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Journal of Mathematical Physics

Generating function of spin correlation functions for kinetic Glauber–Ising model with time-dependent transition rates

М. А. Алиев

Published Aug 1, 2009• 8 citations
Publisher

Abstract

The exact analytical solution in quadratures have been found for the generating function of spin correlation functions for the Reiss variant of the kinetic Ising model that is the extension of the Glauber–Ising model to account for the dependence of heat bath temperature on time.
DOI
10.1063/1.3190489

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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.
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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 time-dependent projection-operator formalism is used to derive a kinetic equation for the single-spin density operator in an Ising system of NNN two-state spins in weak interaction with a heat bath. This equation contains a contribution at the self-consistent-field level and also higher-order contributions due to dynamically generated correlations. A detailed evaluation of the self-consistent-field terms is made. Possibilities for generalizing the method to obtain equations for multispin density operators are discussed.
The wavefunction of a single spin system in a prepared initial state evolves to equilibrium with a heat bath. The average spin q(t)=p↑(t)−p↓(t)q(t) = p_{\uparrow}(t) - p_{\downarrow}(t)q(t)=p↑​(t)−p↓​(t) exhibits a characteristic time for this evolution. With the proper choice of spin flip rates, a dynamical Ising model (Glauber) can be constructed with the same characteristic time for transition of the average spin to equilibrium. The Glauber dynamics are expressed as a Markoff process that possesses many of the same physical characteristics as its quantum mechanical counterpart. In addition, since the classical trajectories are those of an ergodic process (the time averages of a single trajectory are equivalent to the ensemble averages), the surfaces of constant temperature, in terms of the model parameters, may be derived for the single spin system.
In this work we study the Glauber dynamics of the one-dimensional Ising model with nearest-neighbor and next-nearest-neighbor interactions, for which an approximate solution of the magnetization per site is obtained. When the dynamical critical exponent z is investigated following the treatment of Cordery, Sarker, and Tobochnik [Phys. Rev. B 24, 5402 (1981)], our observation shows that its upper-bound value is the same as the known value, thus implying that z is independent of the range of the interaction. We also suggest a high-temperature expansion approximation which is then used to solve the two-dimensional Glauber dynamics governed by a master equation; this solution is compared with that of the decoupling method. The time-delayed correlation function is also calculated.