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Path Integral Approach to Birth-Death Processes on a Lattice for Memory Effects [22]

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Path Integral Approach to Birth-Death Processes on a Lattice for Memory Effects [22] J. Physique 46 (1985) 1469-1483 SEPTEMBRE 1985, 1469 Classification Physics Abstracts 05.40 - 68.70 - 02.50 Path integral approach to birth-death processes on a lattice L. Peliti Dipartimento di Fisica, Universita « La Sapienza », Piazzale Aldo Moro 2, I-00185 Roma and Gruppo Nazionale di Struttura della Materia, Unità di Roma, Roma, Italy (Reçu le 8 mars 1985, accepté le 22 mai 1985) Résumé. 2014 On donne une formulation par intégrales de chemin du formalisme de Fock pour objets classiques, premièrement introduit par Doi, et on l’applique à des processus généraux de naissance et mort sur réseau. L’intro- duction de variables auxiliaires permet de donner une forme Markovienne aux lois d’évolution des chemins aléa- toires avec mémoire et des processus irréversibles d’agrégation. Les théories des champs existantes pour ces pro- cessus sont obtenues dans la limite continue. On discute brièvement des implications de cette méthode pour leur comportement asymptotique. Abstract 2014 The Fock space formalism for classical objects first introduced by Doi is cast in a path integral form and applied to general birth-death processes on a lattice. The introduction of suitable auxiliary variables allows one to formulate random walks with memory and irreversible aggregation processes in a Markovian way, which is treatable in this formalism. Existing field theories of such processes are recovered in the continuum limit. Impli- cations of the method for their asymptotic behaviour are briefly discussed. the which has a renormalizable 1. Introduction. process, perturbation theory around some upper critical dimension D,. A certain number of models of irreversible aggregation For the case of random walks with memory such processes, which lead to the formation of objects field theories have been introduced on a heuristic possessing some sort of scale invariance, have been basis [12, 13, 16]. On the other hand, field theories introduced in the literature since the proposal of describing some form of aggregation processes or Witten and Sander [1]. One of the main reasons of chemical reactions with diffusion have been from time interest appears to be the hope of understanding the to time introduced and discussed in the literature origin of the scale invariance of many natural objects [17-21]. The treatment has been extended to account by widening the scope of methods, like the renor- for memory effects [22]. malization group, which have proven their validity We wish here to bring together these field theories in the description of scale invariance in critical and show how one can systematically take memory phenomena. effects into account, on the basis of the Fock space This program does not yet appear to have reached formalism for classical objects first introduced by its target While the number of model increases, most Doi [23], and later reformulated by Rose [24] and investigations still essentially rely on numerical simu- Grassberger and Scheunert [25]. The method bears lation [2, 3]. Sometimes arguments which link the strong similarity to the Martin-Siggia-Rose formalism evolution equations which define the model to the for classical evolution [26], which has been cast into geometrical properties of the outcoming aggregate a path integral form by De Dominicis [27] and Janssen have appeared [4-8]. Methods which are strictly [28]. It is also related to the Poisson representation related to the renormalization group have met with of birth-death processes introduced by Gardiner and some success in the particular case of growing linear Chaturvedi [29], on which basis Janssen [20] has been aggregates, which may be produced by processes able to give a path integral formulation of some described as random walks with memory [9-13]. chemical reaction models with diffusion. In this case also Flory-like arguments have found some Although such methods have from time to time applicability [14, 15]. One of the reasons of success appeared in literature, we think it worthwhile to has been the introduction of a field theory describing explain them in a slightly different way, which closely Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019850046090146900 1470 follows the treatment of Bargmann-Fock path integrals The factors w(n’ -+ n) are the transition rates from the in standard texts on Quantum Field Theory (see state with n’ particles to the state with n particles. In e.g. [30]). One can thus most clearly derive the exact a process without memory (as we shall suppose definition of the path integral and its properties. throughout this section) they can only depend on One can stress the flexibility of the formalism by a n’, n (and possibly on time). derivation which does not unnecessarily reduce the The macroscopic state identified by { Pn } may be symmetry existent among the various fields which are considered as an element of a real vector space K introduced Some cases which allow for an exact formed by linear combinations of the base states n > treatment are thus most easily spotted and handled corresponding to states with exactly n particles : It is also convenient to treat in some detail the trick which allows one to transform an aggregation process with memory into a process obeying a Mar- kovian evolution via the introduction of equation The space may be given a Hilbert structure by intro- « bookkeeping variables ». This trick has been sug- ducing the scalar product gested by Grassberger [31 ] and has wide applicability. We show how it leads to a sound derivation of field theories of aggregation processes with memory which have appeared in the literature. This is the « exclusive » scalar product in the ter- Once the field theories have been defined, a renor- minology of Grassberger and Scheunert [25]. These malization procedure may be set up to discuss the authors also introduce a different scalar product, asymptotic behaviour of the model. This procedure which they name « inclusive ». This product leads may succeed or fail depending on the availability of to some simplifications in the formalism, which we a small parameter (like the E = 4 - D of static critical shall discuss later. We refer to the appendix for a brief phenomena) which allows one to bring the fixed discussion of the properties of the inclusive scalar points of the renormalization group within the range product. of applicability of perturbation theory. We shall not We can thus introduce the annihilation operator a pursue in detail the renormalization group treatment, satisfying which is rather standard once the correct field theory is identified We shall however mention the reasons leading to difficulties in applying this approach to some popular aggregation models. and the creation operator x defined by We review in section 2 the derivation of the path integral formalism for classical, zero-dimensional, birth-death processes. Some simple examples and the perturbation theory for such processes are discussed With the scalar product (2.3) one may easily check in section 3. The treatment of birth-death processes that x is the Hermitian conjugate of a. The operators without memory on a lattice is sketched in section 4. a, x have the usual commutation relation In section 5 the trick of the « bookkeeping variables » for the treatment of processes with memory is intro- duced and some inferences on the asymptotic beha- viour of a few interesting processes are drawn. While A look at equations (2. 4)-(2. 5) suggests to represent section 6 contains a short conclusion, the discussion je as a Hilbert space of (real) analytic functions of of a technical detail is dealt with in an appendix. the variable z by means of the correspondence 2. Path integrals for birth-death processes. with real Taylor coefficients 0.- We shall however We start by considering zero-dimensional birth-death allow z to take on complex values. processes whose microscopic states are completely The function O(z) associated with the state p ) described by the number n of particles present at any is of course nothing else as the generating function, time t. well known in the theory of birth-death processes Its macroscopic state is therefore described by the [32, 33]. We are thus providing the space of generating probability 0,,(t) of having exactly n particles at time t. functions with a Hilbert space structure. This probability evolves according to a master Let us remark that the scalar product between two equation of the form : arbitrary vectors 14> ), 10 > is given by 1471 By the use of the identity ([33], p. 280) : The kernel of a product AB of operators is given by : which one can easily check by integrating by parts, The kernel of any operator A is easily computable if A we express (2.8) in terms of the generating functions is first expressed as a normal product, i.e. as a poly- O(z), t/J(z) as follows : nomial in a, n with all creation operators on the left of all destruction operators. It is always possible to express any operator in such a form. Given the normal product expression of A : The use of the usual integral representation of the delta function allows finally to express the scalar product the normal kernel 0 of A is defined as (2.8) as an integral : A(z, simply where z takes the place of x and ( of a ; and the kernel A(z, 0 of A is given by The integral over z’ runs over the whole real axis; that over z over any path which passes through zero. A(z, C) = ez’ A(z, C) - (2.19) We shall always understand that they are both taken over the whole real axis. The derivation of equations (2.16)-(2.19) is the same The evolution equation (2.1) may be symbolically as in quantum mechanics and may be found, e.g., in written as follows : reference [30].
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