in Fig. 1.

63

5 Quantum formalism and glassy dynamics

We now consider a quantum system

[p, x] = −ih . (165)

The density operator evolves according to the equation

∂ρ(t)

∂t= − i

h[H(t), ρ(t)] (166)

that is solved by

ρ(t) = U(t, t0)ρ(t0)[U(t, t0)]† = U(t, t0)ρ(t0)U(t0, t) (167)

with

U(t, t0) = T e− ih

∫

t

t0dt′H(t′)

t > t0 . (168)

Averaged observables read

〈A(t)〉 =Tr

A ρ(t)

Trρ(t)=

Tr

A T e−ihHt ρ0 [T e−

ihHt]†

Tr

T e−ihHt ρ0 [T e−

ihHt]†

=Tr

T [e−ihHt]† A T e−

ihHtρ0

Trρ0. (169)

(To ease the notation we do not write the possible time-dependence in the Hamil-tonian and the integral over t in the exponential.) We then use the Heisenbergoperators

AH(t) ≡ [T e−ihHt]† A T e−

ihHt (170)

The underscript H recalls that these are operators in the Heisenberg represen-tation but we shall drop it in the following.

5.0.1 Feynman path integral for quantum time-ordered averages

Feynman’s path-integral represents the (forward) propagator K(x, t;x′, t′) =〈x|e−iH(t−t′)/h|x′〉 as a sum over all paths leading from, in the Lagrangain rep-resentation, x′ = x(t′) to x = x(t) with t > t′ weighted with the exponential ofthe action (times i/h):

K(x, t;x′, t′) =

∫ x(t)=x

x(t′)=x′

Dx e ihS[x,x] (171)

S[x, x] =

∫ t

t′dt′′

m

2x2(t′′) − V [x(t′′)]

. (172)

64

The formalism is well-suited to compute any type of time-ordered correlationsince the evolution is done towards the future. A concise review article on theconstruction of path-integrals with some applications is [17]; several books onthe subject are [18].

5.0.2 The Matsubara imaginary-time formalism

The statistical properties of a canonical quantum system and, in particular,its equilibrium partition function, can also be obtained with the path-integralmethod,

Z = Tre−βH = Tre−iH(t=−iβh)/h =

∫

dx K(x,−iβh;x, 0) . (173)

Having used an imaginary-time t = −iβh corresponds to a Wick-rotation. Thepropagator K is now represented as a path-integral with periodic boundaryconditions to ensure the fact that one sums over the same initial and finitestate. Introducing τ = it, τ : 0 → βh (since t : 0 → −iβh), defining functionsof τ , x(τ) = x(t = −iτ) and dropping the tilde one has

K(x,−iβh;x, 0) =

∫

x(0)=x(βh)

Dx e− 1hSE [x,x] (174)

with the Euclidean action

SE[x, x] =

∫ βh

0

dτ[m

2x2(τ) + V [x(τ)]

]

. (175)

5.0.3 The equilibrium reduced density matrix

We now follow the same route as in the derivation of the classical Langevinequation by coupling the system to an ensemble of quantum harmonic oscilla-tors,

[πa, qb] = −ihδab . (176)

The equilibrium density matrix reads

ρtot(x′′, q′′a ;x

′, q′a) =1

Ztot

〈x′′, q′′a | e−βHtot |x′, q′a〉 , (177)

with the partition function given by Ztot = Tre−βHtot and the trace taken overall the states of the full system. As usual the density matrix can be representedby a functional integral in imaginary time,

ρtot(x′′, q′′a ;x

′, q′a; η) =1

Ztot

∫ x(hβ)=x′′

x(0)=x′

Dx∫ qa(hβ)=q′′a

qa(0)=q′a

Dqa e−1hSe

tot[η] . (178)

65

The Euclidean action SE has contributions from the system, the reservoir, theinteraction and the counterterm: SEtot[η] = SEsyst + SEenv + SEint + SEcounter +∫ βh

0 dτ η(τ)x(τ) and we have included a source η coupled to the particle’s co-ordinate in order to use it to compute expectation values of the kind definedin (169). The environment action is

SEenv =

N∑

α=1

∫ βh

0

dτ

mα

2[qα(τ)]2 +

mαω2α

2[qα(τ)]2

, (179)

that is to say, we choose an ensemble of independent oscillators. For simplicitywe take a linear coupling

SEint =

∫ hβ

0

dτ x(τ)

N∑

a=1

caqa(τ) (180)

but others, as used in the derivation of the Langevin equation, are also possible.As in the classical case, the path integral over the oscillators’ coordinates andpositions is quadratic. The calculation of expectation values involves a traceover states of the full system. For operators that depend only on the particle,as A(x) above, the trace over the oscillators can be done explicitly. Hence, ifone constructs the reduced equilibrium density operator ρred = Trenv ρtot thatacts on the system’s Hilbert space, the expectation value of the observables ofthe system is given by

〈A(x)〉 =Trsyst A(x)ρred

Trρtot

. (181)

In the path-integral formalism this amounts to performing the functional inte-gral over periodic functions qα(τ). From the point of view of the oscillators thesystem’s coordinate is an external τ -dependent force. Using a Fourier represen-tation, qα(τ) =

∑∞n=−∞ qnα e

iνnτ with νn = 2πnβh the Matsubara frequencies, the

integration over the qα(τ) can be readily done [19]. A long series of steps, verycarefully explained in [15] allow one to obtain the reduced density matrix:

ρred(x′′, x′; η) = Trenvρtot(x′′, q′′a ;x

′, q′a; η)

=1

Zred

∫ x(hβ)=x′′

x(0)=x′

Dx e− 1hSEsyst−

1h

∫

hβ

0dτ∫

τ

0dτ ′ x(τ)K(τ−τ ′)x(τ ′)

(182)

where Zred is the partitition function of the reduced system, Zred = Ztot/Zenv

and Zenv the partition function of the isolated ensemble of oscillators. Theinteraction with the reservoir generated a renormalization of the mass – can-celled by the counterterm – but also a retarded interaction in the effective actioncontroled by the kernel

K(τ) =2

πhβ

∞∑

n=−∞

∫ ∞

0

dωS(ω)

ω

ν2n

ν2n + ω2

eiνnτ , (183)

66

with S(ω) the spectral density of the bath, see eq. (20) for its definition. Thelast retarded interaction in (182) remains. The imaginary time dependence ofK varies according to S(ω). Power laws in S lead to power-law decays in Kand thus to a long-range interaction in the imaginary-time direction.

The effect of the quantum bath is more cumbersome than in the classicalcase and it can lead to rather spectacular effects. A well-known example is thelocalization transition, as function of the coupling strength to an Ohmic bath inthe behaviour of a quantum particle in a double well potential [37]. This meansthat quantum tunneling from the well in which the particle is initially locatedto the other one is completely suppressed by sufficiently strong couplings to thebath. In the context of interacting macroscopic systems, e.g. ferromagnets orspin-glasses, the locus of the transition between the ordered and the disorderedphase depends strongly on the coupling to the bath and on the type of bathconsidered [45], as discussed in Section ??.

5.1 Quantum dynamics

We shall see that the distinction between the effect of a reservoir on thestatistical properties of a classical and quantum system is absent from a fullydynamic treatment. In both classical and quantum problems, the coupling to anenvironment leads to a retarded interaction. In classical problems one generallyargues that the retarded interaction can be simply replaced by a local one dueto the very short correlation times involved in cases of interest, i.e one useswhite baths, but in the quantum problems one cannot do the same.

5.1.1 Schwinger-Keldysh path integral

The Schwinger-Keldysh formalism [42, 43] allows one to analyse the real-time dynamics of a quantum system. The starting point is the time dependentdensity operator

ρ(t) = T e−ihHt ρ(0) T e

ihHt . (184)

We have set the initial time to be to = 0. Introducing identities, an element ofthe time-dependent density matrix reads

ρ(x′′, x′; t) =∫ ∞

−∞

dXdX ′ 〈x′′| T e−ihHt |X〉 〈X | ρ(0) |X ′〉

×〈X ′| T eih Ht |x′〉 . (185)

The first and third factors are the coordinate representation of the evolution

operators e−iHt/h and eiHt/h, respectively, and they can be represented as func-tional integrals:

〈x′′|T e−ihHt |X〉 =

∫ x′′

X

Dx+ eihS+

(186)

67

〈X ′|Te ih Ht |x′〉 =

∫ X′

x′

Dx− e−ihS−

. (187)

Interestingly enough, the evolution operator in eq. (248) gives rise to a pathintegral going backwards in time, from x−(t) = x′ to x−(0) = X ′. The fulltime-integration can then be interpreted as being closed, going forwards fromt0 = 0 to t and then backwards from t to t0 = 0. This motivates the name closedtime-path formalism. A doubling of degrees of freedom (x+, x−) appeared and itis intimately linked to the introduction of Lagrange multipliers in the functionalrepresentation of the Langevin dynamics in the classical limit, see Sect. . Theaction of the system has two terms as in (172) one evaluated in x+ and theother in x−.

S±syst[η

+] =

∫ t

0

dt′[ m

2

(

x±(t′))2 − V (x±(t′)) + η±(t′)x±(t′)

]

(188)

where we have introduced two time-dependent sources η±(t), that appear in theexponential with the sign indeicated.

5.1.2 Green functions

We define the position and fermionic Green functions

GBab(t, t′) ≡ −i〈TC xa(t)xb(t

′)〉 (189)

GFab(t, t′) ≡ −i〈TC ψa(t)ψ

†b(t

′)〉 (190)

where a, b = ± and TC is time ordering on the close contour (see App. 6).The definitions (189) and (190) and the fact that the insertion evaluated

at the later time can be put on the upper (+) or lower (−) branch indistinc-tively *** GIVE EXAMPLE WITH DRAWING *** yield the following relationsbetween different Green functions

G++(t, t′) = G−+(t, t′)θ(t− t′) +G+−(t, t′)θ(t′ − t) , (191)

G−−(t, t′) = G+−(t, t′)θ(t− t′) +G−+(t, t′)θ(t′ − t) , (192)

that hold for the bosonic and fermionic cases as well. We thus erase the super-scripts B,F that become superflous. Adding the last two identities one finds

G++ +G−− −G+− −G−+ = 0 for all t and t′ . (193)

In both cases one defines MSRDJ-like fields. For bosons these are√

2 x(t) = x+(t) + x−(t) ,√

2h x(t) = x+(t) − x−(t) , (194)

while for fermions they are√

2 ψ(t) ≡ ψ+(t) + ψ−(t)√

2h ψ(t) ≡ ψ+(t) − ψ−(t)√2 ψ†(t) ≡ ψ+†(t) + ψ−†(t)

√2h ψ†(t) ≡ ψ+†(t) − ψ−†(t) .

68

One then constructs the Green functions

GBxx(t, t′) ≡ −i〈TC x(t)x(t′)〉 =

1

2

(

GB++ +GB−− +GB−+ +GB+−

)

≡ −2i GBK(t, t′)

GBxx(t, t′) ≡ −i〈TC x(t)x(t′)〉 =

1

2h

(

GB++ −GB−− +GB−+ −GB+−

)

≡ −GBR(t, t′)

GBxx(t, t′) ≡ −i〈TC x(t)x(t′)〉 =

1

2h

(

GB++ −GB−− −GB−+ +GB+−

)

≡ − GBA(t, t′)

GBxx(t, t′) ≡ −i〈TC x(t)x(t′)〉 =

1

2h2

(

GB++ +GB−− −GB−+ −GB+−

)

≡ − GB4 (t, t′) = 0 , (195)

and the fermionic ones

GFψψ(t, t′) ≡ −i〈TC ψ(t)ψ†(t′)〉 =1

2

(

GF++ +GF−− +GF−+ +GF+−

)

≡ −2iGFK(t, t′) ,

GFψψ

(t, t′) ≡ −i〈TC ψ(t)ψ†(t′)〉 =1

2h

(

GF++ −GF−− +GF−+ −GF+−

)

≡ −GFR(t, t′) ,

Gψψ(t, t′) ≡ −i〈TC ψ(t)ψ†(t′)〉 =1

2h

(

GF++ −GF−− −GF−+ +GF+−

)

≡ −GFA(t, t′) ,

GFψψ

(t, t′) ≡ −i〈TC ψ(t)ψ†(t′)〉 =1

2h2

(

GF++ +GF−− −GF−+ −GF+−

)

≡ −GF4 (t, t′) = 0 , (196)

From the definition of GBK it is obvious that

GBK(t, t′) = GBK(t′, t) ∈ Re . (197)

Using eq. (193) the ‘rotated’ Green functions are rewritten as

GK =i

2(G++ +G−−) =

i

2(G+− +G−+) , (198)

GR = − 1

h(G++ −G+−) =

1

h(G−− −G−+) , (199)

GA = − 1

h(G++ −G−+) =

1

h(G−− −G+−) . (200)

Using eqs. (191)-(192) one also finds

GR(t, t′) = − 1

h[G+−(t, t′) −G−+(t, t′)] θ(t− t′) , (201)

69

GA(t, t′) =1

h[G+−(t, t′) −G−+(t, t′)] θ(t′ − t) . (202)

which show explicitly the retarded and advanced character of GR and GA, re-spectively. Moreover,

GBR(t, t′) = GBA(t′t) ∈ Re , (203)

GFR(t, t′) = [GFA(t′, t)]∗ , (204)

GFK(t, t′) = [GFK(t′, t)]∗ . . (205)

Inverting the above relations one has

iG++ = GK − ih(GR +GA)/2 , iG+− = GK + ih(GR −GA)/2 ,iG−+ = GK − ih(GR −GA)/2 , iG−− = GK + ih(GR +GA)/2 .

(206)

Going back to an operational formalism

GBR(t, t′) ≡ 2i

hθ(t− t′) 〈[x(t), x(t′)]〉 , (207)

GBK(t, t′) ≡ 〈x(t), x(t′)〉 , (208)

for bosons (see App. 6), where one recognizes the Kubo formulæ in the first twolines, and

GFR(t, t′) ≡ 2i

hθ(t− t′) 〈ψ(t), ψ†(t′)〉 , (209)

GFK(t, t′) ≡ 〈[ψ(t), ψ†(t′)]〉 , (210)

for fermions.

5.1.3 Generic correlations

A generic two time correlation that depends only on the system is given by

〈A(t)B(t′)〉 = Z−1red(0) TrA(t)B(t′)ρred(0) . (211)

Clearly 〈A(t)B(t′)〉 6= 〈B(t′)A(t)〉 and one can define symmetrized and anti-symmetrized correlations:

CA,B(t, t′) = 〈A(t)B(t′) + B(t′)A(t)〉/2 , (212)

C[A,B](t, t′) = 〈A(t)B(t′) − B(t′)A(t)〉/2 , (213)

respectively.Within the Keldysh path-integral representation these correlations can be

written as

CA,B(t, t′) = 〈A[x+](t)B[x+](t′) +B[x−](t′)〉/2 ,

C[A,B](t, t′) = 〈A[x+](t)B[x+](t′) −B[x−](t′)〉/2 , (214)

70

5.1.4 Linear response and Kubo relation

The linear response is defined as the variation of the averaged observable Aat time t due to a change in the Hamiltonian operated at time t′ in such a waythat H → H − fBB. In linear-response theory, it can be expressed in terms ofthe averaged commutator:

RAB(t, t′) ≡ δ〈A(t)〉δfB(t′)

∣

∣

∣

∣

∣

fB=0

=2i

hθ(t− t′)〈[A(t), B(t′)]〉 . (215)

In the case A = x and B = x this implies Rxx(t, t′) = GBR(t, t′). The path-

integral representation is in terms of the Keldysh fields x+, x−:

RAB(t, t′) = i〈A[x+](t)

B[x+](t′) −B[x−](t′)

〉/h . (216)

5.1.5 Quantum FDT

Proofs and descriptions of the quantum fdt can be found in several text-books [30, 44]. We first present a standard derivation that applies to bosonicand fermionic Green functions in the canonical and grand-canonical ensembles.We next show it for generic correlations and linear responses in the bosoniccase. In so doing we recall its expression in the time-domain and in a mixedtime-Fourier notation that gave us insight as to how to extend it to the case ofglassy non-equilibrium dynamics [31].

Canonical ensemble

Let us consider the canonical ensemble and let us write

iG+−(t, t′) = 〈TC φ+(t)φ−(t′)〉 (217)

where φ+ = x+ and φ− = x− (bosons) or φ+ = ψ+ and φ− = ψ−† (fermions).This is equal to

iG+−(t, t′) = (−1)ζ 〈φ−(t′)φ+(t)〉 (218)

with ζ = 1 for fermions and ζ = 0 for bosons. Using the analytic properties ofGreen functions, we have

iG+−(t+ iβh, t′) = (−1)ζ 〈φ−(t′)φ+(t+ iβh)〉 . (219)

In the canonical ensemble, 〈· · ·〉 ∝ Tr · · · ρ0 ∝ Tr · · · e−βH and after expandingφ+(t+ iβh) = ρ0φ

+(t)ρ−10 (in the Heisenberg representation) we get

iG+−(t+ iβh, t′) = (−1)ζTr [φ−(t′)ρ0φ

+(t)]

Trρ0. (220)

71

Using the cyclic property of the trace

iG+−(t+ iβh, t′) = (−1)ζ〈φ+(t)φ−(t′)〉 (221)

and we recognize

G+−(t+ iβh, t′) = (−1)ζ G−+(t, t′) (222)

If the systen has reached equilibrium, time translational invariance implies

G+−(t− t′ + iβh) = (−1)ζ G−+(t− t′) . (223)

After Fourier transforming with respect to t− t′ we get the KMS relation

G+−(ω)eβhω = (−1)ζ G−+(ω) . (224)

Using eqs. (206), we have on the one hand

h[GR(ω) −GA(ω)] = G+−(ω) −G−+(ω) . (225)

Inserting the KMS relation (224) we get

h[GR(ω) −GA(ω)] = G+−(ω)[1 − (−1)ζeβhω] (226)

On the other hand,

GK(ω) =i

2[G+−(ω) +G−+(ω)] =

i

2G+−(ω)[1 + (−1)ζeβhω] . (227)

Combining this relation with eq. (224) we obtain the quantum FDTs

GK(ω) =ih

2[GR(ω) −GA(ω)]

[

1 + (−1)ζeβhω

1 − (−1)ζeβhω

]

(228)

Using GR(ω) −GA(ω) = R(ω) −R∗(ω) = 2iImGR(ω) for bosons and fermions:

GFK(ω) = − ih2

[GFR(ω) −GFA(ω)] tanhβhω

2(229)

GFK(ω) = h ImGFR(ω) tanh βhω2 , (230)

GBK(ω) = − ih2

[GBR(ω) −GBA(ω)] cotanhβhω

2(231)

GBK(ω) = h ImGBR(ω) cotanhβhω2 . (232)

72

In the case of bosons the FDT can be easily extended to

CA,B(ω) = h ImRBAB(ω) cotanhβhω

2. (233)

Grand canonical ensemble

In the grand-canonical ensemble, the proof remains essentially the same.

The initial density operator reads ρ0 ∝ e−βH+βµN , where N is the numberoperator which commutes with H (the number of particules is conserved innon-relativistic quantum mechanics). For concreteness, let us focus on fermionsThe three first steps in the proof presented above imply

GF+−(t+ iβh, t′) = −Tr[

ψ−†(t′)e−βH ψ+(t)eβµN]

/trρ0 (234)

Since H and N commute and since for any operator f(N), one has the property

that ψf(N) = f(N + 1)ψ, we have

ψ+(t)eβµN = eβµ(N+1)ψ+(t) (235)

and so

GF+−(t+ iβh, t′) = −eβµ Tr[

ψ−†(t′)ρ0ψ+(t)

]

/Trρ0 . (236)

At this point, we can follow the same steps as in the proof for the canonicalensemble, namely use the cyclic property of the trace, time-translational in-variance and a Fourier transform, to obtain the fermionic KMS relation in thegrand-canonical ensemble

GF+−(ω)eβhω = −eβµGF−+(ω) . (237)

and the grand-canonical fermionic quantum FDT relation

GFK(ω) = − ih2

[GFR(ω) −GFA(ω)] tanh

(

βhω − µ

2

)

(238)

or, equivalently

GFK(ω) = h ImGFR(ω) tanh(

β hω−µ2

)

(239)

FDT for Generic observables in time-domain

If at time t′ the system is characterized by a density functional ρ(t′), thetwo-time correlation functions are given by eq. (211), (212) and (213). In linearresponse theory RAB(t, t′) and the correlation C[A,B](t, t

′) are related by theKubo formula (215). If the system has reached equilibrium with a heat-bath at

73

temperature T at time t′, the density functional ρsyst(t′) is just the Boltzmann

factor exp(−βHsyst)/Zsyst. It is then immediate to show that, in equilibrium,time-translation invariance, CAB(t, t′) = CAB(t− t′), and the KMS properties

CAB(t, t′) = CBA(t′, t+ iβh) = CBA(−t− iβh,−t′) (240)

hold. Using now these identities and assuming, for definiteness, that t > 0 it iseasy to verify the following equation

CA,B(T ) +ih

2RAB(T ) = CA,B(T ∗) − ih

2RAB(T ∗) , (241)

where T = t + iβh/2. This is a way to express FDT through an analyticcontinuation to complex times.

In terms of the Fourier transform defined in App. ?? the KMS relations readCAB(ω) = exp(−βhω)CBA(−ω) and lead to 2C[A,B](ω) =

(

1 − eβhω)

CAB(ω),

2CA,B(ω) =(

1 + eβhω)

CAB(ω) and C[A,B](ω) = − tanh(βhω/2)CA,B(ω).Back in the Kubo relation this implies

RAB(t− t′) = − ih θ(t− t′)

∫∞

−∞dωπ eiω(t−t′) tanh(βhω/2)CA,B(ω) . (242)

(recall CA,B(ω) = CA,B(−ω).) Using∫∞

0 dt exp(−iωt) = limǫ→0+i

−ω+iǫ =

πδ(ω) − iPω one has

RAB(ω) = − 1

hlimǫ→0+

∫ ∞

−∞

dω′

π

1

ω − ω′ + iǫtanh

βhω′

2CA,B(ω

′) (243)

from which we obtain the real and imaginary relations

ImRAB(ω) =1

htanh

βhω

2CA,B(ω

′) ,

ReRAB(ω) = − 1

hP

∫ ∞

−∞

dω′

π

1

ω − ω′tanh

(

βhω′

2

)

CA,B(ω′) . (244)

If βhω/2 ≪ 1, tanh(βhω/2) ∼ βhω/2 and eq. (242) becomes the classical FDT:

RAB(t− t′) = − 1

kBT

dCAB(t− t′)

dtθ(t− t′) . (245)

5.1.6 The influence functional

As in the derivation of the Langevin equation we model the environmentas an ensemble of many non-interaction variables that couple to the relevantsystem’s degrees of freedom in some convenient linear way. The choice of theenvironment variables depends on the type of bath one intends to consider.Phonons are typically modeled by independent quantum harmonic oscillators

74

that can be dealt with exactly. We then consider a system (x, p) coupled to anenvironment made of independent harmonic oscillators (qa, πa). An element ofthe total density function reads

ρ(x′′, q′′a ;x′, q′a; t) =∫ ∞

−∞

dXdX ′dQadQ′a 〈x′′, q′′a | T e−

ihHtott |X,Qa〉 〈X,Qa| ρtot(0) |X ′, Q′

a〉

×〈X ′, Q′a| T e

ihHtott |x′, q′a〉 . (246)

The first and third factors are the coordinate representation of the evolution

operators e−iHtott/h and eiHtott/h, respectively, and they can be represented asfunctional integrals:

〈x′′, q′′a |T e−ihHtott |X,Qa〉 =

∫ x′′

X

Dx+

∫ q′′a

Qa

Dq+a eihS+

tot (247)

〈X ′, Q′a|Te

ihHtott |x′, q′a〉 =

∫ X′

x′

Dx−∫ Q′

a

q′a

Dq−a e−ihS−

tot . (248)

The action Stot has the usual four contributions, from the system, the reser-voir, the interaction and the counterterm.

As usual we are interested in the dynamics of the system under the effect ofthe reservoir. Hence, we compute the reduced density matrix

ρred(x′′, x′; t) =

∫ ∞

−∞

dqa 〈x′′, qa| ρtot(t) |x′, qa〉 . (249)

5.1.7 Initial conditions

Factorization

The initial density operator ρtot(0) has the information about the initialstate of the whole system. If one assumes that the system and the bath are setin contact at the initial time, the operator factorizes

ρ(0) = ρsyst(0)ρenv(0) . (250)

(Other initial preparations, where the factorization does not hold, can also beconsidered and may be more realistic in certain cases [15] and see below.) If theenvironment is initially in equilibrium at an inverse temperature β,

ρenv(0) = Z−1env e

−β ˆHenv , (251)

(we also include the interaction and the counterterm at the initial time 0 inthis measure, as done for the Langevin process) the dependence on the bath

75

variables is quadratic; they can be traced away to yield the reduced densitymatrix:

ρred(x′′, x′; t) =

∫ ∞

−∞

dX

∫ ∞

−∞

dX ′

∫ x+(t)=x′′

x+(0)=X

Dx+

∫ x−(t)=x′

x−(0)=X′

Dx−

×e ihSeff 〈X | ρsyst(0) |X ′〉 (252)

with the effective action Seff = S+syst−S−

syst+Senv eff. The factor that originatein the bath variables and that remains in the path integral over the systemvariables is called the influence functional and it reads

F [x+, x−] = =

∫

dQadQ′a

∫ q′′a

Qa

Dq+a∫ Q′

a

q′a

Dq−a

eih (Sint[x

+,q+]−Sint[x−,q−]+Scount[x

+]−Scount[x−])

× eih (Senv[q+a ]−Senv[q−a ]) 〈q+a (0)|ρenv|q−a (0)〉

= eihSenv,eff (253)

The last term has been generated by the interaction with the environment andit reads [15]

i

hSenv,eff = −i

∫ T

0

dt′∫ T

0

dt′′[x+(t′) − x−(t′)]

hγ(t′ − t′′)

d

dt′′[x+(t′′) + x−(t′′)]

2

−∫ T

0

dt′∫ T

0

dt′′[x+(t′) − x−(t′)]

h

1

4hν(t′ − t′′)

[x+(t′′) − x−(t′′)]

h. (254)

The noise and dissipative kernels ν and γ are given by

ν(t) =

∫ ∞

0

dω S(ω) coth

(

1

2βhω

)

cos(ωt) , (255)

γ(t) = θ(t)Γ(t) = θ(t)

∫ ∞

0

dωS(ω)

ωcos(ωt) . (256)

One can easily check that ν(t) = ν(−t) and γ(t) is causal; moreover they verifythe bosonic FDT, as they should since the bath was assumed to be in equilib-rium. One can also write the bath-generated action (e

ihSenv,eff ) in the form

Senv,eff ≡ −∑

ab=±

1

2

∫ ∫

dtdt′ xa(t)Σbathab (t, t′)xb(t′) , (257)

with

Σbath++ = +γ ddt′′ − 1

2 iν Σbath+− = +γd

dt′′+

1

2iν , (258)

Σbath−+ = −γ ddt′′ + 1

2 iν Σbath−− = −γ d

dt′′− 1

2iν (259)

76

and Σbath++ + Σbath−− + Σbath+− + Σbath−+ = 0. Although these relation resemble theones satisfied by the Gabs a more careful analysis shows that the matricial Σbath

is more like G−1 than G (see the discussion on the classical dynamics of therandom manifold and the dependence on k; the relation between G and Σ is thesame here, see also the section on MSR variables below where this fact is madeexplicit).

In these equations, as in the classical case, S(ω) is the spectral density ofthe bath:

S(ω) =π

2

Nb∑

a=1

c2amaωa

δ(ω − ωa) , (260)

that can also be taken of the form in (20),

S(ω) = 2γ0ω(ω

ω

)α

e−ω/Λ . (261)

As usual, a counterterm cancels the mass renormalization. The Γ kernel isindependent of T and h and it is thus identical to the classical ones, see eq. (21).The kernel ν does depend on T and h. After a change of variables in the integral,in the cases in which S(ω) ∝ ωα,

ν(t) = γ0ω1−αt−(1+α)g

(

βh

t,Λt

)

(262)

and, now for all S(ω),lim

βh/t→0βhν(t) = 2Γ(t) . (263)

The classical limit is realized at high temperature and/or long times.Figure 13-left shows the time-dependence of the kernels ν in the quantum

Ohmic case for different values of T . In the right panel of the same figure weshow the dependence on α of the kernel ν.

Upper critical initial conditions for the system

Next, we have to choose an initial density matrix for the system. One naturalchoice, having in mind the quenching experiments usually performed in classicalsystem, is the diagonal density matrix

〈X | ρsyst(0) |X ′〉 = δ(X −X ′) (264)

that corresponds to a random ‘high-temperature’ situation and that simplifiesconsiderably the expression in (252). As in the classical stochastic problem,Trρred(0) = 1 and it is trivially independent of disorder. Hence, there is noneed to introduce replicas in such a quantum dynamic calculation.

Equilibrium initial conditions for the system

77

T = 2T = 1T = 0

Ohmic

t− t′

ν

32.521.510.50

25

20

15

10

5

0

-5

Figure 12: The kernel ν in the Ohmic α = 1 case for different values of T(Λ = 5).

α = 1.5α = 1.0α = 0.5

T = 0

t− t′

ν

21.510.50

30

20

10

0

α = 1.0α = 1.5α = 0.5

t− t′

ν

100001000100101

1

0.1

0.01

0.001

0.0001

1e-05

1e-06

1e-07

1e-08

1e-09

1e-10

Figure 13: The kernel ν at T = 0 for different values of α (h = 1 = γ0 = ω = 1and Λ = 5) in linear (left) and logarithmic (right) scales.

78

One might also be interested in using equilibrium initial conditions for theisolated system

〈X | ρsyst(0) |X ′〉 = Z−1sys 〈X | e−βHsyst |X ′〉 . (265)

This factor introduces interesting real-time – imaginary-time correlations.

5.1.8 Transformation to ‘MSR-like fields’

Newton dynamics

We use x± = x± (h/2) x. We first study the kinetic terms:

i

h

∫

dtm

2

(

x2+ − x2

−

)

=i

h

∫

dtm

2

[

(

x+h

2˙x

)2

−(

x− h

2˙x

)2]

(266)

expanding the squares the integrand equals −2h ˙xx and integrating by parts

−∫

dt ix mx . (267)

The potential term

The potential term is treated similarly

− i

h[V (x+) − V (x−)] = − i

h

[

V

(

x+h

2x

)

− V

(

x− h

2x

)]

(268)

Note that this expression is specially simple for quadratic and quartic potentials:

=

−xix quadratic V (y) = y2

−4xix[

x2 − y20 +

(

h2

)2(ix)2

]

quartic V (y) = (y2 − y20)

2

The noise terms

We introduce the variables x and ix in the action terms generated by thecoupling to the environment:

i

hSenv = −

∫ T

0

dt′∫ T

0

dt′′ ix(t′) θ(t′ − t′′)dΓ(t′ − t′′)

dt′x(t′′)

+

∫ T

0

dt′∫ T

0

dt′′ ix(t′)1

4hν(t′ − t′′)ix(t′′)

=

∫ T

0

dt′∫ T

0

dt′′ ix(t′) γ(t′ − t′′) x(t′′)

+

∫ T

0

dt′∫ T

0

dt′′ ix(t′)1

4hν(t′ − t′′)ix(t′′) . (269)

79

If we now write (257) in the rotated basis of the x, ix variables:

Σbathxx (t′ − t′′) = 0 , Σbathxix (t′ − t′′) = γ(t′′ − t′) ddt′ ,

Σbathixx (t′ − t′′) = γ(t′ − t′′)d

dt′′, Σbathixix (t′ − t′′) = 1

2ν(t′ − t′′) . (270)

Note that the matrix Σbath has the structure of the matrix G−1 (in the samebasis)

(

Gxx = K Gxix = RGixx = A Gixix = Q = 0

) (

Σbathxx = ΣbathQ = 0 Σbathxix = ΣbathA

Σbathixx = ΣbathR Σbathxx = ΣbathK

)

5.1.9 Classical limit

In the classical limit, h→ 0, the Schwinger-Keldysh effective action reducesto the one in the functional representation of a Newton classical mechanics(no bath) or a Langevin process with coloured noise (bath). The kinetic termeq. (267) is already in the expected form with no nead to take h → 0. Thepotential term

− i

h[V (x+) − V (x−)] = −V ′(x)ix+O(h) ≃ −V ′(x)ix (271)

upto first order in h (the first term is exact for quadratic Hamiltonians). In thenoise terms we just have to take the classical limit of the kernel ν:

i

hSenv ≃

∫ T

0

dt′∫ T

0

dt′′ ix(t′) γ(t′ − t′′) x(t′′)

+kBT

2

∫ T

0

dt′∫ T

0

dt′′ ix(t′) Γ(t′ − t′′) ix(t′′) (272)

with O(h) corrections

5.1.10 A particle or manifold coupled to a fermionic reservoir

The coupling to leads are modeled by interactions to electron creation anddestruction operators in the form:

Hint =1

N

N∑

k,k′=1

Vkk′ x(

ψ†LkψRk′ + ψ†

RkψLk′)

(273)

L and R are left and right labels associated to two leads. ψ†L(R)k and ψL(R)k are

creation and destruction operators, respectively, acting on the left (L) and right(R) leads. k is the wave-vector of the electron. N is the number of fermionicdegrees of freedom that we take N → ∞. Vkk′ is the coupling constant for thekk′ reservoir degrees of freedom. For simplicy we take Vkk′ = hωcδkk′ .

80

The influence functional is

F [x+, x−] =

∫ Ψ−(∞)=Ψ+(∞)

DψDψ† eih (Sint[x

+,Ψ+]−Sint[x−,Ψ−])

× eih (Senv[Ψ+]−Senv[Ψ−]) 〈Ψ+(0)|ρenv|Ψ−(0)〉

The coordinates Ψ represent the fermionic variables of the reservoirs: Ψ ≡(ψL, ψ

†L, ψR, ψ

†R). We expand the reservoir-system coupling at second order

(first order gives no contribution) to exhibit an effective action Senv,eff by thefollowing procedure

〈e ihSint〉env ≃ 1 + 〈 ihSint〉env +

1

2〈(

i

hSint

)2

〉env ≃ e12〈( ihSint)

2〉env

≡ eihSeff . (274)

We used the notation 〈. . .〉env to represent the functional integral over thefermions evolution. The electron reservoirs’ contribution to the action is

Sint ≡∑

a=±

a

∫

dt hωcxa(t)

1

N

N∑

k=1

(

ψa†LkψaRk + ψa†Rkψ

aLk

)

. (275)

We have

− 1

h2 〈S2int〉env = − 1

h2

∑

a,b=±

ab

∫

dtdt′ (hωc)2xa(t)xb(t′) (276)

× 1

N2

N∑

kk′qq′=1

〈(

ψa†Lk(t)ψaRk′ (t) + L↔ R

)(

ψb†Lq′(t′)ψbRq′ (t

′) + L↔ R)

〉env .

Since left and right reservoirs are independant, operators of left reservoir com-mute with the right ones and the averaging factorizes : 〈TψRψ†

LψLψ†R〉 =

〈TψRψ†R〉〈Tψ†

LψL〉. Conservation of mometum implies 〈Tψkψ†q′〉 = δkq′ 〈Tψkψ†

k〉and similarly 〈Tψk′ψ†

q〉 = δk′q〈Tψk′ψ†k′ 〉. Moreover, we use the fermionic prop-

erty 〈Tψψ〉 = 〈Tψ†ψ†〉 = 0 to finally simplify the expression (276) into

− 1

h2 〈S2int〉env = − 1

h2

∑

a,b=±

ab

∫ ∫

dtdt′ (hωc)2xa(t)xb(t′) (277)

×(

1

N

N∑

k=1

〈ψa†Lk(t)ψbLk(t′)〉env1

N

N∑

k=1

〈ψaRk(t)ψb†Rk(t′)〉env + L↔ R

)

Recalling the definition of the fermionic green functionsGa,b(t, t′) ≡−i 1

N

∑Nk=1〈TC ψak(t)ψ

b†k (t′)〉,

and reconstructing an exponential into an effective action Senv,eff of the generalform

Senv,eff ≡ −∑

ab=±

1

2

∫ ∫

dtdt′ xa(t)Σbathab (t, t′)xb(t′) , (278)

81

we identify the self-energy components

Σbathab (t, t′) ≡ −iabhω2

c

[

GRab(t, t′)GLba(t

′, t) + L↔ R]

. (279)

For a single free fermion, H = hω0ψ†ψ, in equilibirum in the grand-canonical

ensemble, ρ0 ∝ e−β(H−µN), the Green function is

G+−(τ) = inF e−iω0τ G−+(τ) = −i(1 − nF )e−iω0τ (280)

where nF is the Fermi factor nF ≡ [1+eβ(hω0−µ)]−1. After the Keldysh rotationwe have

GK(τ) = −2 tanh[β/2 (hω0 − µ)] e−iω0τ , GR(τ) =i

he−iω0τθ(τ) = G∗

A(−τ) .

The reservoir has a collection of free fermions (N → ∞) with a distribution ofenergy levels, or density of states, F (ω0). Thus,

GK(τ) = −2

∫

dω0 (ω0) tanh[β(hω0 − µ)/2] e−iω0τ

= −2 〈 tanh[β(hω0 − µ)/2] e−iω0τ 〉ω0

GR(τ) =

∫

dω0 (ω0)i

he−iω0τ θ(τ) =

i

h〈 e−iω0τ 〉ω0

θ(τ) (281)

GA(τ) =

∫

dω0 (ω0)−ihe−iω0τ θ(−τ) = − i

h〈 e−iω0τ 〉ω0

θ(−τ)

where we introduced a short-hand notation, the angular brackets, for the inte-gration over energy levels. In the end, the explicit calculation of the kernels Σoriginating from the coupling to the bath yields

ΣbathK (τ) = − (hωc)2

2〈〈

tanh[βL2

(hωL − µL)] tanh[βR2

(hωR − µR)] − 1

× cos[(ωL − ωR)τ ]〉ωL〉ωRΣbathR (τ) = (hωc)

2 1

h〈〈

tanh[βL2

(hωL − µL)] − tanh[βR2

(hωR − µR)]

× sin[(ωL − ωR)τ ]〉ωL 〉ωRθ(τ) (282)

Two important energy scales characterize the density of states, (ω): their‘mean-square-displacement’ or bandwidth, hωF , and whether they have a finitecut-off, hωcut or not. Some typical examples one can use are

(ω) =

ω−1F

√

1 − (ω − ωF )2/ω2F semi-circle hωF , hωcut <∞ ,

ω−1F

√

ωωFe− 1

2

(

ωωF

)2

∼ Oscil. hωF <∞ hωcut → ∞ ,

ω−1F

(

ωωF

)(d−2)/2

Elect. gas hωF → ∞ hωcut → ∞ .

(283)

82

Clearly, the maximal current one can apply to the system is eVmax = hωcut−µ0.Some interesting limiting cases of the kernels Σbath are worth mentioning:• For hωF ≫ hω and kBT . We obtain the Ohmic (∝ ω) behaviour

ImΣbathR (ω) ≃ 2π(hωc)2 1

hρ2(µ0/h+ eV/h) ω (284)

Interesting enough, this expression is independent of T . Simultaneously,

ΣbathK (ω) ≃ 2π(hωc)2 1

hρ2(µ0/h+ eV/h)

eV sinh(βeV ) − hω sinh(βhω)

cosh(βeV ) − cosh(βhω)(285)

• In equilibrium V = 0 one recovers the bosonic FDT relation

ImΣbathR (ω) ≃ 2π(hωc)2 1

hρ2(µ0/h)ω (286)

ΣbathK (ω) ≃ 2π(hωc)2ρ2(µ0/h)ω cothβ

hω

2(287)

eqs. (284) and (285). For hω ≪ kBT ≪ hωF

ΣbathK (ω) ≃ 4π(hωc)2 1

hρ2(µ0/h) kBT . (288)

• At T = 0 and eV ≪ hωF , one derives from eq. (285)

ΣbathK (ω = 0, T = 0, V ) ≃ 2π(hωc)2 1

hρ2(µ0/h)|eV | (289)

This last expression is of the same form as eq. (288) suggesting the defi-nition of an equivalent temperature

kBT∗ ≡ |eV |/2 (290)

A ’FDT like’ relation is verified up to the second order in eVhωF

ΣbathK [T = 0, ω = 0,O(eV

h)2] = h coth(β∗ hω

2)Im ΣbathR [O(ω)1,O(

eV

h)1](291)

5.1.11 Other baths

The electron-phonon interactions, in which the phonons are considered thebath, was dealt with in [13]. Spin baths are reviewed in [14].

83

5.2 Quantum driven coarsening

Take the quantum O(N) model and couple it to two electronic reservoirs. Aswe have seen this environment encompasses as a limit the oscillator case, andallows for the application of a current through the system.

In MSR fields the action reads

i

hStot =

∫

dtddx

i~φ(~x, t)[m~φ(~x, t) + ∇2~φ(~x, t)]

−4i~φ(~x, t) · ~φ(~x, t)

[

1

Nφ2 − φ2

0 −1

N

(

h

2i~φ(~x, t)

)2]

+

∫

dt′[

i~φ(~x, t)γ(t− t′)~φ(~x, t′) + i

~φ(~x, t)

1

4hν(t− t′)

~φ(~x, t′)

]

Apart from the last term in the potential part this action is like the MSR onefor a vector field theory in a strange bath. In the large N limit we assumethat the term between sqaure brackets can be replaced by its average (as in theclassical case) the value of which will be fixed self-consistently. The contribution

〈(iφ)2〉 vanishes due to causality. We are then in presence of an effective classicalfield theory in contact with an environment with quantum character. The mainresults are [8]

• There is a coarsening phase in the (T,Γ, eV, γ0) phase diagram with Γ =h2/m and γ0 the strength of the coupling to the bath. It survives the finitecurrent imposed by the leads. Whether there is a finite critical eVmax (atT = Γ = 0) or the ordering phase extends to infinity in this directiondepends upon the type of electron bath used. See Fig. 14-left.

• The coupling strength, say γ0, between the baths and the system plays avery important role in the determination of the phase diagram. The extentof the ‘sub-critical’ region in which the system will undergo coarseningafter a quench (or annealing) increases for increasing coupling strengthalthough the classical critical temperature is not modified by γ0. Theeffect is highly non-trivial in this very simple model: the action is ‘almost’classical, apart from the last term that arises from the coupling to the bathand involves the kernel ν. Although at long-times this kernel becomesclassical one cannot simply argue that it will not have an effect on thedynamics of the system; on the contrary, its short-time effects alter thedynamics and make the phase transition occur elsewhere.

This effect is similar to the Caldeira-Leggett localisation phenomenon.

A static calculation in which the partition function of the system andthe environment (in the case V = 0) gives the same critical line as thedynamic one for this model. This is not necessarily the case in such mean-field models, as we shall see below with the random manifold.

84

• In the ordering phase the correlations, C(~x, t; ~x, t′), and linear responses,χ(~x, t; ~x, t′), present a separation of two-time scales, similar to the oneencountered in the classical limit with a well-defined plateau in the corre-lation achieved at a value that depends on the parameters (T,Γ, eV, γ0).See Fig. 14-center. In the regime at short time-differences the decay isstationary and this feature is lost at long time differences.

• The relaxation in the stationary regime depends strongly on the param-eters (T,Γ, eV, γ0). The correlations may be non-monotonic and presentoscillations, the frequency of which depend on the values of these param-eters. The correlations and linear-responses are related by the quantumFDT if eV = 0 but they are not if eV 6= 0.

• The study of the correlation functions in the aging regime yields a grow-ing “domain” length R(t;T,Γ, eV, γ0) with a weak dependence upon theparameters T,Γ, eV, γ0 and t1/2 growth in time as in the classical limit.The system continues to grow domains under the drive.

• The scaling functions are the same as in the classical limit and do notdepend on the parameters (T,Γ, eV, γ0). This is an extension of the super-universality hypothesis to the quantum case.

• The relation between the linear-response and the correlation in the agingregime is not the equilibrium FDT but a modified one with an effectivetemperature that diverges in the coarsening regime.

We conclude that the environment plays a dual role: its quantum characterbasically determines the phase diagram but the coarsening process at long timesand large length-scales only ‘feels’ a classical white bath at temperature T ∗,see eq. 290. The two-time dependent decoherence phenomenon (absence ofoscillations, validity of a classical FDT when t/tw = O(1), etc.) is intimatelyrelated to the development of a non-zero (actually infinite) effective temperature,Teff, of the system as defined from the deviation from the (quantum) FDT.Teff should be distinguished from T ∗ as the former is generated not only by theenvironment but by the system interactions as well (Teff > 0 even at T ∗ = 0).Moreover, we found an extension of the irrelevance of T in classical ferromagneticcoarsening (T = 0 ‘fixed-point’ scenario): after a suitable normalization of theobservables that takes into account all microscopic fluctuations (e.g. qEA ) thescaling functions are independent of all parameters including V and Γ. Althoughwe proved this result through a mapping to a Langevin equation that applies toquadratic models only, we expect it to hold in all instances with the same typeof ordered phase, say ferromagnetic, and a long-time aging dynamics dominatedby the slow mo- tion of large domains. Thus, a large class of coarsening systems(classical, quantum, pure and disordered) should be characterized by the samescaling functions.

85

0.2

0.4

0.6

0.8

1

10-1 100 101 102 103

C(t

,t w)

t - tw

(a)

tw = 128256512

1024

0.8

0.9

1

0.55 0.6 0.65

χ

C

(b)

Figure 14: Left: sketch of the phase diagram of quantum O(N) model coupledto two fermionic leads at different chemical potential. Center: the decay of thecorrelation function in the subcritical region. Right: the fluctuation-dissipationplot in the subcritical region. Figures taken from [8].

86

5.3 Quantum aging

The study of quantum random manifold problems with short-ranged corre-lated quenched random potential coupled to an environment made of an ensem-ble of quantum harmonic oscillators was done in a series of papers [31, 32, 5, 3].The interest of this model, apart from its purely theoretical one, resides in its rel-evance to describe pinned charge density wave systems and Wigner crystals [7].The main results that one finds are:

• The statics of the coupled system+bath problem can be studied with thereplica Matsubara technique. One finds a critical surface in the (T/∆0,Γ/∆0, γ0)plane with Γ = h2/m the parameter controlling quantum fluctuations, ∆0

the strength of the random potential, and γ0 the parameter that controlsthe coupling to the bath. The phase transition is of random first ordertype close to the classical critical point, Γ small, and of trully first ordertype when quantum fluctuations are strong, close to T = 0 [32, 5].

• The relaxation in the subcritical region presents two regimes as the onesobserved in the classical case. Short time-difference scales feel the quan-tum fluctuations, they show oscillations and satify the quantum FDT.Long time-differences instead age, as in the classical case and the quantumFDT is modified in a classical looking manner: an effective temperaturecharacterises this regime. The effective temperature is higher than theone of the bath and it is non-zero even when T = 0 [31, 3].

• The effect of the effective temperature resembles strongly decoherence,although taking place in a two-time regime [31].

• The phase diagram depends strongly on the coupling to the environment.For stronger coulpling the ordering region increases. This phenomenonis an extension of the Caldeira-Leggett localization phenomenon to aninteracting case [32].

6 Summary

These lectures dealt with the out of equilibrium dynamics of a system coupled toan environment and set to evolve from an out of equilibrium initial configuration.The dynamics we considered is dissipative both in classical and quantum cases.The lectures were structured around three main ‘chapters’.

In the first one we present a derivation of the Langevin equation to describethe stochastic evolution of a system coupled to an environment and we explainedthe relaxation of a particle in a number of representative cases. We first noticedthat the velocity and position of the particle are two variables that behave verydifferently. In all cases the velocity relatively quickly equilibrated with the bathwhile the coordinate can remain out of equilibrium for much longer. We then

87

focused on several representative potential terms. These are the quadratic case,in which the particle relaxes to equilibrium in a finite relaxation time, flat, inwhich the particle does not equilibrate with the sample and diffuses forever (inthe infinite volume size limit), and a tilted double well in which the particleneeds andexponential time in the height of the barrier to escape the metastablestate. We stressed the importance of distinguishing the relaxation time (timeneeded to reach equilibrium with the bath) and the decorrelation time (timeneeded to decorrelate from a given configuration).

In the second part of the lectures we studied the out of equilibrium dynamicsof macroscopic systems evolving in contact with an environment, thus followinga Langevin-type equation. We described two cases: the dynamics following aphase transition (of second or first order) into an ordered phase that is well-known (using magnetic systems as the paradigm) and the relaxation of glassysystems for which we do not really know the nature of the low-temperaturephase nor the mechanism for the dynamics. To illustrate the two types ofmacroscopic dynamics, on the one hand we used the O(N) ferromagnetic modelin the large N limit for the domain growth in second-order phase transitionsand we explained the nucleation argument in first-order phase transitions. Onthe other hand, we used the random manifold in the limit of an N → ∞ numberof trasverse dimensions as an example for glassy dynamics and we stressed therole played by the proliferation of metastable states in this case. Method-wisewe introduced a path-integral for the generating functional of correlations andresponses which involves the Martin-Siggia-Rose action, and we explained howto derive Dyson equations.

Finally, in the third part of the lectures we introduced quantum fluctuations.We first discussed the path-integral formalism used to deal with the real-timedynamics of quantum dissipative systems, the so-called Schwinger-Keldysh orclose time path path-integral, and we related it, in the classical limit in whichh can be neglected with respect to another scale, to the Martin-Siggia-Roseaction. We dealt with oscillator and fermion representations of the environment,the latter allowing for the study of the effect of a voltage drop accross thesystem, and we derived the influence functional for both of them. We thenwent back to the O(N) and random manifold models, we quantized them, andwe recalled the main features of their quantum dynamics that differ from theclassical ones: the extension of the Caldeira-Leggett localization phenomenonto interacting systems and the fact that the ordered phase is favoured by astronger coupling to the bath; the appearence of first order critical regions closeto the quantum critical point (T = 0) for the random manifold problem; thede-coherence phenomenon that depends on two times, the waiting-time and themeasuring time, in a non-trivial way.

88

A Conventions

A.1 Fourier transform

The convention for the Fourier transform is

f(τ) =

∫ ∞

−∞

dω

2πe−iωτ f(ω) , (292)

f(ω) =

∫ ∞

−∞

dτ e+iωτ f(τ) . (293)

The Fourier transform of the theta function reads

θ(ω) = ivp1

ω+ πδ(ω) . (294)

The convolution is

[f · g](ω) = f ⊗ g(ω) ≡∫

dω′

2πf(ω′)g(ω − ω′) . (295)

A.2 Commutation relations

We defined the commutator and anticommutator: A,B = (AB + BA)/2and [A,B] = (AB −BA)/2.

A.3 Time ordering

We define the time odering operator acting on bosons as

T q(t)q(t′) ≡ θ(t, t′)q(t)q(t′) + θ(t′, t)q(t′)q(t) . (296)

For fermions, we define the time ordering operator as

T ψ(t)ψ(t′) ≡ θ(t, t′)ψ(t)ψ(t′) − θ(t′, t)ψ(t′)ψ(t) , (297)

T ψ(t)ψ†(t′) ≡ θ(t, t′)ψ(t)ψ†(t′) − θ(t′, t)ψ†(t′)ψ(t) , (298)

In both cases θ(t, t′) is the Heaviside-function.We define the time-ordering operator TC on the Keldysh contour in such a

way that times are ordered along it:

TC x+(t)x−(t′) = x−(t′)x+(t) TC x−(t)x+(t′) = x−(t)x+(t′)

TCψ+(t)ψ−(t′) = −ψ−(t′)ψ+(t) TCψ−(t)ψ+(t′) = ψ−(t)ψ+(t′) (299)

for all t and t′.

89

B The instanton calculation

The path-integral formalism yields an alternative calculation of the Kramersescape time, the Arrhenius exponential law and its prefactor that, in principle,is easier to generalize to multidimensional cases. For the sake of simplicity letus focus on the overdamped limit in which we neglect inertia. We first rederivethe Arrhenius exponential using a simplified saddle-point argument, and thenshow how Kramers calculation can be recovered by correctly computing thefluctuations around this saddle point. Starting from the following representationof the probability to reach the top of the barrier from the potential well:

P (xmax, t|xmin) =

⟨

∫ x(t)=xmax

x(0)=xmin

Dx δ(ξ − eq[x])

∣

∣

∣

∣

det

(

δeq[x](t)

δx(t′)

)∣

∣

∣

∣

⟩

ξ

,

and neglecting the determinant (which is justified if one follows the Ito conven-tion), then, for a Gaussian white noise ξ:

P (xmax, t|xmin) =

∫ x(t)=xmax

x(0)=xmin

Dx e−1

4kBT

∫

t

0dt′(x+ dV

dx )2

Expanding the square, we find a total derivative contribution to the integralequal to 2[V (xmax)−V (xmin)], plus the sum of two squares:

∫ t

0dt′[x2+(V ′(x))2].

For small T , the path, x∗, contributing most to the transition probability is suchthat this integral is minimized. Using standard rules of functional derivationone finds

d2x∗

dt′2= V ′(x∗)V ′′(x∗) ⇒ x∗ = ±V ′(x∗).

In order to be compatible with the boundary conditions x∗(0) = xmin and x(t) =xmax, the + solution must be chosen, corresponding to an overdamped motionin the inverted potential −V (x). The ‘action’ of this trajectory is

∫ t

0

dt′[

x∗2 + (V ′(x∗))2]

= 2

∫ t

0

dt′x∗V ′(x∗) = 2[V (xmax) − V (xmin)],

that doubles the contribution of the total derivative above. Hence,

P (xmax, t|xmin) ≈ e−β(V (xmax)−V (xmin)),

independently of t, as in eq. (48). This type of calculation can be readily ex-tended to cases in which the noise ξ has temporal correlations, or non Gaussiantails, and to see how these effects change the Arrhenius result. The calcula-tion of the attempt frequency is done using the standard dilute gas instantonapproximation dveeloped by several authors [33] but we shall not discuss it here.

The path-integral that we have just computed is a sum over the subset ofnoise trajectories that lead from the initial condition to a particular final con-dition that we imposed. Imposing a boundary condition in the future destroysthe causal character of the theory.

90

In a one dimensional problem as the one treated in this Section there isonly one possible ‘reaction path’. In a multidimensional problem, instead, asystem can transit from one state to another following different paths that gothrough different saddle-points. The lowest saddle-point might not be the mostconvenient way to go and which is the most favorable path is, in general, difficultto established.

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