Magnetization distribution in the transverse Ising chain with energy flux

The zero-temperature transverse Ising chain carrying an energy flux j_E is studied with the aim of determining the nonequilibrium distribution functions, P(M_z) and P(M_x), of its transverse and longitudinal magnetizations, respectively. An exact calculation reveals that P(M_z) is a Gaussian both at j_E=0 and j_E not equal 0, and the width of the distribution decreases with increasing energy flux. The distribution of the order-parameter fluctuations, P(M_x), is evaluated numerically for spin-chains of up to 20 spins. For the equilibrium case (j_E=0), we find the expected Gaussian fluctuations away from the critical point while the critical order-parameter fluctuations are shown to be non-gaussian with a scaling function Phi(x)=Phi(M_x/<M_x>)=<M_x>P(M_x) strongly dependent on the boundary conditions. When j_E not equal 0, the system displays long-range, oscillating correlations but P(M_x) is a Gaussian nevertheless, and the width of the Gaussian decreases with increasing j_E. In particular, we find that, at critical transverse field, the width has a j_E^(-3/8) asymptotic in the j_E -> 0 limit.