Path-integral formulation of stochastic processes for exclusive particle systems

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.

Kinetics of the bosonic A+B-->0 reaction with on-site attractive interaction

We investigate kinetics of the uniformly driven bosonic A+B-->0 reaction with on-site attractive interaction in one dimension. In this model, n(i)(lambda) particles from a site with n(i) particles are driven to the right. When particles of opposite species occupy the same site, the reaction takes place instantaneously. Since n(i)(lambda)<n(i) for lambda<1, lambda controls the on-site attractive interaction between like particles. The lambda=0 case corresponds to the hard-core (HC) particle model. With equal initial densities of both species, we numerically confirm that the scaling behaviors of density and lengths are the same as those of the uniformly driven HC particle system. Especially the domain length l satisfies the power law l approximately t(2/3). The kinetics of the reaction is independent of lambda as long as lambda<1. The lambda -independent kinetics results from the lambda -independent collective motions of single species domains.

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.

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

We present the path integral representation of the generating function for classical exclusive particle systems. By introducing hard-core bosonic creation and annihilation operators and appropriate commutation relations, we construct the Fock space structure. Using the state vector, the generating function is defined and the master equation of the system is transformed into the equation for the generating function. Finally, the solution of the linear equation for the generating function is derived in the form of the path integral. Applying the formalism, the equivalence of reaction-diffusion processes of single species and two species is described.

Probing universality classes in solid-on-solid deposition

We consider several stochastic processes corresponding to the same physical solid-on-solid deposition problem. Simplified models presenting the same (conditional) mean and variance for each process are also introduced as well as generalizations in terms of the deposition of blobs and probabilistic deposition rules. We compare the evolution of the roughness as a function of time for a three-parameter family that includes as limit cases the Family model and the Edwards-Wilkinson equation, showing that in all cases the derived models with the same mean and variance are indistinguishable from the originating models in terms of the evolution of the roughness. Finally, we show that although all the models studied belong to the same universality class, some relevant features such as the final surface roughness are reproduced only for models within a restricted class determined by sharing the same (conditional) mean and variance.

Universality classification of restricted solid-on-solid type surface growth models

We consider the restricted solid-on-solid (RSOS) type surface growth models and classify them into dynamic universality classes according to their symmetry and conservation law. Four groups of RSOS-type microscopic models--asymmetric (A), asymmetric-conserved (AC), symmetric (S), and symmetric-conserved (SC) groups--are introduced and the corresponding stochastic differential equations (SDEs) are derived. Analyzing these SDEs using dynamic renormalization group theory, we confirm the previous results that A-RSOS, AC-RSOS, and S-RSOS groups belong to the Kardar-Parisi-Zhang class, the Villain-Lai-Das Sarma class, and the Edwards-Wilkinson class, respectively. We also find that SC-RSOS group belongs to a new universality class featuring the conserved-cubic nonlinearity.

Universality class of the restricted solid-on-solid model with hopping

We study the restricted solid-on-solid (RSOS) model with finite hopping distance l(0), using both analytical and numerical methods. Analytically, we use the hard-core bosonic field theory developed by the authors [Phys. Rev. E 62, 7642 (2000)] and derive the Villain-Lai-Das Sarma (VLD) equation for the l(0)=infinity case, which corresponds to the conserved RSOS (CRSOS) model and the Kardar-Parisi-Zhang (KPZ) equation for all finite values of l(0). Consequently, we find that the CRSOS model belongs to the VLD universality class and that the RSOS models with any finite hopping distance belong to the KPZ universality class. There is no phase transition at a certain finite hopping distance contrary to the previous result. We confirm the analytic results using the Monte Carlo simulations for several values of the finite hopping distance.

Derivation of continuum stochastic equations for discrete growth models

We present a formalism to derive the stochastic differential equations (SDEs) for several solid-on-solid growth models. Our formalism begins with a mapping of the microscopic dynamics of growth models onto the particle systems with reactions and diffusion. We then write the master equations for these corresponding particle systems and find the SDEs for the particle densities. Finally, by connecting the particle densities with the growth heights, we derive the SDEs for the height variables. Applying this formalism to discrete growth models, we find the Edwards-Wilkinson equation for the symmetric body-centered solid-on-solid (BCSOS) model, the Kardar-Parisi-Zhang equation for the asymmetric BCSOS model and the generalized restricted solid-on-solid (RSOS) model, and the Villain-Lai-Das Sarma equation for the conserved RSOS model. In addition to the consistent forms of equations for growth models, we also obtain the coefficients associated with the SDEs.

Two-point correlation functions of the diffusion-limited annihilation in one dimension

Two-point density-density correlation functions for the diffusive binary reaction system A+A--> phi are obtained in one dimension via Monte Carlo simulation. The long-time behavior of these correlation functions clearly deviates from that of a recent analytical prediction of Bares and Mobilia [Phys. Rev. Lett. 83, 5214 (1999)]. An alternative expression for the asymptotic behavior is conjectured from numerical data.

Field theory for reaction-diffusion processes with hard-core particles

We show how to build up a systematic bosonic field theory for a general reaction-diffusion process involving hard-core particles in arbitrary dimension. We discuss a recent approach proposed by Park, Kim, and Park [Phys. Rev. E 62, 7642 (2000)]. As a test bench for our method, we show how to recover the equivalence between asymmetric diffusion of excluding particles and the noisy Burgers equation.