Two-dimensional Einstein manifolds in geometrothermodynamics

TL;DR

This paper explores a class of thermodynamic systems with constant curvature in a geometric framework, identifying solutions that may correspond to physically relevant models like polytropic fluids.

Contribution

It introduces a new class of thermodynamic systems with constant curvature and characterizes solutions on a characteristic circumference, linking geometry to physical relevance.

Findings

Identified a subset of solutions on a circumference in an abstract space.

Connected solutions on the circumference to polytropic fluid models.

Proposed a conjecture on the physical relevance of solutions on the characteristic circumference.

Abstract

We present a class of thermodynamic systems with constant thermodynamic curvature which, within the context of geometric approaches of thermodynamics, can be interpreted as constant thermodynamic interaction among their components. In particular, for systems constrained by the vanishing of the Hessian curvature we write down the systems of partial differential equations. In such a case it is possible to find a subset of solutions lying on a circumference in an abstract space constructed from the first derivatives of the isothermal coordinates. We conjecture that solutions on the characteristic circumference are of physical relevance, separating them from those of pure mathematical interest. We present the case of a one-parameter family of fundamental relations that -- when lying in the circumference -- describe a polytropic fluid.