Self-Similarity in Random Collision Processes

TL;DR

This paper analytically investigates the long-term behavior of collision processes with linear mixing, revealing self-similar velocity distributions with algebraic or stretched exponential tails, influenced by conservation laws and mixing parameters.

Contribution

It provides a detailed analytical study of self-similarity and tail behavior in collision kinetics with variable mixing rules and conservation laws.

Findings

Velocity distributions become self-similar over time.

Tail behaviors include algebraic and stretched exponential forms.

Universal distributions emerge under conservation laws.

Abstract

Kinetics of collision processes with linear mixing rules are investigated analytically. The velocity distribution becomes self-similar in the long time limit and the similarity functions have algebraic or stretched exponential tails. The characteristic exponents are roots of transcendental equations and vary continuously with the mixing parameters. In the presence of conservation laws, the velocity distributions become universal.