Covariant statistical mechanics and the stress-energy tensor

TL;DR

This paper develops a covariant framework for relativistic statistical mechanics, deriving the stress-energy tensor at equilibrium from a thermodynamic potential, linking it to entropy and potentially extending to non-equilibrium scenarios.

Contribution

It introduces a covariant formalism for equilibrium statistical mechanics in relativity, deriving the stress-energy tensor from a thermodynamic potential current and relating it to the entropy current.

Findings

Stress-energy tensor obtained from partition function derivative.

Relates stress-energy tensor to thermodynamic potential current.

Potential extension to non-equilibrium hydrodynamics.

Abstract

After recapitulating the covariant formalism of equilibrium statistical mechanics in special relativity and extending it to the case of a non-vanishing spin tensor, we show that the relativistic stress-energy tensor at thermodynamical equilibrium can be obtained from a functional derivative of the partition function with respect to the inverse temperature four-vector \beta. For usual thermodynamical equilibrium, the stress-energy tensor turns out to be the derivative of the relativistic thermodynamic potential current with respect to the four-vector \beta, i.e. T^{\mu \nu} = - \partial \Phi^\mu/\partial \beta_\nu. This formula establishes a relation between stress-energy tensor and entropy current at equilibrium possibly extendable to non-equilibrium hydrodynamics.