A Schr\"odinger-like equation for the Thermodynamics of a particle in a box

TL;DR

This paper introduces a Schrödinger-like equation for the thermodynamics of a particle in a box, linking quantum mechanics and thermodynamic entropy evolution within a Hamiltonian framework.

Contribution

It proposes a novel Hamiltonian scheme with an angle-dependent potential that captures both mechanical and thermodynamic behavior, including entropy production.

Findings

Thermal conductance matches the quantum of heat conductance G_Q.

Wavefunction solutions encode entropy evolution.

Results agree with classical thermodynamics for non-abrupt changes.

Abstract

The particle in an expanding/contracting 1-dimension box is revisited in action-angle like variables with direct thermodynamic interpretation. An angle dependent potential is proposed accurately describing the mechanical behavior while also capturing thermodynamic evolution -- entropy production -- within a canonical Hamiltonian framework. Heat transfer at constant volume is analyzed, and the derived thermal conductance matches the universal quantum of heat conductance $G_{Q}$ in the quantum limit. Having a Hamiltonian scheme interpretable in thermodynamic terms, a Schr\"odinger-like wave equation is formulated whose wavefunction solutions contain the information about the entropy evolution. The results show exact agreement with 'classical' results for non abrupt changes. Finally, comparisons with a pure quantum mechanical treatment of the wave function in an expanding box confirm consistency in quasi-static regimes and reveal adiabaticity breakdown under far-from-equilibrium conditions.