Exact approach to equilibrium for the glauber model of a one-dimensional Ising system with random ±J bonds

The (discrete-time) Glauber model is considered for a one-dimensional system of spins s_j = ±1 with nearest-neighbor Ising interaction $\text{H}\text{= -Σ}{}_{\mathrm{j}}\text{J}{}_{\mathrm{j}}\text{s}{}_{\mathrm{j}}\text{s}{}_{\mathrm{j+1}}$. The J_j = ±J are treated as random variables with an arbitrary joint probability p(J). The exact time-dependent average 〈s_j〉_t is determined, and from it the “quenched” average $\text{〈〈s}{}_{\mathrm{j}}\text{〉}{}_{\mathrm{t}}\text{〉}{}_{\mathrm{<mtext>av</mtext>}}\text{=Σ}{}_{\mathrm{<mtext>J</mtext>}}\text{p}\text{(}\text{J}\text{)〈s}{}_{\mathrm{j}}\text{〉}{}_{\mathrm{t}}$ is explicitly found.