# Stochastic order parameter dynamics for phase coexistence in heat conduction

## TL;DR

This paper introduces a stochastic order parameter model for phase coexistence in steady heat conduction, revealing how interface temperature deviations occur near equilibrium due to differences in thermal conductivity.

## Contribution

It develops a variational principle based on stochastic dynamics to analyze non-equilibrium phase coexistence, extending thermodynamics frameworks.

## Key findings

- Interface temperature exceeds equilibrium transition temperature when thermal conductivity is lower in the ordered phase.
- Super-heated and super-cooled states emerge near the interface due to conductivity differences.
- The model predicts phase behavior near equilibrium in steady heat conduction conditions.

## Abstract

We propose a stochastic order parameter equation for describing phase coexistence in steady heat conduction near equilibrium. By analyzing the stochastic dynamics with a non-equilibrium adiabatic boundary condition, where total energy is conserved over time, we derive a variational principle that determines thermodynamic properties in non-equilibrium steady states. The resulting variational principle indicates that the temperature of the interface between the ordered region and the disordered region becomes greater (less) than the equilibrium transition temperature in the linear response regime when the thermal conductivity in the ordered region is less (greater) than that in the disordered region. This means that a super-heated ordered (super-cooled disordered) state appears near the interface, which was predicted by an extended framework of thermodynamics proposed in [N. Nakagawa and S.-i. Sasa, Liquid-gas transitions in steady heat conduction, Phys. Rev. Lett. {\bf 119}, 260602, (2017).]