Do Tsallis distributions really originate from the finite baths?

TL;DR

This paper rigorously demonstrates that Tsallis distributions with fat tails originate only from finite baths with negative heat capacity, challenging common assumptions and highlighting implications for statistical mechanics and the q-generalized central limit theorem.

Contribution

It clarifies the conditions under which Tsallis distributions arise from finite baths, specifically linking fat tails to negative heat capacity baths and discussing the violation of equipartition.

Findings

Tsallis distributions with fat tails occur only for finite baths with negative heat capacity.

Finite positive heat capacity baths produce distributions with sharp cut-offs, not fat tails.

The correspondence between Tsallis distributions and finite baths involves a violation of the equipartition theorem.

Abstract

It is often stated that heat baths with finite degrees of freedom i.e., finite baths, are sources of Tsallis distributions for the classical Hamiltonian systems. By using well-known fundamental statistical mechanical expressions, we rigorously show that Tsallis distributions with fat tails are possible \textit{only} for finite baths with constant negative heat capacity while constant positive heat capacity finite baths yield decays with sharp cut-off with no fat tails. However, the correspondence between Tsallis distributions and finite baths holds at the expense of violating equipartition theorem for finite classical systems at equilibrium. Finally, we comment on the implications of the finite bath for the recent attempts towards a $q$-generalized central limit theorem.