Dynamic Behavior of a Set of Weakly Coupled Ising Spins

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Abstract
The master equation of a set of independent equivalent spins contains only one undetermined constant, the rate constant. If one assumes the local field to be altered by the field produced by the two neighboring spins, one can formulate a set of equations for the average of one, two, three, etc., spins. On assuming an Ising interaction between the spins, weak compared to the coupling with the heat bath, we can terminate the hierarchy and solve the problem of a linear chain with periodic boundary conditions by Fourier-transformation. The resulting secular equation determines two sets of relaxation times and two sets of eigensolutions. An explicit solution for the spin averages is given for the initial condition describing a localized excitation. Similarities with, and differences between, this and the random walk problem is pointed out.

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