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

We consider a system in a non-equilibrium steady state by joining two semi-infinite Ising chains coupled to thermal reservoirs with different temperatures, T and T'. 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'=∞ and arbitrary T. We find that, in the T'=∞ sector, the in-flux occurs only at the first spin. In the T<∞ 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/ξ, where ξ is twice the correlation length of the equilibrium Ising chain. As T→0, this decay crosses over to a power law (x^-3) and resembles a "critical" system. Simulations affirm our analytic results.