Kyozi Kawasaki

Theory of Dynamical Behaviors of Ferromagnetic Spins

The dynamical behavior of ferromagnetic spins is studied on the basis of the statistical mechanics of irreversible processes. A macroscopic equation determining the change in time of an inhomogeneous magnetization is derived with explicit expressions for the frequency spectrum and damping constant. With the use of the general expressions thus obtained, the following problems are discussed on the basis of the Heisenberg model of ferromagnetic spins: (1) the pair correlation of spins, (2) the magnetic scattering cross section of neutrons, (3) the frequency spectrum and the clamping constant Of spin waves, (4) the damping of the longitudinal spin component above and below the Curie point. In the low temperature limit, a straightforward reduction of our expressions leads to the spin wave frequency and damping equivalent to Dysons theory of spin wave interactions. A general expression is given for van Hoves parameters describing the asymptotic behavior of the spin pair correlation at large distances. It is shown that the longitudinal spin damping in the low temperature limit exhibits a variety of k dependences, depending upon the relative magnitudes of the spin wave energy Dk2, the effective magnetic field H coming from other than the exchange interaction and the thermal energy keT; in particular, for !JtJ.eH<t:Dk2<t,kBT, the longitudinal damping is proportional to k2, namely, the change in time of the longitudinal spin density obeys the diffusion equation. In the vicinity of the Curie point and in the paramagnetic region, the longitudinal clamping is shown to obey the· diffusion equation. The diffusion constant thus obtained vanishes at the Curie point and is in good agreement with the values observed by Ericson and J acrot for iron above the Curie point.

Application of Mode-Coupling Theory to Nonlinear Stress Tensor in Fluids

In a steady state of fluid the nonlinear dependence of stress tensor on a thermal driving force is investigated by a direct extension of the mode-coupling theory to far-from-equilibrium situation. The steady state is generated from a local equilibrium state having the same average values of gross variables as in the steady state. In the lowest approximation for the mode-coupling term ·the explicit expression of the stress tensor is obtained and evaluated. The effects of the higher mode-coupling approximations on the stress tensor are also briefly examined.

Simple derivations of generalized linear and nonlinear Langevin equations

With the aid of a single operator identity, the derivation of the Mori generalized linear Langevin equation is simplified and a new generalized nonlinear Langevin equation is obtained. The flexibility of the method is stressed which allows the derivation of various generalized nonlinear Langevin equations that can be used as bases for devising approximation schemes such as the mode coupling scheme.

Long time behavior of the velocity autocorrelation function

Abstract The velocity autocorrelation function has been studied using the mode-mode coupling theory of transport phenomena. The result agrees with the molecular-dynamic calculations of Alder and Wainwright.

Growth rate of critical nuclei near the critical point of a fluid

The growth rate of critical nuclei for a fluid near its critical point is discussed on the basis of the stochastic equation for the probability distribution function of the local order parameter, which was derived previously by the author. The growth rate was found to depend on ξ, the range of correlation of the order parameter fluctuation, andR, the radius of critical nuclei, as ξ0R−3, in conformity with dynamical scaling. The rate of nucleation at the liquid-gas transition near the critical point is also discussed on the basis of this result.

Non-hydrodynamical behavior of two-dimensional fluids

Abstract The extended mode coupling theory is used to investigate the non-hydrodynamical behavior of the two-dimensional fluids.

A note on projection operators in non-equilibrium statistical mechanics ☆

Abstract A new projection operator is introduced which is useful for the description of statistical mechanics of arbitrary non-equilibrium states, and exact equations of motion for macroscopic variables are derived using this projection operator.

Contributions to Statistical Mechanics Far from Equilibrium. III Non-Perturbative Method for Steady States

The systematic perturbation scheme for solving stochastic equations for the distribution function of gross variables described in earlier publications of this series is used to develop a non-perturbative self-consistent method for obtaining average values of the gross variables, time correlation functions of fluctuations, non-equilibrium steady state distribution function and the response function to an external disturbance. The analogy of the present problem with that of condensed Bose systems is found to be helpful, and the method requires two kinds of renormalized propagators and three kinds of renormalized vertices to be determined self-consistently. The formalism of Martin, Rose and Siggia on the same problem comes out rather naturally in our approach without introducing their unfamiliar operator. The problem can be simplified so that only one type of propagators and one type of vertices appear if the steady state distribution function is Gaussian.

Contributions to Statistical Mechanics Far from Equilibrium. IV --Improved and Simplified Treatment of Non-Steady States--

The method of dealing with non-steady states presented in Part II of this series is improved and simplified by dissociation of the reference Gaussian distribution function from local equilibrium states. Furthermore, a new approach gives an extra degree of freedom of choosing the variance of the reference distribution function, and its advantage is briefly disxad cussed with an illustrative example. § 1. Introductioh In Part II of this series of papers 1l devoted to statistical mechanics far from equilibrium we have presented a method for obtaining the probability distribution function of gross variables in a non-steady situation. Since then have arisen some circumstances that necessitate improvement and further extension of the methpd. (1) The method of II begins with the local equilibrium distribution function of a Gaussian form but this is not always adequate. For instance, if the process under consideration starts from a thermodynamically unst~ble state (spinodal decomposition2l) or passes through such a state (nucleation 3l), the variance of the Gaussian local equilibrium distribution blows up. (2) With the method of II the Gaussian local equilibrium distribution . furictiqn is implicitly assumed to be also formed by application of an external field and hence II (2 · 8) is assumed for the Gaussian local equilibrium distribution g10 (t). This, however, has the effect of restricting the theory to the cases where the variance of the local equilibrium distribution does not vary in time. Hence the part of II dealing with the cases where the variance changes in time is incomplete and unnecessarily complicated. In the present paper we present a modified version of II by dissociating the starting reference Gaussian distribution function from local equilibrium states. This leaves some. arbitrariness in the formalism which can be fixed later on to our best advantage. Many of the developments of II are still applicable to a

Tricritical dynamics of multicomponent fluids

Tricritical anomalies of transport coefficients of diffusive type in multicomponent fluids are studied theoretically for the thermodynamic model system due to Griffiths (1974). Transport anomalies are characterized by the two critical modes whose decay rates behave as xi -3, xi being the range of correlation of critical fluctuations. Usefulness of the variables having a diagonal susceptibility matrix considered by Tisza (1966), as well as by Gunton and Green (1971) in the present problem is indicated.