# Glauber's Ising chain between two thermostats

## TL;DR

This paper analyzes the energy current in a one-dimensional Ising model with spins connected to two thermostats at different temperatures, using fermionization to compute cumulants and explore various physical regimes.

## Contribution

It provides an exact calculation of the cumulant generating function for the energy current in the Glauber dynamics of the Ising chain with two thermostats, revealing new phenomena and scaling behaviors.

## Key findings

- Explicit formulas for cumulants up to fourth order.
- Identification of regimes with kinetic mean-field effects and zero-temperature dissipation.
- Linear growth of cumulants with system size in the zero-temperature limit.

## Abstract

We consider a one-dimensional Ising model each of whose $N$ spins is in contact with two thermostats of distinct temperatures $T_1$ and $T_2$. Under Glauber dynamics the stationary state happens to coincide with the equilibrium state at an effective intermediate temperature $T(T_1,T_2)$. The system nevertheless carries a nontrivial energy current between the thermostats. By means of the fermionization technique, for a chain initially in equilibrium at an arbitrary temperature $T_0$ we calculate the Fourier transform of the probability $P({\cal Q};\tau)$ for the time-integrated energy current ${\cal Q}$ during a finite time interval $\tau$. In the long time limit we determine the corresponding generating function for the cumulants per site and unit of time $\langle{\cal Q}^n\rangle_{\rm c}/(N\tau)$ and explicitly give those with $n=1,2,3,4.$ We exhibit various phenomena in specific regimes: kinetic mean-field effects when one thermostat flips any spin less often than the other one, as well as dissipation towards a thermostat at zero temperature. Moreover, when the system size $N$ goes to infinity while the effective temperature $T$ vanishes, the cumulants of ${\cal Q}$ per unit of time grow linearly with $N$ and are equal to those of a random walk process. In two adequate scaling regimes involving $T$ and $N$ we exhibit the dependence of the first correction upon the ratio of the spin-spin correlation length $\xi(T)$ and the size $N$.