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2016 | OriginalPaper | Buchkapitel

1. On the Dynamics of Large Particle Systems in the Mean Field Limit

verfasst von : François Golse

Erschienen in: Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity

Verlag: Springer International Publishing

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Abstract

This course explains how the usual mean field evolution partial differential equations (PDEs) in Statistical Physics—such as the Vlasov-Poisson system, the vorticity formulation of the two-dimensional Euler equation for incompressible fluids, or the time-dependent Hartree equation in quantum mechanics—can be rigorously derived from the fundamental microscopic equations that govern the evolution of large, interacting particle systems. The emphasis is put on the mathematical methods used in these derivations, such as Dobrushin’s stability estimate in the Monge-Kantorovich distance for the empirical measures built on the solution of the N-particle motion equations in classical mechanics, or the theory of BBGKY hierarchies in the case of classical as well as quantum problems. We explain in detail how these different approaches are related; in particular we insist on the notion of chaotic sequences and on the propagation of chaos in the BBGKY hierarchy as the number of particles tends to infinity.

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Nächstes Kapitel Continuum Limits of Discrete Models via -Convergence

Illustrations of the Dynamical Theory of Gases, Philosophical Magazine (1860); reprinted in “The Scientific Papers of James Clerk Maxwell”, edited by W.D. Niven, Cambridge University Press, 1890; pp. 377–409.

For each topological space X and each finite dimensional vector space E on $\mathbf {R}$, we designate by C(X, E) the set of continuous functions defined on X with values in E, and by $C_c(X,E)$ the set of functions belonging to C(X, E) whose support is compact in X. For each $n,k\ge 1$, we denote by $C^k_c(\mathbf {R}^n,E)$ the set of functions of class $C^k$ defined on $\mathbf {R}^n$ with values in E whose support is compact in $\mathbf {R}^n$. We also denote $C(X):=C(X,\mathbf {R})$, $C_c(X):=C_c(X,\mathbf {R})$ and $C^k_c(\mathbf {R}^n):=C^k_c(\mathbf {R}^n,\mathbf {R})$.

For each $n\ge 1$, the Schwartz class $\fancyscript{S}(\mathbf {R}^n)$ is the set of real-valued $C^\infty $ functions defined on $\mathbf {R}^n$ all of whose partial derivatives are rapidly decreasing at infinity:

$$ \fancyscript{S}(\mathbf {R}^n):=\{f\in C^\infty (\mathbf {R}^n)\hbox { s.t. }|x|^m{\partial }^{\alpha }f(x)\rightarrow 0\hbox { as }|x|\rightarrow \infty \, \hbox {for all }m\ge 1\hbox { and }{\alpha }\in \mathbf {N}^n\}\, . $$

For each topological space X and each finite dimensional vector space E on $\mathbf {R}$, we denote by $C_b(X,E)$ the set of continuous functions defined on X with values in E that are bounded on X. For each $n,k\ge 1$, we denote by $C^k_b(\mathbf {R}^n,E)$ the set of functions of class $C^k$ defined on $\mathbf {R}^n$ with values in E all of whose partial derivatives are bounded on $\mathbf {R}^n$: for each norm $|\cdot |_E$ on E, one has

$$ C^k_b(\mathbf {R}^n,E):=\{f\in C^k(\mathbf {R}^n,E)\hbox { s.t. }\sup _{x\in \mathbf {R}^n}|{\partial }^{\alpha }f(x)|_E<\infty \hbox { for each }{\alpha }\in \mathbf {N}^n\}. $$

We also denote $C_b(X):=C_b(X,\mathbf {R})$ and $C^k_b(\mathbf {R}^n):=C^k_b(\mathbf {R}^n,\mathbf {R})$.

Henceforth, the set of Borel probability measures on $\mathbf {R}^d$ will be denoted by $\fancyscript{P}(\mathbf {R}^d)$.

Given two measurable spaces $(X,\fancyscript{A})$ and $(Y,\mathscr {B})$, a measurable map ${\varPhi }:\,(X,\fancyscript{A})\rightarrow (Y,\mathscr {B})$ and a measure m on $(X,\fancyscript{A})$, the push-forward of m under ${\varPhi }$ is the measure on $(Y,\mathscr {B})$ defined by the formula

$$ {\varPhi }\#m(B)=m({\varPhi }^{-1}(B)),\quad \hbox { for all }B\in \mathscr {B}. $$

We designate by $w-\fancyscript{P}(\mathbf {R}^d)$ the set $\fancyscript{P}(\mathbf {R}^d)$ equipped with the weak topology of probability measures, i.e. the topology defined by the family of semi-distances

$$ d_\phi (\mu ,\nu ):=\left| \int \limits _{\mathbf {R}^d}\phi (z)\mu (dz)-\int \limits _{\mathbf {R}^d}\phi (z)\nu (dz)\right| $$

as $\phi $ runs through $C_b(\mathbf {R}^d)$.

The reader should be aware of the following subtle point. In classical references on distribution theory, such as [57], the Dirac mass is viewed as a distribution, therefore as an object that generalizes the notion of function. There is a notion of pull-back of a distribution under a $C^\infty $ diffeomorphism such that the pull-back of the Dirac mass at $y_0$ with a $C^\infty $ diffeomorphism $\chi :\,\mathbf {R}^N\rightarrow \mathbf {R}^N$ satisfying $\chi (x_0)=y_0$ is

$$ {\delta }_{y_0}\circ \chi =|{\text {det}}(D\chi (x_0))|^{-1}{\delta }_{x_0}. $$

This notion is not to be confused with the push-forward under $\chi $ of the Dirac mass at ${\delta }_{x_0}$ viewed as a probability measure, which, according to the definition in the previous footnote is

$$ \chi \#{\delta }_{x_0}={\delta }_{y_0}. $$

In particular

$$ \chi \#{\delta }_{x_0}\not ={\delta }_{x_0}\circ \chi ^{-1} $$

unless $\chi $ has Jacobian determinant 1 at $x_0$.

Monge’s original problem was to minimize over the class of all Borel measurable transportation maps $T:\,\mathbf {R}^d\rightarrow \mathbf {R}^d$ such that $T\#\mu =\nu $ the transportation cost $\int \nolimits _{\mathbf {R}^d}|x-T(x)|\mu (dx).$

The notation $\mathscr {L}^d$ designates the Lebesgue measure on $\mathbf {R}^d$.

This last statement is not completely correct, as P.-L. Lions recently proposed a well defined mathematical object that would play the role of a “symmetric function of infinitely many variables that is slowly varying in each variable”: see [72] and Sect. 1.7.3.

For the case of the Boltzmann-Grad limit for a system of N hard spheres, the infinite hierarchy cannot be derived rigorously from the Liouville equation by passing to the limit in the sense of distributions in each equation of the (finite) BBGKY hierarchy: see the discussion on pp. 74–75 in [26]. The infinite Boltzmann hierarchy is derived by a different, more subtle procedure that is the core of the Lanford proof—see Sect. 4.4 in [26].

For each locally compact topological space X and each finite dimensional vector space E on $\mathbf {R}$, we denote by $C_0(X,E)$ the set of continuous functions on X with values in E that converge to 0 at infinity. We set $C_0(X):=C_0(X,E)$.

For each $n,k\ge 1$ and each finite dimensional vector space E on $\mathbf {R}$, we denote by $C^k_0(\mathbf {R}^n,E)$ the set of functions of class $C^k$ defined on $\mathbf {R}^n$ with values in E all of whose partial derivatives converge to 0 at infinity. In other words,

$$ C^k_0(\mathbf {R}^n,E):=\{f\in C^k(\mathbf {R}^n,E)\hbox { s.t. }{\partial }^{\alpha }f(x)\rightarrow 0\hbox { as }|x|\rightarrow \infty \hbox { for each }{\alpha }\in \mathbf {N}^n\}. $$

We denote $C^k_0(\mathbf {R}^n):=C^k_0(\mathbf {R}^n,\mathbf {R})$.

Since $(\fancyscript{P}(Q),{\text {dist}}_{LP})$ is compact, any element U of $C(\fancyscript{P}(Q))$ is bounded on $\fancyscript{P}(Q)$, so that

$$ \sup _{\mu \in \fancyscript{P}(Q)}|U(\mu )|<\infty . $$

This normalization condition is not satisfied by “generalized eigenfunctions” of an operator with continuous spectrum. Consider the two following examples, where ${\mathfrak {H}}=L^2(\mathbf {R})$.

(a) Let $\mathcal{H}=-\tfrac{1}{2}\frac{d^2}{dx^2}+\tfrac{1}{2}x^2$ (the quantum harmonic oscillator), which has discrete spectrum only. The sequence of eigenvalues of $\mathcal{H}$ is $n+\tfrac{1}{2}$ with $n\in \mathbf {N}$. Besides ${\text {Ker}}(\mathcal{H}-(n+\tfrac{1}{2})I)=\mathbf {C}h_n$ for each $n\in \mathbf {N}$, with

$$ h_n(x):=\frac{1}{\sqrt{2^nn!}\pi ^{1/4}}e^{-x^2/2}H_n(x),\quad \hbox { where }H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}. $$

The function $H_n$ is the nth Hermite polynomial, and the sequence $(h_n)_{n\ge 0}$ is a Hilbert basis of $\mathfrak {H}$. In particular one has the orthonormality condition

$$ \int \limits _\mathbf {R}h_n(x)h_m(x)dx={\delta }_{mn},\quad \hbox { for all }m,n\ge 0 $$

where ${\delta }_{mn}$ is the Kronecker symbol (i.e. ${\delta }_{mn}=0$ if $m\not =n$ and ${\delta }_{mn}=1$ if $m=n$).

(b) Let $P=-i\frac{d}{dx}$ (the momentum operator), which has continuous spectrum only. The spectrum of P is the real line $\mathbf {R}$, and the generalized eigenfunctions of P are the functions $e_k:\,x\mapsto e_k(x):=e^{i2\pi kx}$. For each $k\in \mathbf {R}$, one has $Pe_k=2\pi ke_k$ but $e_k\notin \mathfrak {H}$. However one has the “formula” analogous to the orthonormality condition in case (a):

$$ \int \limits _{\mathbf {R}}\overline{e_k(x)}e_l(x)dx={\delta }_0(k-l) $$

where ${\delta }_0$ is the Dirac mass at the origin. The integrand on the left hand side of the equality above is not an element of $L^1(\mathbf {R})$, and the integral is not a Lebesgue integral. The identity above should be understood as the Fourier inversion formula on the class $\fancyscript{S}'(\mathbf {R})$ of tempered distributions on the real line $\mathbf {R}$.

In the case (a), if the system is in an eigenstate corresponding with the eigenvalue $n+\tfrac{1}{2}$ of the operator $\mathcal{H}$, its wave function is of the form $\psi ={\omega }h_n$ with $|{\omega }|=1$, so that $\Vert \psi \Vert _{\mathfrak {H}}=1$.

In the case (b), if the system is in an eigenstate corresponding with the element k of the spectrum of the operator P, it cannot be described by any wave function in $\mathfrak {H}$, but only by a generalized eigenfunction of P, that does not belong to $\mathfrak {H}$.

In the discussion below, we shall never consider quantum states described by generalized eigenfunctions as in (b), but only quantum states corresponding with normalized wave functions.

A mild solution of the Cauchy problem

$$ \left\{ \begin{aligned}{}&\dot{u}(t)=Au(t)+F[u(t)], \\&u{\big |}_{t=0}=u^{in}, \end{aligned}\right. $$

where A generates a strongly continuous contraction semigroup on a Banach space X and $F:\,X\rightarrow X$ is a continuous map, is an X-valued continuous function $I\ni t\mapsto u(t)\in X$ defined on $I=[0,\tau ]$ with $\tau \in [0,+\infty ]$ that is a solution of the integral equation

$$ u(t)=e^{tA}u^{in}+\int \limits _0^te^{(t-s)A}F[u(s)]ds $$

for each $t\in I$.

For linear operators A, B on the Hilbert space $\mathfrak {H}$, we designate by [A, B] their commutator in other words $[A,B]:=AB-BA$. There is obviously a difficulty with the domain of [A, B] viewed as an unbounded operator on $\mathfrak {H}$ if A and B are unbounded operators on $\mathfrak {H}$. This difficulty will be deliberately left aside, as we shall mostly consider $B\mapsto [A,B]$ as an unbounded operator on $\fancyscript{L}(\mathfrak {H})$, which is different matter.

For each topological space X and each vector space E on $\mathbf {R}$ equipped with the norm $\Vert \cdot \Vert _E$, we designate by C(X, E) the set of continuous maps from X to E, and we denote

$$ C_b(X,E):=\{f\in C(X,E)\hbox { s.t. }\sup _{x\in X}\Vert f(x)\Vert _E<\infty \}. $$

In other words, $C_b(X,E)$ is the set of bounded continuous functions defined on X with values in E.

The BBGKY hierarchy has been derived from the N-particle Schrödinger equation in the case where the potential V belongs to $L^\infty (\mathbf {R}^d)$—see Theorem 1.10.1. However its validity in the sense of distributions under the assumptions of Theorem 1.10.2 results from the same arguments as those used in the derivation of the infinite hierarchy and presented below.

This problem was mentioned to me by M. Pulvirenti in October 2013.

After completion of these lecture notes, there has been some progress on this problem: see [45, 47]. Earlier results on this problem can be also found in [50] and in [85].

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Titel: On the Dynamics of Large Particle Systems in the Mean Field Limit
verfasst von: François Golse
Verlag: Springer International Publishing
Buch: Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity
Print ISBN: 978-3-319-26882-8

Electronic ISBN: 978-3-319-26883-5

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-319-26883-5_1

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