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 ^(o)_n ∼P ₁(σ;t) —P ^(o)₁ (σ);C _n(σ ⁽ⁿ⁾; t) —C ^(o)_n (σ ⁽ⁿ⁾) ∼ <σ>;Q _n(σ ⁽ⁿ⁾); —Q ^(o)_n (σ ⁽ⁿ⁾) ∼ [P ₁(σ;t) —P ^(o)₁ (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 ^(o)₁ (σ)]ⁿ and that then-spin correlation function <o⁽ⁿ⁾> withn > 2 decays to a functional of lower-order correlation functions as <σ>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.”