Michio Tokuyama

Statistical-Mechanical Theory of Random Frequency Modulations and Generalized Brownian Motions

It is shown that the Heisenberg equation of motion can be transformed exactly into the reduced form dA(t)/dt= [i.Q- S:</!(s)ds] ·A(t) +g(t), where A(t) is an arbitrary column vector of dynamical variables, .Q IS a frequency matrix determining collective oscillations, and g (t) IS a fluctuating force. <P (t) differs from the memory function and is related to the time-correlation function of g(t) by </J(t) = (g(t), g*(O)) ·(A (t), A* (0)) _,_ This equation of motion is shown to give a statistical-mechanical theory of random frequency modulations, including Kubos theory of the stochastic Liouville equation, and to lead to a new theory of generalized Brownian motions which complements a memory-function theory given by Mori.

Statistical-mechanical theory of coarsening of spherical droplets

A new statistical-mechanical theory of diffusion-controlled droplet coarsening is presented. With the aid of a scaling expansion method, a spatial graining is carried out in a manner consistent with an expansion in droplet volume fraction φ to obtain kinetic equations for a single distribution function of droplets. It is shown that there two characteristic stages of coarsening, depending on their space-time scales; an intermediate stage and a late stage. In both stages, new kinetic equations are systematically derived to order φ. These equations have two terms at order φ; a collisionless drift term and a collision term. The collision term is shown to be different from the conventional encounter integral discussed by Lifshitz and Slyozov since the former is of order φ and describes distant (soft) collisions, while the latter is of order φ and describes close (hard) collisions. It is shown in both stages that the mean droplet radius increases as the cube root of the time (t13). A scaling behavior of the distribution function is also found in both stages. In particular, in the late stage this scaling behavior is shown to coincide with that obtained by Lifshitz and Slysov in the limit φ→O. It is also pointed out that a naive expansion in powers of φ breaks down due to the long -range nature of interactions among droplets through diffusions. Fluctuations around the kinetic equations are also explicitly explored. They are shown to be nonthermal fluctuations generated by the soft collision process and to be small. Especially, in the late stage they are shown to obey a linear Gaussian Markov process, satisfying a fluctuation-dissipation relation of the second kind.

Finite volume fraction effects on Ostwald ripening

Abstract We investigate the effect of a finite droplet volume fraction on the Ostwald ripening on the basis of the statistical theory recently developed by us. The theory takes into account both the competitive growth and soft-collision effect of droplets arising from statistical correlations among them. The Lifshitz-Slyozov-Wagner scaling law is found to hold. The scaled droplet size distribution function and the average droplet size are determined self-consistently, up to order Q 1 2 , Q being the volume fraction, by numerically solving the kinetic equation of the theory for long times. The results are in excellent agreement with those of the computer by Voorhees and Glicksman within the data scatter, which is not the case if soft-collision processes are omitted as in all the previous theories.

On the theory of concentrated hard-sphere suspensions

A systematic theory for the dynamics of hard-sphere suspensions of interacting Brownian particles with both hydrodynamic and direct interactions is presented. A generalized diffusion equation is derived for concentrated suspensions. The volume fraction (φ) dependence of the short- and long-time self-diffusion coefficients are thus explored from a unifying point of view. The long-range hydrodynamic interactions due to the Oseen tensor are shown to play a crucial role in both coefficients, while the short-range hydrodynamic interactions just lead to corrections. The importance of the correlation effects between particles due to the long-range hydrodynamic interactions is also stressed. The nonlocal correlation effect is an important factor, leading to the behavior of the long-time self-diffusion coefficient (DSL) as DSL ∼ (1 − φ/φ0)2 near the volume fraction of φ0 = 0.5718. The direct interactions are also found to be drastically reduced by the short-range hydrodynamic interactions.

Computer modelling of Ostwald ripening

Abstract On the basis of the multiparticle diffusion equation in the Ostwald ripening, we construct a new effective model by using three simplifications: 1. (1) explicit consideration of the screening effect of the diffusion field, 2. (2) the dimensional reduction, 3. (3) the expansion in the volume fraction. We simulate this model for various values of the volume fraction, and obtain the droplet size distribution functions, coarsening rates, the standard deviations and the skewness of the distribution function, which are compared with those of the earlier theories and the recent direct computer simulations of the multiparticle diffusion equation. The present results are in good agreement with those of the theory by Tokuyama and Kawasaki, which have pointed out the importance of the soft-collision processes.

Nonequilibrium statistical description of anomalous diffusion

In this paper, from the unifying viewpoint we will cover our recent work on the nonequilibrium statistical description of anomalous diffusion and application of this theory to explaining late experiment. We will study the motion of a particle under the influence of a random force modeled as Gaussian colored noise with arbitrary correlation and with/without external field. In the very general case, the generalized Langevin equation is presented. We obtain the variances of displacement, velocity and cross variance between displacement and velocity, their asymptotic and crossover behavior. The exact equations for the joint and marginal probability density functions, and their solutions are obtained. Finally the anomalous diffusion is described in the framework of nonequilibrium statistical mechanics. The experimental results (Skjeltorp et al., Phys. Rev. E 58 (1998) 4229) can well be explained by our theory presented in this paper.

Fractal dimensions for diffusion-limited aggregation

Abstract A mean-field theory is proposed for fractal dimensions of diffusion-limited aggregates grown on a substrate surface of arbitrary dimensionality. The results are in good agreement with those of the computer simulations for all dimensionalities.

Dynamics of diffusion‐controlled reactions among stationary sinks: Scaling expansion approach

Diffusion‐controlled reactions in a nondilute sink system are rigorously studied with the aid of a scaling expansion method. A space‐time coarse graining is carried out in a manner consistent with an expansion in sink concentration to obtain macroscopic transport equations from microscopic equations. It is shown that, beyond the lowest order in sink concentration, the macroscopic transport equation for the reaction–diffusion process cannot be written in a conventional local form in space and time since a nonlocal contribution in space becomes important. Properties of the fluctuations around the macroscopic motion are also explicitly explored, and they are shown to be small in comparison with the macroscopic motion for three‐dimensional systems with an appropriate choice for the size of a sink radius and obey a Gaussian process. An absorption process whose characteristic length is much longer than that of the reaction–diffusion process is also investigated, and a local damping equation in space and time is...

Kinetic equations for Ostwald ripening

A new viewpoint on the kinetics of Ostwald ripening is presented by studying the kinetic equation recently derived for the late stage of phase separation. It is shown that the average droplet radius grows as t13 and the number density of droplets decays as t-1. An important effect of a soft (distant) collision on coarsening is discussed. Thus the relative droplet size distribution function is found to obey a second-order differential equation. The coarsening rate is also expressed in terms of the distribution function, leading to a dependence on the volume fraction of the minority phase.

The time dependent behavior of the ostwald ripening for the finite volume fraction

Abstract On the basis of the theory of Tokuyama and Kawasaki on the Ostwald ripening, we investigate the time dependent behavior from the late stage into the scaling region. We calculate numerically the transient behavior of the droplet size distribution function and the average droplet radius for various initial conditions. Moreover we compare the obtained results with the experiments in which the transient behavior of the Al-Li, Ni-Al and Ni-Si alloy systems were investigated for various ageing temperatures. The present theoretical results are in good agreement with these experimental results.