Canonical quantization for Equilibrium Thermodynamics

TL;DR

This paper introduces a novel canonical quantization framework for equilibrium thermodynamics, treating thermodynamic variables as quantum operators, and demonstrates its application to various gases, revealing new quantum-like features in thermodynamics.

Contribution

It develops a formalism applying Dirac's constrained systems to thermodynamics, enabling operator treatment of thermodynamic variables and uncovering quantum analogs in classical thermodynamic systems.

Findings

Emergence of Schrödinger-like equation with entropy as time

Restoration of Hermiticity via pseudo-Hermitian framework

Derivation of thermodynamic uncertainty relations

Abstract

We formulate a canonical quantization of Equilibrium Thermodynamics by applying Dirac's theory of constrained systems. Thermodynamic variables are treated as conjugate pairs of coordinates and momenta, allowing extensive and intensive quantities to be promoted to operators in a Hilbert space. The formalism is applied to the ideal gas, the van der Waals gas, and the photon gas, illustrating both first- and second-class quantization procedures. For the ideal gas, a Schr\"odinger-like equation emerges in which entropy plays the role of time, and the wave function acquires a phase determined by the internal energy. A pseudo-Hermitian framework restores Hermiticity of the temperature operator and establishes the equivalence among constraint realizations. The approach naturally leads to thermodynamic uncertainty relations and suggests extensions to quantum and topological phase transitions, as well as black-hole and non-equilibrium thermodynamics.