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Journal of Physics A Mathematical and General

Correlation functions in a one-dimensional kinetic Ising model

Semen I Kuchanov, М. А. Алиев

Published Dec 21, 1997• 5 citations
Publisher

Abstract

In the limit of the strong magnetic field , the closed hierarchy of kinetic equations for correlation functions of an arbitrary order is derived for a one-dimensional kinetic Ising model. This hierarchy allows asymptotically exact solution in quadratures. For the first time the closed set of equation for generating functions of the domain distribution of the identically oriented spins have been obtained. The relations between kinetic parameters that make it possible to find exact solutions in quadratures are indicated.

DOI
10.1088/0305-4470/30/24/013

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In the limit of the strong magnetic field , the closed hierarchy of kinetic equations for correlation functions of an arbitrary order is derived for a one-dimensional kinetic Ising model. This hierarchy allows asymptotically exact solution in quadratures. For the first time the closed set of equation for generating functions of the domain distribution of the identically oriented spins have been obtained. The relations between kinetic parameters that make it possible to find exact solutions in quadratures are indicated.
The one-dimensional Ising model with arbitrary nearest neighbor interactions and a magnetic field is formulated in terms of the spin correlation function relations derived in the preceding article I. The local magnetization and the two-spin correlation functions are shown to satisfy a set of coupled algebraic summation equations. The solution of this set is given in terms of a power series in hyberbolic tangents of the magnetic field strength, for chains of arbitrary length. For cases in which the interaction constants are random variables, explicit expressions are derived for the configurational averaged magnetization and the two-spin correlation functions to the lowest order in the magnetic field.
A one-dimensional kinetic Ising model with domain wall dynamics corresponding to diffusion-limited annihilation in the presence of particle sources is considered. The spin-spin correlation function C, is calculated exactly for a nonequilibrium steady state which can be brought close to a critical point of the system. We find that in contrast to equilibrium situations, C, contains two distinctly diverging correlation lengths ~ ~ ~ 42, where ~2 is a length closely related to the nonequilibrium kinetics of the system.
The kinetic Glauber–Ising spin chain is one of the very few exactly solvable models of non-equilibrium statistical mechanics. Nevertheless, existing solutions do not yield tractable expressions for two-time correlation and response functions of observables involving products of more than one or two spins. We use a new approach to solve explicitly the full hierarchy of differential equations for the correlation and response functions. From this general solution follow closed expressions for arbitrary multispin two-time correlation and response functions, for the case where the system is quenched from equilibrium at T i > 0 to some arbitrary T ⩾ 0. By way of application, we give the results for two- and four-spin two-time correlation and response functions. From the standard mapping, these also imply new exact results for two-time particle correlation and response functions in one-dimensional diffusion limited annihilation.
A single-spin transition critical dynamics is used to investigate the three-dimensional kinetic Ising model on an anisotropic cubic lattice. We first derive the fundamental dynamical equations, and then linearize them by a cutoff approximation. We obtain the approximate solutions of the local magnetization and equal-time pair correlation function in zero field. In which the axial-decoupling terms , and as higher infinitesimal quantity are ignored, where . We think that it is reasonable as the temperature of the system is very high. The result of what we obtain in this paper can go back to the one-dimensional Glauber's theory as long as .
A spin system with unperturbed Hamiltonian ${\mathcal{H}}_{0}=\frac{1}{2}\ensuremath{\Sigma}\ensuremath{\Sigma}{B}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}{{\ensuremath{\sigma}}_{x}}^{k}\ensuremath{-}\ensuremath{\gamma}H\ensuremath{\Sigma}{{\ensuremath{\sigma}}_{x}}^{j}$ relaxing via the spin-lattice coupling $G=\frac{1}{2}\ensuremath{\Sigma}\ensuremath{\Sigma}{C}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}({{\ensuremath{\sigma}}_{y}}^{k}+{{\ensuremath{\sigma}}_{z}}^{k})$ is studied by means of the general density-matrix theory of magnetic relaxation. By making some assumptions about the magnitude and time constants of the lattice correlation functions 〈Cij(t)Ckl(0)〉〈{C}_{\mathrm{ij}}(t){C}_{\mathrm{kl}}(0)〉〈Cij​(t)Ckl​(0)〉, a master equation is obtained. It agrees at high temperatures with a master equation previously suggested by Glauber for the one-dimensional nearest-neighbor case. At high temperatures the magnetic moment relaxes with a single relaxation time, and the spin pair-correlation functions satisfy a closed set of equations. At low temperatures, however, the equations for the magnetization and the correlation functions are coupled to higher-order moments.