Rigorous results in non-extensive thermodynamics

TL;DR

This paper rigorously analyzes non-extensive thermodynamics for quantum systems, establishing conditions for the existence and uniqueness of equilibrium density matrices based on the entropic parameter q.

Contribution

It provides a mathematical proof for the equilibrium density matrix in non-extensive thermodynamics and clarifies how the parameter q influences existence and uniqueness.

Findings

Existence of equilibrium density matrix depends on q and Hamiltonian eigenvalue divergence.

Unique equilibrium density matrix exists if q<2.

High energy levels have zero occupancy in equilibrium.

Abstract

This paper studies quantum systems with a finite number of degrees of freedom in the context of non-extensive thermodynamics. A trial density matrix, obtained by heuristic methods, is proved to be the equilibrium density matrix. If the entropic parameter q is larger than 1 then existence of the trial equilibrium density matrix requires that q is less than some critical value q_c which depends on the rate by which the eigenvalues of the hamiltonian diverge. Existence of a unique equilibrium density matrix is proved if in addition q<2 holds. For q between 0 and 1, such that 2<q+q_c, the free energy has at least one minimum in the set of trial density matrices. If a unique equilibrium density matrix exists then it is necessarily one of the trial density matrices. Note that this is a finite rank operator, which means that in equilibrium high energy levels have zero probability of occupancy.