An asymptotic expansion of the nonlinear master equation

Recently, a nonlinear master equation has been suggested to account for the effect of diffusion in the fluctuations of nonlinear systems away from equilibrium. An asymptotic expansion of the solutions of this master equation in the inverse of the diffusion constant is presented. The applicability of the method is illustrated with several examples of model chemical reactions.     

Diffusion equation for the quasi-incoherent motion of Frenkel excitons in molecular crystals

In a recent paper a master equation describing the quasi-incoherent motion of Frenkel excitons in molecular crystals has been derived within the Haken-Strobl model for the coupled coherent and incoherent motion of Frenkel excitons. Starting from this master equation and using only the translational symmetry of the crystal lattice, for crystals with one molecule in the unit cell a diffusion equation is derived. For crystals with several molecules in the unit cell instead of a diffusion equation one obtains a set of diffusion-like equations. These equations are solved explicitly for the case of two molecules in the unit cell and asymptotic expressions are discussed. It is shown that this asymptotic behaviour is again described by a diffusion equation.     

Some comments on nonequilibrium phase transitions in chemical systems

Two new approaches for investigating critical fluctuations near an instability point of unstable chemical models are proposed. The master equation approach is used. For a homogeneous system without the effect of diffusion, three single-component chemical systems exhibiting critical behavior are considered. The cumulant functions are expanded in a small parameter-the inverse size of the system-and singular perturbation solutions of the master equation are developed. Exponents describing the divergence of the second-order variance are found to be classical. For a system including diffusion effects, spatial correlations for a quasi-one-dimensional case are investigated by considering scale transformation behavior within the multivariate master equation formalism.This work was supported in part by NSF grants MPS-7411925 and CHE 76-05583.     

Explicit solution of diffusion master equation under the action of linear resonance force via the thermal entangled state representation

Using the well-behaved features of the thermal entangled state representation, we solve the diffusion master equation under the action of a linear resonance force, and then obtain the infinitive operator-sum representation of the density operator. This approach may also be effective for treating other master equations. Moreover, we find that the initial pure coherent state evolves into a mixed thermal state after passing through the diffusion process under the action of the linear resonance force.     

First passage time problems for a class of master equations with separable kernels

We discuss first passage time problems for a class of one-dimensional master equations with separable kernels. For this class of master equations the integral equation for first passage time moments can be transformed exactly into ordinary differential equations. When the separable kernel has only a single term the equation for the mean first passage time obtained is exactly that for simple diffusion. The boundary conditions, however, differ from those appropriate to simple diffusion. The equations for higher moments differ slightly from those for simple diffusion. Analysis is presented, of a generalization of a model of a random walk with long-range jumps first investigated by Lindenberg and Shuler. Since the equations can be solved exactly one can study the behavior of boundary conditions in the continuum limit. The generalization to a larger number of terms in the separable kernel leads to higher order equations for the first passage time moments. In each case, boundary conditions can be found directly from the original master equation.     

多变量Master方程及其波动解

本文讨论了线性反应过程:A→X多变量Master方程的波动解。除了可以由反应扩散方程得到的密度波以及由之耦合而成的多波解外,不可能得到类似于文献[2]中得到的涨落波分支。在考虑压力效应后,情况基本不变。本文讨论了由于化学反应及扩散的空间尺度(自由程)一般差别很大,多变量Master方程所存在的问题。 关键词：     

Some comments on approximations to the master equation

A common strategy for approximating a master equation is to replace it by a diffusion-like equation. Many methods for deriving the form of such an equation have been suggested in the literature. We compare two of these in the light of an example in which the master equation can be solved exactly. One of these is the van KampenΩ-expansion, which generally does not give a useful approximation to the equilibrium solution, and the second is a technique which preserves the noise-free dynamics and gives the correct equilibrium solution. It is shown that the second moment calculated in the latter approximation is not an accurate one at short times. The difficulty is the restriction of the approximating equation to the diffusion form.     

Master equation with two times: Diffusion in external field

The master equation for diffusion involving two times applies to the problem of diffusion in a time-dependent (in general inhomogeneous) external field. We consider the case of the quasi Fokker-Planck approximation, when the probability transition function for diffusion (PTD-function) does not possess a long tail in coordinate space and can be expanded as the function of instantaneous displacements. For relatively weak external field the linear expansion of the PTD function leads to a simple generalization of diffusion equation, containing the retardation factors.     

Relaxation dynamics of a quantum Brownian particle in an ideal gas

We show how the quantum analog of the Fokker-Planck equation for describing Brownian motion can be obtained as the diffusive limit of the quantum linear Boltzmann equation. The latter describes the quantum dynamics of a tracer particle in a dilute, ideal gas by means of a translation-covariant master equation. We discuss the type of approximations required to obtain the generalized form of the Caldeira-Leggett master equation, along with their physical justification. Microscopic expressions for the diffusion and relaxation coefficients are obtained by analyzing the limiting form of the equation in both the Schr?dinger and the Heisenberg picture.     

A stochastic derivation of the multivariate master equation describing reaction diffusion systems

We derive the multivariate master equation describing reaction diffusion systems from a discrete form master equation in phase space, assuming that the elastic collisions of the chemically active substances with the inert carrier gas have relaxed. In this state of collisional equilibrium the stochastic operator modelling the displacement of the particles between spatial cells reduces to the random wall operator and the reactive collision term yields the usual birth and death operator. Correlation functions are derived and their validity is discussed.     

Analytical and Numerical Studies of the One-Dimensional Spin Facilitated Kinetic Ising Model

The one-dimensional spin facilitated kinetic Ising model is studied analytically using the master equation and by simulations. The local state of the spins (corresponding to mobile and immobile cells) can change depending on the state of the neighbored spins, which reflects the high cooperativity inherent in glassy materials. The short-time behavior is analyzed using a Fock space representation for the master equation. The hierarchy of evolution equations for the averaged spin state and the time dependence of the spin autocorrelation function are calculated with different methods (mean-field theory, expansion in powers of the time, partial summation) and compared with numerical simulations. The long-time behavior can be obtained by mapping the one-dimensional spin facilitated kinetic Ising model onto a one-dimensional diffusion model containing birth and death processes. The resulting master equation is solved by van Kampen's size expansion, which leads to a Langevin equation with Gaussian noise. The predicted autocorrelation function and the global memory offer in the long-time limit a screened algebraic decay and a stretched exponential decay, respectively, consistent with numerical simulations.     

化学反应体系中涨落的时间空间关联(Ⅰ)——涨落、扩散和波

本文着重分析了如何描述化学反应体系中密度涨落的空间关联,对照反应扩散方程,我们在密度涨落分布函数所满足的Master方程中引入了扩散项或压力项,对无限介质,解Master方程,得到了一系列波,第一支波是与反应扩散方程的解一致,描述了平均密度起伏在空间的传播,在气体中就是声波,其它各支波,则描述了局部密度涨落高阶矩的变化在空间的传播(平均密度不改变),其实就是局部机率分布函数畸变后的传播过程,我们称之为涨落波,文中还讨论了如何求解线性初值问题。 关键词：     

Generalized nonlinear master equations for local fluctuations: Dimensionality expansions and augmented mean field theory

By application of a projection operator technique we derive a formally exact generalization of the nonlinear mean field master equation introduced recently for the study of local fluctuations in a reacting medium. Our starting point is a phenomenological cell master equation. The results of our theory are applicable to the theory of a fluctuating hydrodynamic reacting system. The mean field equation is placed on a firm theoretical foundation by showing it to be the lowest order approximation in an expansion in the dimensionality of the physical space keeping the product of the number of nearest neighbors (an increasing function of dimensionality) and the typical diffusion coefficient constant. A more accurate nonlinear master equation that allows for the correlation and fluctuations in the environment of a given volume element is derived in the form of an augmented mean field equation.Work supported in part by a grant from the National Science Foundation.     

THE GENERALIZED MASTER EQUATION THEORY OF ELECTRON SCAVENGING IN AMORPHOUS SOLIDS

Electron scavenging in amorphous solids is analyzed by using diffusion controlled reaction model. In terms of stochastic process theory, the process is an age-dependent branching process which is described by linear death process of generalized master equation.The variation of number of trapped electron with time, N(t), is calculated with Ngai's fractional exponential waiting time density for the time between hops. Quantitative comparison with Miller's pulse radiolysis experiments on frozen 6M NaOH is made and the agreement is fairly well. The rigour and simplicity in mathematics of the generalized master equation method developed here are in sharp contrast to the master equation method in which quantitative calculation can hardly be done.     

Diffusing opinions in bounded confidence processes

We study the effects of diffusing opinions on the Deffuant et al. model for continuous opinion dynamics. Individuals are given the opportunity to change their opinion, with a given probability, to a randomly selected opinion inside an interval centered around the present opinion. We show that diffusion induces an order-disorder transition. In the disordered state the opinion distribution tends to be uniform, while for the ordered state a set of well defined opinion clusters are formed, although with some opinion spread inside them. If the diffusion jumps are not large, clusters coalesce, so that weak diffusion favors opinion consensus. A master equation for the process described above is presented. We find that the master equation and the Monte Carlo simulations do not always agree due to finite-size induced fluctuations. Using a linear stability analysis we can derive approximate conditions for the transition between opinion clusters and the disordered state. The linear stability analysis is compared with Monte Carlo simulations. Novel interesting phenomena are analyzed.     

Dual instantons

A completely positive master equation describing quantum dissipation for a Brownian particle is derived starting from microphysical collisions, exploiting a recently introduced approach to subdynamics of a macrosystem. The obtained equation can be cast into Lindblad form with a single generator for each Cartesian direction. Temperature dependent friction and diffusion coefficients for both position and momentum are expressed in terms of the collision cross section.     

Decay rate with coordinate-dependent diffusion coefficients

Using a master equation for the reduced density matrix of open quantum system, the influence of coordinate-dependent microscopical diffusion coefficients on the decay rate from a potential well is studied. For different temperatures, frictions, heights of barrier and ratios of stiffnesses of the potential in the minimum and on the top of the barrier, the quasistationary decay rates are obtained with the sets of coordinate-dependent and -independent microscopical diffusion coefficients, and coordinate-dependent phenomenological diffusion coefficients.     

Unstable state dynamics: A systematic evaluation of the master equation

The master equation for diffusion in a bistable potential is evaluated systematically in terms of the small-noise parameter for the case where the system is initially at the unstable state. The expansion is valid for all times, that is in the initial and intermediate as well as in the final regime. The theory does not involve free fitting parameters and is easily generalized to more complicated processes.