Gaston Giroux

Compactness of the fluctuations associated with some generalized nonlinear Boltzmann equations

In this paper, we develop a new approach to obtain the compactness of the fluctuation processes for Boltzmann dynamics. Our method is applicable to Kacs model, already studied by Uchiyama, but it covers many other cases. A novelty worth mentioning is the use of the weak topology of a Hilbert space

A functional central limit theorem for a nonequilibrium model of interacting particles with unbounded intensity

Under suitable physically reasonable initial assumptions, a functional central limit theorem is obtained for a nonequilibrium model of randomly interacting particles with unbounded jump intensity. This model is related to a nonlinear Boltzmann-type equation.

Cutoff-type Boltzmann equations: Convergence of the solution

Using an approach similar to Tanakas we prove the convergence toward equilibrium for general classes of models which correspond to Boltzmann equations of the cutoff type. A major step consists in showing a convex inequality involving the Kantorovich-Vasherstein metric. This requires assumptions on the interacting kernels. These assumptions are very natural from a physical point of view. In particular, our classes include models recently developed by physicists to study relaxation of closed oscillator systems.

A geometric rate of convergence to the equilibrium for Boltzmann processes with multiple particle interactions

We construct Boltzmann processes using the formalism of random trees. We are then able to extend previous results about convergence toward the equilibrium law to interactions involving random numbers of particles. We even show a geometric rate of convergence for an extended class of processes, especially for those having a scaling invariant interaction mechanism.

The Convergence of the Solution of a Boltzmann Type Equation Related to Quantum Mechanics

McKean (1966), Tanaka (1978), and Sznitman (1984) have obtained existence, uniqueness and asymptotic results for the solution of a Boltzmann type equation, for the cases of Kac’s caricature, Maxwell’s gas and Boltzmann’s gas, respectively. Their methods use Wild’s sums. Here we adapt Tanaka’s method for his asymptotic result to show, with the help of Wild’s sums, the convergence toward the geometric equilibrium of the solution of a Boltzmann type equation related to the Bose-Einstein statistic (r = 1) of quantum mechanics.