Cercignani conjecture is true for Smoluchowski coagulation equation

TL;DR

This paper demonstrates that the Cercignani conjecture is valid for the Smoluchowski coagulation equation by proposing an information entropy framework based on statistical physics, linking it to equilibrium particle distributions.

Contribution

It introduces a new entropy-based approach to prove the Cercignani conjecture for the Smoluchowski coagulation equation, extending its validity to two-body collision systems.

Findings

Normalized particle size distribution is log-normal at equilibrium.

The algebraic mean volume assumption is validated.

Cercignani conjecture holds for Smoluchowski and Boltzmann equations.

Abstract

In the present study, the information entropy for smoluchowski coagulation equation is proposed based on the statistical physics. and the normalized particle size distribution is a log-normal function at equilibrium from the principle of maximum entropy and moment constraint. the parameters in the particle size distribution are determined as simple constants, the result reveals that the assumption that algebraic mean volume be unit in self-preserving hypothesis is reasonable. based on the present definition of the information entropy, the cercignani conjecture holds naturally for smoluchowski coagulation equation. Together with the fact that the conjecture is also true for Boltzmann equation, cercignani conjecture holds for any two-body collision systems, which will benefit the understanding of Brownian motion and molecule kinematic theory, such as the stability of the dissipative system, and the mathematical theory of convergence to thermodynamic equilibrium.