Sharp Uniform-in-Time Propagation of Chaos for the Kac model of a Maxwellian gas

Emanuele Dolera (Università di Pavia)

Abstract: We establish a uniform-in-time propagation of chaos (POC) with the optimal quantitative rate $O(1/N)$ for any $k$-marginal of the Kac $N$-particle Markov process on the energy sphere $\mathbb{S}^{N-1}(\sqrt{N})$. Throughout, we refer to the Kac process as a 1-dimensional caricature of a spatially homogeneous Maxwellian gas.

Considering initial data obtained by conditioning the product measure $\gamma_0^{\otimes N}$ onto $\mathbb{S}^{N-1}(\sqrt{N})$, we prove the quantitative bound
$$
\mathcal{W}_1\left(\mu_t^{(k,N)}; \gamma_t^{\otimes k}\right) \leq C\frac{k^{1+\delta}}{N}
$$
for all $t \geq 0$ and any $k \in \{1, \dots, N\}$, with some $\delta \in [0,1]$. Here, $\mu_t^{(k,N)}$ is the joint distribution of $k$ particles in the Kac dynamics, $\gamma_t$ is the solution to the mean-field (non-linear) Kac equation with initial datum $\gamma_0$, and $\mathcal{W}_1$ denotes the 1-Wasserstein distance. The constant $C$ relies on the quantitative bounds of $\gamma_0$ (moments, Sobolev regularity, and potential structural assumptions), remaining strictly independent of $t, k$, and $N$. We highlight a critical trade-off regarding the regularity of $\gamma_0$: under a strong Holley-Stroock condition, the strictly optimal rate (corresponding to $\delta = 0$) is achieved. Conversely, relaxing these assumptions to accommodate a broader class of initial data inherently entails $\delta > 0$, quantifying a slower mitigation of dynamical correlations. This quantitative approach to the POC naturally inscribes itself in the foundational program on rigorous kinetic limits, offering a sharp, uniform-in-time perspective on the decay of correlations in interacting particle systems.

Our result highlights that the finite-size correction to chaos does not deteriorate in time, contrasting with previous bounds that exhibit exponential or algebraic temporal growth.
The $O(k/N)$ rate is strictly optimal and reflects the intrinsic correlations induced by the conservation of energy in finite systems. Crucially, this work clarifies the distinction between the propagation of chaos, inherently governed by a $1/N$ scaling, and empirical-measure fluctuations of order $1/\sqrt{N}$, which pertain to a fundamentally different probabilistic regime.
In comparison with the existing literature, our estimate sharpens the quantitative rates established in [2,4]. Moreover, it reveals a strong structural connection between the exact expression of the spectral gap for the Kac $N$-particle dynamics, obtained in [1], and the optimal exponential rate of relaxation to equilibrium for solutions of the Kac-Boltzmann equation, achieved in [3].

[1] Carlen, E.A., Carvalho, M.C., and Loss, M. (2003). Determination of the spectral gap for Kac’s master equation and related stochastic evolution. Acta Math. 191, 1-54

[2] Cortez, R. (2016). Uniform propagation of chaos for Kac’s 1D particle system. J. Stat. Phys. 165, 1102-1113

[3] Dolera, E., Gabetta, E., and Regazzini, E. (2009). Reaching the best possible rate of convergence to equilibrium for solutions of Kac’s equation via central limit theorem. Ann. Appl. Probab. 19, 186-209

[4] Mischler, S. and Mouhot, C. (2013). Kac’s Program in Kinetic Theory. Invent. Math. 193, 1-147