SINGULAR PERTURBATION EXPANSION FOR MARKOVIAN MASTER EQUATIONS

Asymptotic properties of the master equations for chemical reactive systems whose macroacopic rate equations have more than one stationary state are discussed Itsing generating function method. The systematic singular perturbation expansion method for equation of generating function is generalized to include the case of multi-stationary system beyond the bifurcation point and the following conclusions are proved: For such systems, there is a Gaussian fluctuation before next genuine bifurcation point] a critical fluctuation at a genuine bifurcation point; and a macroscopic fluctuation when the system is on the coexistence line which is determined by the master equation. Furthermore, the relation between the cumulant of generat-ing function and the stochastic potentia 1 is also established. Our discussion, however, is limited to homogeneous system of one variable only.     

A stochastic approach to the kinetic theory of gases

A theory of fluctuations in non-equilibrium diluted gases is presented. The velocity distribution function is treated as a stochastic variable and a master equation for its probability is derived. This evolution equation is based on two processes: binary hard sphere collisions and free flow. A mean-field approximation leads to a non-linear master equation containing explicitly a parameter which represents the spatial correlation length of the fluctuations. An infinite hierarchy of equations for the successive moments is found. If the correlation length is sufficiently short a truncation after the first equation is possible and this leads to the Boltzmann kinetic equation. The associated probability distribution is Poissonian. As to the fluctuation of the macroscopic quantities, an approximation scheme permits to recover the Langevin approach of fluctuating hydrodynamics near equilibrium and its fluctuation-dissipation relations.     

Efficient simulations of gas-grain chemistry in interstellar clouds

Chemical reactions on dust grains are of crucial importance in interstellar chemistry because they produce molecular hydrogen and various organic molecules. Because of the submicron size of the grains and the low flux, the surface populations of reactive species are small and strongly fluctuate. Under these conditions rate equations fail and the master equation is needed for modeling these reactions. However, the number of equations grows exponentially with the number of reactive species, severely limiting its feasibility. Here we present a method which dramatically reduces the number of equations, thus enabling the incorporation of the master equation in models of interstellar chemistry.     

Exact Fokker-Planck equations from stochastic master equations

A method for deriving exact Fokker-Planck equations from stochastic master equations by expanding the probability distribution in terms of Poisson distribution is given. It is applied to two non-linear chemical processes to obtain the steady state distributions.     

Microdynamics and time-evolution of macroscopic non-Markovian systems. II

We study the time-evolution of the joint and the conditional probability of macroscopic variables of a closed system from a microscopic point of view. We derive an exact generalized master equation for their time rate of change which consists of two parts, one instantaneous and local in state space, the other retarded and nonlocal in state space. It is represented by stochastic operators depending both on the initial preparation and on the initial macrodistribution, which reflects the non-Markovian character of the process. The connection with the time-evolution of the single-event probability is discussed.Work Supported by the Swiss National Science Foundation     

非平衡系统Master方程的稳定性

针对非平衡统计中出现的多元线性Master方程,利用“熵产生”和“剩余熵产生”的概念讨论了Master方程在线性平衡区和非线性远离平衡区的稳定性问题。从而得到与Prigo-gine宏观热力学理论中一致的结果。此外还提出了Master方程所决定的“概率流”的概念,给出了概率流分解的具体解析表达式。 关键词：     

Analytical results for a non-Markovian process of gene expression with positive and negative feedbacks

Gene expression is a very complex process and involves many small biochemical reaction steps, resulting in a non-Markovian discrete stochastic process due to molecular memory between individual reactions. At present, this process is successfully investigated by generalized chemical master equation models. However, these models do not consider the role of feedback networks in gene expression. How the interaction between feedbacks and molecular memory affects gene expression still remains not well understood. Here, we establish generalized chemical master equation models of gene expression with positive and negative feedbacks. Assuming that the process of producing proteins follows an Erlang probability distribution, we obtain the analytical expression for this model in a steady state, as well as the measure of the noise of protein numbers. We further find that molecular memory competes with the positive feedback in suppressing the noise of the protein number. For our model with a negative feedback, molecular memory can strengthen the intensity of suppressing this noise. These interesting results imply that molecular memory are as important as the feedbacks to affect gene expression.     

Chemical reactions and fluctuations

Lattice systems with one species diffusion-reaction processes under local complete exclusion rules are studied analytically. We discuss a rigorously derived Fokker-Planck equation for a so-called pseudo-probability. This probability distribution depends on continuous variables in contrast to the original discrete master equation, and their stochastic dynamics may be interpreted as a substitute process which is completely equivalent to the original lattice dynamics. Especially, averages and correlation functions of the continuous variables are connected to corresponding lattice quantities by simple relations. Although the substitute process for diffusion-reaction systems with exclusion rules has some similarities to the well known substitute process for the same system without exclusion rules, their exist a set of remarkable differences. The given approach is not only valid for the discussed single species processes. We give sufficient arguments that arbitrary combinations of uni-molecular and bimolecular lattice reactions under complete local exclusions may be described in terms of our approach.     

Some properties of stochastic equations for coupled chemical reactions far from equilibrium

Solutions of master equations for coupled chemical reactions far from equilibrium with one varying molecule species are studied and used for getting information about nonlinear Fokker-Planck equations and slow time-dependent processes such as extinction to an absorbing state and transition between several steady states. The Fokker-Planck equation solution is compared to that of the master equation in a relative sense and it is shown that they agree quite well in some important situations but that in general the cases can deviate considerably, when, e.g., accounting for the mutual importance of two probability maxima.     

Fluktuationen von Besetzungszahlen

Kinetic equations for the fluctuations and the fluctuation spectrum of the occupation numbers in a nonequilibrium stationary state are derived within a stochastic theory. The probability of changes of the distribution in infinitesimal time intervals is given by the “Stoßzahlansatz”. It is assumed for the decoupling of the higher correlations, that the occupation numbers are macroscopic variables.     

Quantum-statistical approach to gross properties of peripheral collisions between heavy nuclei

Non-equilibrium quantum-statistical mechanics is applied to peripheral collisions between heavy nuclei (A?40) where a large number of degrees of freedom are involved during the process. By eliminating the relative motion from the explicit consideration, the transitions between different channels are determined by a Liouville equation with timedependent coupling matrix elements. The introduction of subsets of channels (coarse graining) leads to the definition of macroscopic variables which correspond to observable quantities. The equation of motion for the macroscopic variables become irreversible by assuming the values of the coupling matrix elements to be randomly distributed. The validity and possible applications of the resulting master equations are discussed.     

Turbulence in a Maxwell fluid

Three-dimensional homogeneous isotropic turbulence is formulated in terms of a discrete stochastic process in Fourier space. The time-dependent joint probability distribution of the stochastic Fourier modes is governed by a multivariate master equation. It is demonstrated that the characteristic functional of the stochastic process obeys the Hopf functional equation. As a first application of the method stochastic simulations of the Burgers's turbulence model are performed and shown to yield the expected energy spectrum.     

Stationary Response of Lotka-Volterra System with Real Noises

A stochastic version of Lotka-Volterra model subjected to real noises is proposed and investigated. The approximate stationary probability densities for both predator and prey are obtained analytically. The original system is firstly transformed to a pair of It o stochastic differential equations. The Ito formula is then carried out to obtain the It o stochastic differential equation for the period orbit function. The orbit function is considered as slowly varying process under reasonable assumptions. By applying the stochastic averaging method to the orbit function in one period, the averaged Ito stochastic differential equation of the motion orbit and the corresponding Fokker-Planck equation are derived. The probability density functions of the two species are thus formulated. Finally, a classical real noise model is given as an example to show the proposed approximate method. The accuracy of the proposed procedure is verified by Monte Carlo simulation.     

Binomial moment equations for stochastic reaction systems

A highly efficient formulation of moment equations for stochastic reaction networks is introduced. It is based on a set of binomial moments that capture the combinatorics of the reaction processes. The resulting set of equations can be easily truncated to include moments up to any desired order. The number of equations is dramatically reduced compared to the master equation. This formulation enables the simulation of complex reaction networks, involving a large number of reactive species much beyond the feasibility limit of any existing method. It provides an equation-based paradigm to the analysis of stochastic networks, complementing the commonly used Monte Carlo simulations.     

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.