Deformed Density Matrix and Generalized Uncertainty Relation in Thermodynamics

TL;DR

This paper proposes a generalized thermodynamic uncertainty relation incorporating an additional energy term, leading to a lower temperature limit, by deforming the statistical density matrix analogous to quantum mechanics at Planck scale.

Contribution

It introduces a deformed statistical density matrix, called the statistical density pro-matrix, and demonstrates its analogy to quantum density matrices at Planck scale.

Findings

Existence of a lower limit of inverse temperature.

Explicit deformation of the canonical Gibbs distribution.

Analogy between quantum and statistical density matrices at Planck scale.

Abstract

A generalization of the thermodynamic uncertainty relations is proposed. It is done by introducing of an additional term proportional to the interior energy into the standard thermodynamic uncertainty relation that leads to existence of the lower limit of inverse temperature. The authors are of the opinion that the approach proposed may lead to proof of these relations. To this end, the statistical mechanics deformation at Planck scale. The statistical mechanics deformation is constructed by analogy to the earlier quantum mechanical results. As previously, the primary object is a density matrix, but now the statistical one. The obtained deformed object is referred to as a statistical density pro-matrix. This object is explicitly described, and it is demonstrated that there is a complete analogy in the construction and properties of quantum mechanics and statistical density matrices at Plank scale (i.e. density pro-matrices). It is shown that an ordinary statistical density matrix occurs in the low-temperature limit at temperatures much lower than the Plank's. The associated deformation of a canonical Gibbs distribution is given explicitly.