Steady-state properties of coupled hot and cold Ising chains

TL;DR

This paper derives and analyzes the steady-state energy flux and correlations in two coupled Ising chains at different temperatures, providing explicit formulas, asymptotic behaviors, and simulation validation.

Contribution

It offers a complete derivation and detailed characterization of energy flux and correlations in coupled Ising chains at arbitrary temperatures, extending previous results.

Findings

Energy flux decays exponentially into the cold bath for high hot temperature

Flux transitions to power law decay as cold temperature approaches zero

Monte Carlo simulations confirm analytical and asymptotic results

Abstract

Recently, the author and Zia (2010) reported on exact results for a far-from-equilibrium system in which two coupled semi-infinite Ising chains at temperatures $T_h$ and $T_c$, with $T_h>T_c$, establish a flux of energy across their junction. This paper provides a complete derivation of those results, more explicit expressions for the energy flux, and a more detailed characterization of the system at arbitrary $T_c$ and $T_h$. We consider the two-point correlation functions and the energy flux $F(x)$ between each spin, located at integer position $x$, and its associated heat bath. In the $T_h \rightarrow \infty$ limit, the flux $F(x)$ decays exponentially into the cold bath (spins with $x=1,2,...$) for all $T_c>0$ and transitions into a power law decay as $T_c \rightarrow 0$. We find an asymptotic expansion for large $x$ in terms of modified Bessel functions that captures both of these behaviors. We perform Monte Carlo simulations that give excellent agreement with both the exact and asymptotic results for $F(x)$. The simulations are also used to study the system at arbitrary $T_h$ and $T_c$.