On the entropy of classical systems with long-range interaction

TL;DR

This paper explores the form and stability of entropy in classical long-range interacting systems using the Vlasov equation, revealing that stationary states maximize Boltzmann-Gibbs entropy with implications for ensemble equivalence.

Contribution

It demonstrates that stationary states of long-range systems are entropy maxima under Vlasov dynamics, introducing an entropy function dependent on infinite initial-condition-related parameters.

Findings

Stationary states correspond to entropy maxima.

Entropy depends on an infinite set of Lagrange multipliers.

Discussion of ensemble inequivalence and temperature.

Abstract

We discuss the form of the entropy for classical hamiltonian systems with long-range interaction using the Vlasov equation which describes the dynamics of a $N$-particle in the limit $N\to\infty$. The stationary states of the hamiltonian system are subject to infinite conserved quantities due to the Vlasov dynamics. We show that the stationary states correspond to an extremum of the Boltzmann-Gibbs entropy, and their stability is obtained from the condition that this extremum is a maximum. As a consequence the entropy is a function of an infinite set of Lagrange multipliers that depend on the initial condition. We also discuss in this context the meaning of ensemble inequivalence and the temperature.