Dilational Symmetry-Breaking in Thermodynamics

TL;DR

This paper derives a general thermodynamic formula linking energy, pressure, and microscopic scales for non-relativistic and relativistic systems, highlighting the connection with dimensional analysis and the virial theorem.

Contribution

It introduces a novel general formula for thermodynamic trace relations based on dimensional analysis, applicable to both non-relativistic and relativistic systems, including those with anomalous scale development.

Findings

Derived formulas for thermodynamic traces in various systems.

Connected dimensional analysis with the virial theorem.

Demonstrated the formulas through multiple case studies.

Abstract

Using thermodynamic relations and dimensional analysis we derive a general formula for the thermodynamical trace $2\mathcal{E}-DP$ for non-relativistic systems and $\mathcal{E-DP}$ for relativistic systems, where $D$ is the number of spatial dimensions, in terms of the microscopic scales of the system within the grand canonical ensemble. We demonstrate the formula for several cases, including anomalous systems which develop scales through dimensional transmutation. Using this relation, we make explicit the connection between dimensional analysis and the virial theorem. This paper is focused mainly on the non-relativistic aspects of this relation.