Generating function, path integral representation, and equivalence for stochastic exclusive particle systems

One-dimensional annihilating random walk with long-range interaction

We study the annihilating random walk with long-range interaction in one dimension. Each particle performs random walks on a one-dimensional ring in such a way that the probability of hopping toward the nearest particle is W=[1-ɛ(x+μ)^{-σ}]/2 (the probability of moving away from its nearest particle is 1-W), where x is the distance from the hopping particle to its nearest particle and ɛ, μ, and σ are parameters. For positive (negative) ɛ, a particle is effectively repulsed (attracted) by its nearest particle and each hopping is generally biased. On encounter, two particles are immediately removed from the system. We first study the survival probability and the mean spreading behaves in the long-time limit if there are only two particles in the beginning. Then we study how the density decays to zero if all sites are occupied at the outset. We find that the asymptotic behaviors are classified by seven categories: (i) σ>1 or ɛ=0, (ii) σ=1 and 2ɛ>1, (iii) σ=1 and 2ɛ=1, (iv) σ=1 and 2ɛ<1, (v) σ<1 and ɛ>0, (vi) σ=0 and ɛ<0, and (vii) 0<σ<1 and ɛ<0. The asymptotic behaviors in each category are universal in the sense that μ (and sometimes ɛ) cannot affect the asymptotic behaviors.

Master equations and the theory of stochastic path integrals

This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon expanding the forward and the backward path integrals around stationary paths, we then discuss and extend a recent method for the computation of rare event probabilities. Besides, we also derive path integral representations for processes with continuous state spaces whose forward and backward master equations admit Kramers-Moyal expansions. A truncation of the backward expansion at the level of a diffusion approximation recovers a classic path integral representation of the (backward) Fokker-Planck equation. One can rewrite this path integral in terms of an Onsager-Machlup function and, for purely diffusive Brownian motion, it simplifies to the path integral of Wiener. To make this review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory and do not assume any knowledge of the latter. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit.

Irreversible nA+mB→0 reaction of driven hard-core particles

We investigate the kinetics of general two species annihilation nA+mB→0 of driven hard-core (HC) particles with N=n+m in one dimension. With uniform drift velocity, all particles are driven to the right. HC exclusion forbids the interchange of any particles and restricts the number of particles on a site to 0 or 1. The reaction is classified into two classes, the symmetric and the asymmetric reaction. The symmetric reaction means both nA+mB→0 and mA+nB→0 , while the asymmetric reaction means only nA+mB→0 for a given (n,m) pair. As N increases, the trains of particles causing the reaction rarely form. Hence, for sufficiently large N, particles are evenly distributed before the reaction, so one expects a crossover N(c) above which the kinetics follows the classical mean-field rate equation. We show the existence of N(c) and that the kinetics for N<N(c) is the same as that of A+B→0 of driven HC particles as in the reactions with the isotropic diffusion. However, compared to the isotropic cases, N(c) and the kinetics for N≥N(c) are shown to be completely changed by the interplay of the drift and HC exclusion, and strongly depend on the reaction symmetry. We also show that densities decay as t(-1/N) which cannot be explained by the classical mean-field rate equation. Instead the kinetics is explained analytically by a variant theory.

Mechanism of murine epidermal maintenance: cell division and the voter model

The dynamics of a genetically labeled cell population may be used to infer the laws of cell division in mammalian tissue. Recently, we showed that in mouse tail skin, where proliferating cells are confined to a two-dimensional layer, cells proliferate and differentiate according to a simple stochastic model of cell division involving just one type of proliferating cell that may divide both symmetrically and asymmetrically. Curiously, these simple rules provide excellent predictions of the cell population dynamics without having to address the cells' spatial distribution. Yet, if the spatial behavior of cells is addressed by allowing cells to diffuse at random, one deduces that density fluctuations destroy tissue confluence, implying some hidden degree of spatial regulation of cell division. To infer the mechanism of spatial regulation, we consider a two-dimensional model of cell fate that preserves the overall population dynamics. By identifying the resulting behavior with a three-species variation of the voter model, we predict that proliferating cells in the basal layer should cluster. Analysis of empirical correlations of cells stained for proliferation activity confirms that the expected clustering behavior is indeed seen in nature. As well as explaining how cells maintain a uniform two-dimensional density, these findings present an interesting experimental example of voter-model statistics in biology.

Nontrivial critical crossover between directed percolation models: effect of infinitely many absorbing states

At nonequilibrium phase transitions into absorbing (trapped) states, it is well known that the directed percolation (DP) critical scaling is shared by two classes of models with a single (S) absorbing state and with infinitely many (IM) absorbing states. We study the crossover behavior in one dimension, arising from a considerable reduction of the number of absorbing states (typically from the IM-type to the S -type DP models) by following two different (excitatory or inhibitory) routes which make the auxiliary field density abruptly jump at the crossover. Along the excitatory route, the system becomes overly activated even for an infinitesimal perturbation and its crossover becomes discontinuous. Along the inhibitory route, we find a continuous crossover with universal crossover exponent phi approximately=1.78(6), which is argued to be equal to nu||, the relaxation time exponent of the DP universality class on a general footing. This conjecture is also confirmed in the case of the directed Ising (parity-conserving) class. Finally, we discuss the effect of diffusion on the IM-type models and suggest an argument why diffusive models with some hybrid-type reactions should belong to the DP class.

Monte Carlo simulations of bosonic reaction-diffusion systems

An efficient Monte Carlo simulation method for bosonic reaction-diffusion systems which are mainly used in the renormalization group (RG) study is proposed. Using this method, one-dimensional bosonic single species annihilation model is studied and, in turn, the results are compared with RG calculations. The numerical data are consistent with RG predictions. As a second application, a bosonic variant of the pair contact process with diffusion (PCPD) is simulated and shown to share the critical behavior with the PCPD. The invariance under the Galilean transformation of this boson model is also checked and discussion about the invariance in conjunction with other models are in order.