A theorem allowing the derivation of deterministic evolution equations from stochastic evolution equations. III The Markovian-non-Markovian mix

The proof of a theorem that allows one to construct deterministic evolution equations from a set, with two subsets, containing two types of discrete stochastic evolution equation is developed. One subset evolves Markovianly and the other non-Markovianly. As an illustrative example, the deterministic evolution equations of quantum electrodynamics are derived from two sets of Markovian and non-Markovian stochastic evolution equations, of different type, after an average over realization, using the theorem. This example shows that deterministic differential equations that contain both first-order and second-order time derivatives can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations. Two explicit examples, the first containing updating rules that depend on one previous time step and the second containing updating rules that depend on two previous time steps, are given in detail in order to show step by step the linear transformations that allow one to obtain the deterministic differential equations.     

On the nonergodicity of the transverse magnetization in the transverse Ising model

The nonergodic behavior exhibited by the transverse spin correlation function _q=0 ^xx (t) the transverse Ising model obtained as the solution of approximate kinetic equations (derived on the basis of Résibois and De Leener's method), is shown to be an intrinsic property of the model and not the result of the approximations made in the derivation of the kinetic equations.Chargé de Recherches au Fonds National Belge de la Recherche Scientifique.     

On the derivation of quantum kinetic equations. II. Nonlocal Uehling-Uhlenbeck equation

The binary and triple collision terms of the quantum kinetic equation derived previously are analyzed in the weak coupling approximation. In this approximation the equation appears to be a nonlocal Markovian extension of the kinetic equation due to Uehling and Uhlenbeck. After linearization, its relationship with non-Markovian formulations found in the literature is studied.This investigation is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (ZWO).     

The evolution of the resonant mode in plasma-maser interaction

We study the evolution of the resonant waves considering its interaction with the nonresonant waves and the plasma particles due to plasma-maser effect. The nonlinear dielectric function of the resonant wave is calculated and is found to consist of two parts: the direct and the polarization coupling terms. On the other hand, the nonlinear dielectric function of the nonresonant wave consists only of the direct coupling term. The significance of our results is discussed.     

On a simple derivation of master equations for diffusion processes driven by white noise and dichotomic Markov noise

A very simple way is presented of deriving the partial differential equations (the master equations) satisfied by the probability density for certain kinds of diffusion processes in one dimension, in which the driving term is a Gaussian white noise, or a dichotomic noise, or a combination of the two. The method involves the use of certain ‘formulas of differentiation’ to derive the equations obeyed by the characteristic functions of the processes concerned, and thence the corresponding master equations. The examples presented cover a substantial number of diffusion processes that occur in physical modelling, including some master equations derived recently in the literature for generalizations of persistent diffusion.     

On the derivation of thermohydrodynamic equations from the Boltzmann equations

The Boltzmann equation deals with a distributionf(x, ), wherex denotes the space variable and is the momentum. The hydrodynamic equations deal with-moments of the distribution. The paper deals with the derivation of the hydrodynamic equations in the case that the collision kernel is Maxwellian, i.e., independent of the velocity. For such a kernel, a computational tool, based on the theory of representations of the orthogonal group, is developed. With this tool it is possible to derive systems of equations for any number of moments. The construction of closed systems is based on asymptotic estimates for solutions of Boltzmann equations. These show that, in some definite sense, an approximating system involving moments of high order is more accurate than a system of lower order.     

Studies in nonlinear stochastic processes. I. Approximate solutions of nonlinear stochastic differential equations by the method of statistical linearization

A method based on a variational procedure is presented which provides simple and useful approximate solutions to a wide variety of nonlinear stochastic differential equations. This method of statistical linearization is most successful when the stochasticity of the differential equation is due to excitations which are normally distributed or harmonic with random phase. Effects due to deviations from normality can be corrected for in a systematic fashion. Comments regarding existence and uniqueness are given and some error bounds arising from the use of statistical linearization are computed.Supported by the National Science Foundation under Grants NSF GP 32031X and MPS72-04353 A03.     

Time relaxation of the solutions of master equations for large systems

The time relaxation behavior of the solutions of certain classes of discrete master equations is studied in the limit of an infinite number of states. Depending on the range of the transition matrix, a relaxation behavior is found reaching from at ^–1/2 law for short range, over enhanced relaxation to an exponential relaxation for the extreme long-range case. The behavior in the limit of a continuous family of states is also discussed.     

On the derivation of the Schrödinger equation from stochastic mechanics

It is shown that the existing formulations of stochastic mechanics are not equivalent to the Schrödinger equation, as had previously been believed. It is argued that this is a reflection of fundamental inadequacies in the physical foundations of stochastic mechanics.Some relatively minor difficulties with the demonstration of equivalence are already known for the special case in which the nodal surface separates the manifold of the diffusion into disjoint components.^(1,11) The problems described in this paper are much more general and quite unrelated.     

On the derivation of quantum kinetic equations. I. Collision expansions

A straightforward scheme for deriving quantum kinetic equations is presented. It is based on Bogoliubov's initiai condition of vanishing correlations in the infinite past and consists in the elimination of an initial oneparticle Wigner function between two nonlinear functionals. By performing the elimination to second order in the density the quantum analog of the Choh-Uhlenbeck three-particle collision term is obtained. The scheme may be extended to include relativistic particles as well as particles with internal degrees of freedom.This investigation is part of the research program of the Stichting voor fundamenteel onderzoek der materie (FOM), which is financially supported by the Organisatie voor zuiver-wetenschappelijk onderzoek (ZWO).     

A unified approach for deriving kinetic equations in nonequilibrium statistical mechanics. I. Exact results

A unified method for deriving exact kinetic equations for dynamical quantities of a many-body system is presented. The well-known results of Mori and Zwanzig are recovered as special cases. Furthermore, it is shown that they differ only by the way in which the system is prepared at the initial time. Connections between this method and others recently developed are also discussed.     

非线性Schroedinger方程组的精确解组

运用Maple语言程序，在没有假设的条件下，得到了具有耦合特性的非线性Schrfidinger方程组的行波精确解组及其约束条件方程，它们的表达式涵盖了所有的耦合解组与非耦合解组，具有任意性。耦合解组的算例函数及其特性分析，解释了α螺旋蛋白质螺旋链运动模型的行波孤立子解的耦合效应，揭示了增加、稳定和控制蛋白质活性和功能的方向。文章的研究方法，为求解耦合的非线性微分方程组的行波精确解组探索了蹊径。     

Existence theory of the linear equations appearing in the Chapman-Enskog solutions to two kinetic equations for liquids

The linear operators appearing in the Chapman-Enskog solutions to Kirkwood's Fokker-Planck kinetic equation and to Rice and Allnatt's kinetic equation are studied in this article. Existence proofs are given for the linearized Chapman-Enskog equations involving either the Fokker-Planck or the Rice-Allnatt operators. It is shown that the Fokker-Planck and Rice-Allnatt operators, defined in the domain appropriate to kinetic theory, are essentially self-adjoint. It is also shown that the spectrum of either of these operators coincides with the spectrum of the self-adjoint extension of the corresponding operator.Sloan Foundation Fellow 1968–70. Guggenheim Fellow 1969–70.     

On the problem of constraints in minimally coupled relativistic wave equations for particles of unique mass

We study the problem of a possible change in the number of constraints in linear relativistic wave equations (-iβ _μ ∂ ^μ +m)ψ=0 for particles of unique mass, on introduction of minimal coupling to an external electromagnetic field. Complementing our earlier work in which we obtained conditions for non-loss of constraints in equations characterised by the minimalβ-algebraβ ₀ ⁵ =β ₀ ³ we derive here the conditions for such theories not to generate more constraints than in the free case. The results are illustrated by considering specific equations and a fallacy in certain conclusions of Kobayashi and Shamaly on this problem is pointed out.     

Relationships between a microscopic parameter and the stochastic equations for interface's evolution of two growth models

The relationship between a microscopic parameter p, that is related to the probability of choosing a mechanism of deposition, and the stochastic equation for the interface's evolution is studied for two different models. It is found that in one model, that is similar to ballistic deposition, the corresponding stochastic equation can be represented by a Kardar-Parisi-Zhang (KPZ) equation where both λ and ν depend on p in the following way: ν(p) = νp and λ(p) = λp ^3/2. Furthermore, in the other studied model, which is similar to random deposition with relaxation, the stochastic equation can be represented by an Edwards-Wilkinson (EW) equation where ν depends on p according to ν(p) = νp ². It is expected that these results will help to find a framework for the development of stochastic equations starting from microscopic details of growth models. Received 26 August 2002 / Received in final form 20 November 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: ealbano@inifta.unlp.edu.ar     

On the statistical dynamics of the Lorenz model

We use the theory of stochastic differential equations with rapidly fluctuating coefficients to study the statistical dynamics of the Lorenz model in the turbulent region. On the assumption that the system is ergodic we are able to calculate self-consistently several basic statistical quantities in terms of the parameters of the model. Our results are in good agreement with numerical computations.Supported financially by the Summer Study Program in Geophysical Fluid Dynamics at the Woods Hole Oceanographic Institution.     

Revealing of the simplest mechanisms of a structure formation

In this paper, results of investigations of the simplest mechanisms of a structure formation are presented. In frameworks of the suggested model the main attention was focused on such characteristics as wiring of the system, clusters formation, dynamics of the wiring. The idea to take into account an influence of the environment factor is employed in the proposed model. Investigations of systems with such principle of a structure formation reveal that the system's dynamics has typical features of self-organized criticality phenomenon. In the avalanche-like processes, which occur in the wiring dynamics, a power law was found with the index close to 1.4. It is independent on the environment factor (which in a sense can be considered as system parameter). The system wiring is approximated pretty well by the Gaussian distribution. The size of the system does not play any role in the dynamics of the model. Received 10 March 1999 and Received in final form 24 May 1999     

Moment equations for a spatially extended system of two competing species

The dynamics of a spatially extended system of two competing species in the presence of two noise sources is studied. A correlated dichotomous noise acts on the interaction parameter and a multiplicative white noise affects directly the dynamics of the two species. To describe the spatial distribution of the species we use a model based on Lotka-Volterra (LV) equations. By writing them in a mean field form, the corresponding moment equations for the species concentrations are obtained in Gaussian approximation. In this formalism the system dynamics is analyzed for different values of the multiplicative noise intensity. Finally by comparing these results with those obtained by direct simulations of the time discrete version of LV equations, that is coupled map lattice (CML) model, we conclude that the anticorrelated oscillations of the species densities are strictly related to non-overlapping spatial patterns.