computer simulation of soft condensed matter and other ... · computer simulation of soft condensed...

Computer simulation of condensed matter Microscopic description: particle-based methods Phenomenologic description: stochastic

Computer simulation of soft condensed matterand other complex systems

Guido Germano

Fachbereich 15 und WZMW, Philipps-Universität Marburg, GermanyDiSEI, Università del Piemonte Orientale “Amedeo Avogadro”, Novara, Italy

Basque Center for Applied Mathematics, 28 February 2012

Guido Germano Condensed matter and complex systems


Overview

1 Computer simulation of condensed matter

2 Microscopic description: particle-based methodsDiscotic liquid crystals in cylindrical confinement

3 Phenomenologic description: stochastic processesAnisotropic diffusionAnomalous diffusionOther complex systems: finance

4 Stochastic solution of the space-time fractional diffusion equationUncoupled continuous-time random walk (CTRW)Standard and anomalous diffusion

5 Stochastic calculus for uncoupled continuous-time random walksDefinition of the stochastic integralMonte Carlo simulation



Use of HPC at the FZ Jülich in 2011



Computer simulation of condensed matter

Different methods for different length and time scales:

Continuum mechanics, finite difference methods(engineering).Particle-based methods (physics, chemistry, biology):

Molecular simulation with classical mechanicsat different levels of accuracy: coarse-grained oratomic-detail.Electronic-detail calculation with quantum mechanicsat different levels of accuracy: DFT, MO, relativistic, ...

Combining methods to link different length and time scales is along-standing goal.


Computer simulation of condensed matter Microscopic description: particle-based methods Phenomenologic description: stochasticDiscotic liquid crystals in cylindrical confinement

Microscopic description: quantum mechanics

In traditional QM methods based on molecular orbital (MO)theory, the unknown for an n-electron system is thewavefunction

Ψ(r1, r2, . . . , rn;R1,R2, . . . ,Rm).

It results from the solution of the time-dependent Schrödingerequation

i~∂Ψ

∂t= HΨ

or the time-independent Schrödinger equation

HΨ = EΨ.

In density functional theory (DFT), only the electron density isconsidered:

ρ(r) =∫

Ψ2(r, r2, . . . , rn) dr2 . . . drn.



Microscopic description: classical mechanics

With classical mechanics, electrons are neglected and the timeevolution of the nuclear coordinates is given by Newton’s law

mR = − ∂

∂RE(R)

usually in Hamilton’s canonical formulation (p = ∂L/∂q)

q =∂H∂p

, p = −∂H∂q

.

Except for ab initio MD, the energy is given by empiricalfunctions, e.g.:

E(R) = 4ǫ[( σ

R

)12−( σ

R

)6]

E(R) =12

k(R − R0)2.



Symplectic integrators, e.g. velocity Verlet

The time evolution of a function f (p, q) is given by the Liouvilleequation:

df (p, q)dt

= F · ∂f∂p

+ q · ∂f∂q

= {f ,H} = iLf

f (t) = eiLt f (0)

ei(L1+L2)∆t ∆t→0∼ eiL1∆t/2eiL2∆teiL1∆t/2

eh∂/∂x f (x) h→0∼ f (x + h)

The three pieces of the classical short-time propator in theTrotter-Suzuki approximation can be cast one-to-one into code:

vvec += fvec*tstep_half;rvec += vvec*tstep;fvec = forces();vvec += fvec*tstep_half;



A realistic force field (AMBER, Kollman et al., 1984–)

EA =∑

b

12

kb(Rb − R0b)

2 +∑

a

12

ka(ϑa − ϑ0a)

2

+∑

d

{∑

m

kdm(1 + cos(mϕd − ϕ0dm))

+ f

{4εd

[(σd

Rd

)12

−(σd

Rd

)6]

Q1Q2

4πǫ0Rd

}}

+∑

n

{4εn

[(σn

Rn

)12

−(σn

Rn

)6]+

Q1Q2

4πǫ0Rn

}

Index b runs over bonds, a over angles, d over dihedrals, mover integers from 1 to a maximum depending on d , and n overatom pairs that are nonbonded or more than 3 bonds apart; kb,R0

b , ka, ϑ0a, kdm, ϕ0

dm, f , εi , σi and Qi are parameters of the forcefield.



Another functional form

EG =∑

b

12

kb(Rb − R0b)

2 +∑

a

ka[1 − cos(ϑa − ϑ0a)]

+∑

d

∑

m

kdmcosm(ϕm − ϕ0m)

+∑

n

{4εn

[(σn

Rn

)12

−(σn

Rn

)6]+

Q1Q2

4πǫ0Rn

}

Index b runs over bonds, a over angles, d over dihedrals, mover integers from 1 to a maximum depending on d , and n overatom pairs that are nonbonded or more than 4 bonds apart; kb,R0

b , ka, ϑ0a, kdm, ϕ0

dm, εi , σi and Qi are parameters of the forcefield.



Coarse-grained models based on (soft) rigid bodies



Gay-Berne potential with Bates-Luckhurst extensionUij(r ij , ei , ej) = 4ε(r ij , ei , ej)

[12

ij (r ij , ei , ej)− 6ij (r ij , ei , ej)

]

ij(r ij , ei , ej) =rij − σ(r ij , ei , ej) + ξσ0

ξσ0, ξ = min(κ, 1)

σ(r ij , ei , ej) = σ0

{1 − χ

2

[(r ij · ei + r ij · ej)

2

1 + χ ei · ej+

(r ij · ei − r ij · ej)2

1 − χ ei · ej

]}− 12

ε(r ij , ei , ej) = ε0[ε1(ei , ej)]ν [ε2(r ij , ei , ej)]

µ

ε1(ei , ej) =[1 − (χ ei · ej)

2]− 1

2

ε2(r ij , ei , ej) = 1 − χ′

2

[(r ij · ei + r ij · ej)

2

1 + χ′ ei · ej+

(r ij · ei − r ij · ej)2

1 − χ′ ei · ej

]

χ = (κ2 − 1)/(κ2 + 1), κ = σface−face/σedge−edge (for disks)

χ′ = (κ′1/µ − 1)/(κ′1/µ + 1), κ′ = εedge−edge/εface−face (for disks)



Soft potentials for axially symmetric rigid bodies

U=∑

i<j

Uij(r ij , ei , ej)=∑

i<j

Uij(rij , r ij ·ei , r ij ·ej , ei ·ej)=∑

i<j

Uij(rij , ci , cj , cij)

f ij = −∂Uij

∂r ij= −∂Uij

∂rij

drij

dr ij− ∂Uij

∂ci

dci

dr ij− ∂Uij

∂cj

dcj

dr ij− ∂Uij

∂cij

dcij

dr ij

= −∂Uij

∂rijr ij −

∂Uij

∂ci

ei − ci r ij

rij− ∂Uij

∂cj

ej − cj r ij

rij

=1rij

[(∂Uij

∂cici +

∂Uij

∂cjcj −

∂Uij

∂rijrij

)r ij +

∂Uij

∂ciei +

∂Uij

∂cjej

]= −f ji

g ij = −∂Uij

∂ei= −∂Uij

∂rij

drij

dei− ∂Uij

∂ci

dci

dei− ∂Uij

∂cj

dcj

dei− ∂Uij

∂cij

dcij

dei

= −∂Uij

∂cir ij −

∂Uij

∂cijei 6= −g ji

τ ij = −ei ×∂Uij

∂ei= ei × g ij 6= −τ ji



Coarse graining



Coarse-grained models of hexyloxycyanobiphenyl(6OCB)



Details for the more general biaxial case can be read inM. P. Allen, G. Germano, “Expression for forces and torques inmolecular simulations using rigid bodies”, Molecular Physics104, 3225–3235 (2006).




Production is done with GBmega, a paralleldomain-decomposition Fortran/MPI program:M. R. Wilson, M. P. Allen, M. A. Warren, A. Sauron, W. Smith,Journal of Computational Chemistry 18, 478–488 (1997).Apart from extending this legacy code, all programming of mygroup is done with C++/MPI.




Production is done with GBmega, a paralleldomain-decomposition Fortran/MPI program:M. R. Wilson, M. P. Allen, M. A. Warren, A. Sauron, W. Smith,Journal of Computational Chemistry 18, 478–488 (1997).Apart from extending this legacy code, all programming of mygroup is done with C++/MPI.

It would be nice if every kind of numeric software could bewritten in C++ without loss of efficiency, but unless somethingcan be found that achieves this without compromising the C++type system it may be preferable to rely on Fortran, assembleror architecture-specific extensions. Bjarne Stroustrup



Numeric libraries for scientific computing in ANSI C++with multidimensional array classes (including vector and matrix)

1. Roldan Pozo, TNT — The Template Numerical Toolkit,http://math.nist.gov/tnt: supersedes Lapack++ (thatincorporates BLAS++), IML++, SparseLib++, and MV++.

2. Todd Veldhuizen, Blitz++, “the library that thinks it is acompiler”, www.oonumerics.org/blitz: “performance on parwith Fortran 77/90” using expression templates (1994);dormant since 2005 (Intel Compiler 8).

3. Scott Haney et al., POOMA — Parallel Object-OrientedMethods and Applications, http://acts.nersc.gov/pooma:goals similar to Blitz++, includes classes for partialdifferential equations and parallel architectures.

4. Michael Lehn, FLENS — Flexible Library for EfficientNumerical Solutions, http://flens.sourceforge.net: usesBLAS/Lapack.



Numeric libraries for scientific computing in ANSI C++with multidimensional array classes (including vector and matrix)

5. Boost, www.boost.org: Peer-reviewed libraries, some ofwhich are included in the C++ Standards Committee’sLibrary Technical Report as a step towards becoming partof a future C++ standard.

6. Jeremy Siek et al., MTL — The Matrix Template Library,www.osl.iu.edu/research/mtl: BLAS support.

7. Joerg Walter, Mathias Koch et al., uBLAS: BLAS support.

8. GSL — GNU Scientific Library, www.gnu.org/software/gsl.BLAS support, no overloaded operators.

9. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P.Flannery, Numerical Recipes in C++, 3rd ed. (2007),www.nr.com.



Contact with continuum methods

Frank elastic free energy for the nematic phase (1958)

F =12

∫ {K1 [∇·n(r)]2+K2 [n(r)·∇×n(r)]2+K3 [n(r)×(∇×n(r))]2

}dr

splay twist bend

The reference state has n(r) = constant. The elastic constantscan be found from molecular simulation and used asparameters in a continuous description of the mesophasethrough its director field n.

G. Germano, M. P. Allen, A. J. Masters, “Simultaneouscalculation of the helical pitch and the twist elastic constant inchiral liquid crystals from intermolecular torques”, Journal ofChemical Physics 116, 9422–9430 (2002).



Discotic liquid crystals in cylindrical confinementSupramolecular architecture of LC nanorods

C. Stillings, E. Martin, M. Steinhart, R. Pettau,J. Paraknowitsch, M. Geuss, J. Schmidt, G. Germano,H. W. Schmidt, U. Gösele, J. H. Wendorff, Molecular Crystalsand Liquid Crystals, 495, 285[637]–293[645] (2008).

Template with nanopores

Top view (left) and section (right).



Experimental preparation

Wetting (a, b) and filling (c, d) of the pores, dissolution of the template



Nanorods after the dissolution of the template.



X-ray data and their interpretation.



Structural information from X-ray data.



Experimentalist’s picture of a core-shell model.



Adamantane-pentakis(butyloxy)- andhexakis(butyloxy)triphenylene

OBu

OBu

OBu

BuO

BuO

O

OOBu

OBu

OBu

OBu

BuO

BuO



Atom-surface potential (Steele)

The integral over an infinite half-space for Lennard-Jonesatoms is easy and leads from 6–12 to 3–9 powers;W. A. Steele, Surface Science 36, 317–352 (1973).There are a couple of extensions of this for molecular potentialswith slab boundary conditions, i.e. with the system confinedbetween two parallel walls, also called a slit pore.



Molecule-surface potential (Wall-Cleaver)

Uiw(riw, θiw) = αǫw(θiw)

[9

10(1 − χ2)ν

] 12

ρiw(riw, θiw) =riw − σw(θiw) + σ0

σ0

σw(θiw) = σ0[1 − χ cos2 θiw]− 1

2 , εw(θiw) = ε0[1 − χ′ cos2 θiw]µ

where riw is the particle-wall distance, θiw is the angle betweenthe disc axis and the normal to the wall, i.e. cos θiw = ei · nw,and α is the strength of the particle-wall interaction.This is our own modification for disclike molecules of a potentialoriginally proposed for rodlike molecules by G. D. Wall,D. J. Cleaver, Physical Review E 56, 4306–4316 (1997).



Another molecule-surface potentialBellier-Castella et al.

Uiw(riw, θiw) = ǫw[1 + AP2(cos θiw)]

[2

15ρ−9

iw (riw, θiw)− ρ−3iw (riw, θiw)

]

ρiw(riw, θiw) =riw − rshift(θiw)

ξσ0

where P2 is the second Legendre polynomial, εw is an energyfactor, and the parameter A ∈ [−0.5, 1] determines the tiltangle, ranging from an edge-on (A = −0.5) to a face-on (A = 1)orientation of discs with respect to the surface.L. Bellier-Castella, D. Caprion, J.-P. Ryckaert, Journal ofChemiical Physics 121, 4874–4883 (2004).



Atom-surface potential with surface curvature

The integral over an infinitely thick region outside a cylinder ofradius R is awkward even for Lennard-Jones atoms; it leads toelliptic type integrals and hypergeometric functions F (a, b; c, x):

Up,qiw (riw) =2π2ρwε0Cp,q

×[(q − 5)!!(q − 2)!!

σq0Rq−3

(R2 − r2ic)

q−3F(

3 − q2

,5 − q

2; 1;

( ric

R

)2)

− (p − 5)!!(p − 2)!!

σp0Rp−3

(R2 − r2ic)

p−3F(

3 − p2

,5 − p

2; 1;

( ric

R

)2)]

,

where ric = R − riw, Cp,q = qq−p

(qp

) pq−p , and p, q are the

exponents of the distance; in the Lennard-Jones caseC6,12 = 4. G. Jiang, J. Zhang, X. Zhang, W. Wang, ANZIAMJournal 46, E70–E84 (2004).


Computer simulation of condensed matter Microscopic description: particle-based methods Phenomenologic description: stochasticAnisotropic diffusion Anomalous diffusion Other complex systems:

Anisotropic diffusion in discotic liquid crystals

OPentyl

OPentyl

OPentyl

OPentyl

PentylO

PentylO

Hexakis(pentyloxy)triphenylene, a platelike molecule.



Isotropic phase P∗ = 200 and T ∗ = 13



Nematic phase at P∗ = 200 and T ∗ = 12



Columnar phase at P∗ = 200 and T ∗ = 11



Determination of the diffusion constant

Diffusion equation (Fick’s 2nd law)

∂ρ(r, t)∂t

= D∂2ρ(r, t)∂r2

Slope of mean square displacement vs. time (Einstein relation)

D = limτ→∞

16Nτ

N∑

i=1

〈|r i(t + τ)− r i(t)|2〉t

Integral of the velocity autocorrelation (Green-Kubo relation)

D =1

3N

N∑

i=1

∫ ∞

0〈v i(t + τ) · v i(t)〉t dτ



Anisotropic case

The diffusion tensor is diagonal in the molecular referenceframe at time t :

D =

Dxx Dxy Dxz

Dyx Dyy Dyz

Dzx Dzy Dzz

→

D⊥ 0 00 D⊥ 00 0 D‖

D‖ = limτ→∞

12Nτ

N∑

i=1

〈|r i(t + τ) · ei(t)− r i(t) · ei(t)|2〉t

D⊥ = limτ→∞

12Nτ

N∑

i=1

〈|r i(t + τ) · e⊥i(t)− r i(t) · e⊥i(t)|2〉t



Single-file diffusion

Particles cannot go past each other:�

The MSD of single particles is proportional to τ1/2:

N∑

i=1

〈|ri(t + τ)− ri(t)|2〉t ∝ τ1/2

The center of mass of the whole system still obeys the Einsteinrelation:

r2 = 2Dt

P. M. Richards, Physical Review B 16, 1393 (1977).



Snapshot of an isotropic phase



Isotropic diffusion in an isotropic phase

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40 50 60 70 80 90 100

MS

D

t

parallel, P = 200, T = 13senkrecht, P = 200, T = 13

parallel, P = 150, T = 11senkrecht, P = 150, T = 11



Snapshot of a nematic phase



Anisotropic diffusion in a nematic phase

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40 50 60 70 80 90 100

MS

D

t

senkrecht, P = 200, T = 12parallel, P = 200, T = 12



Snapshot of a columnar phase



Anisotropic diffusion in a columnar phase

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70 80 90 100

MS

D

t

parallel, P = 150, T = 10parallel, P = 200, T = 11

parallel, P = 150, T = 9senkrecht, P = 150, T = 10senkrecht, P = 200, T = 11

senkrecht, P = 150, T = 9



Transition from single-file to standard diffusion

0.01

0.1

1

1 10 100

MS

D

t

P = 150, T = 10

N = 280N = 2240

N = 17920N = 143360

tt0.5



Space-time fractional diffusion equation

The standard diffusion equation can be generalized to

∂β

∂tβuX (x , t) = D

∂α

∂|x |αuX (x , t)

uX (x , 0+) = δ(x), x ∈ R, t ∈ R+.

Riesz space-fractional derivative of order α ∈ (0, 2]:

dα

d |x |α f (x) = F−1k [−|k |α f (k)](x).

Caputo time-fractional derivative of order β ∈ (0, 1]:

dβ

dtβf (t) = L−1

s [sβ f (s)− sβ−1f (0+)](t).



Stochastic solutions by continuous-time random walks

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

0 0.5 1 1.5 2

x(t)

t

α = 1.7, β = 0.8α = 2.0, β = 1.0α = 1.0, β = 0.9

The scale parameters are linked by γαx = γβt with γt = 0.001.The jumps become larger with smaller α and larger γx ; thewaiting times become longer with smaller β and larger γt .



Statistical mechanics: condensed matter and complexsystems

The most important journals are1 Physical Review E — Statistical, Nonlinear, and Soft

Matter Physics2 Physica A: Statistical Mechanics and its Applications3 European Physical Journal B — Condensed Matter and

Complex Systems

Complex systems include econophysics, financial markets,networks, traffic, language evolution, pattern formation,time-series analysis, flocking, foraging, population dynamics,opinion dynamics, multi-agent systems, etc.



Diffusion in finance

In modern finance theory, stock prices S(t) are modelledcustomarily with geometric Brownian motion (W (t) is theWiener process):

dS(t) = µS(t)dt + σS(t)dW (t).

This has many convenient mathematical properties, but is notvery realistic, as has been pointed out already a long time ago:B. Mandelbrot, “The variation of certain speculative prices”,Journal of Business 36, 394–419 (1963).



Search for realistic high-frequency stock priceprocesses beyond geometric Brownian motion


Computer simulation of condensed matter Microscopic description: particle-based methods Phenomenologic description: stochasticUncoupled continuous-time random walk (CTRW) Standard and

Continuous-time random walks

A CTRW is a pure jump process; it consists of a sequence ofindependent identically distributed (IID) random jumps (events)ξi separated by IID random waiting times τi , i = 1, . . . , n, withi , n ∈ N,

tn =n∑

i=1

τi , τi = ti − ti−1, τi ∈ R+,

so that the position X (t) of the random walker at timet ∈ [tn, tn+1) is

X (t) def= SN(t)

def=

N(t)∑

i=1

ξi , ξi ∈ Rd .



Sample paths of continuous-time random walks

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

0 0.5 1 1.5 2

x(t)

t

α = 1.7, β = 0.8α = 2.0, β = 1.0α = 1.0, β = 0.9

The scale parameters are linked by γαx = γβt with γt = 0.001.The jumps become larger with smaller α and larger γx ; thewaiting times become longer with smaller β and larger γt .



Markov and Lévy properties of uncoupled CTRWs

A CTRW is right-continuous with left limits.




A CTRW is right-continuous with left limits.The assumption of IID jumps and waiting times means thatthe joint probability density function (PDF) of any pair ofjumps and waiting times does not depend on i , i.e.,ϕ(ξi , τi) = ϕ(ξ, τ).




A CTRW is right-continuous with left limits.The assumption of IID jumps and waiting times means thatthe joint probability density function (PDF) of any pair ofjumps and waiting times does not depend on i , i.e.,ϕ(ξi , τi) = ϕ(ξ, τ).A CTRW is called uncoupled if the joint PDF ϕ(ξ, τ)factorizes into its marginal PDFs for jumps λ(ξ) andwaiting times ψ(τ), i.e., ϕ(ξ, τ) = λ(ξ)ψ(τ).




A CTRW is right-continuous with left limits.The assumption of IID jumps and waiting times means thatthe joint probability density function (PDF) of any pair ofjumps and waiting times does not depend on i , i.e.,ϕ(ξi , τi) = ϕ(ξ, τ).A CTRW is called uncoupled if the joint PDF ϕ(ξ, τ)factorizes into its marginal PDFs for jumps λ(ξ) andwaiting times ψ(τ), i.e., ϕ(ξ, τ) = λ(ξ)ψ(τ).An uncoupled CTRW is Markovian if and only if itsmarginal waiting time distribution is exponential, i.e.,ψ(τ) = exp(−τ/γt)/γt ; in this case it is called a compoundPoisson process (CPP).




A CTRW is right-continuous with left limits.The assumption of IID jumps and waiting times means thatthe joint probability density function (PDF) of any pair ofjumps and waiting times does not depend on i , i.e.,ϕ(ξi , τi) = ϕ(ξ, τ).A CTRW is called uncoupled if the joint PDF ϕ(ξ, τ)factorizes into its marginal PDFs for jumps λ(ξ) andwaiting times ψ(τ), i.e., ϕ(ξ, τ) = λ(ξ)ψ(τ).An uncoupled CTRW is Markovian if and only if itsmarginal waiting time distribution is exponential, i.e.,ψ(τ) = exp(−τ/γt)/γt ; in this case it is called a compoundPoisson process (CPP).A CPP is not only a Markov, but also a Lévy process , i.e.,it has independent and time-homogeneous (stationary)increments.



Semi-Markov property of uncoupled CTRWs

In general a CTRW is a semi-Markov process , i.e., for anyA ⊂ R

d and t > 0

P(Sn ∈ A, τn ≤ t |S0, . . . ,Sn−1, τ1, . . . , τn−1)

= P(Sn ∈ A, τn ≤ t |Sn−1).

If we fix the position Sn−1 = y of the random walker at timetn−1, the probability on the right is independent of n.



The zoo of stochastic processes

Source: R. L. Schilling, Lectures on Lévy processes, TU Dresden 2009, p. 6.



Montroll-Weiss equation

In the generic coupled case, where the law of (ξi , τi) is given bya joint PDF ϕ(ξ, τ), we can rewrite Sn = Sn−1 + ξn as

P(Sn ∈ A, τn ≤ t |Sn−1) =

∫

A

∫ t

0ϕ(x − Sn−1, τ) dτdx .



Montroll-Weiss equation

In the generic coupled case, where the law of (ξi , τi) is given bya joint PDF ϕ(ξ, τ), we can rewrite Sn = Sn−1 + ξn as

P(Sn ∈ A, τn ≤ t |Sn−1) =

∫

A

∫ t

0ϕ(x − Sn−1, τ) dτdx .

Montroll and Weiss (1965) wrote this as an integral equation forthe PDF pX (x , t) of finding X = Sn in position x at time t ,

pX (x , t) = δ(x)Ψ(t) +∫

Rd

∫ t

0ϕ(ξ, τ)pX (x − ξ, t − τ) dτdξ,

where Ψ(t) = 1 −∫ t

0 ψ(τ) dτ is the complementary cumulativedistribution function for the waiting times, also called survivalfunction. This equation can be solved in the Fourier-Laplacedomain, but the inverse transforms are possible only in theuncoupled case, and yield a series.



Choice of waiting-time and jump marginal densities

The marginal jump PDF is a symmetric Lévy α-stable functionwith order α ∈ (0, 2] and scale parameter γx ∈ R+:

λ(ξ) = Lα(ξ).

The marginal waiting-time PDF is the derivative of aMittag-Leffler function with order β ∈ (0, 1] and scaleparameter γt ∈ R+:

ψ(τ) = − ddτ

Ψ(τ) = − ddτ

Eβ(−(τ/γt)β)

A motivation is the behaviour in the diffusive limit

γx → 0, γt → 0 with γαx /γβt = D.



Standard diffusion equation

The well-known solution of the Cauchy problem

∂

∂tuX (x , t) = D

∂2

∂x2 uX (x , t)

uX (x , 0+) = δ(x), x ∈ R, t ∈ R+,

is the one-point PDF of the Wiener process X (t) = W (t),

uW (x , t) =1√

4πDte−x2/(4Dt),

i.e. a normal distribution N(µ, σ2) with µ = 0 and σ2 = 2Dt .



Properties of diffusion processes

Let u(x , t) be the solution of a second order parabolic partialdifferential equation. Its properties are:

1 Conservation of the total quantity:∫ +∞−∞ u(x , t) dx =

∫ +∞−∞ u(x , 0+) dx , ∀t ∈ R+.






∫ +∞−∞ u(x , 0+) dx , ∀t ∈ R+.

2 Conservation of the non-negativity:u(x , 0+) ≥ 0, ∀x ∈ R ⇒ u(x , t) ≥ 0, ∀x ∈ R, ∀t ∈ R+.






∫ +∞−∞ u(x , 0+) dx , ∀t ∈ R+.

2 Conservation of the non-negativity:u(x , 0+) ≥ 0, ∀x ∈ R ⇒ u(x , t) ≥ 0, ∀x ∈ R, ∀t ∈ R+.

3 Spreading law for t → ∞:σ2(t) =

∫ +∞−∞ x2u(x , t) dx ∼ 2Dt ,

or more generally, if there is a drift µ(t) =∫ +∞−∞ xu(x , t) dx ,

σ2(t) =∫ +∞−∞ [x − µ(t)]2u(x , t) dx ∼ 2Dt .



Anomalous vs. standard diffusion

Some processes have only properties 1 and 2;their variance does not exhibit linear growth for t → ∞.This is called anomalous diffusion.





In sub-diffusion, the variance grows more slowly thanlinearly.






In super-diffusion, the variance grows faster than linearly,or is infinite.






In super-diffusion, the variance grows faster than linearly,or is infinite.

Classes of sub- and super-diffusive processes can bedescribed by fractional diffusion equations, that generalizethe standard diffusion equation solved by the one-pointPDF of the Wiener process.



Space-time fractional diffusion equation

The standard diffusion equation can be generalized to

∂β

∂tβuX (x , t) = D

∂α

∂|x |αuX (x , t)

uX (x , 0+) = δ(x), x ∈ R, t ∈ R+.

Riesz space-fractional derivative of order α ∈ (0, 2]:

dα

d |x |α f (x) = F−1k [−|k |α f (k)](x).

Caputo time-fractional derivative of order β ∈ (0, 1]:

dβ

dtβf (t) = L−1

s [sβ f (s)− sβ−1f (0+)](t).



Symmetric Lévy α-stable distribution

The Lévy α-stable function is a generalization of a Gaussian,the latter being a special case for α = 2, and is best defined asthe inverse Fourier (or cosine) transform of its characteristicfunction exp(−|γxk |α):

Lα(ξ) = F−1k

(e−|γx k |α

)(ξ) =

1π

∫ ∞

0e−(γx k)α cos(ξk) dk .

However, there are series expressions for the Lévy function too:

Lα(ξ) = − 1πξ

∞∑

n=1

Γ(n/α+ 1)n!

sin(nπ

2

)(−ξ)n, α ∈ (1, 2]

Lα(ξ) = − 1πξ

∞∑

n=1

Γ(nα+ 1)n!

sin(nπα

2

)(−ξ−α)n, α ∈ (0, 1]



One-parameter Mittag-Leffler function

Eβ(z) =∞∑

n=0

zn

Γ(βn + 1)

with

Eβ(−Dtβ) = L−1s

[sβ−1

D + sβ

](t), t ∈ R+.

For β = 1 the Mittag-Leffler function is a standard exponential:

E1(z) =∞∑

n=0

zn

Γ(n + 1)=

∞∑

n=0

zn

n!= ez .

Other special cases:

E1/2(z) = exp(z2)erfc(−z), E0(z) = (1−z)−1, E2(z) = cosh(√

z).



Asymptotic behaviour of the Mittag-Leffler function

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1 1 10 100 1000

Ψ(t

)

t

β = 0.9

Eβ(-tβ), stochasticEβ(-tβ), analytical

exp(-tβ/Γ(1+β))t−β/Γ(1−β)

The Mittag-Leffler function is halfway between a stretchedexponential (Weibull function) and a power law with index β:

Eβ

(− tβ

)∼

{exp

(−tβ/Γ(1 + β)

)for t → 0+,

t−β/Γ(1 − β) for t → ∞.



Empirical evidence of ML waiting times in finance

100

101

102

103

104

105

10−6

10−5

10−4

10−3

10−2

10−1

100

τ {s}

Ψ(τ

)

100

101

102

103

104

105

10−6

10−5

10−4

10−3

10−2

10−1

100

τ {s}

Ψ(τ

)

Survivalfunctions for BTP futures traded at LIFFE with delivery dateJune (left) and September (right) 1997; in both casesβ = 0.96, γt = 13 s. From M. Raberto, E. Scalas, R. Gorenflo,F. Mainardi, “The waiting time distribution of LIFFE bondfutures”, APFA2, Liège, 13–15/07/2000, arXiv:cond-mat/0012497; see also same authors, Physica A 287, 468 (2000).



Lévy α-stable PDF Lα(ξ/γx)

Chambers, Mallows, Stuck, J. Am. Stat. Assoc. 71, 340 (1976):

ξ = γx

( − log u cosφcos((1 − α)φ)

)1−1/α sin(αφ)cosφ

, φ = π

(v − 1

2

).

For α = 2 this gives Box-Muller: ξ = 2γx√− log u sinφ.



Lévy α-stable PDF Lα(ξ/γx)

Chambers, Mallows, Stuck, J. Am. Stat. Assoc. 71, 340 (1976):

ξ = γx

( − log u cosφcos((1 − α)φ)

)1−1/α sin(αφ)cosφ

, φ = π

(v − 1

2

).

For α = 2 this gives Box-Muller: ξ = 2γx√− log u sinφ.

Mittag-Leffler PDF −dEβ

(− (τ/γt)

β)/dτ

Kozubowski, Rachev, J. Comput. Anal. Appl. 1, 177 (1999):

τ = −γt log u(

sin(βπ)tan(βπv)

− cos(βπ))1/β

.

For β = 1 this gives the exponential distribution: τ = −γt log u.



Solution of the space-time fractional diffusion equation

In the Fourier-Laplace domain

uX (k , s) =sβ−1

D|k |α + sβ.

Because

L−1s

[sβ−1

D|k |α + sβ

](t) = Eβ(−D|k |αtβ)

in the space-time domain

uX (x , t) = t−β/α Gα,β(xt−β/α),

with the time-independent Green function

Gα,β(ξ) = F−1κ [Eβ(−D|κ|α)] (ξ).

where κ = ktβ/α and α ∈ (0, 2], β ∈ (0, 1].Guido Germano Condensed matter and complex systems


Monte Carlo approximation of the Green function

A stochastic solution of the FDE can be obtained from thediffusive limit of a properly scaled CTRW with a symmetricLévy α-stable distribution of jumps and a one-parameterMittag-Leffler distribution of waiting times.





In the diffusive limit γx and γt vanish with γαx /γβt = D;

the histogram of the PDF pX (x , t ;α, β, γx , γt) of finding theCTRW X in position x at time t converges weakly to theGreen function of the FDE uX (x , t ;α, β), weakly because,for any finite value of γx and γt , pX (x , t ;α, β, γx , γt) alwayshas a singularity at x = 0 that is absent in uX (x , t ;α, β).





In the diffusive limit γx and γt vanish with γαx /γβt = D;

the histogram of the PDF pX (x , t ;α, β, γx , γt) of finding theCTRW X in position x at time t converges weakly to theGreen function of the FDE uX (x , t ;α, β), weakly because,for any finite value of γx and γt , pX (x , t ;α, β, γx , γt) alwayshas a singularity at x = 0 that is absent in uX (x , t ;α, β).

For α = 2 and β = 1, one recovers the Green functionuW (x , t) of the standard diffusion equation, i.e. theone-point PDF of the Wiener process.



FDE solution uX (x , t) for α = 1.8, β = 0.9.



2 4

6 8

10

t

-3 -2 -1 0 1 2 3

x

0

0.2

0.4

0.6

0.8

pγx,γt(x,t; α,β)

PDF pX (x , t ;α, β, γx , γt) with α = 1.7, β = 0.8, γt = 0.1,γx = γ

β/αt . The crest at x = 0 is the survival function

Ψ(t) = Eβ

(−(t/γt)

β).



0.1

0.2

-3 -2 -1 0 1 2 3

p γx,

γ t(x,t;

α,β

)

x

t = 2.0, α = 2.0, β = 1.0

u(x,t; α,β)γt = 0.01 γt = 0.80 γt = 1.00 γt = 1.20 γt = 1.40

Convergence

of pγx ,γt (x , t ;α, β) to uX (x , t , α, β) at t = 2.0.



0

0.1

0.2

0.3

-3 -2 -1 0 1 2 3

p γx,

γ t(x,t;

α,β

)

x

α = 1.7, β = 0.8

u(x,t; α,β)γt = 0.0001 γt = 0.0010 γt = 0.0100 γt = 0.1000 γt = 0.3000 γt = 0.5000

Convergence




0

0.1

0.2

0.3

0.4

-3 -2 -1 0 1 2 3

p γx,

γ t(x,t;

α,β

)

x

α = 1.0, β = 0.9u(x,t; α,β)γt = 0.004 γt = 0.600 γt = 0.800 γt = 1.000 γt = 1.200

Convergence




0.001

0.01

0.1

0.001 0.01 0.1 1

max

x ≠

0 |p

γ x,γ

t(x,t;

α,β

) -

u(x,

t; α,

β)|

γt

α = 1.7, β = 0.8α = 1.0, β = 0.9α = 2.0, β = 1.0

Convergence of maxx 6=0 |pγx ,γt (x , t ; α, β)− uX (x , t ; α, β)| forselected values of α and β when γx , γt → 0 with γαx = γβt .



CPU time for 100 million samples

Pentium Athlon Opteron Power4+Gaussian 16 12 11 19Lévy 73 66 52 95Exponential 16 11 12 20Mittag-Leffler 52 44 36 72

CPU time in seconds needed to generate 108 pseudorandomnumbers with different probability distributions on differentarchitectures: an Intel Pentium IV operating at 2.4 GHz, anAMD Athlon 64 X2 “Toledo” Dual-Core at 2.2 GHz, an AMDOpteron 270 at 2.0 GHz, and an IBM Power4+ at 1.7 GHz. Onthe first three architectures we used the Intel C++ compiler withthe -O3 optimisation option; on the fourth, we used the IBM xlCcompiler with the -O5 option.



CPU times for 10 million Monte Carlo runs

α β γt n tCPU/sec2.0 1.0 0.010 200 3372.0 1.0 0.001 2000 33621.7 0.8 0.010 74 4371.7 0.8 0.001 470 2895

Average number n of jumps per run and total CPU time tCPU inseconds for 107 runs with t ∈ [0, 2] on an AMD Athlon 64 X2Dual-Core at 2.2 GHz using the Intel C++ compiler and the -O3-static optimization options.



References

D. Fulger, E. Scalas, G. Germano, “Monte Carlo simulationof uncoupled continuous-time random walks andstochastic solution of the space-time fractional diffusionequation”, Phys. Rev. E 77, 021122 (2008).



References


G. Germano, M. Politi, E. Scalas, R. L. Schilling,“Stochastic calculus for uncoupled continuous-timerandom walks”, Phys. Rev. E, 79, 066102 (2009).



References


G. Germano, M. Politi, E. Scalas, R. L. Schilling,“Stochastic calculus for uncoupled continuous-timerandom walks”, Phys. Rev. E, 79, 066102 (2009).

G. Germano, M. Politi, E. Scalas, R. L. Schilling,“Stochastic integrals on uncoupled continuous-timerandom walks”, Comm. Nonlin. Sci. Numer. Simul.,in press (2009).


Computer simulation of condensed matter Microscopic description: particle-based methods Phenomenologic description: stochasticDefinition of the stochastic integral Monte Carlo simulation

Definition of a stochastic integral driven by a CTRW

The result of a stochastic integral depends on where theintegrand is evaluated with respect to the increment. This canbe expressed with a parameter a ∈ [0, 1] that interpolateslinearly between Y (t−i ) = Y (ti−1) and Y (ti):

Ja(t)def=

∫ t

0Y (sa) dX (s) =

N(t)∑

i=1

Y (tai )ξi

=

N(t)∑

i=1

[(1 − a)Y (t−i ) + aY (ti)][X (ti)− X (t−i )].



Definition of a stochastic integral driven by a CTRW

The previous equation can be rearranged to

Ja(t) = J1/2(t) +(

a − 12

)[X ,Y ](t),

where

[X ,Y ](t) def=

N(t)∑

i=1

[X (ti)− X (t−i )][Y (ti)− Y (t−i )]

is the covariation (or cross variation) of X (s) and Y (s) fors ∈ [0, t ]. When Y (s) = X (s), the covariation [X ,X ](t) is calledquadratic variation and written shorthand [X ](t).



Ito and Stratonovich integrals

Thus each member of the family of stochastic integrals witha ∈ [0, 1] can be obtained adding a “compensator” to theStratonovich integral J1/2(t) = S(t) =

∫Y (s) ◦ dX (s):

Ja(t) = S(t) +(

a − 12

)[X ,Y ](t).





∫Y (s) ◦ dX (s):

Ja(t) = S(t) +(

a − 12

)[X ,Y ](t).

The Stratonovich integral J1/2(t) = S(t) corresponds tothe symmetric variant of Heaviside’s unit step function,H(t) = (sgn t+1)/2, and is particularly appealing becauseit can be computed according to the usual rules of calculus.





∫Y (s) ◦ dX (s):

Ja(t) = S(t) +(

a − 12

)[X ,Y ](t).

The Stratonovich integral J1/2(t) = S(t) corresponds tothe symmetric variant of Heaviside’s unit step function,H(t) = (sgn t+1)/2, and is particularly appealing becauseit can be computed according to the usual rules of calculus.

The Ito integral J0(t) = I(t) =∫

Y (s−) dX (s) = S(t)−[X ,Y ](t)/2, corresponding to the left-continuous variant ofHeaviside’s step function, has the advantage of being amartingale .



Monte Carlo simulation

The definition of a stochastic integral on a CTRW is exactwithout the need for a limit: the number of jumps N(t)between 0 and t is a random finite integer.





Stochastic integrals on a CTRW can be easily calculatedby a Monte Carlo simulation.





Stochastic integrals on a CTRW can be easily calculatedby a Monte Carlo simulation.

The following figures show histograms from 1 million MonteCarlo realizations of X (t), I(t), S(t) and [X ](t), wheret = 1 and Y (t) = X (t) is a symmetric CTRW with jump andtime scale parameters γαx = γβt .



Relation between X , [X ],S, I

The PDF of S(t) = X 2(t)/2 can be worked out from the PDF ofX (t) by the transformation

pS(s, t) =∑

i

pX (xi(s), t)

∣∣∣∣dxi(s)

ds

∣∣∣∣ ,

where the sum is over all xi that yield the same s. For s = x2/2this is x1,2 = ±

√2s and thus

pS(s, t) = 2pX (√

2s, t)/√

2s, s > 0.



Relation between X , [X ],S, I

The PDF of S(t) = X 2(t)/2 can be worked out from the PDF ofX (t) by the transformation

pS(s, t) =∑

i

pX (xi(s), t)

∣∣∣∣dxi(s)

ds

∣∣∣∣ ,

where the sum is over all xi that yield the same s. For s = x2/2this is x1,2 = ±

√2s and thus

pS(s, t) = 2pX (√

2s, t)/√

2s, s > 0.

As seen before I(t) = S(t)− [X ](t)/2; if the dependence of Sand [X ] is small, the PDF of I can be approximated by theconvolution of the PDF of S with the PDF of [X ] mirroredaround zero and scaled to half its width:

pI(x , t) ≃ 2∫ +∞

−∞pS(x + 2x ′, t)p[X ](−2x ′, t) dx ′.



0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 2.0, β = 1.0γt = 0.1

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)

0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 2.0, β = 1.0γt = 0.01

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)

0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 2.0, β = 1.0γt = 0.001

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)

0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 2.0, β = 1.0γt = 0.0001

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)



0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 1.9, β = 1.0γt = 0.1

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)

0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 1.9, β = 1.0γt = 0.01

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)

0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 1.9, β = 1.0γt = 0.001

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)

0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 1.9, β = 1.0γt = 0.0001

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)



0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 1.9, β = 0.9γt = 0.1

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)

0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 1.9, β = 0.9γt = 0.01

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)

0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 1.9, β = 0.9γt = 0.001

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)

0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 1.9, β = 0.9γt = 0.0001

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)



0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 2.0, β = 0.9γt = 0.1

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)

0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 2.0, β = 0.9γt = 0.01

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)

0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 2.0, β = 0.9γt = 0.001

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)

0

1

2

3

-3 -2 -1 0 1 2 3

Pro

babi

lity

dens

ity fu

nctio

n

x

α = 2.0, β = 0.9γt = 0.0001

pX(x)uX(x)pS(x)

2uX|dX/dS|(x)pI(x)

2pS(x)∗ p[X](-2x)p[X](-x)u[X](-x)



Quadratic variation of the solution of the FDE

p[X ](k , t) =

∞∑

n=0

P(n, t)pnξ2(k)

= Eβ[−(t/γt)β(1 − pξ2(k))].

As the jumps ξ follow a Lévy α-stable distribution, forx → ∞, pξ2(x) ∼ x−α/2−1, and the sum of ξ2

i converges to thepositive stable distribution with index α/2, whose characteristicfunction is

L+α/2(k) = exp

(−(iγxk)α/2

).

Inserting this distribution in the previous equation, thecontinuous limit yields the following characteristic function forthe quadratic variation:

u[X ](k , t) = Eβ[−Dtβ(−ik)α/2].



Quadratic variation of the solution of the FDE

For α = 2, inverting the Fourier transform, one gets

u[X ](x , t) = t−β Mβ(xt−β),

where Mβ(u) is the Mainardi-Wright function

Mβ(u) = F−1κ [Eβ(iDκ)] (u).


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