Asymptotic properties of markovian master equations

An asymptotic method of analysis of fluctuations in systems far from equilibrium is developed. A systematic singular perturbative expansion of the equation for the generating function is set up, using as smallness parameter the inverse of the size of the system. Static and time-dependent properties are analysed before, near and at a bifurcation point, both for homogeneous and inhomogeneous fluctuations. The connection with critical phenomena near equilibrium is also discussed.     

Dynamics of the Schlögl models on lattices of low spatial dimension

The steady states and dynamics of the two Schlögl models on one- and two-dimensional lattices are studied using master equation techniques in tandem with simulations. It is found that the classic bistable behavior of model II is modified to monostable behavior at low dimension. An explanation of this modification is proposed in terms of the effective potential that appears in the dynamical equations on considering the significant effect of fluctuation correlations. The behavior can be modeled by replacing the transient average fluctuation correlation by its asymptotic value plus Gaussian white noise and analyzing the resulting effective potential obtained from the Fokker-Planck equation with multiplicative noise. For model I the transcritical bifurcation point is shifted to lower values of the forward ratek ₂ of the second step of the reaction scheme and this shift can also be explained via an effective potential as a function of the average asymptotic fluctuation correlation. Further addition of noise to the asymptotic value is irrelevant for this model since the noise term in the corresponding Fokker-Planck equation turns out to be purely additive.     

Generating Functionals in Nonequilibrium Statistical Mechanics

The main ideas and methods of calculations within the framework of the generating functional technique are considered in a systematical way. The nonequilibrium generating functionals are defined as functional mappings of the nonequilibrium statistical operator and so appear to be dependent on a certain set of macroscopic variables describing the nonequilibrium state of the system. The boundary conditions and the differential equation of motion for the generating functionals are considered which result in an explicit expression for the nonequilibrium generating functionals in terms of the so-called coarse-grained generating functional being the functional mapping of the quasiequilibrium statistical operator. Various types of integral equations are derived for the generating functionals which are convenient to develop the perturbation theories with respect to either small interaction or small density of particles. The master equation for the coarse-grained generating functionals is obtained and its connection with the generalized kinetic equations for a set of macrovariables is shown. The derivation of the generalized kinetic equations for some physical systems (classical and quantum systems of interacting particles, the Kondo system) is treated in detail, with due regard for the polarization effects as well as the energy and momentum exchange between the colliding particles and the surrounding media.     

THE GROWTH RATE AND STATISTICAL FLUCTUATION OF BOSE-EINSTEIN CONDENSATE FORMATION

Using the generating function method to solve the master equation of Bose-Einstein condensate and to evaluate the growth rate, statistical fluctuation of condensate atoms, we find out that there is a plateau in the growth rate curve and a super-Poisson distribution observed.     

Fluctuation in superradiance

The fluctuation of the intensity of superradiant pulse is calculated by applying a general asymptotic evaluation method of quantal macrovariables to a superradiant master equation. It is shown that the fluctuation of superradiant intensity is proportional to the cube of the number of the relevant atoms. The time-dependence of the proportional coefficient has been evaluated explicitly. It is predicted that there occurs an enhancement of fluctuation of superradiant intensity at an intermediate time. The intensity and fluctuation of higher order are also calculated. General expressions of expansion coefficients of a generating function are also given rigorously.     

Additive noise-induced Turing transitions in spatial systems with application to neural fields and the Swift-Hohenberg equation

This work studies the spatio-temporal dynamics of a generic integral-differential equation subject to additive random fluctuations. It introduces a combination of the stochastic center manifold approach for stochastic differential equations and the adiabatic elimination for Fokker-Planck equations, and studies analytically the systems’ stability near Turing bifurcations. In addition two types of fluctuation are studied, namely fluctuations uncorrelated in space and time, and global fluctuations, which are constant in space but uncorrelated in time. We show that the global fluctuations shift the Turing bifurcation threshold. This shift is proportional to the fluctuation variance. Applications to a neural field equation and the Swift-Hohenberg equation reveal the shift of the bifurcation to larger control parameters, which represents a stabilization of the system. All analytical results are confirmed by numerical simulations of the occurring mode equations and the full stochastic integral-differential equation. To gain some insight into experimental manifestations, the sum of uncorrelated and global additive fluctuations is studied numerically and the analytical results on global fluctuations are confirmed qualitatively.     

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：     

Steady state solutions of master equations without detailed balance

Steady state solutions of master equations with one variable are constructed. The method of solution is based on a transformation of the original equation for the probability into one for a slowly varying function. The method is of general applicability and is particularly useful in obtaining solutions in the case where detailed balance does not hold. Examples of such systems in chemical reaction models and the two photon laser are discussed.On leave from the University of Waikato, Hamilton, New Zealand     

介观化学反应体系中非平衡相变最可几路径的熵产生

针对一个双稳的介观化学反应体系计算了在非平衡相变时最可几路径的熵产生. 利用概率产生函数和程函近似,将化学主方程转化为经典的哈密顿-雅可比方程并通过相空间的零能轨线找到双稳态之间转变的最可几路径. 通过计算前向和逆向最可几路径的熵产生,发现在共存点系统熵变和介质熵变都为零,而在非共存点系统熵变和介质熵变皆不为零.     

Superradiance and energy transfer within a system of atoms

We investigate cooperative effects in energy relaxation and energy transfer for N atoms in a thermal radiation field with superradiance master equations as well as a closed set of coupled moment equations. Both spatially large and spatially small systems are considered. For small systems nonlinear rate equations for the energy are related to the moment equations. Symmetry of the small system to interchanging atoms is used to incorporate off-diagonal solutions of the superradiance master equation in expressions for the probability of the transfer of energy from one group of atoms to another. The long time excitation probability for initially unexcited atoms is large and strongly correlated. Cooperative processes in a large system which fall off with the distance between a cooperating pair of atoms include energy loss and transfer terms in the master equation. The energy transfer is oscillatory in time. Energy relaxation is shown by numerical solution to become cooperative in a very sudden manner as the scale of the atomic system is decreased through the resonant wavelength.     

非平衡系统的概率模型及Master方程的建立

针对非平衡态统计中出现的非线性及多元线性Master方程,明确提出了一般的概率假定;应用概率论的方法建立了一般的非线性及多元线性Master方程;讨论了它们与宏观动力学方程的一致性问题;最后应用多元线性Master方程讨论了Brusselator的涨落,并对其耗散结构的产生作了初步解释。 关键词：     

Quantum Brownian motion with large friction

Quantum Brownian motion in the strong friction limit is studied based on the exact path integral formulation of dissipative systems. In this limit the time-nonlocal reduced dynamics can be cast into an effective equation of motion, the quantum Smoluchowski equation. For strongly condensed phase environments it plays a similar role as master equations in the weak coupling range. Applications for chemical, mesoscopic, and soft matter systems are discussed and reveal the substantial role of quantum fluctuations.     

A derivation of the Lorenz equations for some unstable dispersive physical systems

The amplitude equations which are valid in the neighbourhood of the bifurcation point of a class of dispersively unstable physical systems when small dissipation is included are shown to be transformable to the Lorenz equations. There is a strong connection with systems which yield equations solvable by the inverse scattering transform when damping is excluded and spatial variation included. Two examples are given in very brief detail: (1) a two-layer model for baroclinic instability, (2) the laser equations giving rise to the SIT equations.     

Time-dependent probability density of statistical mechanics

We study fluctuations around nonequilibrium steady states of some model nonlinear chemical systems. A previous result of Nicolis and Prigogine states that the mean square fluctuation computed from a master equation in the space of internal states of the reacting species is identical to that calculated from Einstein's fluctuation formula. Our analysis of fluctuations based on that master equation leads with the assumption of local equilibrium to a result identical to that obtained from a master equation for the total concentration of the reacting species, which is different from the equilibrium (Einstein relation) result. Nicolis and Prigogine approximated one term in their master equation, and a discussion of this approximation is presented. The master equation without this approximation yields at equilibrium the result expected on the basis of Einstein's formula.Work supported in part by the National Science Foundation and Project SQUID, Office of Naval Research.     

Theory of open systems I

A new method of treating open systems is presented. The normal treatment using the generalized master equation with the projection of Argyres and Kelley is meaningful only if the state of the reservoir never deviates appreciably from the reference state which appears in the projection. Otherwise, one must make at least a partial resummation of the perturbative expansion of the kernel of the generalized master equation. The present method avoids the introduction of a projection operator and allows us to overcome such resummation difficulties. It is based on an integrodifferential equation for the statistical operator of the composite system, which naturally provides a hierarchy of equations involving the statistical operator ?(t) of the open system and suitable quantities describing higher and higher order bath-system correlations. Treating the deviations of the bath from its initial equilibrium or stationary state as expansion parameters, one gets an approximation scheme, each step of which gives a closed system of equations for ?(t) and a suitable set of correlation quantities.Eliminating such quantities one obtains a closed linear integrodifferential equation for ?(t). The zeroth approximation in the deviations coincides with the Born approximation of the generalized master equation which uses the projection of Argyres and Kelley.On the other hand, even the first approximation is equivalent to the resummation of infinite contribution of the Born series of such a generalised master equation. When it is suitable, the concentration of the bath can also be used as an expansion parameter to handle the hierarchy.     

Fluctuations of a macro-spin in a superradiant system

Fluctuation phenomena in a superradiating atomic system are investigated in the framework of the generalized phase space method. An operator form of the superradiant master equation is mapped onto ac-number space. Evolution of the most probable path and small deviations from the path are determined. Fluctuations around the path are solved in a closed form. A remarkable enhancement of fluctuation is observed and this is recognized as a sort of anomalous fluctuation around an unstable point.     

Dynamics of fluctuations in a reactive system of low spatial dimension

We study, using master equation techniques, the time evolution of the average concentration and fluctuations in the two-speciesn-molecule reactionA+(n-1)XnX in one dimension described by a Glauber-type dynamical lattice model for the specific casesn=2 (bimolecular) andn=3 (trimolecular). The evolution is found to be quite different from that described by the Mean-Field equations even for the bimolecular case, where the steady state is meanfield. For the trimolecular process, the values of fluctuation correlations in the nonequilibrium steady state are well predicted by the fixed points of the dynamical equations obtained from the master equation. In addition, three-point fluctuation correlations are found to play an important role in both processes and are accounted for by an extended Bethe-type ansatz. The bimolecular system shows no memory effects of initial conditions, while the trimolecular system is characterized by memory effects in terms of the average concentration, fluctuations as well as the entropy. The spatial decay of fluctuation correlations is found to be short range at the steady state for the trimolecular system.