Mean-field limit and Vlasov equation for a Gaussian-thermostatted Lorentz gas

Introduction

The derivation of kinetic equations from microscopic particle systems is a central problem in mathematical physics. In the mean-field regime, a large class of interacting systems is expected to be described, as the number of particles $N \to\infty$, by nonlinear transport equations of Vlasov type. This scenario has been rigorously established in various settings, starting from the classical works of Braun–Hepp [11] and Dobrushin [25] (see also [39, 27] for the general theory), and [28] for the relation between empirical measures and Vlasov hierarchies. Important extensions of the classical theory to rough or singular interaction forces were obtained in [30, 32, 38]. A distinct line of work derives Vlasov equations from fermionic quantum many-body dynamics in a joint mean-field and semiclassical regime, see, e.g., [4] and references therein. Taken together, the above works support the general principle that a microscopic system with mean-field interactions should, in the many-particle limit, be governed by a macroscopic equation of Vlasov type.

In the present paper we investigate this principle for a classical mechanical system whose one-particle dynamics is a billiard flow. The resulting specular collisions and grazing trajectories introduce singularities that constitute the main novelty of the problem.

We study a thermostatted system of particles evolving in a dispersing billiard (Sinai billiard) under the action of a constant¹ external electric field. The dynamics is that of a Lorentz gas with specular reflections at the obstacles, coupled through a Gaussian thermostat which enforces conservation of the total kinetic energy. The case of a single moving particle is a canonical model of deterministic nonequilibrium statistical mechanics. For sufficiently weak fields, it admits a physical nonequilibrium steady state (NESS) with well-understood statistical properties; in particular, its transport, response and entropy-production mechanisms have been studied extensively [7, 12, 19, 20, 22, 23]. In the many particles case, although the particles don’t interact directly, the friction coefficient depends on their empirical current and therefore couples all their motions through a mean field while preserving the total kinetic energy. Formal analysis and numerical evidence point to a self-consistent kinetic evolution and to a stable small-field distribution of particle speeds [10]. However, a rigorous derivation of this macroscopic evolution from the deterministic many-particle billiard dynamics has remained unavailable.

From the mathematical point of view, the thermostat leads to a nonlinear Vlasov-type equation for the one-particle distribution, while the obstacles introduce specular reflections and grazing collisions. The billiard flow is only piecewise smooth, and its derivative is unbounded near grazing trajectories [21]. As a consequence the Cauchy–Lipschitz theory does not apply to the characteristics, and the classical Dobrushin stability argument [25] is not available. There are results on Vlasov equations with specular reflection in bounded domains, mainly for Vlasov–Poisson-type systems, such as the boundary regularity results in [29], the global existence results [31], and the more recent global well-posedness result [17]. In these works the behaviour near grazing velocities plays a crucial role, and conditions such as flatness of the datum near the grazing set are imposed, which do not applied in the present setting. The closest rigorous analogue of our setting is the thermostatted kinetic model of [6], where the Gaussian thermostat also generates a mean-field interaction but collisions are modeled by a stochastic mechanism instead of deterministic reflections. To the best of our knowledge, a rigorous mean-field limit for a deterministic thermostatted Lorentz gas has not been established before.

Our approach is mainly dynamical and it is based on the observation that the nonlinearity is of rank one, i.e. all particles feel the same friction coefficient $a(t)$, and when the field $E_0$ is constant the dynamics depends on $a$ only through its primitive $A(t) = \int_0^t a$. For a given history $A$ the one-particle dynamics is a non-autonomous billiard flow $\Phi_A^t$, with explicit affine free flights, and it turns out that the kinetic equation is equivalent to the scalar fixed point problem

where $\widetilde{X}^{A}_{t}(z)-x$ is the displacement of the particle started at $z=(x,v)$, measured in the universal cover.² The right-hand side involves positions only, which are continuous in time, so that no trace theory at the boundary is needed. The $N$-particle system satisfies exactly the same identity with $\mu_0$ replaced by the initial empirical measure.

The results follow the classical scheme of a mean-field limit.

• Particle system (Theorem 3.4). For almost every initial configuration the $N$-particle dynamics is well defined and unique, and its empirical measure $\delta_{N,t}$ is a weak solution of the kinetic equation.

• Kinetic equation (Theorem 3.5). For every absolutely continuous initial datum $\mu_0$ with unit kinetic energy there exists a weak solution $\mu_t$. If $\mu_0$ has a bounded density supported in $\{c_0 \leq|v| \leq L\}$ and $T$ is such that all speeds remain large compared with $\sqrt{|E_0|/\kappa_{\min}}$, where $\kappa_{\min}$ is the minimal curvature of the obstacles, the weak solution is unique among absolutely continuous weak solutions. Under finite horizon (see § 8), the upper bound $L$ can be replaced by an exponential tail of the initial density (see Remark 8.3) compatible with the predictions of [8].

• Mean-field limit (Theorem 3.6). In the regime of uniqueness, if the initial empirical measures converge to $\mu_0$ then $\delta_{N,t}$ converges to $\mu_t$ for every $t$. Without the uniqueness assumption, the empirical measures are relatively compact and all their limit points are weak solutions with initial datum $\mu_0$.

The idea of the proof in the regime $\{c_0 \leq|v| \leq L\}$ is actually naive, its implementation is nevertheless quite technical. It suffices to compare two histories on a short time window after the last time at which they agree. In the fast regime each particle has at most one collision in such a window, and for a single collision the loss of regularity is a factor $1/\nu$, where $\nu$ is the normal velocity at the collision. It turns out that this singularity is balanced exactly because the Lebesgue measure of the initial conditions is transported to the flux measure,³ which has the factor $\nu$. Hence $1/\nu$ is integrable, with an explicit bound, and the nearly grazing collisions gives a higher order term. Thanks to this observation, we can prove the result in this regime without requiring hyperbolicity, transversality of singularity manifolds or flatness of the datum near the grazing set.

Without the upper bound on the velocities, a very fast particle has many collisions in a window, and the singular factors of the collisions multiply and the above argument fails. However, if the table has finite horizon, we show that the Lebesgue average of their product grows at most exponentially in the number of collisions, uniformly in the history, by following one-parameter families of initial data and counting the pieces into which the collisions cut them. An exponential tail of the initial density, compatible with previous numerical results, compensates this growth (Theorem 8.1).

The main question left open is global uniqueness, that is, uniqueness without the lower cut-off on the speeds.⁴ Slow particles may return several times to the same obstacle in a short time, hopping along its boundary, and for such trajectories a different argument is needed (we discuss this in Remark 7.8, where we explain why we expect uniqueness to hold in general). Other natural questions are the extension

the right-hand side is the average work performed by the external field up to time $t$; under the normalizing the kinetic energy, self-consistency identifies it with the thermostat coefficient $A(t)$. to non-constant fields (we discuss this possibility in § 9), quantitative rates and propagation of chaos, and the long-time behaviour and stationary states of the limit equation, in connection with the nonequilibrium steady states studied in [6, 7, 8, 9, 10].

Finally, the thermostat couples the particles through a global observable, which places the model in the broader circle of ideas surrounding globally coupled dynamical systems and self-consistent transfer operators, initiated by Keller [33, 34]. For general results on self-consistent transfer operators see [26, 40]; for hyperbolic globally coupled systems and their statistical properties see [3], and for bifurcations and phase transitions in mean-field coupled chaotic systems see [2]. Recent developments concerning the differential and stability of self-consistent transfer operators are in [15, 16]. We develop this point of view in Appendix B. All of the aforementioned works treat the uniform hyperbolic case, while the non-uniformly hyperbolic case has proven to be essential in the understanding of physically-relevant models. In particular, an interesting connection to the derivation of macroscopic transport and stochastic behaviour from deterministic chaotic dynamics is the investigation of fast–slow partially hyperbolic systems (see, for example, [14, 24, 35] and the references therein); a first evidence of that is the recent remarkable result [13] in which the heat equation has been derived for a harmonic chain under fast deterministic chaotic forcing, with no introduction of randomness. Those works provide a complementary dynamical-systems perspective on the emergence of effective transport laws, but they still concern smooth dynamics and we hope that the present paper can be a useful step forward to the study of more general discontinuous cases.

Structure of the paper. The length of the paper is largely due to the fact that it is entirely self-contained, although the proofs of some lemmas follow a standard approach, particularly those relating to Sinai billiards.

Section 2 introduces the particle system and the kinetic equation, and Section 3 collects the definitions and a summary of the results. In Section 4 we construct the billiard flow for a given history and prove that almost every point is regular. Section 5 contains the fixed point formulation, the existence theorem and the equivalence between weak and Lagrangian solutions, Section 6 the mean-field limit, and Section 7 the uniqueness theorem in the fast regime. In Sections 8–8.4 we remove the upper bound on the velocities for tables with finite horizon, and Section 9 collects further questions. The paper concludes with two appendices: Appendix A is the technical proof of Lemma 5.8; Appendix B explores the connection with self-consistent transfer operators.

The model

Microscopic dynamics

Let $\mathbb{T}^{2} = \mathbb{R}^{2}/\mathbb{Z}^{2}$ and $\mathbf{D} := \mathbb{T}^{2} \setminus\bigcup_{i=1}^{r} Q_i$, where $Q_1,\ldots,Q_r$ are open, strictly convex subsets of $\mathbb{T}^{2}$ with pairwise disjoint closures and $C^{3}$ boundaries of positive curvature. The one-particle phase space is $\Omega:= \mathbf{D} \times\mathbb{R}^{2}$. We consider $N$ particles of unit mass with positions $x_i \in\mathbf{D}$ and velocities $v_i \in\mathbb{R}^{2}$, subject to a constant field $E_0 \in\mathbb{R}^{2} \setminus\{0\}$ and to the Gaussian thermostat. Setting

the equations of motion in the interior of $\mathbf{D}$ are

and when a particle reaches $\partial\mathbf{D}$ at a point $x$ its velocity is reflected specularly,

where $n(x)$ is the unit normal at $x \in\partial\mathbf{D}$ pointing into $\mathbf{D}$. A direct computation gives $\frac{d}{dt}U_N = 2J_N - 2\alpha_N U_N = 0$, and reflections preserve $|v_i|$, so the total kinetic energy is conserved. The flow is not Hamiltonian: the divergence of the vector field in (1) is $-(2N-1)\alpha_N$ (it is however conformally symplectic, see [36]). The collisions with the obstacles do not correlate the particles and the interaction is due to the thermostat alone.

We fix the energy per particle to be one, i.e. we work on the invariant shell

This is the normalisation compatible with a nontrivial limit $N\to\infty$ in which the empirical measure $\frac{1}{N}\sum_i\delta_{(x_i,v_i)}$ converges to a probability measure with finite second moment in velocity. On the shell (3) the thermostat coefficient is an empirical average, $\alpha_N=\frac{1}{N}\sum_j\langle E_0,v_j\rangle$, and (1) reads

The kinetic equation

Equation (4) has the mean-field form

with

Replacing the empirical average by an average with respect to the law $f_t$ of one particle leads formally to the Vlasov-type equation

with initial datum $f(0,\cdot)=f_0$ and the specular boundary condition