C. W. Gardiner

Correlations in stochastic theories of chemical reactions

A comprehensive study of correlations in linear and nonlinear chemical reactions is presented using coupled chemical and diffusion master equations. As a consequence of including correlations in linear reactions the approach to the steady-state Poisson distribution from an initial non-Poissonian distribution is given by a power law rather than the exponential predicted by neglecting correlations. In nonlinear reactions we show that a steadystate Poisson distribution is achieved in small volumes, whereas in large volumes a non-Poissonian distribution is built up via the correlation. The spatial correlation function is calculated for two examples, one which exhibits an instability, the other which exhibits a second-order phase transition, and correlation length and correlation time are calculated and shown to become infinite as the critical point is approached. The critical exponents are found to be classical.

The poisson representation. I. A new technique for chemical master equations

We introduce a new technique for handling chemical master equations, based on an expansion of the probability distribution in Poisson distributions. This enables chemical master equations to be transformed into Fokker-Planck and stochastic differential equations and yields very simple descriptions of chemical equilibrium states. Certain nonequilibrium systems are investigated and the results are compared with those obtained previously. The Gaussian approximation is investigated and is found to be valid almost always, except near critical points. The stochastic differential equations derived have a few novel features, such as the possibility of pure imaginary noise terms and the possibility of higher order noise, which do not seem to have been previously studied by physicists. These features are allowable because the transform of the probability distribution is a quasiprobability, which may be negative or even complex.

Stochastic Analysis of a Chemical Reaction with Spatial and Temporal Structures

A stochastic analysis of the spatial and temporal structures in the Prigogine-Lefever-Nicolis model (the Brusselator) is presented. The analysis is carried out through a Langevin equation derived from a multivariate master equation using the Poisson representation method, which is used to calculate the spatial correlation functions and the fluctuation spectra in the Gaussian approximation. The case of an infinite three-dimensional system is considered in detail. The calculations for the spatial correlation functions and the fluctuation spectra for a finite system subject to different kinds of boundary conditions are also given.

The escape time in nonpotential systems

A generalisation of Kramers method is developed for computing the escape time in non potential systems. The method is applied to i) moderate friction case in the Kramers problem; ii) a certain two dimensional system treated by Caroli et. al. using a perturbative method.

The Poisson Representation. II. Two-Time Correlation Functions

Basic formulas for the two-time correlation functions are derived using the Poisson representation method. The formulas for the chemical system in thermodynamic equilibrium are shown to relate directly to the fluctuationdissipation theorems, which may be derived from equilibrium statistical mechanical considerations. For nonequilibrium systems, the formulas are shown to be generalizations of these fluctuation-dissipation theorems, but containing an extra term which arises entirely from the nonequilibrium nature of the system. These formulas are applied to two representative examples of equilibrium reactions (without spatial diffusion) and to a nonequilibrium chemical reaction model (including the process of spatial diffusion) for which the first two terms in a systematic expansion for the two-time correlation functions are calculated. The relation between the Poisson representation method and Glauber-SudarshanP-representation used in quantum optics is discussed.

Higher order corrections to the Dicke superradiant phase transition

Abstract The phase transition in the Dicke model for superradiance obtained by Hepp and Lieb and Wang and Hioe is modified by eliminating the rotating wave approximation.

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.

Local and global fluctuations in a soluble model of a non-equilibrium chemically reacting system

Abstract By use of coupled chemical and diffusion master equations we derive exact formulae for correlations in a certain non equilibrium chemical reaction. We show that the probability distribution is poissonian in small volumes, and non poissonian in large volumes. We show also that as a point of instability is reached, the correlation length becomes infinite, in a manner reminiscent of a second order phase transition.

Approximations employed in spontaneous emission theory

Abstract It is shown that the Wigner-Weisskopf, rotating wave, and ladder approximations used in spontaneous emission theory are closely related.

Diffusive Traversal Time of a One-Dimensional Medium

It is shown that the mean time to traverse a medium in a rangea, b fromb toa is given byT= (a−b)2/3D, whereD is the diffusion coefficient.