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

We have studied the relaxation of then-spin correlation function and distribution functionPn(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:Pn(σ(n);t) —Pn(o) ∼P1(σ;t) —P1(o)(σ);Cn(σ(n); t) —Cn(o)(σ(n)) ∼ ;Qn(σ(n)); —Qn(o)(σ(n)) ∼ [P1(σ;t) —P1(o)(σ)]n; where the superscript zero denotes the equilibrium function. These results imply thatPn(σ(n);),n> 2, decays to a functional of lower-order distribution functions as [P1(σ;) —P1(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 functionPn(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 temperatureTo 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.”