Propagation of Chaos for a Thermostated Kinetic Model

Abstract

We consider a system of N point particles moving on a d-dimensional torus $\mathbb{T}^{d}$. Each particle is subject to a uniform field E and random speed conserving collisions $\mathbf{v}_{i}\to\mathbf{v}_{i}'$ with $|\mathbf{v}_{i}|=|\mathbf{v}_{i}'|$. This model is a variant of the Drude-Lorentz model of electrical conduction (Ashcroft and Mermin in Solid state physics. Brooks Cole, Pacific Grove, 1983). In order to avoid heating by the external field, the particles also interact with a Gaussian thermostat which keeps the total kinetic energy of the system constant. The thermostat induces a mean-field type of interaction between the particles. Here we prove that, starting from a product measure, in the limit N→∞, the one particle velocity distribution f(q,v,t) satisfies a self consistent Vlasov-Boltzmann equation, for all finite time t. This is a consequence of “propagation of chaos”, which we also prove for this model.

Appendix: Properties of the Boltzmann Equation

In this appendix, we prove existence and uniqueness for our Boltzmann equation. As in the proof of Theorem 1.3, we do this first in the spatially homogeneous case, where the notation is less cumbersome, and then explain how to extend the argument to phase space.

As we have indicated, the key is the fact that the current satisfies an autonomous equation. Let $\tilde{\mathbf {j}}_{0}$ be computed from f ₀(v), the initial data for (1.6). Then solve for $\tilde{\mathbf {j}}(t)$. Using this expression for $\tilde{\mathbf {j}}$ in (1.6) we obtain the equation

$$ \frac{\partial }{\partial t}f(\mathbf{v},t) + {\rm div} \bigl({\bf Y}(\mathbf{v},t)f(\mathbf{v},t) \bigr) = \mathcal{C}f(\mathbf{v},t) , $$

(A.1)

where $\mathcal{C}$ denotes the collision operator on the right side of (1.6), and where ${\bf Y}(\mathbf{v},t)$ is the vector field

$$ {\bf Y}(\mathbf{v},t) = \mathbf {E}-\frac{\mathbf {E}\cdot\tilde{\mathbf {j}}(t)}{\tilde{u}} \mathbf{v} , $$

(A.2)

which is linear, and hence globally Lipschitz. Let Φ _t (v) denote the flow corresponding to this vector field, so that v(t):=Φ _t (v) is the unique solution to $\dot {\mathbf{v}}(t) = {\bf Y}(\mathbf{v}(t),t)$ with v(0)=v. Calling g(v,t) the push-forward of f(v,t) under the flow transformation Φ _t , that is g(v,t)=det(DΦ _t (v))f(Φ _t (v),t), where DΦ _t is the Jacobian of Φ _t , we deduce that f(v,t) satisfies (A.1) if and only if g(v,t) satisfies

$$ \frac{\partial }{\partial t}g(\mathbf{v},t) = \widetilde {\mathcal{C}}_t g( \mathbf{v},t) , $$

(A.3)

where

$$\widetilde {\mathcal{C}}_tg(\mathbf{v}) = \det(D\varPhi_{-t}) \bigl[ \mathcal{C} \bigl( \det(D\varPhi_t) g\circ \varPhi_t \bigr) \bigr]\circ \varPhi_{-t} . $$

Under our hypotheses, $\widetilde {\mathcal{C}}_{t}$ is bounded for each t, and depends continuously on t, and hence existence and uniqueness of solutions of (A.3) is straightforward to prove. It is also straightforward to prove that the current generated by the solution f(v,t) satisfies the differential equation and initial conditions of Lemma 1.2 and hence it equals $\tilde{\mathbf {j}}$.

To obtain the same result in the spatially inhomogeneous case, we replace the vector field ${\bf Y}$ in (A.2) by

$$ {\bf Y}(\mathbf{q},\mathbf{v},t) = \biggl(\mathbf{v} , \mathbf {E}-\frac{\mathbf {E}\cdot\tilde{\mathbf {j}}(t)}{\tilde{u}} \mathbf{v} \biggr) , $$

(A.4)

and let Φ _t be the phase space flow it generates. Again, the vector field is linear and hence globally Lipschitz which guarantees the existence, uniqueness and smoothness of flow. Now the same argument may be repeated.