Two interacting Ising chains in relative motion

We consider two parallel cyclic Ising chains counter-rotating at a relative velocity v, the motion actually being a succession of discrete steps. There is an in-chain interaction between nearest-neighbor spins and a cross-chain interaction between instantaneously opposite spins. For velocities v>0 the system, subject to a suitable markovian dynamics at a temperature T, can reach only a nonequilibrium steady state (NESS). This system was introduced by Hucht et al., who showed that for v=\infty it undergoes a para- to ferromagnetic transition, essentially due to the fact that each chain exerts an effective field on the other one. The present study of the v=\infty case determines the consequences of the fluctuations of this effective field when the system size N is finite. We show that whereas to leading order the system obeys detailed balancing with respect to an effective time-independent Hamiltonian, the higher order finite-size corrections violate detailed balancing. Expressions are given to various orders in 1/N for the interaction free energy between the chains, the spontaneous magnetization, the in-chain and cross-chain spin-spin correlations, and the spontaneus magnetization. It is shown how finite-size scaling functions may be derived explicitly. This study was motivated by recent work on a two-lane traffic problem in which a similar phase transition was found.