Path integrals and perturbation theory for stochastic processes

73 Brazilian Journal of Physics, vol. 33, no. 1, March, 2003 Path Integrals and Perturbation Theory for Stochastic Processes Ronald Dickman and Ronaldo Vidigal Departamento de F sica, ICEx, Universidade Federal de Minas Gerais, 30123-970, Belo Horizonte, MG, Brasil Received on 27 August, 2002 We review and extend the formalism introduced by Peliti, that maps a Markov process to a pathintegral representation. After developing the mapping, we apply it to some illustrative examples: the simple decay process, the birth-and-death process, and the Malthus-Verhulst process. In the rst two cases we show how to obtain the exact probability generating function using the path integral. We show how to implement a diagrammatic perturbation theory for processes that do not admit an exact solution. Analysis of a set of coupled Malthus-Verhulst processes on a lattice leads, in the continuum limit, to a eld theory for directed percolation and allied models. 1 Introduction It is often noted that nonequilibrium statistical mechanics lacks the comprehensive formalism of ensembles that has proved so useful in equilibrium. The reason is that equilibrium statistical mechanics treats stationary states for a special class of systems, that possess detailed balance. This allows one to bypass the dynamics, and study stationary properties directly. For systems out of equilibrium, we generally do not have such a shortcut, and must deal with the full dynamical problem, even if our goal is only to obtain stationary properties. In the case of stochastic systems with a discrete state space, the fundamental description is given by the master equation, which governs the evolution of the probability distribution. This class of problems includes a wide range of systems of current interest, that exhibit phase transitions or scale invariance far from equilibrium: driven lattice gases, birth-and-death processes such as directed percolation or the contact process, sandpile models, and interface growth models. One of the more powerful tools for studying stochastic models is a formalism that maps the process to a path-integral representation. This mapping generates an e ective action that can be studied using the tools of equilibrium statistical physics, for example, the renormalization group. Several methods for mapping a stochastic process to an equilibrium-like action have been proposed [1-7]. In this article we review the method developed by Peliti [8], and apply it to some simple stochastic processes. This method has several advantages. With it, one can map any birth-anddeath process to a path-integral representation without ambiguity. In particular, the step of writing a Langevin equation, and of postulating noise autocor- relations, does not arise in this formalism. Thus it provides a direct path from the model of interest to an e ective action, and (in the continuum limit), to its eld theory, without the uncertainties that often attend the speci cation of the noise term [9]. A second advantage, which we explore in detail, is that it leads to a systematic perturbative analysis for Markov processes. The principal aim of this article is to acquaint the reader with the formalism and provide a set of worked examples whose mastery will allow one to apply the method to problems at the frontier of research. While Peliti's article [8] provides an excellent exposition of the mapping, we include, for completeness, a derivation of the central formulas. Our development of the perturbation theory di ers somewhat from Peliti's. Most of the applications discussed are also new. The balance of this article is structured as follows. In Sec. II we derive the path-integral representation, starting from the master equation. Sec. III presents an application to the simple decay process, and expressions for two-time joint probabilities. In Sec. IV we begin our discussion of diagrammatic perturbation theory for the probability generating function, which is illustrated with a pedagogical example. This is extended in Sec. V where we analyze the birth-and-death process using perturbation theory. In Sec. VI a perturbation expansion for moments of the distribution is developed, which turns out to be much simpler than that for the full generating function. This method is applied to the Malthus-Verhulst process in Sec. VII. In Sec. VIII we illustrate another application of the formalism, showing how the path-integral description for a lattice of coupled Malthus-Verhulst processes leads, in the continuum limit, to a eld theory for directed percolation. Sec. IX presents a brief summary. 74 2 Ronald Dickman and Ronaldo Vidigal From the master equation to a path integral In this section we recapitulate Peliti's derivation of the path integral mapping. We consider Markov processes in continuous time, and with a discrete state space n = 0; 1; 2; :::. (We may think of n as the size of a certain population.) The probability pn(t) of state n at time t is governed by the master equation [10, 11, 12]: X X dpn (t) = pn (t) wmn + wnm pm (t) ; dt m m (1) where wmn is the rate for transitions from n to m. (We study stationary stationary Markov processes, i.e., time-independent transition rates.) We now associate a vector jni in a Hilbert space with each state n, and for convenience de ne the inner product so: hmjni = n!Æm;n : (2) The identity may then be written as X 1 = n1! jnihnj : (3) n (All sums run from zero to in nity unless otherwise speci ed.) A probability distribution n (n  0, P n n = 1), may be represented as a linear combination of basis states: X ji = n jni : (4) n The Hilbert space formalism is useful because it provides a simple way to express the evolution in terms of creation () and annihilation (a) operators, which we de ne via: ajni = njn 1i  jni = jn +1i : (5) These relations imply that [a; ] = 1. (Note that while we make use of many pieces of notation familiar from quantum mechanics, there are fundamental di erences. Expected values, for example, are linear, not bilinear, in ji.) The mean population size is given by E[n(t)] = h jaj(t)i ; (6) where X h j  m1! hmj (7) m is the the projection onto all possible states. Central to our analysis will be the probability generating function (PGF), X t (z)  pn(t)zn : (8) n We denote P the PGF corresponding to state ji as (z ) = n n z n . (Note that (1) = 1 by normalization.) Next consider the inner product between states ji and j i: hj i = 1 hjnihnj i = X n!n n : n! n n X (9) We can write this in terms of the corresponding PGFs if we note the identity Z dzz n  d dz m Æ (z ) = n!Æn;m ; (10) which is readily proved, integrating by parts. (Unless otherwise speci ed, all integrals are over the real axis.) Then we have hj i = =  Z  d dz(z ) Æ (z ) dz Z dzdz 0 0 izz0 2 (z) (iz )e ; (11) where we used the integral representation of the Æ function. For birth-and-death processes, it is always possible to write the master equation i (...truncated)