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(Mx) of its transverse and longitudinal magnetizations, respectively. An exact calculation reveals that P(M(z)) is a Gaussian both at j(E)=0 and at j(E) not equal to 0, and the width of the distribution decreases with increasing energy flux. The distribution of the order-parameter fluctuations, P(Mx), 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)/)=P(Mx) strongly dependent on the boundary conditions. When j(E) not equal to 0, the system displays long-range, oscillating correlations but P(Mx) 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(-3/8)(E) asymptotic in the j(E)-->0 limit.