Pathwise convergence of the hard spheres Kac process

Abstract

We derive two estimates for the deviation of the N -particle, hardspheres Kac process from the corresponding Boltzmann equation, measured in expected Wasserstein distance. Particular care is paid to the long-time properties of our estimates, exploiting the stability properties of the limiting Boltzmann equation at the level of realisations of the interacting particle system. As a consequence, we obtain an estimate for the propagation of chaos, uniformly in time and with polynomial rates, as soon as the initial data has a k th moment, k > 2. Our approach is similar to Kac's proposal of relating the long-time behaviour of the particle system to that of the limit equation. Along the way, we prove a new estimate for the continuity of the Boltzmann flow measured in Wasserstein distance.

Large Deviations of Kac's Conservative Particle System and Energy Non-Conserving Solutions to the Boltzmann Equation: A Counterexample to the Predicted Rate Function

“…and by the choice (322), we have proven the same bound for the limit inferior over the full sequence N ∈ N. The lower bound is independent of U ⊃ A Θ , and so we have proven the claim (26).…”

“…We now prove (26,27). For the first item, let U ⊃ A Θ be any open set, and S ′ ⊂ N a subsequence such that (322) lim…”

Kac's Process with Hard Potentials and a Moderate Angular Singularity

“…The sense in which Kac first proposed to relate his stochastic process to Boltzmann's equation is through the propagation of chaos: he proposed that, if V N t is a labelled Kac process, with symmetric initial conditions and µ 0 ∈ S is such that marginal distribution of (V 1 0 , ..., V k 0 ) is approximately µ ⊗k 0 , then this approximation is propagated through time: the law of (V 1 t , ...V k t ) is approximately φ t (µ 0 ) ⊗k , for any fixed k, t in the regime where N is large, and where the approximation is understood as the weak topology of measures on (R d ) k . This chaoticity property is equivalent to the convergence of the empirical measures [34]; quantitatively, the same arguments as in [29,20] show how the conclusion of Theorem 3 can be viewed as a quantitative estimate of this approximation. We now mention some existing works in this direction: i).…”

“…In the case of Maxwell molecules, including the noncutoff case, we refer the reader to [37,36]; in the case of hard spheres (γ = 1, ν < 0), let us mention the works [3,10,25,26,28]. Most recently, Mischler and Mouhot [29] prove very strong 'twice-differentiability' and exponential stability of the Boltzmann equation in the hard-spheres case measured in total variation distance, and the author obtained a uniform-in-time Wasserstein stability result in a previous work [20].…”

“…Regarding the case of hard spheres, Mischler and Mouhot [29] obtained results decaying as (log N ) −r for some r > 0. Norris [30] obtained results with optimal N dependence, replacing the right-hand side in Theorem 3 with the optimal N dependence N −1/d but which are not uniform in time, and the author [20] obtained results with close-to-optimal N dependence and which are uniform in time.…”