many-particle systems and kinetic theorymar 26, 2012 · classical mechanics classical mechanics...
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Many-particle systems and kinetic theory
Clement Mouhot, University of Cambridge
Spring school “Kinetic theory and fluid mechanics”Lyon, March 26th to 30th, 2012
Joint works w/. Mischler, Wennberg, Marahrens
The problem at hand
I How to derive rigorously macroscopic evolution equations interms of the microscopic laws?
I → Foundation of continuum mechanics (Hilbert 6-th pb)
I Stastitical mechanics and kinetic theory for large number ofparticles as an intermediate step
I → Foundation of kinetic theory?
I A reduced description can be achieved only assuming someform of randomness and decorrelation on the particle system→ molecular chaos
I Mathematical challenge: prove its propagation along the flow→ propagation of chaos
I Randomness can be assumed only initially (deterministicfoundation) or in each step of the evolution (probabilisticfoundation)
Plan
I. From microscopic to macroscopic evolutions
II. Weak coupling (Vlasov) limit and empirical measure
III. Probabilistic foundation of kinetic theory
IV. Statistical stability and quantitative chaos
Classical mechanics
Classical mechanics rests on the fundamental laws of dynamics
Force 1
Force 2
Acceleration ("trend to move")
Isaac Newton (2d law): Sum of all forces applied to a body isproportional to its mass and to its acceleration
Particle systems
Many bodies (atoms, electrons, grains, stars. . . )Fluid: N ∼ 1024 Avogadro numberPlasma of the solar kernel: N ∼ 1032
Galaxies: N ∼ 1011 stars in the MilkywayInteractions: collision, electro-magnetic forces, gravitation forces
Microscopic description
Microscopic description: Fundamental law of dynamics to eachparticle → trajectories of all particles.
Position X(t) Vitesse V(t)
Resists to analysis as soon as N ≥ 3 (!), and extremely hard tocomputer because of large N
Macroscopic description
Macroscopic description: Fundamental laws of dynamics oninfinitesimal volume elements of a continuum (fluid)
force 1
force 2
Acceleration
→ hydrodynamical (partial differential) equations:Leonhard Euler (1707–1783) in 1755 for non-viscous fluidsClaude-Louis Navier (1785–1836) in 1821 and George Stokes(1819–1903) in 1845 for viscous fluids
Mesoscopic description
Between the microscopic and macroscopic levels of description→ Kinetic theory (mesoscopic level)Describe proportions of particles with given position and velocityImportant thing is not which particle but how many of them
x x+dx
Proportion of particules in [x,x+dx]
f (t, x , v) distribution of particles at x and with velocity v (cf.statistics of a population)
Importance of kinetic theory
I Incorporate details of microscopic dynamics and still muchsimpler than microscopic level (average behavior)
I Model fluids out of the local thermodynamical equilibrium:plasmas, rarefied gas, galaxies, planetary rings. . .
I Entropy and irreversibility (2d principle of thermodynamics)can be “proved” at this level
I Natural intermediate step between micro. & macro. inHilbert’s 6-th problem: “Axiomatize mechanics”
I However as we shall discuss below, microscopic dynamics isreversible (!) which seems contradictory
From Maxwell to Boltzmann
I Maxwell 1867 derived the distribution of a gas at equilibrium(maxwellian=gaussian) and the form of the collision operator
I Boltzmann 1872: “it has not yet been proved that, for anyinitial state of the gas, it must approach the limit distributiondiscovered by Maxwell”
I To answer this question, Boltzmann derived the so-calledBoltzmann equation (see below) on which he proves(formally) the H-theorem, that is the growth of the entropy
I [Cf. spectral gap estimates, hypocoercivity, exponentialH-theorem]
I But he also introduces a microscopic interpretation of theentropy and a tentative explanation to the macroscopicirreversibility
Irreversibility according to Boltzmann (I)
Representation in terms of a “factorization” of a dynamics
M0
M1
W
W
W0
W1
t
S = k logW
Irreversibility according to Boltzmann (II)
I Phase space W : all possible microscopic configurations(positions and velocities of all particles)
I Right hand side: macroscopic configurations (distributions, ortemperatures. . . )
I Microscopic evolution Hamiltonian which preserves volume
⇒ |W1| ≥ |W0|I Possible that several microscopic states evolves toward the
same macroscopic state ⇒ |W1| > |W0|I Boltzmann’s entropy: S(M) := k log W (M): increases with
time
I No contradiction (loss in factorized-hidden degrees offreedom)
I → irreversibility behavior through our “macroscopicfiltering-blurring glasses”
Irreversibility according to Boltzmann (III)
Implicitly nontrivial “factorization” assumption of the dynamics:X ∈W (M)⇒ Tt(X ) ∈W (Ft(M)) Tt ,Ft micro./macro. semigroups
M0
M1
W
W
W0
W1
t
S = k logW
W ′1
M ′1 6= M1
Compulsory for macroscopic evolution laws (closed equation)
Irreversibility according to Boltzmann (IV)
How to justify this “factorization”:
I Boltzmann’s idea of molecular chaos (“Stosszahlansatz”)
I Roughly speaking: for certain initial data (low correlations),the very low correlations are mostly preserved with times andthe Poincare recurrence time is “sent to ∞” as N → +∞
I At least the time scale of such spurious “reversiblefluctuations” remains out of the range of observations
I This small region in the phase space gives birth to a“forward” factorization for t ≥ 0 and a “backward”factorization for t ≤ 0
Hilbert’s (1862–1943) 6-th problem (ICM Paris 1900)
The investigations on the foundations of geometry suggest theproblem: To treat in the same manner, by means of axioms, thosephysical sciences in which mathematics plays an important part; inthe first rank are the theory of probabilities and mechanics. [. . . ]
It is therefore very desirable that the discussion of the foundationsof mechanics be taken up by mathematicians also. ThusBoltzmann’s work on the principles of mechanics suggests theproblem of developing mathematically the limiting processes, theremerely indicated, which lead from the atomistic view to the laws ofmotion of continua.
The Boltzmann equation (1872)
∂t f︸︷︷︸time change
+ v · ∂x f︸ ︷︷ ︸space change
= Q(f , f )︸ ︷︷ ︸collision operator
on f (t, x , v) ≥ 0
I Partial differential equation: relates infinitesimal changes ofseveral variables
I Transport term v · ∂x : straight line along velocity v
I Collision operator Q(f , f ): bilinear, acting on v only, integral
Q(f , f )(v) =
∫v∗
∫collisions
[f (v ′)f (v ′∗)︸ ︷︷ ︸
(v ′,v ′∗)→(v ,v∗)
− f (v)f (v∗)︸ ︷︷ ︸(v ,v∗)→...
]B
The Vlasov-Poisson equation (1938)
Landau equation 1936 (“diffusive variant of Boltzmann equation”):
∂t f︸︷︷︸time change
+ v · ∂x f︸ ︷︷ ︸space change (inertia)
= Q(f , f )︸ ︷︷ ︸Landau collision operator
But plasmas and galaxies mostly non-collisional→ Vlasov-Poisson equation (Vlasov 1938):
∂t f︸︷︷︸time change
+ v · ∂x f︸ ︷︷ ︸space change (inertia)
− E · ∂v f︸ ︷︷ ︸velocity change (field E )
= 0
Mean-field interaction (electric, gravitation)
E = ψ ∗ ρ ρ[f ] =
∫v
f
Microscopic Hamiltonian dynamics
I Binary interactions through a potential ψ depending only onthe distance between two interacting bodies
I External forces with some potential φ(t, position).
I Hamilton equations (Newton laws)
1 ≤ i ≤ N,dxidt
=∂HN
∂vi
dvidt
= −∂HN
∂xi
HN =N∑i=1
v 2i
2︸ ︷︷ ︸kinetic energy
+∑i<j
ψ(xi − xj)︸ ︷︷ ︸interaction energy
+N∑i=1
φ(t, xi ).︸ ︷︷ ︸potential energy
I This corresponds to the following ODE’s
1 ≤ i ≤ N, xi = vi , vi = −∑i 6=j
∇xψ(xi − xj)−∇xφ(xi ).
The N-body Liouville equation (I)
Statistical solution to the previous ODEs (distribution oftrajectories):
∂FN
∂t+
N∑i=1
(∂HN
∂vi· ∂FN
∂xi− ∂HN
∂xi· ∂FN
∂vi
)= 0
on the joint microscopic probability distribution function FN
Liouville theorem
For any t ∈ R one has FN(t, St(X ,V )) = FN(0,X ,V ), where St
is the flow of the Hamilton equations, and St preserves volume.
Consequence: statistical Casimir invariants (for Θ : R 7→ R+)∫R2dN
Θ(
FN(t,X ,V ))dX dV =
∫R2dN
Θ(
FN(0,X ,V ))dX dV
including Boltzmann entropy for Θ(r) = r log r
The N-body Liouville equation (II)
Proof: Differentiate in time FN(t, St(X ,V )) = FN(t,Xt ,Vt):(∂
∂tFN
)(t,Xt ,Vt) +
(∂
∂tXt
)·(∂
∂XFN
)(t,Xt ,Vt)
+
(∂
∂tVt
)·(∂
∂VFN
)(t,Xt ,Vt) = 0
which means, using the equations on Xt and Vt :(∂
∂tFN
)(t,Xt ,Vt) +
(∂
∂VH
)·(∂
∂XFN
)(t,Xt ,Vt)
−(∂
∂XH
)·(∂
∂VFN
)(t,Xt ,Vt) = 0
which is the desired equation at the point (t,Xt ,Vt).
The N-body Liouville equation (III)
Then compute time derivative of J(t,X ,V ) := det∇X ,V St(X ,V ):
d
dtJ(t,X ,V ) =
[∑i
(∂2HN
∂xi∂vi− ∂2HN
∂vi∂xi
)]J(t,X ,V ) = 0
Together with J(0,X ,V ) = det Id = 1, it yields J(t,X ,V ) ≡ 1One deduces by change of variable∫
R2dN
Θ(
FN(t,X ,V ))dX dV =
∫R2dN
Θ(
FN(0,X ,V ))dX dV
→ conservation of Lebesgue norms, Boltzmann entropy. . .This reflects the time-reversibility of the Liouville equation:invariance under the change of variable (t,X ,V ) 7→ (−t,X ,−V )Cf. reversibility of Newton laws at microscopic level
The N-body Liouville equation (IV)
Observe the microscopic conservation of the Hamiltonian
d
dtHN(Xt ,Vt) =
HN ,HN
(Xt ,Vt) = 0 (Poisson bracket)
→ preservation of energy along the microscopic trajectories
The counterpart at a statistical level (statistical Hamiltonian) is
d
dtHN(FN(t, ·, ·)) :=
∫R2dN
HN(X ,V )FN(t,X ,V ) dX dV = 0
The BBGKY hierarchy (I)
I N-particle Liouville equation allows for consideringsuperpositions of all trajectories at the same time, stillcontains same amount of information as the Newton equations
I Simplify description of the system by throwing awayinformation: (Hopefully) the system is described by a oneparticle distribution (first marginal):
f N1 (t, x , v) :=
∫R2d(N−1)
FN(t,X ,V ) dx2 dx3 . . . dxN dv2 . . . dvN
(Observe that it still depends on N!)
I Why the marginal according to the first variable? ConsiderFN symmetric (invariant under permutations) byindistinguability of the particles
The BBGKY hierarchy (II)
I How can we obtain an equation for how f N1 evolves?
I Integrate the N-body Liouville equation
∂FN
∂t+
N∑i=1
(∂HN
∂vi· ∂FN
∂xi− ∂HN
∂xi· ∂FN
∂vi
)= 0
according to X (1) := (x2, x3, . . . , xN), V (1) = (v2, v3, . . . , vN)
I The first term is∫R2d(N−1)
∂FN
∂tdX (1) dV (1) =
∂
∂t
[∫R2d(N−1)
FN dX (1) dV (1)
]=∂f N
1
∂t.
The BBGKY hierarchy (III)
I The second term is the sum over i of∫R2d(N−1)
∂H
∂vi·∂FN
∂xidX (1) dV (1) =
∫R2d(N−1)
vi∂FN
∂xidX (1) dV (1)
=
∫Rd(N−1)
vi
(∫Rd(N−1)
∂FN
∂xidX (1)
)dV (1) =
v1∂f N
1
∂x1i = 1
0 i ≥ 2
by Stoke’s TheoremI Finally, the third term is the (negative of) the sum over i of∫
R2d(N−1)
∂H
∂xi
∂FN
∂vidX (1) dV (1)
=
∫R2d(N−1)
∑j 6=i
∂
∂xi(ψ(xi − xj))
∂FN
∂vi+
∂
∂xi(φ(t, xi ))
∂FN
∂vi
dX (1) dV (1)
=: Ai + Bi
The BBGKY hierarchy (IV)
I Ai and Bi are zero for i 6= 1 by Stoke’s Theorem againI For i = 1
A1 =N∑j=2
∫R2d(N−1)
∂
∂x1(ψ(x1 − xj)
∂FN
∂v1dX (1) dV (1)
= (N − 1)
∫R2d(N−1)
∂
∂x1ψ(x1 − x2)
∂FN
∂v1dX (1) dV (1)
by symmetry of FN and thus
A = (N − 1)
∫R2d
∂
∂x1(ψ(x1 − x2))
∂f N2
∂v1dx2 dv2
by defining the second marginal
f N2 (t, x1, x2, v1, v2) :=
∫R2d(N−2)
FN dX (2) dV (2)
where X (k) and V (k) omits the first k coordinates
The BBGKY hierarchy (V)
I Bi = 0 for i 6= 1 by Stoke’s Theorem
I For i = 1
B =
∫Rd(N−1)
∂
∂x1φ(t, x1)
∂FN
∂v1dX (1) dV (1)
=
(∂
∂x1φ(t, x1)
)∫R2d(N−1)
∂FN
∂v1dX (1) dV (1)
=
(∂
∂x1φ(t, x1)
)∂
∂v1
(∫R2d(N−1)
FN dX (1) dV (1)
)=
∂φ
∂x1
∂f N1
∂v1.
I Thus we obtain the following equation for the one marginaldistribution
∂f N1
∂t+v
∂f N1
∂x−∂φ∂x
∂f N1
∂v−(N−1)
∫R2
∂
∂x(ψ(x−y))
∂f N2
∂v(x , y , v ,w) dy dw = 0
with the substitutions x1 → x , x2 → y , v1 → v , v2 → w
The BBGKY hierarchy (VI)
I How can we interpret this equation?
I Binary collisions ⇒ evolution of first marginal (f N1 ) depends
on second marginal f N2 : interactions = correlations!
I Similarly f N2 ’s evolution depends on f N
3 and so on:
∂f N1
∂t= L1(f N
1 ) + B1(f N2 )
. . .∂f N
k
∂t= Lk(f N
k ) + Bk(f Nk+1)
. . .∂fN∂t
=∂FN
∂t=
HN ,FN
I This is the BBGKY hierarchy (Bogoliubov, Born, Green,Kirkwood, Yvon) or “Bogoliubov approach” for
f N1 , f N
3 , . . . , f Nk , . . . , f
NN = FN .
The Many-particle or “Thermodynamic” Limit
I Goal of thermodynamical limit: perform N →∞ and recoverclosed equations on reduced distributions, ideally on the firstmarginal f N
1 ∼ f1 as N ∼ ∞I In view of the equation on f N
1 and assuming low correlations itis natural to ask whether
f N2 = f N
1 ⊗ f N1 := f N
1 (t, x , v)f N1 (t, y ,w)?
I However the probability independence assumption is alwaysfalse for interacting particle systems, due to interactions!
I What Boltzmann understood (and Kac formulatedmathematically as we shall see) was that (hopefully) thiscould hold in the limit as N →∞
f N2 ∼ f N
1 ⊗ f N1 as N → +∞ (”near-product structure”)
This is the idea of molecular chaos of Boltzmann.
The scaling (regime) of the limit
I Observe finally that there needs to be some kind of scaling sothat the factor (N − 1) does not blow-up in the equaton
∂f N1
∂t+v
∂f N1
∂x−∂φ∂x
∂f N1
∂v−(N − 1)
∫R2
∂
∂x(ψ(x−y))
∂f N2
∂v(x , y , v ,w)dy dw = 0
I In general one has consider the non-zero radius r of particlesand some assumptions of the form
1. N 1 which means mathematically N →∞2. Fixed volume V, so because N →∞ take r = r(N)→ 03. Some version of molecular chaos4. Some way of scaling the interaction, i.e. ψ = ψN
I Given different choices in how to make these assumptions wearrive at different models. This is how the kinetic equationsare derived and obtained from physics, at least formally.
Weak coupling / mean-field / Vlasov limit (I)
I Discovered by Jeans 1915 and Vlasov 1938
I Describe binary interactions through their collective effect
I Adapted to long-range interactions: Coulomb or Newtonfields, but not hard spheres!
I Mathematically, we let ψN(z) = ψ(z)/N and r = r(N)→ 0
such that Nr3
V 1 (dilute gas)
I Force between two particles is O(1/N), hence the action ofone particle becomes negligible in the limit
I However a given particle feels the interaction of N − 1 otherparticles, hence it feels a force of O(N−1
N ) = O(1).
I This “mean-field approach” has found much wider applicationin many areas (biology, sociology, etc.)
Weak coupling / mean-field / Vlasov limit (II)
I Vlasov equation for f = lim f N1 as N →∞
∂f
∂t+v
∂f
∂x−∂φ∂x
∂f
∂v−∫
∂
∂x(ψ(x − y))
∂f
∂v(x , v)f (y ,w)dydw︸ ︷︷ ︸
(∇xψ∗ρf )·∇v f
= 0
obtained from equation on f N1 as N →∞
I Vlasov-Poisson equation when ψ Coulomb or Newtonpotential: ∆Ψf = ±(ρf − ρ0) with ψf = ψ ∗ ρf
I Proof of the mean-field limit known when ψ regular as weshall see: Braun-Hepp-Dobrushin
I For Coulomb-Newton interaction potential major openproblem, best result so far [Hauray-Jabin-2007].
Structure of the mean-field Vlasov equation (I)
I The mean-field Vlasov equation is still time-reversible: iff = f (t, x , v) is a solution, then g(t, x , v) := f (−t, x ,−v) isalso a solution, inherited from microscopic reversibility
I Nonlinear equation due to mean-field term (∇xψ ∗ ρf ) · ∇v f
I But still (mean-field) Hamiltonian structure
∂f
∂t+
(∂Ef
∂v· ∂f
∂x− ∂Ef
∂x· ∂f
∂v
)= Ef , f = 0
with the microscopic mean-field Hamiltonian function
Ef (t, x , v) :=|v |2
2+ Ψf (t, x).
I Ef obtained by considering one particle evolving in themean-field and forgetting the nonlinearity: characteristicsmethod f (t, St(x , v)) = f (0, x , v) but St depends on f !
Structure of the mean-field Vlasov equation (II)
I Statistical mean-field Hamiltonian
H(f ) :=
∫R2d
f
( |v |22
+Ψf (t, x)
2
)dx dv
Observe the factor 1/2 as compared to Ef : nonlinearity!
I “Hamiltonian PDE”:
d
dtH(f ) = 0 ”conservation along the PDE flow”
I In case Ψf satisfies the Poisson equation one gets
H(f ) =
∫R2d
f|v |2
2dx dv ±
∫Rd
|∇xΨf |22
dx
where the + corresponds to plasmas and the − to galaxies:cf. focusing or defocusing PDEs (Schrodinger eq. . . )
I Extensions: magnetic plasmas (Vlasov-Maxwell), gravitationalgases with relativity (Vlasov-Einstein). . .
Boltzmann-Grad / collisional limit (I)
I Short-range interactions and no external force or boundary
I Assume N →∞ (infinite number of particles)
I Finite volume V and r(N)→ 0
I Each particle performs O(1) collisions per unit of time
Nr(N)2 = O(1)
I Then mean-free path `(N) = V/(Nr(N)2) = O(1) r(N)and interparticle distance (V/N)1/3 → 0 and larger thanradius of particles (“continuum of point particle in the limit”)
I Mass m(N)→ 0 with Nm(N) = O(1): average mass densityρ ∼ Nm(N)/V = O(1)
I Assume some form of molecular chaos f N2 ∼ f N
1 ⊗ f N1 : however
here this cannot be true before and after collision, and thisnotion refines into pre-collisional and post-collisional chaos
Boltzmann-Grad / collisional limit (II)
I Only very very rough sketch of the formal derivation: muchmore complicated than the (complicated) mean-field limit!
I Start from the N-body Liouville equation
∂tFN + V · ∇XFN = 0
on the domain
ΩN := ∀i 6= j , |xi − xj | ≥ 2r(N)
I Corresponds to a “wall” potential: mathematically involvesboundary integral terms (additional difficulty)
I We then consider again the one-particle distribution
f N1 (t, x1, v1) :=
∫R2d(N−1)
FN(t,X ,V ) dx2 dx3, . . . dxN dv2 . . . dvN
Boltzmann-Grad / collisional limit (III)
I Search for an evolution equation on it by integrating
∂t fN
1 + v1 · ∇x1f N1 = −
N∑j=2
∫X (1),V (1)∈Ω
(1)N
vj · ∇xj FN
= (N − 1)r(N)2[ ∫
+f N2 (x1, x2, v1, v2)|(v1 − v2) · ω12|dσ12 dv2
−∫−
f N2 (x1, x2, v1, v2)|(v1−v2)·ω12|dσ12
]+ cancelling or negligeable terms
(factor r(N)2 comes from surface element of the sphere)
I ω12 outer normal to the sphere |x1 − x2| = 2r(N)
I dσ12 surface element on the same sphere
I∫
+ surface term for outgoing collisions (v1 − v2) · ω12 ≥ 0
I∫− surface term for ingoing collisions (v1 − v2) · ω12 ≤ 0
Boltzmann-Grad / collisional limit (IV)
I We have neglected multiple collisions (more than binary)which have zero measure in the limit, and used cancellation ofthe surface terms not involving x1 (thanks to the reversibilityof the collisions)
I Then in∫
+ and∫− one has to express outgoing velocities in∫
+ in terms of the ingoing velocities in∫−
I This is where a time arrow is introduced
I Choice seems innocent and arbitrary at microscopic level butcannot be reversed after the limit N → +∞ has been taken
I Cf. in the limit binary collision trajectories no morewell-defined between point particles: statistical outcomes
I Other choice (expressing pre-collisional velocities in terms ofpost-collisional ones) would lead to a backward Boltzmannequation, with a minus in front of the collision operator
Boltzmann-Grad / collisional limit (V)
I Using previous assumptions go back to
∂t fN
1 + v1 · ∇x1f N1
= (N − 1)r(N)2
[∫+
f N2 (x1, x2, v1, v2)|(v1− v2) ·ω12|dσ12 dv2
−∫−
f N2 (x1, x2, v1, v2)|(v1−v2)·ω12|dσ12
]+ cancelling or negligeable
I Use in∫
+ with ω12 = (x1 − x2)/(2r(N))
v +1 := v1−ω12 (ω12 · (v1 − v2)) , v +
2 := v1+ω12 (ω12 · (v1 − v2))
I If propagation of pre-collisional chaos holds f N2 ∼ f N
1 ⊗ f N1
and scaling Nr(N)2 = O(1): hard spheres Boltzmann eq.
∂f
∂t+v ·∇x f =
∫Sd−1×Rd
(f (v ′)f (v ′∗)− f (v)f (v∗)
)|(v−v∗)·ω|dv∗ dω
Structure of the Boltzmann equation (I)
I∂f
∂t+ v · ∇x f = Q(f , f )
I Q(f , f ) bilinear integral operator acting on v only (so it islocal in t and x), representing interactions between particles:
Q(f , f )(v) :=
∫v∗∈R3
∫ω∈S2
[f (v ′∗)f (v ′)︸ ︷︷ ︸“appearing”
− f (v)f (v∗)︸ ︷︷ ︸“dissapearing”
] B(v − v∗, ω)︸ ︷︷ ︸collision kernel (≥ 0)
dω dv∗
I Velocity collision rule ((d − 1) free parameters → ω):
v ′ := v − (v − v∗, ω)ω, v ′∗ := v∗ + (v − v∗, ω)ω
I One has (microscopic conservation laws)
v ′ + v ′∗ = v + v∗, |v ′∗|2 + |v ′|2 = |v |2 + |v∗|2
Structure of the Boltzmann equation (II)
I For ω ∈ Sd−1 the map (v , v∗) 7→ (v ′, v ′∗) has Jacobian −1I We deduce for a test function ϕ(v)∫
Rd
Q(f , f )ϕ(v) dv
=1
4
∫R2d×Sd−1
[f ′f ′∗−ff∗]B(v−v∗, ω)(ϕ+ϕ∗−ϕ′−ϕ′∗)dω dv∗ dv
I Choosing correctly ϕ we deduce∫Rd
Q(f , f )
1v|v |2
dv = 0
I This implies formally
d
dt
∫R2d
f
1v|v |2
dv dx = 0
Structure of the Boltzmann equation (III)
I Choosing ϕ = log f we obtain the H-theorem
d
dtH(f ) =
d
dt
∫R2d
f log f dv dx = −D(f ) ≤ 0
I The entropy production is
D(f ) = −∫R2d
Q(f , f ) log f dx dv
=
∫R2d×Sd−1
[f ′f ′∗ − ff∗] logf ′f ′∗ff∗
B(v − v∗, ω)dx dv ≥ 0
with cancellation only at Maxwellian local equilibriumsatisfying ff∗ = f ′f ′∗ everywhere
Mf =ρ
(2πT )d/2e−|v−u|
2/2T
I Time-irreversible equation and mathematical basis forstudying relaxation to equilibrium (2-d law of thermodynamic)
Proving the Boltzmann-Grad limit (I)
Cercignani 1972: The apparently paradoxical connection between thereversible nature of the basic equations of classical mechanics and theirreversible features of the gross description of large systems of classicalparticles satisfying those equations, came under strong focus with thecelebrated H-theorem of Boltzmann and the related controversiesbetween Boltzmann on one side and Loschmidt and Zermelo on the other.Although much has been done to elucidate the paradox and reconcile thetwo apparently conflicting facts, the question even now is not fullysettled. In fact, the Bogolioubov approach and related procedures, quiteapart from the troubles connected with the fornalism, leave too manyquestions unanswered to be considered definitive.In particular, it is not clear whether an averaging is taking place duringthe duration and over the region of a molecular collision. This averagingis related to another controversial point, i.e., whether irreversibility canappear only through the intervention of a stochastic or random model orcan be a consequence of the progressive weakening of the property ofcontinuous dependence on initial conditions.
Proving the Boltzmann-Grad limit (II)
I Second viewpoint related to picture of irreversibility accordingto Boltzmann: blurring and scales of description
I Best (and astonishing) result so far Lanford 1973:convergence for short time (less than mean free time)
I Conceptually based on expansion of the solution in terms ofthe initial data but hard and technical: see recent preprintGallagher-Saint-Raymond-Texier
I First viewpoint leads to the question of probabilisticfoundation of kinetic theory (Kac 1956): randomness in theevolution itself and probabilistic methods
Plan
I. From microscopic to macroscopic evolutions
II. Weak coupling (Vlasov) limit and empirical measure
III. Probabilistic foundation of kinetic theory
IV. Statistical stability and quantitative chaos
N-particle trajectories and empirical measure (I)
I We consider the weak coupling (Vlasov) limit without externalforce and without boundary
I N-particle trajectories
1 ≤ i ≤ N,dxidt
=∂HN
∂vi
dvidt
= −∂HN
∂xi
I N-particle Hamiltonian
HN(X ,V ) =N∑i=1
v 2i
2+
1
N
∑i<j
ψ(xi − xj)
I This corresponds to the following ODE’s
1 ≤ i ≤ N, xi = vi , vi = − 1
N
∑i 6=j
∇x ψ(xi − xj)
N-particle trajectories and empirical measure (II)
I Denote ZNt = (XN
t ,VNt ) and zi (t) = (xi (t), vi (t))
I Denote the flow SNt (Z ) = ZN
t
I Denote E = R2d
I Assume ψ : Rd → R is W 2,∞ (regularity of the interaction)
I Assume ∇x ψ(0) = 0 (could be relaxed)
I Then by Cauchy-Lipschitz (Picard-Lindelof) theorem: the flowSNt exists globally in time and is C 1(R× EN ; EN)
I We define the empirical distribution associated with a pointZN of the phase space as
µNZN =1
N
N∑i=1
δxi ,vi
N-particle trajectories and empirical measure (III)
I It is a probability measure on E , i.e. the phase space of oneparticle: it provides a great tool for embedding the N-particledynamics into the phase space of the limit equation
I Denoting µN(t,ZN0 ) to be the empirical distribution
associated with the point ZNt = SN
t (ZN0 ) at each time t, this
defines a trajectory in the space P(E )
I It is then equivalent to know µN(t,ZN0 ) or to know the
trajectory of each of the particles considered
I µNt ∈ P(E ) should not be confused with the joint microscopicdistribution FN
t ∈ P(EN)
I Observe (we shall come back to this point) that if eachparticle zi was drawn at random according to a law f , thenthe empirical measure at Z would be a random measuresampling the distribution f
Empirical distribution solutions to the Vlasov equation (I)
A crucial property uncovered by Dobrushin is that the empiricaldistribution following the microscopic trajectories is a weaksolution to the nonlinear Vlasov equation
Lemma
Let ZNt be the solutions to the microscopic equations with initial
data ZN0 , then the corresponding empirical distribution µNt satisfies
∂µNt∂t
+ v · ∇xµNt −∇xΨ[µnt ](t, x) · ∇vµ
Nt = 0
in the weak sense with
Ψ[µNt ](t, x) :=
∫Eψ(x − y) dµNt (y ,w)
Be careful to not confuse the variable x , v of the equation with thepoints ZN
t defining the empirical distribution at each time
Empirical distribution solutions to the Vlasov equation (II)
Observe that from the regularity of ψ the potential
Ψ[µ](t, x) :=
∫Eψ(x − y) dµ(y ,w)
is defined as a C 2 function for µ ∈ P(E )→ no problem for defining weak measure solutions by dualityProof: in the sense of distribution for a test function ϕ ∈ C∞c (E )
∂t〈µNt , ϕ〉 = ∂t
(1
N
N∑i=1
ϕ(xi , vi )
)
=1
N
∑i=1
(∂xi∂t· ∇xϕ(xi , vi ) +
∂vi∂t· ∇vϕ(xi , vi )
)
=1
N
N∑i=1
vi · ∇xϕ(xi , vi )−1
N
N∑j=1
∇x ψ(xi − xj) · ∇vϕ(xi , vi )
(j = i added for free in the last sum as ∇x ψ(0) = 0)
Empirical distribution solutions to the Vlasov equation (III)
∂t〈µNt , ϕ〉
=1
N
N∑i=1
vi · ∇xϕ(xi , vi )−1
N
N∑j=1
∇x ψ(xi − xj) · ∇vϕ(xi , vi )
= 〈µNt , v · ∇xϕ〉 − 〈µNt ,Ψ[µNt ]∇vϕ〉
= −〈v · ∇xµNt , ϕ〉+ 〈Ψ[µNt ] · ∇vµ
Nt , ϕ〉
which concludes the proof:
〈∂tµNt + v · ∇xµNt −Ψ[µNt ] · ∇vµ
Nt , ϕ〉 = 0
In the case where ∇x ψ(0) 6= 0 there is an additional termvanishing as O(1/N) in the limit N →∞:
∂µNt∂t
+ v ·∇xµNt −∇xΨ[µnt ](t, x) ·∇vµ
Nt =
1
N2
N∑i=1
∇x ψ(0) · ∇vδzk
Empirical distribution solutions to the Vlasov equation (IV)
I Hence we have shown that for smooth interactionsmicroscopic trajectories are embedded in the sense of measuresolutions to the limit nonlinear mean-field equation
I Provided that, as we shall see for such smooth interactions,we can prove uniqueness and stability of such solutions, wesee that the microscopic dynamics is fully included in the setof measure solutions to the limit PDE
I In other words, the limit nonlinear PDE in fact still containsas much information as all the N-particle Hamiltonian systemwhen considered in the space of measure
I Surprising fact possible due to the purely deterministicdynamics and mean-field scaling
I From a mathematical viewpoint the frontier betweenmany-particle point mechanics and statistical lies in thefunctional setting: any space better or equal to L1 rules out“trajectory solution” and means statistical physics
Limit of the empirical distributions (I)
Again a key interesting property of empirical distributions µNt isthat they all live in the same space P(E ) for any N ≥ 1Hence we can study the convergence of (µNt )N≥1 in P(E )Let us prove a tightness property on this sequence
Lemma
Assume that initially
supN≥1
Nsupi=1|vN
i | ≤ M < +∞
then
supN≥1
supt≥0
∫E|v |2 dµNt (x , v) <∞
We shall use the conservation of the microscopic Hamiltonian
Limit of the empirical distributions (II)
From the conservation of the Hamiltonian we have for any t ≥ 0
1
NHN(XN
t ,VNt ) =
1
NHN(XN
0 ,VN0 ) ≤ 1
2supN≥1
Nsupi=1|vN
i ,0|2 + ‖ψ‖L∞
Then we have∫E|v |2 dµNt (x , v) =
1
N
N∑i=1
|vNi ,t |2 ≤ 2
1
NHN(t) + 2‖ψ‖L∞
which yields the result∫E|v |2 dµNt (x , v) ≤ M2 + 4‖ψ‖L∞
Limit of the empirical distributions (III)
Proposition
Assume µN0 → fin weak-* in M(E ) and 1N HN(ZN
0 ) = O(1) asN ∼ ∞, then µNt is relatively compact in C (R+; w ∗ −M(E ))(uniformly on compact sets of R+) and any cluster point f is ameasure solution to the Vlasov equation
∂f
∂t+ v · ∇x f −∇xΨ[f ](t, x) · ∇v f = 0
in the weak sense with
Ψ[f ](t, x) :=
∫Eψ(x − y) df (y ,w)
and initial data fin.
Limit of the empirical distributions (IV)
I µN(t, x , v) sequence weak-* compact in M(R+ × E )
I Up to subsequence µNt → ft weak-* in M(R+ × E )
I From the tightness property we deduce that∫Rd
dµNt (x , v)→∫Rd
dft(x , v)
weak-* in M(R+ × Rd).
I Using bounds provided by the Hamiltonian one getsequicontinuity (equi-Lipschitz) on t 7→ 〈µNt , ϕ〉 for ϕ ∈ C 1
c oncompact subsets of R+
I By Ascoli’s theorem Ψ[µNt ]→ Ψ[ft ] uniformly on compactsubsets of R+ × Rd
I Hence one deduce the convergence of the nonlinear term∇xΨ[µNt ] · ∇vµ
Nt which concludes the proof
Strong solutions and weak-strong uniqueness for theVlasov equation
We want now to use the rigidity of the limit in order tocharacterize the limit we have obtained by compactness:
Theorem
Assume 0 ≤ fin ∈ C 1(E ) with∫E
(1 + |v |2)fin(x , v)dx dv < +∞.
Then the Vlasov equation has a unique solution f ∈ C 1(R+ × E )with initial data fin.Moreover any g ∈ C (R+,w ∗ −M(E )) that solves the Vlasov inthe sense of distribution on R∗+ × E with initial data g|t=0 = fincoincides with f on R∗+ × E .
Easy proof (characteristics method)
Mean-field limit
Collecting the previous results we obtain finally
Theorem (Neunzert, Braun-Hepp, Dobrushin, Maslov)
Consider a sequence ZN0 , N ≥ 1 of initial configurations so that:
I µN0 → fin weak-* in M(E )
I fin ∈ C 1(E )
I HN(ZN0 ) = O(N).
Then µNt → ft weak-* in M(E ) uniformly on compact sets of R+,with ft is the solution of the Vlasov equation with initial data fin.
Remarks:- Hence the proof of weak-* stability of the limit Vlasov equation.- However weak-strong stability principle is enough.- When the strategy based on empirical distribution cannot beused, a similar conceptual strategy is to prove uniqueness ofsolutions to the whole infinite BBGKY hierarchy
Quantitative mean-field limit (Dobrushin’s estimate) (I)
I Let us present a beautiful and powerful idea of Dobrushin:performing quantitative estimates of measure stability for theVlasov equation in Wasserstein distance
I Strong and weak topologies on P(E )
I Canonical distance M1 for the strong topology
I But many distances for the weak topology and choice crucial
I Observe that strong distance is too crude for handling Diracmasses ‖δx − δy‖ = 2 1x 6=y . . .
I One wants a distance sensitive to the distance betweenpositions and velocities of particles as encoded in theempirical distribution
Quantitative mean-field limit (Dobrushin’s estimate) (II)
I A convenient set of distances is provided by optimal transport
Wp(µ, ν) =
(inf
π∈Π(µ,ν)
∫E×E|Z − Z ′|p dπ(Z ,Z ′)
)1/p
where Π(µ, ν) set of probability on E × E with marginals µand ν (“coupling”)
I In terms of probability
Wp(µ, ν) =
(inf
(Z ,Z ′)∼π∈Π(µ,ν)E(|Z − Z ′|p
)1/p
I Duality formula for p = 1: Monge-Kantorovich-Rubinsteindistance
W1(µ, ν) = sup‖ϕ‖Lip≤1
∫Eϕ(dµ− dν).
I Observe that Wp(δx , δy ) = |x − y |p (sensitive to the distance)
Quantitative mean-field limit (Dobrushin’s estimate) (III)
Theorem (Dobrushin and implicit in Braun-Hepp)
Consider a transport equation
∂tµ+ (K ∗z µ) · ∇zµ = 0 with ∇ · K = 0
on P(E ) with K uniformly Lipschitz on z ∈ E , then
W1(µt , νt) ≤ e2‖K‖LiptW1(µ0, ν0).
Characteristics are given by
z(t, z0, µ0) = (K ∗ µt)(z(t, z0, µ0)), z(0, z0, µ0) = z0
and the measure is constant along the characteristics:
µt = z(t, ·, µ0)∗µ0 image measure
z(t, z0, µ0) =
∫E
K(z(t, z0, µ0)− z(t, z ′0, µ0)
)dµ0(z ′0)
Quantitative mean-field limit (Dobrushin’s estimate) (III)
Sketch of the proof:
I Consider some coupling π0 ∈ Π(µ0, ν0) and its image πt underthe map (z0, z
′0) 7→ (z(t, z0, µ0), z(t, z ′0, ν0)).
I Then πt ∈ Π(µt , νt) from the conservation alongcharacteristics
I We define
Ξ(t) :=
∫E×E|z0 − z ′0| dπt
=
∫E×E|z(t, z0, µ0)− z(t, z ′0, ν0)|dπ0(z0, z
′0)
and we have W1(µt , νt) ≤ Ξ(t): choice of a particularcoupling at time t (transported from coupling at time 0)
Quantitative mean-field limit (Dobrushin’s estimate) (IV)
I Observe that
z(t, z0, µ0) = z0+
∫ t
0
∫E
K (z(s, z0, µ0)− z(s, z0, µ0)) ds dµ0(z0)
= z0+
∫ t
0
∫E×E
K (z(s, z0, µ0)− z(s, z0, µ0)) ds dπ0(z0, z′0)
I Similarly
z(t, z ′0, ν0) = z ′0+
∫ t
0
∫E
K(z(s, z ′0, ν0)− z(s, z ′0, ν0)
)ds dν0(z ′0)
= z ′0+
∫ t
0
∫E×E
K(z(s, z ′0, ν0)− z(s, z ′0, ν0)
)ds dπ0(z0, z
′0)
Quantitative mean-field limit (Dobrushin’s estimate) (V)
I We deduce the estimate on difference of trajectories
|z(t, z0, µ0)− z(t, z ′0, ν0)| ≤ |z0 − z ′0|+ ‖K‖Lip×∫ t
0
∫E×E
∣∣∣ [z(s, z0, µ0)− z(s, z0, µ0)]
−[z(s, z ′0, ν0)− z(s, z ′0, ν0)
] ∣∣∣ds dπ0(z0, z′0)
≤ |z0 − z ′0|+ ‖K‖Lip∫ t
0
∣∣∣z(s, z0, µ0)− z(s, z ′0, ν0)∣∣∣ds
+ ‖K‖Lip∫ t
0
∫E×E
∣∣∣z(s, z0, µ0)− z(s, z ′0, ν0)∣∣∣ ds dπ0(z0, z
′0)
Quantitative mean-field limit (Dobrushin’s estimate) (VI)
I We then integrate this inequality in z0, z′0:
Ξ(t) ≤∫E×E|z0 − z ′0| dπ0(z0, z
′0)
+ ‖K‖Lip∫ t
0
∫E×E
∣∣∣z(s, z0, µ0)− z(s, z ′0, ν0)∣∣∣ds dπ0(z0, z
′0)
+ ‖K‖Lip∫ t
0
∫E×E
∣∣∣z(s, z0, µ0)− z(s, z ′0, ν0)∣∣∣ ds dπ0(z0, z
′0)
which means
Ξ(t) ≤ Ξ(0) + 2‖K‖Lip∫ t
0Ξ(s) ds.
Quantitative mean-field limit (Dobrushin’s estimate) (VI)
I We deduce that
Ξ(t) ≤ e2‖K‖LiptΞ(0).
I Finally by taking the infimum over π0 ∈ Π(µ0, ν0) we have
W1(µ0, ν0) = infπ0∈Π(µ0,ν0)
∫E×E|z0 − z ′0|dπ0(z0, z
′0)
= infπ0∈Π(µ0,ν0)
Ξ(0)
we recover W1(µ0, ν0) on the right-hand side.
I Using that W1(µt , νt) ≤ Ξ(t) for any choice of π0 we deducethe result
W1(µt , νt) ≤ e2‖K‖LiptW1(µ0, ν0).
Consequences of Dobrushin’s estimate
I We can deduce uniqueness of solutions to the Vlasov equationin the class C (R+; w ∗ −M(E ))
I It is much more precise: we have a Lipschitz property of theflow of the Vlasov equation in Wasserstein distance
I We can now use the quantitative stability estimate: considerµN0 → f0 weak-* in P(E ) and the solutions µNt and ftconstructed before, then
W1(µNt , ft) ≤ e2‖K‖LiptW1(µN0 , f0).
Hence the convergence of the initial configuration to somedistribution implies the mean-field limit to the correspondingsolution for all times, with quantitative rates.
I First example of the use of statistical stability
Plan
I. From microscopic to macroscopic evolutions
II. Weak coupling (Vlasov) limit and empirical measure
III. Probabilistic foundation of kinetic theory
IV. Statistical stability and quantitative chaos
Deriving the Boltzmann equation from a stochastic process
I Kac 1956 proposed a tentative probabilistic foundation ofkinetic theory: start from a Markov jump process, and try toprove the limit towards the spatially homogeneous Boltzmannequation
I Going back to the idea of Boltzmann of “stosszahlansatz” heformulated a notion of chaos
I (f N)N≥1 symmetric probabilities on EN , where E polish space(phase space): the sequence is said f -chaotic if
f N ∼ f ⊗N when N →∞
for some given one-particle probability f on E
I Meaning? Convergence in the weak measure topology for anymarginal depending on a finite number of variables→ low correlation assumption
Reduction of the problem to propagation of chaos
I Clear since Boltzmann that if f N = f ⊗N tensorized duringsome time interval then f satisfies the limit nonlinearBoltzmann equation during this time interval
I Kac remarks that although the “tensorization” property is notpropagated (interactions!) the weaker property of chaoticitycan be propagated (hopefully!) in the correct scaling limit
I Hence many-particle limit is reduced to proving thepropagation of chaos & Kac proves it on a baby model
I The framework set by Kac is our starting point
I Note that the limit performed in this setting is different fromthe Boltzmann-Grad limit: mean-field limit
The notion of chaos and how to measure it I
I f N ∈ Psym(EN) is f -chaotic, f ∈ P(E ), if for any ` ∈ N∗ andany ϕ ∈ Cb(E )⊗` there holds
limN→∞
⟨f N , ϕ⊗ 1N−`
⟩=⟨
f ⊗`, ϕ⟩
which amounts to the weak convergence of any marginals
I This can be expressed for instance in Wasserstein distance:
limN→∞
W1
(Π`
(f N), f ⊗`
)= 0
where Π` denotes the marginal on the ` first variables
I This is Kac’s original definition of chaos that we shall callfinite-dimensional chaos
The notion of chaos and how to measure it II
I Quantitative chaos: f N is f -chaotic with rate ε(N), whereε(N)→ 0 when N →∞ if for any ` ∈ N∗ there existsK` ∈ (0,∞) such that
W1
(Π`f
N , f ⊗`)≤ K` ε(N)
I Statements with other metrics: for some normed space ofsmooth functions F ⊂ Cb(E ) (to be precised) and for any` ∈ N∗ there exists K` ∈ (0,∞) such that for any ϕ ∈ F⊗`,‖ϕ‖F ≤ 1, there holds∣∣∣⟨Π`f
N − f ⊗`, ϕ⟩∣∣∣ ≤ K` ε(N)
The notion of chaos and how to measure it III
I Observe that in the latter statements the number of variables` considered in the marginal is kept fixed as N goes to infinity.A stronger notion of infinite-dimensional chaos would be
limN→∞
W1
(f N , f ⊗N
)N
= 0
I Or in a quantitative manner
W1
(f N , f ⊗N
)N
≤ ε(N)
for some ε(N)→ 0 as N →∞I This amounts to say that one can prove a linear control on K`
in terms of ` in the previous statements. Variants for othermetrics could also be considered.
The notion of chaos and how to measure it IV
I Finally inspired by Kac and following[Carlen-Carvalho-Leroux-Loss-Villani-2010] one can formulatea stronger notion of (infinite-dimensional) entropic chaos:
1
NH(
f N)
N→∞−−−−→ H (f )
with obvious quantitative versions
I Infinite-dimensional chaos requires the use of some “distance”additive according to the tensorization (cf. extensivity)
I Entropic chaos important as it corresponds to the derivationof Boltzmann’s entropy from the many-particle systementropies (see later)
The propagation of chaos
I Let us introduce some time evolution
I Consider a sequence of symmetric N-particle densities
f Nt ∈ C ([0,∞); Psym(EN))
and a 1-particle density of the expected mean field limit
ft ∈ C ([0,∞); P(E ))
I Notion of propagation of chaos on a time interval t ∈ [0,T ]:
f0-chaoticity of (f N0 )N≥1 implies ft-chaoticity of (f N
t )N≥1
Kac’s program I
I Goal: derive the spatially homogeneous Boltzmann Eq. andH-theorem from a many-particle Markov jump process.
I The process is studied through its master equation (theequation on the law of the process).
I This amounts intuitively to consider the spatial variable as ahidden variable inducing ergodicity and markovian propertieson the velocity variable.
I Although the latter point has not been proved so far to ourknowledge, it is worth pointing out that it is at the same avery natural guess and an extremely interesting (and probablyalso difficult) open problem.
Kac’s program II
“This formulation led to the well-known paradoxes which were fully
discussed in the classical article of P. and T. Ehrenfest. These writers
made it clear (a) that the “Stosszahlansatz” cannot be strictly derivable
from purely dynamic considerations and (b) that the “Stosszahlansatz”
has to be interpreted probabilistically. [. . . ]The “master equation”
approach which we have chosen seems to us to follow closely the
intentions of Boltzmann.”
Interpretation not clear! Cf. Cercignani 1972, Lanford 1973
But fascinating more tractable question: if we have to introducestochasticity, at least can we keep it under control all along theprocess of many-particle limit and relate it to the dissipativity ofthe limit equation
Recalls on Markov processes (I)
I Markov process with discrete time and finite phase space:encoded in a transition matrix P and state at time k:
ρ(k + 1) = Pρ(k) =
M∑j=1
Pijρj(k)
i=1,...,M
I Transition matrix P satisfies P ≥ 0 and
∀j = 1, . . . ,M,
M∑i=1
Pij = 1
I Markov process ergodic if for all i 6= j , ∃Pij > 0.
I Perron-Frobenius implies existence of invariant measure γ, itis unique and never zero with ergodicity
I Markov process reversible if symmetric in L2(γ−1) (differentfrom time-reversibility of the master equation!)
Recalls on Markov processes (II)
I Infinite continuous phase space: replace P by an operatorI Continuous in time Markov process: exponential random time
in order to preserve Markov property
P(T ≥ t) =e−at
aand E(T ) =
1
aso that
P(n jumps before t) =(at)n
n!e−at and
ρ(t) =∑n≥0
P(n jumps before t)Pnρ(0)
=∑n≥0
(at)n
n!e−atPnρ(0) = eta(P−Id)ρ(0)
I Differentiating the latter equation in time we obtain theMaster equation (Kolmogorov forward equation)
∂tρ = a(P − Id)ρ
Kac’s setting
I Simplify collisions: one-dimensional velocitiesI One-dimensional collision is trivial with momentum and
energy conservation: drop momentum conservationI Draw pairs (i , j), exponential time and perform the collision
v ′i = vi cos θ + vj sin θ, v ′j = −vi sin θ + vj cos θ
I Energy is preserved, normalize it as (∑N
i=1 v 2i )/N = 1
I In order to maintain O(1) collisions happening per unit oftime, scale the random exponential time so that E(T ) = 1/N
∂FN
∂t= N(P − Id)FN = LNFN
I Jump process on SN−1(√
N) (Kac’s walk):
∂FN
∂t=
2
(N − 1)
∑i<j
∫ 2π
0
(FN(. . . , v ′i (θ), . . . , v ′j (θ), . . . )− FN
) dθ
2π
Kac’s main theorem (I)
Kac’s main propagation of chaos theorem
For the model above, (f Nt )N≥1 propagates chaos: if at time t = 0
f Nk (t = 0)
N→∞−→ f ⊗k1 (t = 0), ∀k ≥ 1
then ∀t > 0f Nk (t)
N→∞−→ f ⊗k1 (t), ∀k ≥ 1,
where f Nk (t) marginals of the solution to the master equation
Remarks:- Enough to do the proof on [0,T ] with T but independent of thesolution- Statement not quantitative and certainly error growing in time,we shall come back to this
Kac’s main theorem (II)
Sketch of the proof:Master equation ∂t f
N = LN f N where LN is a bounded symmetricoperator on L2(SN−1(
√N)):
f Nt = etL
Nf0 =
∞∑k=0
tk
k!(LN)k f N
0 .
Define for g1 = g1(v1) bounded function (of one variable)
g2 = g2(v1, v2) =
∫ 2π
0dθ[g1(v1 cos θ + v2 sin θ)− g1(v1)
]and similarly for k ≥ 2 and gk = gk(v1, . . . , vk) bounded function
gk+1(v1, . . . , vk+1) =k∑
j=1
∫ 2π
0[gk(v1, . . . , vj cos θ + vk+1 sin θ, . . . vk)− gk(v1, . . . , vk)
]dθ
Kac’s main theorem (III)
Claim: for N ≥ k , N → +∞∫f N0 (LN)kg1 ∼
∫f N0,k gk+1
Hence initial chaos ⇒ in RHS replace f N0,k by f ⊗k0,1 as N → +∞
Main induction step: prove
∫f N0 LNgk ∼
∫f N0,k gk+1
∫f N0 LNgk =
∫f N0
[2
(N − 1)
∑i<j∫ 2π
0(gk⊗1N−k)(v1, . . . , v
′i , . . . , v
′j , . . . , vN)−gk(v1, . . . , vk)
]dθ.
Kac’s main theorem (IV)
In order to understand the calculation for general k , let us firstlook at the case k = 2. From the definition of LN , we have
LNg2(v1, v2) =2
(N − 1)
∫ 2π
0
[g2(v1 cos θ + v2 sin θ,
−v1 sin θ + v2 cos θ)− g2(v1, v2)]dθ
+2
(N − 1)
N∑j=3
∫ 2π
0
[g2(v1 cos θ + vj sin θ, v2)− g2(v1, v2)
]dθ
+2
(N − 1)
N∑j=3
∫ 2π
0
[g2(v1, v2 cos θ + vj sin θ)− g2(v1, v2)
]dθ.
Easy induction bound ‖LNgk‖∞ ≤ Ck‖gk‖∞ and therefore∣∣LNgk∣∣ ≤ C kk!
Kac’s main theorem (V)
Back to the proof of the claim∫
f N0 (LN)kg1 ∼
∫f N0,k gk+1:
- If i , j ≥ k the whole thing is zero.
- Terms i , j = 1, . . . , k are of order O(k2
N
).
- Denoting Rkθ (vi , vj) = (v1, . . . , vi cos θ + vj sin θ, . . . , vk):
=N∑
j=k+1
k∑i=1
2
(N − 1)
⟨f N0 ,
∫ 2π
0[gk(Rθ(vi , vj))− gk(v1, . . . , vk)] dθ
⟩+O
(k2
N
)
=N − k
(N − 1)
⟨f N0 ,
k∑i=1
∫ 2π
0[gk(Rθ(vi , vk+1))− gk(v1, . . . , vk)] dθ
⟩+O
(k2
N
)
= 〈f N0 , gk+1〉+ O
(k2
N
)+ O
(k
N
)
Kac’s main theorem (VI)
Define now the first order marginal in dual sense,
I1 := 〈f Nt , g〉 = 〈eLN t f N
0 , g〉 =∑k≥0
tk
k!〈f N
0 , (LN)kg〉,
where we have used the symmetry of LN .From previous calculations and initial chaos assumption
I1(t)N→∞∼
∑k≥0
tk
k!〈f ⊗k0,1 , gk+1〉.
Then consider similarly the 2-marginal I2(t) = 〈f Nt , g(v1)h(v2)〉
and denote Dϕk = ϕk+1:
I2(t) ∼∑k≥0
tk
k!〈f ⊗k0,1 ,D
k(gh)〉
Kac’s main theorem (VII)
Crucial point is the following derivation-like formula:
D(gh) = gDh + Dgh
We deduce
I2(t) ∼∑k≥0
k∑k1=0
C k1k
tk
k!〈f ⊗k0,1 , (Dk1g)(Dk−k1h)〉
and from the initial chaos
I2(t) ∼∑k≥0
k∑k1=0
C k1k
tk
k!
(〈f ⊗k1
0,1 ,Dk1g〉)(〈f ⊗k−k1
0,1 ,Dk−k1h〉)
which yields a Cauchy product∑k1+k2=k≥0
tk1tk−k1
k1!(k − k1)!
(〈f ⊗k1
0,1 ,Dk1g〉)(〈f ⊗k−k1
0,1 ,Dk−k1h〉)∼ I g1 (t)I h1 (t)
Painful but easy to extend argument to any Ik(t), k > 0. . .
Kac’s open problem 1
I McKean 1967 extends Kac’s argument to the real geometry ofcollision but for cutoff Maxwell molecules
I But main physical interaction: hard spheres
I This seemingly technical issue is in fact related to deepdifficulty for dealing with jump process whose jump times lawdepends on the velocity variables: “The above proof suffers from
the defect that it works only if the restriction on time is
independent of the initial distribution. It is therefore inapplicable to
the physically significant case of hard spheres because in this case
our simple estimates yield a time restriction which depends on the
initial distribution.”
I Propagation of chaos for the hard spheres collision process?Incomplete attempt Grunbaum 1971 (important inspiration!)and first proof Sznitman 1984 but with no rate
Kac’s open problem 2
I Spirit of the previous question: going beyond the limit ofKac’s original combinatorial insight in order to deal withrealistic collision processes
I Hence seems very natural to ask whether one can provepropagation of chaos for the true Maxwell molecules collisionprocess?
I The difficulty lies now in the fact that the particle system canundergo infinite number of collisions in a finite time interval,and no “tree” representation of solutions is available.
I This is related with the physical interesting situation oflong-range interactions, as well as with the mathematicalinteresting framework of fractional derivative operator andLevy walk.
Kac’s open problem 3
I Kac then discusses Boltzmann H-theorem in line with hisoriginal goal is the derivation of Boltzmann’s equation andmolecular chaos
I Ergodic property of the Markov process under consideration→ infinite number of Liapunov functions, including the L2
norm and Boltzmann’s entropy
I In contrast with it, the limit equation admits only (in general)the Boltzmann entropy as a Liapunov function.
I Kac then heuristically conjectures H(f Nt )/N → H(ft) along
time, which would recover Boltzmann’s H-theorem from themonotonicity of H(f N)/N for the Markov process.
I “If the above steps could be made rigorous we would have a
thoroughly satisfactory justification of Boltzmann’s H-theorem”
I In our notation the question is can one prove propagation ofentropic chaos along time?
Kac’s open problem 4 I
I He finally discusses relaxation times, with the goal of derivingrelaxation times of the limit equation from the many-particlesystem
I Needs relaxation times independent of the number of particles
I First step proposed by Kac: L2 spectral gap of the process“Surprisingly enough this seems quite difficult and we have not
succeeded in finding a proof. Even for the simplified model we have
been considering, the question remains unsettled although we are
able to give a reasonably explicit solution of the master equation.”
I Triggered beautiful works: Carlen-Carvalho-Loss 2003, see alsoDiaconis-Saloff-Coste 2000, Janvresse 2001, Maslen 2003,Carlen-Geronimo-Loss 2011. . . who fully solved this question
Kac’s problem 4 II
I But no hope of passing to the limit N →∞ in this spectralgap estimate, even if the spectral gap is independent of N.The L2 norm is catastrophic in infinite dimension:
‖f ⊗N‖L2 ∼ CN geometric growth
I Therefore following quite closely the intention of Kac, wereframe the question in a setting which “tensorizes correctly inthe limit N →∞”
I In our notation: can one prove relaxation times independentof the number of particles on
I normalized Wasserstein distance W (f N ,γN )N ?
I normalized relative entropy H(f N |γN )N ?
where γN denotes the N-particle invariant measure?
Convergence of empirical distribution and chaos (I)
Convergence result about µNt implies propagation of chaos from
Empirical measure and chaos (equivalence)
(1) for each ε > 0 and ϕ ∈ Cc(E )
FN(
ZN ∈ E ;∣∣∣〈µNZN − f , ϕ〉
∣∣∣ ≥ ε)→ 0
(2) For any k ≥ 1, we have f Nj → f ⊗k as N →∞
Thus one can recover the chaoticity of the joint microscopicdistribution using the convergence of the empirical measure
Convergence of empirical distribution and chaos (II)
First observe that (1) and L∞ bound on 〈µNZN − f , ϕ〉 implies∫
EN
∣∣∣〈µNZN − f , ϕ〉∣∣∣r dZN → 0 for all r ≥ 1
Case r = 1 yields 〈f N1 , ϕ〉 → 〈f , ϕ〉
Case r = 2 yields by expanding
1
N〈f N
1 , ϕ2〉+N(N − 1)
N2〈f N
2 , ϕ⊗ ϕ〉 − 2〈f , ϕ〉〈f N1 , ϕ〉+ 〈f , ϕ〉2 → 0
which implies (using the case r = 1) 〈f N2 , ϕ⊗ ϕ〉 → 〈f , ϕ〉2
Higher-order marginals are treated similarlyConversely
FN(
ZN ∈ E ;∣∣∣〈µNZN − f , ϕ〉
∣∣∣ ≥ ε) ≤ 1
ε2
∫EN
∣∣∣〈µNZN − f , ϕ〉∣∣∣r dZN
and the RHS goes to zero by inspecting the previous calculationand using the convergence f N
1 → f and f N2 → f ⊗ f
Plan
I. From microscopic to macroscopic evolutions
II. Weak coupling (Vlasov) limit and empirical measure
III. Probabilistic foundation of kinetic theory
IV. Statistical stability and quantitative chaos
What we prove I
In short we answer the four open problems formulated above.
I Propagation of chaos with quantitative rates for collision jumpprocesses in the case of “hard spheres” and long-rangeinteraction (“Maxwell molecules without cutoff”)
I Most importantly estimates uniform in time:⇒ a “reversed” (see later) answer to Kac’s problem
I Infinite-dimensional chaos (in Wasserstein or entropic form)
I Quantification of relaxation times independently of thenumber of particles (in Wasserstein or entropic form)
I New method whose philosophy is a numerical analystpertubation intuition: identify (1) consistency estimates and(2) stability estimates on the limit PDE[No compactness argt. or expansion in terms of initial data]
What we prove II
I Uniform in time finite-dimensional chaos
supt≥0
W1
(Π`f
Nt , f
⊗`t
)≤ K` ε(N)
for fixed `, and ε polynomial (Maxwell molecules) or power oflogarithm (hard spheres)
I More refined (but less understandable!) versions with otherweak distances→ e.g. with negative Sobolev space distance and on finitetime intervals optimal rate of CLT for Maxwell molecules
supt∈[0,T ]
dist(
Π`fNt , f
⊗`)≤ K`,T
N1/2
What we prove III
I Infinite-dimensional chaos
supt≥0
W1
(f Nt , f
⊗Nt
)N
≤ K ε(N)
and entropic chaos
∀ t ≥ 0,H(f Nt
)N
N→∞−−−−→ H(f )
I Estimates on relaxation times indep. of N
∀N ≥ 1,W1
(f Nt , γ
N)
N≤ β(t) with β(t) −−−−→
t→+∞0
(also in entropic form for Maxwell molecules)Rk: Rate β(t) not optimal
The Markov process I
I Start from Markov process (VNt ) on (Rd)N
I Scale time t → t/N in order that the number of interactionsis of order O(1) on finite time interval
I Denote by f Nt the law of VNt and SN
t the associated semigroup
I Master equation on f Nt = SN
t f0 in dual form
∂t〈f Nt , ϕ〉 = 〈f N
t ,GNϕ〉
(GNϕ)(V ) =1
N
N∑i ,j=1
Γ (|vi − vj |)∫Sd−1
b(cos θij)[ϕ∗ij − ϕ
]dσ
where ϕ∗ij = ϕ(V ∗ij ) and ϕ = ϕ(V ) ∈ Cb(RNd) (see next slidefor V ∗ij )
The Markov process II (short-range interaction)
(i) for any i 6= j , draw a random time TΓ(|vi−vj |) of collision(exponential law of parameter Γ(|vi − vj |)), then choose thecollision time T1 and the colliding couple (vi0 , vj0) s.t.
T1 = TΓ(|vi0−vj0 |) := min1≤i 6=j≤N
TΓ(|vi−vj |);
(ii) then draw σ ∈ Sd−1 according to the law b(cos θij), wherecos θij = σ · (vj − vi )/|vj − vi |;
(iii) the new state after collision at time T1 becomes
V ∗ij = (v1, . . . , v∗i , . . . , v
∗j , . . . , vN),
v∗i =vi + vj
2+|vi − vj |σ
2, v∗j =
vi + vj2− |vi − vj |σ
2.
The Markov process IIII Process invariant under velocities permutations and preserves
momentum and energy at any jump
N∑j=1
v∗j =N∑j=1
vj and |V ∗|2 =N∑j=1
|v∗j |2 =N∑j=1
|vj |2 = |V |2
I Hence process on RdN but can be restricted to the manifold
SN :=
N∑j=1
|vj |2 = E ,N∑j=1
vj = 0
I As a consequence∫
RdN
N∑j=1
vj
f Nt (dV ) =
∫RdN
N∑j=1
vj
f N0 (dV ),
∀φ : R+ → R+,
∫RdN
φ(|V |2) f Nt (dV ) =
∫RdN
φ(|V |2) f N0 (dV )
The Markov process IV (expected limit PDE)
I (Expected) limit nonlinear semigroup SNLt (f0) := ft for any
f0 ∈ P2(Rd) (probabilities with bounded second moment)
∂t f = Q(f , f ) with B(v − v∗, σ) = Γ(|v − v∗|)b(cos θ)
Q(f , f )(v) :=
∫v∗∈Rd
∫σ∈Sd−1
(f (v ′∗)f (v ′)−f (v)f (v∗)
)B(v − v∗, σ)
(nonlinear spatially homogeneous Boltzmann equation)
I Conservation of momentum and energy for t ≥ 0∫Rd
v ft(dv) =
∫Rd
v f0(dv),
∫Rd
|v |2 ft(dv) =
∫Rd
|v |2 f0(dv).
The collision operator Q
I Kernel B := Γ(|v − v∗|) b(cos θ): physical information aboutmolecular interaction (different from fluid mechanics model)
I Hard spheres: Γ(Z ) = Z and b = 1 in dimension 3
I Long-range interactions (inverse-power laws): Γ(Z ) = Zβ,β ∈ (−d , 1) and b(cos θ) ∼ C θ−(d−1)−α, θ ∼ 0, α ∈ (0, 2)
I Intuition: Γ ∼ polynomial growth or decay of the coefficentsin a PDE, and order of singularity of b ∼ (fractional) order ofdifferentiation (cf. Levy processes)
I Our theorems cover (in d = 3): γ = 1 and α = d − 1 (hardspheres) and γ = 0 and α = 1/2 (Maxwell molecules)
The key finite-dimensional uniform in time chaos estimates
f0 ∈ P(Rd) with compact support
(i) Hard spheres: for any ` ∈ N∗ and any N ≥ 2`:
supt∈(0,∞)
sup‖ϕ‖
Lip0(Rd )⊗`≤1
⟨Π`
[SNt
(f ⊗N0
)]− SNL
t (f0)⊗`, ϕ⟩≤ ` ε(N)
with ε(N)→ 0 and Lip0(Rd) means Lipschitz vanishing at O
(ii) Maxwell molecules: ∀ ` ∈ N∗, N ≥ 2`, η << 1:
supt∈(0,∞)
sup‖ϕ‖F⊗`≤1
⟨Π`
[SNt
(f ⊗N0
)]− SNL
t (f0)⊗`, ϕ⟩≤ `2 Cη
N1
2(d+4)−η
F :=
ϕ : Rd → R; ‖ϕ‖F :=
∫Rd
(1 + |ξ|4) |ϕ(ξ)| dξ <∞,
Extensions
(i) Cutoff Maxwell molecules (γ = 0 and α = d − 1):
supt∈(0,T )
sup‖ϕ‖
Hs (Rd )⊗`≤1
⟨Π`
[SNt
(f ⊗N0
)]− SNL
t (f0)⊗`, ϕ⟩≤ `2 Cs,T
N12
with ` ∈ N∗, N ≥ 2`, s ∈ (d/2, d/2 + 1) and T ∈ (0,∞)
(ii) Compact support assumption can be relaxed up to anadditionnal error term by conditioning to SN (worse rate)
(iv) Surprising feature: seemingly yields a proof withoutcomputation of chaoticity of the steady states (Poincare’slemma), e.g. first marginal of γN the uniform distribution onthe sphere SN(
√N) converges to a gaussian γ. . . in fact
conditionning to a sphere already makes use of it!
Functional diagram
EN/ΣN
πNE =µN·
Liouville / Kolmogorov//
observables..
Psym(EN)
πNP
duality // Cb(EN)oo
RN
PN(E ) ⊂ P(E )”Super-Liouville” // P(P(E ))
duality // Cb (P(E ))oo
πNC
KK
I E = Rd (or Polish space)
I ΣN N-permutation group
I Psym(EN) symmetric probabilities
Inspiration: Grunbaum 1971, some intersection with idea inKolokoltsov 2010
Maps of the diagram
I µNV denotes the empirical measure:
µNV =1
N
N∑i=1
δvi , V = (v1, . . . , vN)
I PN(E ) = µNV , V ∈ EN ⊂ P(E )
I ∀V ∈ EN/SN , πNE (V ) := µNV
I ∀Φ ∈ Cb (P(E )) , ∀V ∈ EN ,(πNC Φ
)(V ) := Φ
(µNV)
I ∀φ ∈ Cb(EN), ∀ f ∈ P(E ), RN [φ](f ) :=⟨f ⊗N , φ
⟩I ∀Φ ∈ Cb(P(E )), ∀ f N ∈ Psym(EN),
⟨πNP f N ,Φ
⟩=⟨f N , πNC Φ
⟩
Evolution N-particle semigroups
I Process (VNt ) on EN = trajectories: stochastic ODEs (Markovprocess), or deterministic ODEs (Hamiltonian flow).Flow commutes with permutations (part. indistinguishable)
I Corresponding linear semigroup SNt on Psym(EN):
∂t fN = AN f N , f N ∈ Psym(EN),
Forward Kolmogorov equation or Liouville equation
I Dual linear semigroup TNt of SN
t :
∀ f N ∈ P(EN), φ ∈ Cb(EN),⟨
f N ,TNt (φ)
⟩:=⟨
SNt (f N), φ
⟩Semigroup of the observables: ∂tφ = GN(φ), φ ∈ Cb(EN).
Evolution limit semigroups
I (Nonlinear) semigroup SNLt on P(E ) solution to
∂t ft = Q(ft), f0 = f .
I Pushforward linear semigroup T∞t on Cb(P(E )):
∀ f ∈ P(E ), Φ ∈ Cb(P(E )), T∞t [Φ](f ) := Φ(
SNLt (f )
)solution to the linear evolution equation on Cb(P(E )):
∂tΦ = G∞(Φ) with generator G∞.
I T∞t can be interpreted physically as the semigroup of theevolution of observables of the nonlinear Boltzmann equation.
Interpretation of the pushforward semigroup I
I Given a nonlinear ODE V ′ = F (V ) on Rd , one can define (atleast formally) the linear Liouville transport PDE
∂tρ+∇v · (F ρ) = 0,
where ρt(v) = V ∗t (ρ0) = ρ0 V−tI Dual viewpoint of observables: for φ0 function defined on Rd ,
evolution φt(v) = φ0(Vt(v)) = V ∗t φ0 = φ0 Vt solution tothe linear PDE
∂tφ− F · ∇vφ = 0,
I Duality relation:〈φt , ρ0〉 = 〈φ0, ρt〉
Interpretation of the pushforward semigroup II
I Go “one level above”: replace Rd by H = P(E ):The infinite dimensional “ODE” V ′ = Q(V ) on H yields firstthe abstract transport equation
∂tπ +∇ · (Q(v)π) = 0, π ∈ P(H)
and second the abstract dual observable equation
∂tΦ− Q(v) · ∇Φ = 0, Φ ∈ Cb(H).
I Provide intuition and formal formula for the generator“G∞Φ = Q(v) · ∇Φ”: but requires to develop abstractdifferential calculus on Cb(H) to give sense to this heuristic.
I Note that for a dissipative equation, no reversed“characteristics” and observable viewpoint more natural
Diagram of connection between the two dynamics
PNt on EN/ΣN
µNV
observables // TNt on Cb(EN)
RN
PN(E ) ⊂ P(E )observables // T∞t on C (P(E ))
πNC
KK
SNLt on P(E )
”observables”
OO
The metric issue I
I Fundamental space of connection Cb(P(E ))I At the topological level there are two canonical choices:
(1) strong (total variation)(2) weak topology.
I Two different sets: Cb(P(E ),w) ⊂ Cb(P(E ),TV )
I ‖Φ‖L∞(P(E)) does not depend on the choice of topology onP(E ), and induces a Banach topology on the space Cb(P(E )).
The metric issue II
I The transformations πNC and RN satisfy:∥∥∥πNC Φ∥∥∥L∞(EN)
≤ ‖Φ‖L∞(P(E)) and ‖RN [φ]‖L∞(P(E)) ≤ ‖φ‖L∞(EN).
I πNC is well defined from Cb(P(E ),w) to Cb(EN), but ingeneral, it does not map Cb(P(E ),TV ) into Cb(EN) sinceV ∈ EN 7→ µNV ∈ (P(E ),TV ) is not continuous
I RN is well defined from Cb(EN) to Cb(P(E ),w), andtherefore also from Cb(EN) to Cb(P(E ),TV ): for anyφ ∈ Cb(EN) and for any sequence fk → f weakly, we havef ⊗Nk → f ⊗N weakly, and then RN [φ](fk)→ RN [φ](f ).
The metric issue III
I Different possible metric structures inducing the weaktopology not seen at the level of Cb(P(E ),w). However anydifferential structure strongly depends on this choice.
I Define weak metrics on P(E ) by restricting from larger spaces(possibly with moments constraints).
I Ex. 1: Dual-Holder (or Zolotarev’s) distances
[ϕ]s := supx ,y∈E
|ϕ(y)− ϕ(x)|distE (x , y)s
, s ∈ (0, 1], [ϕ]Lip := [ϕ]1.
and
∀ f , g ∈ P1(E ), [g − f ]∗s := supϕ∈C0,s
0 (E)
〈g − f , ϕ〉[ϕ]s
.
The metric issue IV
I Ex. 2: Wasserstein distances
∀ f , g ∈ Pq(E ), Wq(f , g) := infπ∈Π(f ,g)
∫E×E
distE (x , y)q π(dx , dy),
I Ex. 3: Fourier-based norms
∀ f ∈ T P . . . , |f |s := supξ∈Rd
|f (ξ)||ξ|s , s ∈ (0, 2].
I Ex. 4: Negative Sobolev norms For any s ∈ (d/2, d/2 + 1):
∀ f ∈ T P . . . , ‖f ‖H−s(Rd ) :=
∥∥∥∥∥ f (ξ)
|ξ|s
∥∥∥∥∥L2
.
Differential calculus on C (P(Rd )) I
I Less than one derivative: C 0,θΛ (G1, G2) for some metric spaces
G1 and G2, some weight function Λ : G1 7→ R∗+ and someθ ∈ (0, 1], defined by
∀ f1, f2 ∈ G1 distG2(S(f1),S(f2)) ≤ C Λ(f1, f2) distG1
(f1, f2)θ,
with Λ(f1, f2) := maxΛ(f1),Λ(f2).I C 1,θ
Λ (G1; G2)) defined as the space of continuouslydifferentiable functions from G1 to G2, whose derivativesatisfies some weighted θ-Holder regularity.
I Nice usual composition rules. . .
Differential calculus on C (P(Rd )) II
I Well suited to deal with the different objects we have(1-particle semigroup, polynomial, generators, . . . ) oncerestricting to correct “layer” by fixing moments of measures
I Main novelty is the use of this differential calculus to statedifferential stability conditions on the limiting semigroup
I Roughly speaking the latter measures how this limitingsemigroup handles fluctuations around chaoticity
I Corner stone of our analysis, and the hardest part to prove foruniform in time results. However even for finite timepropagation of chaos, this kind of estimates seems mostly new.
Differential calculus on C (P(Rd )) III
I As a first application of this setting one can make rigorous theheuristic derivation of the generator G∞:
I Under natural regularity assumptions on the limit semigroupSNLt and the operator Q Holder, then the generator G∞ of the
pushforward semigroup T∞t exists as an unbounded linearoperator.
I We moreover have the formula
∀Φ ∈ C 1,δ(P...;R), ∀ f ∈ P...,
(G∞Φ) (f ) := 〈DΦ[f ],Q(f )〉 .
Assumptions I
(A1) On the N-particle system.(i) Energy bounds:
∀N ≥ 1, supp f Nt ⊂ EN :=
V ∈ EN ; MN
me(V ) ≤ E
. (1)
where MNm = 1
N
∑Ni=1 m(vi )
(ii) Integral moment bound:
∀N ≥ 1, sup0≤t<T
⟨f Nt ,M
Nm1
⟩≤ CN
T ,m1. (2)
(iii) Support moment bound at initial time:
supp f N0 ⊂
V ∈ EN ; MN
m3(V ) ≤ CN
0,m3
. (3)
Assumptions II
(A2) Existence of the generator of the pushforward semigroup.For some weak distant distG1 on probability induced from G1
(and some constraints)
(i) SNLt acting on probability is continuous (for the metric
distG1), uniformly in time.
(ii) Q is δ-Holder continuous (δ ∈ (0, 1)) from probability intoG1, for the metric distG1 .
(iii) For any f ∈ PG1 , for some τ > 0 the application[0, τ)→ PG1 , t 7→ SNL
t (f ) is C 1,δ([0, τ); PG1), withS(f )′(0) = Q(f ).
Assumptions III
(A3) Convergence of the generators.For the weight function Λ1(f ) = 〈f ,m1〉 and for somefunction ε2(N) going to 0 as N goes to infinity, we assumethat the generators GN and G∞ satisfy∥∥∥∥(MN
m1
)−1(GN πNE − πNE G∞
)Φ
∥∥∥∥L∞(EN)
≤ ε(N) [Φ]C1,θ
Λ1(PG1
),
where MNm =
1
N
N∑i=1
m(vi ) and ε(N)→ 0
Assumptions IV
(A4) Differential stability of the limiting semigroup.
We assume that the flow SNLt is C 1,θ
Λ2(PG1 ,PG2) in the sense
that there exists C∞T > 0 such that∫ T
0
([SNL
t ]C1,θ
Λ2(PG1
,PG2)
+ [SNLt ]1+θ′
C0,θ′′Λ2
(PG1,PG2
)
)dt ≤ C∞T ,
where θ ∈ (0, 1) is the same as in (A3), (θ′, θ′′) = (θ, 1) or
(θ′, θ′′) = (1, (1 + θ)/2), Λ2 = Λ1/(1+θ′)1 and G2 ⊃ G1 are
some normed spaces.
Assumptions V
(A5) Weak stability of the limiting semigroup.We assume that for some probabilistic space PG3(E ) (withweight and constraints) and for any T > 0 there exists aconcave function ΘT : R+ → R+ such that we have
∀ f1, f2 ∈ PG3(E )
sup[0,T )
distG3
(SNLt (f1), SNL
t (f2))≤ ΘT (distG3(f1, f2)) .
Statement I
For any N, ` ∈ N∗, with N ≥ 2`, and ϕ ∈ (F1 ∩F2 ∩F3)⊗` (whereFi are the dual of Gi )
sup[0,T )
∣∣∣∣⟨(SNt (f N
0 )−(
SNLt (f0)
)⊗N), ϕ
⟩∣∣∣∣≤ C
[`2 ‖ϕ‖∞
N+ CN
T ,m1C∞T ε2(N) `2 ‖ϕ‖F2
2⊗(L∞)`−2
+` ‖ϕ‖F3⊗(L∞)`−1 ΘT
(W1,PG3
(πNP pN
0 , δf0
))],
where W1,PG3is the Monge-Kantorovich distance in P(PG3(E )):
W1,PG3
(πNP pN
0 , δf0
)=
∫EN
distG3(µNV , f0) pN0 (dV ).
Statement II
I Assume furthermore (f N0 )N≥1 is f0-chaotic, i.e.
Wθ3,PG3(πNP pN
0 , δf0)→ 0, then pNt is pt-chaotic in the
quantified way above (chaos propagation)
I Treatment of the N-particles system as a perturbation (in avery degenerated sense) of the limiting problem, minimizeassumptions on the many-particle systems in order to avoidcomplications of many dimensions dynamics.
I In the applications worst decay rate in the right-hand side isalways the last one, which deals with the chaoticity of theinitial data (law of large number in probability space)
I Fluctuations estimates explicit in terms of the constant in theassumptions, therefore if these constants are uniform in times,so are the chaos propagation estimates
Scheme of the proof I
For ϕ ∈ (F1 ∩ F2 ∩ F3)⊗`, break up the term to be estimated intothree parts:∣∣∣⟨(SN
t (f N0 )− (S∞t (f0))⊗N
), ϕ⊗ 1⊗N−`
⟩∣∣∣ ≤≤∣∣∣⟨SN
t (f N0 ), ϕ⊗ 1⊗N−`
⟩−⟨
SNt (f N
0 ),R`ϕ µNV
⟩∣∣∣ (=: T1)
+∣∣∣⟨f N
0 ,TNt (R`
ϕ µNV )⟩−⟨
f N0 , (T∞t R`
ϕ) µNV )⟩∣∣∣ (=: T2)
+∣∣∣⟨f N
0 , (T∞t R`ϕ) µNV )
⟩−⟨
(S∞t (f0))⊗`, ϕ⟩∣∣∣ (=: T3)
Scheme of the proof II
We deal separately with each part step by step:
I T1 controlled by a classical purely combinatorial arguments[In some sense price to pay for using the injection πNE basedon empirical measures]
I T2 controlled thanks to the consistency estimate (A3) on thegenerators, the differential stability assumption (A4) on thelimiting semigroup and the moments propagation (A1)
I T3 controlled in terms of the chaoticity of the initial data,which is propagated thanks to the weak stability assumption(A5) on the limiting semigroup (and (A1)-(iii))
Step 1: Estimate of the first term T1
Let us prove that for any t ≥ 0 and any N ≥ 2` there holds
T1 :=∣∣∣⟨SN
t (f N0 ), ϕ⊗ 1⊗N−`
⟩−⟨
SNt (f N
0 ),R`ϕ µNV
⟩∣∣∣ ≤ 2 `2 ‖ϕ‖L∞(E `)
N.
Since SNt (f N
0 ) is a symmetric probability measure, consequence of:Lemma: For any ϕ ∈ Cb(E `) we have
∀N ≥ 2`,
∣∣∣∣(ϕ⊗ 1⊗N−`)sym− πNR`
ϕ
∣∣∣∣ ≤ 2 `2 ‖ϕ‖L∞(E `)
N
and for any symmetric measure pN ∈ P(EN) we have∣∣∣R`ϕ(µNV )〉 − 〈pN , ϕ〉
∣∣∣ ≤ 2 `2 ‖ϕ‖L∞(E `)
N.
Step 2: Estimate of the second term T2 I
Let us prove that for any t ∈ [0,T ) and any N ≥ 2` there holds
T2 :=∣∣∣⟨f N
0 ,TNt
(R`ϕ µNV
)⟩−⟨
f N0 ,(
T∞t R`ϕ
) µNV
⟩∣∣∣≤ CN
T ,m2C∞T ‖ϕ‖∞,F2
2⊗(L∞)`−2 `2 ε(N).
We start from the following identity
TNt πN − πNT∞t = −
∫ t
0
d
ds
(TNt−s πN T∞s
)ds
=
∫ t
0TNt−s
[GNπN − πNG∞
]T∞s ds.
Step 2: Estimate of the second term T2 II
From assumptions (A1) and (A3), we have for any t ∈ [0,T )∣∣∣⟨f N0 ,TN
t
(R`ϕ µNV
)⟩−⟨
f N0 ,(
T∞t R`ϕ
) µNV
⟩∣∣∣≤∫ T
0
∣∣∣∣⟨MNm1
SNt−s
(f N0
),(
MNm1
)−1 [GNπN − πNG∞
] (T∞s R`
ϕ
)⟩∣∣∣∣ ds
≤(
sup0≤t<T
⟨pNt ,M
Nm1
⟩) (∫ T
0
∥∥∥∥(MNm1
)−1 [GNπN − πNG∞
] (T∞s R`
ϕ
)∥∥∥∥∞
ds
)
≤ ε(N) CNT ,m
∫ T
0
[T∞s R`
ϕ
]C1,θ
Λ1(PG1
)ds
with[T∞s
(R`ϕ
)]C1,θ
Λ22
(PG2)≤[SNLt
]C1,θ
Λ2(PG1
,PG2)
∥∥∥R`ϕ
∥∥∥C1,θ(PG2
).
Step 3: Estimate of the third term T3 I
Let us prove that for any t ≥ 0, N ≥ `
T3 :=
∣∣∣∣⟨f N0 ,(
T∞t R`ϕ
) µNV
⟩−⟨(
S∞t (f0))⊗`
, ϕ
⟩∣∣∣∣ ≤≤ [Rϕ]C0,1 ΘT
(W1,PG3
(πNP pN
0 , δf0
)).
We shall proceed as in Step 2, using that:
I suppπNP f N0 ⊂ K := f ∈ PG3 ; Mm3(f ) ≤ CN
0,m3 thanks to
assumption (A1),
I SNLt satisfies some Holder like estimate uniformly on K and
[0,T ) thanks to assumption (A5),
I R`ϕ ∈ C 0,1(PG3 ,R) because ϕ ∈ F⊗`3 .
Step 3: Estimate of the third term T3 II
We deduce thanks to the Jensen inequality (for a concave function)
T3 =∣∣∣⟨pN
0 ,R`ϕ
(SNLt (µNV )
)⟩−⟨
f N0 ,R`
ϕ
(SNLt (f0)
)⟩∣∣∣=
∣∣∣⟨pN0 ,R
`ϕ
(SNLt (µNV )
)− R`
ϕ
(SNLt (f0)
)⟩∣∣∣≤
[Rϕ]C0,1(PG3
)
⟨pN
0 , distG3
(SNLt (f0), SNL
t (µNV ))⟩
≤[Rϕ]C0,1(PG3
)
⟨pN
0 , ΘT
(distG3(f0, µ
NV ))⟩
≤[Rϕ]C0,1(PG3
)ΘT
(⟨pN
0 , distG3(f0, µNV )⟩
≤[Rϕ]C0,1(PG3
)ΘT
(W1,PG3
(πNP f N
0 , δf0
))
Application to the Boltzmann equation I
I There are several interesting steps suggested by our abstractframework in order to apply it to a PDE, in order to prove theabstract assumptions.
I This leads to new interesting problems of a priori estimates inweak measure distances
I Establish the stability estimates (A4) (differentiability of theflow) and (A5) (Holder stability of the flow)→ choose correct metrics for each
I Metric chosen for (A4) impacts and constrained byconsistency estimate (A3)→ total variation metric for hard spheres→ Fourier-based weak metric for Maxwell molecules
I (A5) was proved recently for hard spheres in Fournier-CM2009
Application to the Boltzmann equation II
I Making these estimates uniform in time requires more work,and relies on the most recent a priori estimates onhomogeneous Boltzmann equation:- appearance of exponential moments for hard spheres byMischler-CM 2006- contraction in Fourier-based distance metric for Maxwellmolecules by Toscani, Carrillo et al.
I Another interesting related question arising for optimal rate:dictated by LLN/CLT at initial time, non-trivial pb in generalin Banach setting (sampling of a distribution by empiricalmeasures)
I Other applications inelastic collisions, Fokker-Planck, jump +diffusion, Vlasov in Mischler-CM-Wennberg
The notion of Statistical stability
I In Braun-Hepp-Dobrushin Lipschitz estimate C 0,1 on S∞tI Here crucial point is higher than C 1 differentiability of the
flow in terms of the initial data
I Done in weak distance in order to handle empirical measure,but playing with mollification and interpolation: possible touse stability in stronger space (Marahrens-M. onhydrodynamic limit)
I Differentiability C 2 of S∞t in terms of initial data ⇔propagation of regularity C 2 of the pushforward semigroupT∞t ⇔ propagation of “negative” regularity C−2 for thestatistical flow (T∞t )∗ on P(P(E ))
I Statistical stability: controls fluctuations in perturbation ofT∞t by TN
t around chaos
I Possible to reframe it in the framework of BBGKY hierarchy:correct assumption in order to prove quantitative estimates ofstability on the BBGKY hierarchy
From finite-dimensional to infinite-dimensional chaos
I Recent result Hauray-Mischler (HAL preprint): for anyf ∈ P(Rd) and sequence f N ∈ Psym(Rd) we have
W1
(f N , f ⊗N
)N
≤ C
(W1
(Π2
[f N], f ⊗2
)α1
+1
Nα2
)for some constructive constant C , α1, α2 > 0.
I Idea of the proof: pass “through” a Hilbert negative Sobolevsetting in order to make use of cancellations, and pass“through” generalized Wasserstein distance in P(P(E )), andestimate error (change of norm, combinatorial)
I Morally: the 2-particle correlation measure is enough tocontrol the N-particle correlation measure once correctlyscaled (extensivity)
Propagation of entropic chaos I
I Maxwell molecules (with or without cutoff) or hard spheres
I Initial data f with exponential moment bounds
I Sequence of N-particle initial data (f N0 )N≥1 constructed by
conditioning to SN .
I Then if the initial data is entropy-chaotic in the sense
1
NH(
f N0 |γN
)N→+∞−−−−−→ H (f0|γ)
with
H(
f N0
):=
∫SN
f N0 log
f N0
γNdV
then the solution is also entropy chaotic for any later time:
∀ t ≥ 0,1
NH(
f Nt |γN
)N→+∞−−−−−→ H (ft |γ) .
I Derivation of the H-theorem
Propagation of entropic chaos II
Sketch of the proof
d
dt
1
NH(
f Nt |γN
)= −DN
(f Nt |γN
)DN
(f N)
:=1
2N2
∫SN
∑i 6=j
∫Sd−1
(f N(rij ,σ(V ))− f N(V )
)log
f N(rij ,σ(V )
f N(V )B
Hence
∀ t ≥ 0,1
NH(
f Nt |γN
)+
∫ t
0DN
(f Ns
)ds =
1
NH(
f N0 |γN
)and at the limit
∀ t ≥ 0, H (ft |γ) +
∫ t
0D∞ (fs) ds = H (f0|γ)
Propagation of entropic chaos III
Sketch of the proof - bisThen prove that the many-particle relative entropy and entropyproduction functionals defined above are lower semi-continuous inP(P(Rd)) in terms of f N : if (f N)N≥1 is f -chaotic then (known)
lim infN→∞
1
NH(
f N |γN)≥ H(f |γ) ≥ 0
and (∼new)
lim infN→∞
DN(
f N)≥ D∞(f ) ≥ 0
If we assume furthermore that
1
NH(
f N0 |γN
)→ H (f0|γ)
at initial time, then it implies the convergence of the functionals attime t and concludes the proof
Many-particle relaxation times I
Maxwell molecules or hard spheres and conditionned initial data.Then
∀N ≥ 1, ∀ t ≥ 0,W1
(f Nt , γ
N)
N≤ β(t)
for some β(t)→ 0 as t →∞, where γ gaussian equilibrium withenergy E and γN uniform probability measure on SN(
√NE)
Sketch of the proofWrite
W1
(f Nt , γ
N)
N≤
W1
(f Nt , f
⊗Nt
)N
+W1
(f ⊗Nt , γ⊗N
)N
+W1
(γ⊗N , γN
)N
Many-particle relaxation time II
Sketch of the proof - bisThen
W1
(γ⊗N , γN
)N
≤ α(N)→ 0 as N →∞by explicit computation, and
W1
(f ⊗Nt , γ⊗N
)N
≤W1 (ft , γ)→ 0 as t →∞
by CM 2006, which implies
W1
(f Nt , γ
N)
N≤ α(N) + β(t)
From the L2 spectral gap estimate in Carlen-Geronimo-Loss andCarlen-Carvalho-Loss one can deduce
∀N ≥ 1, ∀ t ≥ 0,W1
(f Nt , γ
N)
N≤ CN e−λ t
Conclusion by optimization
Many-particle relaxation rate in the H-theorem I
In the case of Maxwell molecules, and assuming moreover that theFisher information of the initial data f0 is finite:∫
Rd
|∇v f0|2f0
dv < +∞,
the following estimate on the relaxation induced by the H-theoremuniformly in the number of particles also holds:
∀N ≥ 1,1
NH(
f Nt |γN
)≤ β(t)
for some function β(t)→ 0 as t →∞.
Many-particle relaxation rate in the H-theorem II
Sketch of the proof
First prove propagation of the Fisher information
∀ t ≥ 0,I(f Nt |γN
)N
≤ I(f N0 |γN
)N
with
I(
f N |γN)
:=
∫SN
∣∣∇f N∣∣2
f NdV .
then use the HWI interpolation inequality on the manifold SN
1
NH(
f N |γN)≤ W2
(f N , γN
)√
N
√I (f N |γN)
N− K
2NW2
(f N , γN
)2
≤ W2
(f N , γN
)√
N
√I (f N |γN)
N.
Going back to the probabilistic interpretation
I We prove quantitative LLN in P(P(E )), i.e. deviationestimates in P(P(E ))
I Possible to obtain CLT in P(P(E )) (“gaussian” = solution tothe linearized flow + Ornstein-Uhlenbeck noise)
I Large deviation?
I How is it possible to have uniform in time convergence?Stochastic trajectories departs from deterministic trajectorieslike their variance!
I For Brownian motion (diffusion): time-scale O(√
N). . .
I Here at the level of the laws! Ergodicity time-scale wins overtime-scale of the effect of trajectories fluctuations
Perspectives
I In progress with Mischler: new Liapunov function for someinelastic collision operator plus diffusion by mean-field limit?[Cf. Original goal of Kac: recover new information on thelimiting equation from the many-particle Markov process]
I Inspiration for work in progress withBodineau-Lebowitz-Villani about new relative entropies fornonlinear diffusion with inhomogeneous Dirichlet conditions
I Investigations on hydrodynamical limit by such a quantitativemethod with Daniel Marahrens