Decay of correlations. III. Relaxation of spin correlations and distribution functions in the one-dimensional ising lattice

Abstract

We have studied the relaxation of then-spin correlation function <σ (n)> and distribution functionP n(σ (n);t) for the Glauber model of the one-dimensional Ising lattice. We find that new combinations of correlation functions (C-functions) and distribution functions (Q-functions) are more useful in discussing the relaxation of this system from initial nonequilibrium states than the usual cumulants and Ursell functions used in our papers I and II. The asymptotic behavior of theP, C, andQ functions are:P n(σ (n);t) —P n (o) ∼P 1(σ;t) —P 1 (o) (σ);C n(σ (n); t) —C n (o) (σ (n)) ∼ ;Q n(σ (n)); —Q n (o) (σ (n)) ∼ [P 1(σ;t) —P 1 (o) (σ)]n; where the superscript zero denotes the equilibrium function. These results imply thatP n(σ (n);),n> 2, decays to a functional of lower-order distribution functions as [P 1(σ;) —P 1 (o) (σ)]n and that then-spin correlation function withn > 2 decays to a functional of lower-order correlation functions as n. This result for the distribution functionP n(σ (n);),n> 2, is identical with the results obtained in papers I and II for initially correlated, noninteracting many-particle systems in contact with a heat bath and for an infinite chain of coupled harmonic oscillators. As a special example, we study the relaxation of the spin system when the heat-bath temperature is changed suddenly from an initial temperatureT o to a final temperatureT. We obtain the interesting result that the spin system is not canonically invariant, i.e., it cannot be characterized by a time-dependent “spin temperature.”