Energy flux near the junction of two Ising chains at different temperatures

TL;DR

This paper derives exact results for energy flux in a non-equilibrium steady state of two joined Ising chains at different temperatures, revealing a decay profile that transitions from exponential to power law near zero temperature.

Contribution

It provides an exact analytical solution for energy flux and correlation profiles in a non-equilibrium Ising chain system with different thermal reservoirs.

Findings

Energy flux occurs only at the first spin in the infinite temperature sector.

The out-flux decays exponentially with distance, with a decay length related to the equilibrium correlation length.

At zero temperature, the decay crosses over to a power law, indicating critical behavior.

Abstract

We consider a system in a non-equilibrium steady state by joining two semi-infinite Ising chains coupled to thermal reservoirs with {\em different} temperatures, $T$ and $T^{\prime}$. To compute the energy flux from the hot bath through our system into the cold bath, we exploit Glauber heat-bath dynamics to derive an exact equation for the two-spin correlations, which we solve for $T^{\prime}=\infty$ and arbitrary $T$. We find that, in the $T'=\infty$ sector, the in-flux occurs only at the first spin. In the $T<\infty$ sector (sites $x=1,2,...$), the out-flux shows a non-trivial profile: $F(x)$. Far from the junction of the two chains, $F(x)$ decays as $e^{-x/\xi}$, where $\xi$ is twice the correlation length of the {\em equilibrium} Ising chain. As $T\rightarrow 0$, this decay crosses over to a power law ($x^{-3}$) and resembles a "critical" system. Simulations affirm our analytic results.