Tsallis thermostatistics for finite systems: a Hamiltonian approach

TL;DR

This paper demonstrates that finite systems with Hamiltonians obeying a generalized homogeneity follow Tsallis nonextensive thermostatistics, but revert to Boltzmann-Gibbs statistics in the thermodynamic limit, supported by numerical simulations.

Contribution

It establishes a rigorous connection between Hamiltonian structure and Tsallis entropy for finite systems, explaining the origin of nonlinearity and power-law behavior.

Findings

Finite systems follow Tsallis statistics if Hamiltonians are generalized homogeneous.

In the thermodynamic limit, Boltzmann-Gibbs statistics is recovered.

Numerical simulations confirm the theoretical results.

Abstract

We show that finite systems whose Hamiltonians obey a generalized homogeneity relation rigorously follow the nonextensive thermostatistics of Tsallis. In the thermodynamical limit, however, our results indicate that the Boltzmann-Gibbs statistics is always recovered, regardless of the type of potential among interacting particles. This approach provides, moreover, a one-to-one correspondence between the generalized entropy and the Hamiltonian structure of a wide class of systems, revealing a possible origin for the intrinsic nonlinear features present in the Tsallis formalism that lead naturally to power-law behavior. Finally, we confirm these exact results through extensive numerical simulations of the Fermi-Pasta-Ulam chain of anharmonic oscillators.