Master Equation and Two Heat Reservoirs

TL;DR

This paper investigates a spin-flip process influenced by two heat reservoirs at different temperatures, deriving a master equation solution that reveals a generalized Fermi-Dirac distribution and a novel first-order transition due to a new symmetry.

Contribution

It introduces a master equation approach for a two-reservoir spin system, deriving a generalized Fermi-Dirac distribution and revealing a symmetry-induced first-order transition.

Findings

Generalized Fermi-Dirac distribution with an effective temperature $T_e$

Derivation of a free energy functional exhibiting a third order term

Identification of a symmetry leading to a first-order transition

Abstract

We analyze a simple spin-flip process under the presence of two heat reservoirs. While one flip process is triggered by a bath at temperature $T$, the inverse process is activated by a bath at a different temperature $T ^{\prime}$. The situation can be described by using a master equation approach in a second quantized Hamiltonian formulation. The stationary solution leads to a generalized Fermi-Dirac distribution with an effective temperature $T_e$. Likewise the relaxation time is given in terms of $T_e$. Introducing a spin-representation we perform a Landau expansion for the averaged spin $<\sigma>$ as order parameter and consequently, a free energy functional can be derived. Owing to the two reservoirs the model is invariant with respect to a simultaneous change $\sigma \leftrightarrow - \sigma $ and $ T \leftrightarrow T ^{\prime}$. This new symmetry generates a third order term in the free energy which gives rise a dynamically induced first order transition.