Tomoji Yamada

Pattern Formation in Oscillatory Chemical Reactions

The present paper deals with the problem of pattern formation which takes place in a self-oscillatory continuous medium. Specifically, a mathematically tractxad able two-component field of purely dissipative nature will be considered throughout for primary understanding of some typical wave phenomena of dissipative origin. As an important example of such systems one may mention a two-component reacxad tion-diffusion system driven far from equilibrium, to which a large part of this paper has been devoted. Let a chemically reacting system be represented by a macroscopic dynamical system. Then its fundamental state variables may be taken to be the concenxad trations of reactants. If the concentrations of some species are externally conxad trollable so as to be kept constant in time, or if their time-variation is very slow as compared with the processes of interest, these quantities may be treated as the time-independent parameters that prevent the system from going to thermal equixad librium. It often occurs that a certain parameter has a threshold value for the appearance of the concentration oscillations of limit-cycle type. 1J Our main concern is the spatia-temporal organization closely connected with this type of oscillation. In particular, we shall discuss in detail that our model system can display some well-known wave phenomena such as the circular and spiral pattern formation observed in a malonic acid-bromate system or analogous reaction systems. 2 l~ 5 l

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.

Theoretical Study of a Chemical Turbulence

Recently Kuramoto and one of the present authors have carried out a computer simulaxad tion for a chemically oscillating system and found a turbulence-like behavior similar to the hydrodynamic turbulence. The steady turbulent state of this system is theoretically studied. It is shown that there exist two characteristic regions of wavenumber k. One is a cascade region with ks~1, and the other is a dissipative region with kS> 1, where s is a characteristic length which is much larger than the reaction mean free path lr. Over these two regions = = =

Spiral Waves in a Nonlinear Dissipative System

We have shown recentlyll that spiral waves similar in many respects to the observed ones in the Belousov-ZhabotinskyZaikin reaction~ 1 can arise in a self-oscillatory medium. Two basic assumptions, namely, the existence of a stable limit cycle and slow spatial variation of phase, led to a simple time-evolution equation for the phase in a closed form. In the course of the analysis, however, it became clear that the latter assumption leads to a contradiction in describing the central region of the spiral. Though this difficulty was avoided by a phenomenological cutoff procedure, a more adequate nonperturbative approach was hoped to be developed, which we shall do here for a simple two-component model. Let x and y be concentrations (measured from an unstable steady state), and define w=p exp (ifJ) =.x+iy. The model equation we shall now consider

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

The Dispersion of Cyclotron Waves in Metals with Fermi Surfaces Slightly Deviated from Sphere or Ellipsoid

The dispersion and the attenuation of the cyclotron (or high frequency) waves in metals are calculated. It is shown that there are an upper and a lower limits in the wave numbers of the propagating waves in metals with the Fermi surfaces deviated from ellipsoid even if the relaxation times are infinite. The line shapes of the surface impedances are also calculated and show anomalies concerning with the extremum orbit and the limiting point resonances.

Field-theoretic approach to critical dynamics far from equilibrium

The field-theoretic approach to critical phenomena is extended to deal with critical dynamics far from equilibrium. In particular, the macroscopic evolution equation for the average order parameter is derived in a manner parallel to the derivation of the equation of state. The method is illustrated by deriving the scaled macroscopic equation of motion for the timedependent Ginzburg-Landau model near the critical point for dimensionality near four.

A New Perturbation Approach to Highly Nonlinear Chemical Oscillation with Diffusion Process

We present in this short note a simple perturbation scheme to derive a reduced description of the dynamics of reaction-diffusion systems deep in a temporally ordered state. This work was motivated by Ortoleva and Ross phase-wave theory of heterogeneous reaction-diffusion system in which the degree of heterogeneity is taken as a small parameter.!) In contrast to their work, however, we consider a homogeneous system and the amplitude and the instantaneous frequency are expanded in powers of spacederivatives. We restrict our consideration to a twocomponent reaction -diffusion system because in this case one may give an explicit definition of phase and amplitude so that the theory may be developed in a most unambiguous way. We start with the equations

Landau-Lifshitz Equation of Motion for Ferromagnetic Systems

The purpose of this letter is to propose a new microscopic derivation of the LandauLifshitz equation for ferromagnets1> which are used to study the motion of the Bloch wall and the ferromagnetic resonances. This is done by using a new method similar to Kawabatas recent derivation of a LandauLifshitz type equation for a spin interacting with its surroundings.2> Our derivation is more general than that of Callen which is limited to the spin-wave region.3> We consider an isotropic Heisenberg spin system whose exchange interaction is modulated by the thermal sound waves. Namely, we start with the following Hamiltonian:

Nonlinear Diffusion in Critical Binary Mixture

In the present note we consider the diffusion current in a critical binary mixture from the viewpoint of the nonlinear transport. As was shown by Kawasaki and ·Gunton1> in their study of the nonlinear shear viscosity a rem~rkable feature arises in the ·nonlinear transport; the nonlinear transport coefficient is :ilonanalytic with respect to the thermal ·driving force. This peculiar effect is due to.· the existence of the ·diffusion mode which brings the socalled long time tail into the time correlation function of transverse velocity. Thus, the formal power series expansion of the nonlinear transport coefficient with respect to the thermal driving force breaks down. The resummation of this series expansion Dl.ay give the convergent result and at the same time the nonanalytic transport c::~efxad ficient. Physically this nonanalyticity originates from the fact that in small wave numbers the decay of the transverse velocity interferes with the velocity gradient.U In the present reference system the similar effect can be expected since the decay of the local concentration and that of the transverse velocity are of diffusion type. The present study will be made in the framework of the linearized hydrodynamic equations which can be derived from the mode-coupling theory.2> The steady concentration gradient is present in the xdirection and its magnitude is denoted as d; Then, the linearized hydrodynamic equations are easily obtained as