From a kinetic equation to a diffusion under an anomalous scaling

Abstract

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process (K(t), i(t), Y(t)), where (K(t), i(t)) is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance, and Y(t) is an additive functional of K(t). We prove that under an anomalous rescaling Y converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately rescaled solution of the Boltzmann equation converges to a diffusion equation.

References in corpus (3)

• A momentum conserving model with anomalous thermal conductivity in low dimension
• Energy transport in stochastically perturbed lattice dynamics
• Macroscopic energy diffusion for a chain of anharmonic oscillators

Cited by in corpus (6)

• Superdiffusion of energy in a chain of harmonic oscillators with noise
• Time evolution of the Luttinger model with nonuniform temperature profile
• Diffusive Heat Waves in Random Conformal Field Theory
• Steady states and universal conductance in a quenched Luttinger model
• From normal diffusion to superdiffusion of energy in the evanescent flip noise limit
• The kinetic exclusion process: a tale of two fields