Kinetic models with randomly perturbed binary collisions

TL;DR

This paper introduces a class of kinetic equations with random binary collisions, analyzing their equilibrium states and showing their relevance in econophysics and gas dynamics.

Contribution

It develops conditions for existence and uniqueness of equilibrium profiles in kinetic models with random collisions, linking econophysics and gas theory.

Findings

Existence and uniqueness of equilibrium profiles established.

Equilibrium profiles are solutions to multiple evolution problems.

Models applicable to wealth redistribution and granular gases.

Abstract

We introduce a class of Kac-like kinetic equations on the real line, with general random collisional rules, which include as particular cases models for wealth redistribution in an agent-based market or models for granular gases with a background heat bath. Conditions on these collisional rules which guarantee both the existence and uniqueness of equilibrium profiles and their main properties are found. We show that the characterization of these stationary solutions is of independent interest, since the same profiles are shown to be solutions of different evolution problems, both in the econophysics context and in the kinetic theory of rarefied gases.