Self-similarity and power-like tails in nonconservative kinetic models

TL;DR

This paper investigates the large-time behavior of nonconservative kinetic models, revealing the emergence of power-law tails and self-similarity in solutions, supported by asymptotic analysis and numerical simulations.

Contribution

It introduces a novel analysis of kinetic models with time-varying temperature, demonstrating the formation of power-law tails and quasi-stationary states through asymptotic limits.

Findings

Power-law tails emerge in kinetic models with decreasing or increasing temperature.

Asymptotic limits lead to Fokker-Planck equations describing long-term behavior.

Numerical results confirm the theoretical predictions of Pareto-like distributions.

Abstract

In this paper, we discuss the large--time behavior of solution of a simple kinetic model of Boltzmann--Maxwell type, such that the temperature is time decreasing and/or time increasing. We show that, under the combined effects of the nonlinearity and of the time--monotonicity of the temperature, the kinetic model has non trivial quasi-stationary states with power law tails. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution. The same idea is applied to investigate the large-time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and/or decreasing of the mean wealth. In this last case, the large-time behavior of the solution shows a Pareto power law tail. Numerical results confirm the previous analysis.