Improved Approximation of Infinite Thermostat by Finite Reservoir Using the 3D Kac Model

In this paper, we study a system of $M$ particles interacting with a reservoir of $N$ particles, where $N » M$, and compare this setup to one where the $M$-particle system interacts with a thermostat of infinite particles. Our goal is to prove a suitable upper bound, uniform in time, on the distance between the states of these two setups, given an initial Maxwellian state for both the reservoir and thermostat. Previous work has analyzed this problem using the one-dimensional Kac Model of gas collisions and an $L^2$ norm to define distance; the result was a bound which scaled with $M/\sqrt{N}$. In this paper, we use the $L^2$ norm and the three-dimensional generalization of the Kac Model to prove a bound whose long-term behavior scales with $M/N$.

💡 Research Summary

This paper presents a significant mathematical advancement in understanding how well a finite particle reservoir can approximate an infinite thermodynamic thermostat. The core problem is comparing two scenarios: a system of M particles interacting with a finite reservoir of N particles (where N » M), referred to as the R-system, versus the same M-particle system interacting with an infinite heat bath (the T-system). The goal is to establish a uniform-in-time upper bound on the distance between the states of these two setups, assuming both the reservoir and thermostat start in a Maxwellian (Gaussian) equilibrium state.

The authors employ the three-dimensional Kac model as their foundational framework. This model describes binary collisions between particles as a stochastic process that conserves both energy and momentum, a key distinction from the simpler one-dimensional Kac model used in prior work which only conserved energy. The dynamics are formulated using infinitesimal generators: L_S for collisions within the M-system, L_R for collisions within the N-reservoir, L_I for collisions between the system and the reservoir, and L_B (or its equivalent form L_T) for interactions between the system and the infinite thermostat.

The analysis is conducted within a weighted L² space, L²(ℝ^(3(M+N)), Γ), where the weight Γ is the product of Gaussian functions for all velocity components. The initial joint state is factored as f₀(v,w) = h₀(v) Γ(v,w). The time evolution for the R-system and T-system are given by f_t = e^(Lt) f₀ and ˜f_t = e^(˜Lt) f₀, respectively, which translate to h_t = e^(Lt) h₀ and ˜h_t = e^(¯Lt) h₀ in the factored form. The objective thus becomes bounding the norm difference ||e^(Lt) h₀ - e^(¯Lt) h₀||.

The proof hinges on three pivotal lemmas. Lemma 1 introduces the averaging operator R, which averages a function h(v) over all possible system and reservoir velocity configurations that share the same total energy and total momentum as the input (v,w). It proves a crucial inequality: ||R