Summary of a paper. Gardiner, C. W., & Chaturvedi, S. (1977). The Poisson Representation. I. A New Technique for Chemical Master Equations. Journal of Statistical Physics, 17(6), 429–468. (original). Rights: © publisher — not reproduced. This is a short account of it in our own words; the work itself is not reproduced here.

The Poisson Representation. I. A New Technique for Chemical Master Equations

What this covers

This paper introduces a technique for solving chemical master equations — the exact stochastic description of molecule-number fluctuations in reacting systems — by expanding the probability distribution as a (possibly complex-valued) superposition of Poisson distributions. It speaks to nonequilibrium statistical mechanics and stochastic chemical kinetics.

The question

By the mid-1970s, fluctuations in chemical reactions were handled either heuristically, via Langevin equations whose noise strength was assumed Gaussian and fixed by equilibrium fluctuation-dissipation arguments, or exactly but awkwardly, via the master equation itself, attacked by van Kampen's system-size expansion or cumulant methods. None of the exact routes were easy to use in practice, especially with several reacting and diffusing species, and the Langevin route was not obviously valid away from equilibrium. The authors wanted a method converting a master equation into a Fokker–Planck or stochastic differential equation exactly, not as an approximate by-product of a large-volume limit, and tractable for multivariable, spatially extended systems.

The approach

They represent the probability distribution over molecule numbers as an integral of multivariate Poisson distributions weighted by a function \(f\) of continuous auxiliary variables — the "Poisson representation." Substituting this ansatz into the master equation and integrating by parts turns it into a Fokker–Planck equation for \(f\) with no approximation: reactions of at most bimolecular order give drift and diffusion terms only, and general reactions give derivatives of some higher but finite order. The device that makes this work is that \(f\) is a quasiprobability: it can be negative or complex, and its natural integration domain can be a contour in the complex plane rather than the real line, yet its moments equal exactly the factorial moments of the true molecule-number distribution. The authors first show, from statistical mechanics alone, that ideal reacting systems in thermodynamic equilibrium (grand canonical ensemble) always have exactly Poissonian steady states, which motivates the Poisson basis generally. Where the Fokker–Planck equation is second order, it is converted, via the Itô formalism, into an equivalent stochastic differential equation solvable iteratively as a power series in the inverse square root of system volume, giving a systematic asymptotic expansion of moments. For reactions with trimolecular steps, which produce higher-order Fokker–Planck equations, they extend this with a new "higher-order noise" source — apparently not attempted before, and tolerable only because quasiprobabilities need not stay positive.

What it found

Ideal reacting systems at thermodynamic equilibrium have exactly Poissonian steady-state distributions, recovering the law of mass action along the way. Any reversible reaction with a single pathway, and any reaction obeying detailed balance, also reaches a Poissonian steady state, and the authors give an example where this holds even without detailed balance, with a net probability flux still present. For purely bimolecular reactions, the exact Fokker–Planck noise term vanishes whenever the deterministic rate equations reach steady state, again forcing a Poissonian limit. Worked examples — a linear reversible reaction, a bimolecular second-order phase transition model, a reaction with a negative diffusion coefficient, and a trimolecular model — reproduce results obtained previously by other methods (agreeing with McNeil and Walls, and with Mori and McNeil's critical dimension of \(4\) for a reaction-diffusion phase transition) and extend them with exact moment formulas in confluent hypergeometric and Bessel functions. Diffusion between spatial cells, being linear, contributes no noise of its own: all stochasticity comes from the chemical step, unlike conventional Langevin treatments where diffusion also adds noise. The Gaussian (linearized) approximation holds almost everywhere except near critical points; for the spatial phase-transition model it stays valid for dimension \(d>4\) even arbitrarily close to the critical point, but breaks down for \(d<4\), where corrections to the concentration variance diverge as cell size shrinks. Higher-order noise, worked out for the trimolecular example, affects the skewness but contributes negligibly to the mean and variance at leading orders in the volume expansion.

Limits and context

The authors flag the higher-order noise construction as heuristic, introduced because no prior theory existed for it, and say giving it a rigorous footing is work for mathematicians, not done here. The continuum limit of their cell-based treatment of spatial diffusion gives a second-order correction to the mean concentration that diverges as cell size shrinks for spatial dimension below four (logarithmically at exactly four); they call this "disturbing," trace it to the artificial locality built into the cell model, and state that fixing it needs a more microscopic, nonlocal account of diffusive reaction that they do not supply. They also note the integration contour defining the quasiprobability is not always unique, that the wrong choice gives a finite but inadmissible solution, and that picking the right one needs a separate analyticity argument, carried out case by case in an appendix rather than as a general rule. The method is restricted to master equations with finitely separated states and polynomial transition rates. The paper argues against treating Langevin-style Gaussian noise and the system-size expansion as the general or sufficient tools for nonequilibrium reaction fluctuations, positioning its exact Fokker–Planck derivation as a replacement rather than a supplement. A companion paper, Part II, is said to treat two-time correlation functions, outside this paper's scope.

Citation

Gardiner, C. W., & Chaturvedi, S. (1977). The Poisson Representation. I. A New Technique for Chemical Master Equations. Journal of Statistical Physics, 17(6), 429–468. https://doi.org/10.1007/BF01014349