Stochastic Kinetic Equation Research Articles

Microstructural evolution during the precipitation of ordered intermetallics in multiparticle coherent systems

Abstract Shape transition and spatial rearrangement of ordered particles during precipitation in a multiparticle coherent cubic system are examined in two dimensions by computer simulation. A general phenomenological kinetic model, obtained by incorporating the transformation-induced elastic strain into the stochastic field kinetic equations, is employed. Without any a priori assumptions on particle shapes and their spatial distributions, it is shown that the interaction between particles plays an essential role not only in the development of spatial correlation but also in the shape evolution of individual particles. In the case of a low particle density, an individual shape transition sequence from circle → square (with rounded comers) → concave ‘square’ → square or rectangle is observed. In the case of a high particle density, however, no concave ‘square’ is formed and many irregular particle shapes and complex multiparticle arrangements are found instead owing to the intensive interaction between part...

Kinetics of chiral symmetry breaking in crystallization

Results of our study of the kinetics of stirred crystallization that produces large (greater than 99%) asymmetry are presented. The change of concentration with time as the crystallization progresses is obtained for stirred and unstirred crystallizations. A sharp difference in the two concentration vs time curves, due to secondary nucleation in the stirred system, can be observed. Experimental results are compared with the predictions of a set of stochastic kinetic equations. It is found that the proposed kinetics can quite accurately reproduce the experimental data if it is assumed that secondary nuclei are produced in stirred systems only when the parent crystal reaches a minimum size

The Einstein-Smoluchowski promeasure and the stochastic Boltzmann-Peierls equation

The essential properties of a spatially homogeneous gas consisting of quasiparticles (phonons, magnons, rotons, etc.) are usually described in terms of the one-point distribution function f. This is the last in a series of papers (Physica A 180 (1992) 309, 336), the overall objective of which is the construction of the Einstein-Smoluchowski promeasure μ ɛ in order to formulate and develop a theory dealing with the equilibrium fluctuations of f. The basic idea was the inversion of Boltzman's relation connecting entropy and probability. As a consequence, the derivation of μ ɛ started by considering an infinite-dimensional analog of Einstein's thermodynamic theory of equilibrium fluctuations. Now, investigating a situation in which the time evolution of f is governed by the linear stochastic differential Boltzman-Peierls equation, the object here is to find the conditions under which the stationary measure for the aforesaid equation can be obtained from the Einstein-Smoluchowski promeasure μ ɛ . In agreement with such a general program, the fluctuation-dissipation theorem is analysed and some evidence of the necessity for an algebraic approach to the stochastic kinetic equations is also presented. And finally, the paper discusses the time-correlation function formalism and shows that this formalism is consistent with the universal ideas of Einstein. To summarize, the present paper demonstrates how the Einstein-Smoluchowski promeasure μ ɛ can be used to study some problems in the extended context of the time-dependent theory of equilibrium fluctuations.

Universal mapping for particle dynamics in the wave-packet field and region of stochasticity

The exact universal mapping for particle dynamics in the wide wave-packet field is obtained. The region of stochastic motion and kinetic equation are determined. It is shown that a limitation on the stochastic heating does exist in the packets with discrete spectrum. Exact conditions for applicability of the quasilinear plasma theory are established.

Some methods for non-Markovian stochastic point-reactor kinetic equations

In this paper we discuss some methods used by several authors (Saito, 1979; Quabili and Karasulu, 1979) to solve stochastic point-reactor kinetic equations with a non-white noise. In order to test the validity of these methods a simple exact model without delayed neutrons is considered. The main point of the paper concerns the validity of the causal relation, which is shown not to hold when the noise is a non-white external one. It is shown that the equation obtained by Saito (1979) for the mean power is valid for weak noises. Adversely, the equations obtained by Quabili and Karasulu (1979) for the mean, variance and correlation function of the power reactor, are shown to be inconsistent as regards the non-Markovian terms.

Density fluctuations in simple stochastic reactor models

We present a method to study density fluctuations in a nuclear reactor simultaneously taking into account intrinsic fluctuations usually modelled by a master equation and parametric noise usually modelled by stochastic kinetic equations. The mean density and density-fluctuations variance are calculated for a simple point-reactor model in the presence of Gaussian white-noise fluctuations in the fission, capture and source event rates. They are also calculated in the case of dichotomous noise fluctuations in the source event rate. We find ‘crossed fluctuation’ contributions to the density fluctuations that only appear when considering simultaneously intrinsic and parametric noise. These contributions allow us to distinguish between reactivity fluctuations due to fission-rate and capture-rate fluctuations.

Study of stochastic reactor kinetic equation

By using the diagrammatic technique, an integral equation for power spectral density of the system having the space-time dependent reactor noise, is derived. The method has been applied to the random vibration of absorber plate and to the noise field in a boiling water reactor. The integral equation for a point reactor is investigated numerically by assuming the correlation function of reactivity noise. It was found that when the reactivity noise has an oscillatory behavior and a long correlation time, the linear response theory is not sufficient.

Uniform treatment of fluctuations at critical points

A generalized critical point is characterized by the vanishing of certain linear relationships. In particular, the dynamics near such a point are non-linear. In this paper, we study fluctuations at such points of spatially homogeneous systems. We discuss thermodynamic critical points as a special case; but the main emphasis is on stochastic kinetic equations. We show that fluctuations at a critical point cannot be characterized by a Gaussian density, but more complicated densities can be used. The theory is applied to the critical harmonic oscillator.

Generalized Phase-Space Method in the Langevin-Equation Approach

In this article is proposed a method of constructing the quantal Boltzmann-Langevin equation, i.e. the stochastic kinetic equation for a suitably chosen statistical operator with a fluctuating force term. This method is based on the use of the generalized phase-space method, which makes calculations very simple. As an example, the method of spin coherent states is applied to a model of a localized single spin placed under actions of its magnetic environment, and the Bloch-Langevin equation, i.e. the well-known Bloch equation for a spin magnetic moment with a fluctuating force term, is derived. The drift and the relaxation terms contained in the Boltzmann-Langevin and the Bloch-Langevin equations are completely in agreement with those previously obtained. An expression of the fluctuation-dissipation theorem generalized so as to include non-linear dissipation terms is given.

Propagation of an unstable spectrum of plasmons in the presence of random inhomogeneities

A one-dimensional stochastic kinetic equation for the propagation of unstable plasmons (Langmuir modes) in the presence of large scale random spatial fluctuations is presented, within the frame of the WKB approximation, and then solved. Using probabilistic arguments, an effective growth rate is evaluated, which turns out to be much smaller than the plasmon growth rate in the absence of fluctuations. The theory is then applied to the beam-plasma kinetic instability and to the stimulated photon-plasmon interaction. The former case exhibits a shift of the turbulent plasmon spectrum toward larger wavenumbers. The latter case shows a strong reduction in the efficiency of the stimulated process (i.e., much smaller growth rate) and the removal of the strong asymmetry inherent in this problem.

Kinetic equations for plasmas subjected to a strong time-dependent external field: Part 4. Dynamics of stochastic correlations

In close analogy with part 1, the dynamics of stochastic correlations is used to reformulate the preceding treatment of turbulent equations. The result is an exact and closed equation for an average function, involving only asymptotic stochastic correlations, and no initial fluctuation. This stochastic kinetic equation is valid for all initial conditions, and the corresponding function is a significant particular projection on an independent subspace R(t).

Statistics and Critical Points of Polymerisation Processes

The paper continues the line begun by the author in two earlier papers (1965) in which the equilibrium solution was obtained, under detailed balance assumptions, to a stochastic kinetic equation describing a polymerisation process. The solution is in the form of an integral which has an obvious steepest-descent evaluation in the thermodynamic limit if density is below the critical gelation value. The paper is concerned with evaluation above the critical point. With a change of variables, onset of criticality manifests itself as a switching of appropriate saddle-points from one value (density-dependent) to another (fixed). The fact that the second saddle-point is fixed has as consequence that the sol-component has density and constitution independent of the overall density p when p exceeds the critical gelation value. Results on the constitution of the gel-component are also obtained. POLYMER SIZES; CLUSTERING; CRITICAL POINTS Summary The paper is concerned with the equilibrium statistics of a clustering or polymerisation process. Sections 1-5 review the methods and conclusions of two earlier papers (Whittle (1965a) and (1965b)), Section 6 is devoted to the statistics of the process when in the condensed or gel state, and in Section 7 we consider a number of potential generalisations and applications of these ideas. A stochastic kinetic equation (3) is proposed which, under detailed balance assumptions (4), permits a Poisson equilibrium solution. From this the relevant general solution (11) is generated. However, there is still a combinatorial problem to be solved, which is reduced, by appeal to considerations of partial balance, to solution of the functional equation (22), the appropriate solution being given by (26). Steepest descent evaluation of the integral (11) fails when the process is in the condensed or gel state. This difficulty is circumvented in Section 6 by a change of variable. In the transformed integrals onset of criticality manifests itself as a switching of appropriate saddle-points from one (density-dependent) value to another (fixed) value. 199 This content downloaded from 157.55.39.191 on Tue, 11 Oct 2016 04:32:08 UTC All use subject to http://about.jstor.org/terms