Externally driven one-dimensional Ising model

A one dimensional kinetic Ising model at a finite temperature on a semi-infinite lattice with time varying boundary spins is considered. Exact expressions for the expectation values of the spin at each site are obtained, in terms of the time dependent boundary condition and the initial conditions. The solution consists of a transient part which is due to the initial condition, and a part driven by the boundary. The latter is an evanescent wave when the boundary spin is oscillating harmonically. Low- and high-frequency limits are investigated with greater detail. The total magnetization of the lattice is also obtained. It is seen that for any arbitrary rapidly varying boundary conditions, this total magnetization is equal to the the boundary spin itself, plus essentially the time integral of the boundary spin. A nonuniform model is also investigated.