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Journal of Mathematical Physics

Dynamic Behavior of a Set of Weakly Coupled Ising Spins

Paul H. E. Meijer, Tomoyasu Tanaka, Jeremiah H. Barry

Published Jul 1, 1962• 14 citations
Publisher

Abstract

The master equation of a set of independent equivalent spins contains only one undetermined constant, the rate constant. If one assumes the local field to be altered by the field produced by the two neighboring spins, one can formulate a set of equations for the average of one, two, three, etc., spins. On assuming an Ising interaction between the spins, weak compared to the coupling with the heat bath, we can terminate the hierarchy and solve the problem of a linear chain with periodic boundary conditions by Fourier-transformation. The resulting secular equation determines two sets of relaxation times and two sets of eigensolutions. An explicit solution for the spin averages is given for the initial condition describing a localized excitation. Similarities with, and differences between, this and the random walk problem is pointed out.

DOI
10.1063/1.1724282

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The master equation of a set of independent equivalent spins contains only one undetermined constant, the rate constant. If one assumes the local field to be altered by the field produced by the two neighboring spins, one can formulate a set of equations for the average of one, two, three, etc., spins. On assuming an Ising interaction between the spins, weak compared to the coupling with the heat bath, we can terminate the hierarchy and solve the problem of a linear chain with periodic boundary conditions by Fourier-transformation. The resulting secular equation determines two sets of relaxation times and two sets of eigensolutions. An explicit solution for the spin averages is given for the initial condition describing a localized excitation. Similarities with, and differences between, this and the random walk problem is pointed out.
A modified version of Glauber's one-dimensional spin relaxation model is applied to a spin-1 Ising chain in order to study the time dependence of the system in the weak-coupling limit. The individual spin-1 Ising particles are assumed to interact with the heat bath, which causes them to change their states randomly in time. Coupling between the particles is introduced through the assumption that the transition probabilities for any one spin-1 Ising particle depends on the state of the neighboring spin-1 Ising particles. A special assumption about the rate constants is chosen such that the average values of the dipole moment will return to the equilibrium value. We establish the system of rate equations for average values of the dipole and quadruple moments, as well as their coupling. The Ising interaction between the spin-1 particles is assumed to be weak compared to the coupling with the heat reservoir. In this way we can terminate the hierarchy and solve the problem of a linear chain with periodic boundary conditions, using the Fourier transformation. The resulting secular equation determines two sets of relaxation times and two sets of eigenvectors. From this equation both relaxation times are determined by a perturbation method.
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 two-dimensional (and to some degree three-di- mensional) version of Glauber's one-dimensional relaxation model is described. The model is constructed to yield the Ising model of cooperative phenomena at equilibrium. A complete hi- erarchy of differential equations for multispin correlation func- tions is constructed. Some remarks are made concerning the solu- tion of them for the initial value problem of determining the relaxation of an initial set of distributions. Investigations of the equilibrium thermodynamic properties of the Ising model have expanded our intuition on the general na- ture of phase transitions. They have also provided an accurate representation of the behavior of certain real systems well char- acterized by the model. Dynamical models of cooperative phe- nomena have proven harder to develop. Glauber (1) introduced an interesting master equation for the dynamics of the Ising spin variables on a one-dimensional chain. This equation is constructed so that the spin-flipping process at a given time is determined by two competing influences: (i) temperature ef- fects which at higher temperatures tend to randomize the sys- tem, and (ii) correlation effects of a given with its neighbors which tend to align a given with its neighbors. Although some literature has been generated for the investigation of two- dimensional and three-dimensional generalizations of the Glauber model (cf. Kawasaki (2) and Matsudaira (3, 4)), a de- tailed analysis of the appropriate hierarchy of rate equations for the dynamics of correlation functions has not yet been given. The purpose of this report is to make a contribution to such a program.
We present an alternate method to close the master equation representing the continuous time dynamics of interacting Ising spins. The method makes use of the theory of random point processes to derive a master equation for local conditional probabilities. We analytically test our solution studying two known cases, the dynamics of the mean-field ferromagnet and the dynamics of the one-dimensional Ising system. We present numerical results comparing our predictions with Monte Carlo simulations in three different models on random graphs with finite connectivity: the Ising ferromagnet, the random field Ising model, and the Viana-Bray spin-glass model.
We study energy and spin transport in a one-dimensional Ising chain which is connected to two separate heat baths on both sides. By applying the Born-Markov approximation, within the global approach, we derive the Markovian master equation of the system, and also the explicit form of the Lindblad operators and the steady state. Thereafter, we investigate the behavior of energy and spin dynamics of the system in the global regime. Finally, we solve the problem with the local approach, and we show that the results are not the same for both approaches.
We consider the stochastic dynamics of Ising ferromagnets (either pure or random) near zero temperature. The master equation satisfying detailed balance can be mapped onto a quantum Hamiltonian which has an exact zero-energy ground state representing the thermal equilibrium. The largest relaxation time teqt_{eq}teq​ governing the convergence towards this Boltzmann equilibrium in finite-size systems is determined by the lowest non-vanishing eigenvalue E1=1/teqE_1=1/t_{eq}E1​=1/teq​ of the quantum Hamiltonian HHH. We introduce and study a real-space renormalization procedure for the quantum Hamiltonian associated to the single-spin-flip dynamics of Ising ferromagnets near zero temperature. We solve explicitly the renormalization flow for two cases. (i) For the one-dimensional random ferromagnetic chain with free boundary conditions, the largest relaxation time teqt_{eq}teq​ can be expressed in terms of the set of random couplings for various choices of the dynamical transition rates. The validity of these RG results in d=1d=1d=1 is checked by comparison with another approach. (ii) For the pure Ising model on a Cayley tree of branching ratio KKK, we compute the exponential growth of teq(N)t_{eq}(N)teq​(N) with the number NNN of generations.