Exact Solutions of Low-Dimensional Reaction-Diffusion Systems
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a r X i v :c o n d -m a t /966224v11J ul1996International Journal of Modern Physics B,f c World Scientific Publishing Company EXACT SOLUTIONS OF LOW-DIMENSIONAL REACTION-DIFFUSION SYSTEMS Ant´o nio M.R.Cadilhe,wrence Glasser and Vladimir Privman Department of Physics,Clarkson University Potsdam,New York 13699-5820,USA We briefly review some common diffusion-limited reactions with emphasis on results for two-species reactions with anisotropic hopping.Our review also covers single-species reactions.The scope is that of providing reference and general discussion rather than de-tails of methods and results.Recent exact results for a two-species model with anisotropic hopping and with ‘sticky’interaction of like particles,obtained by a novel method which allows exact solution of certain single-species and two-species reactions,are discussed.Keywords :diffusion-limited reactions,exact solutions,hopping anisotropy 1.Introduction There has been much interest in the study of nonequilibrium systems through repre-sentation by simple stochastic models.1-3In particular,low-dimensional diffusion-limited reactions are tractable and in many cases,exactly solvable,examples of nonequilibrium systems.Since the early works of Ovchinnikov and Zeldovich,4Tou-ssaint and Wilczek,5and Torney and McConnell 6who pointed out the importance of fluctuations of particle density in the kinetics of low-dimensional reactions,a host of results have been published.1-69Despite their simplicity,common diffusion-limited reactions can describe complex phenomena.However,only recently attention has been turned to the effect of biased diffusion on these reactions.7-13,21Here,we review some common diffusion-limited reactions with emphasis on the effects of bi-ased diffusion in two-species reactions.Single-species reactions will also be briefly reviewed.The asymptotic diffusion-limited reaction behavior is associated with diffusion being the slowest,‘rate-determining’process in the system.For dimensions lower than the upper critical dimension,a mean-field theory using the classical,Smolu-chowski-type rate equations—see Kang and Redner 14—breaks down since it does not take into account effects of spatial inhomogeneities of particle concentration.Specifically,the classical mean-field theory predicts,in one dimension (1d ),for the kinetics of single-species reactions—A +A →A and A +A →∅(inert)—an asymptotic decay of the concentration proportional to t −1instead of the exact result t −1/2.10-13,15-44That diffusion is the mechanism leading to anomalously slow kinetics can be illustrated for the annihilation reaction,A +A →∅,on a one-dimensional (1d )lattice,as follows.After a time t ,a particle will diffuse over a12 A.M.R.Cadilhe,M.L.Glasser and V.Privman√distance of orderℓ=Exact solutions of low-dimensional reaction-diffusion systems3 reported in the literature.27,37-39Next,we focus on analytical results for single-and two-species reactions.2.Some common diffusion-limited reactionsThe kinetics of diffusing-particle single-species1d systems reacting via the processes of coalescence(A+A→A)or annihilation(A+A→∅)on particle encounters, is now well understood.For these two basic processes the concentration decays for large times according to C(t)∼t−1/2(in1d).In general,C(t)∼t−d/2for d<2,with a logarithmic correction factor at d=d c=2,and C(t)∼t−1for d>2.5,10,15-17,19-26,31,35,36Here d c denotes the upper critical dimension.The steady state,assuming additional processes,e.g.,particle creation or back-reaction, has also been studied by several authors.16,26,28,30,36In particular,R´a cz23obtained exact results by mapping interfaces of the kinetic Glauber-Ising model to particles in the A+A↔∅system.Amar and Family42pointed out that the Ising-model to reaction-dynamics mapping should be used with caution owing to the presence of subtle correlations in the initial conditions for non-zero values of the magnetization. Multiparticle kA→∅reactions,and other variants of single-species reactions have also attracted attention.12,18,32,54,55For k>3,a mean-field approximation applies in1d.For the borderline case of k=3,Monte-Carlo simulations43and refined mean-field treatment56have recently succeeded in demonstrating the logarithmic correction in the concentration decay.The kinetic behavior of two-species reactions is far richer than that of the single-species reactions.Three two-species models belonging to different universal-ity classes are discussed below.Although there exist other variants of two-species models,33,55,57-64we restrict the present survey to these few examples.In the ‘standard’AB model particles hop isotropically on a d-dimensional lattice to any of their neighboring sites.On encounters,AB particle pairs annihilate while for AA and BB pairs the two particles‘bounce off’each other.This models a hard-core interaction whereby each site can be occupied by at most one particle.The‘standard’AB model shows the non-mean-field concentration decay C(t)∼t−d/4for d<4and the mean-field decay C(t)∼t−1for d>d c=4,for equal con-centrations of the two species.5,35,45,46,55,65-67For unequal(initial)concentrations,√the concentration of the minority species decays exponentially as C(t)∼exp(−λ,compared to134 A.M.R.Cadilhe,M.L.Glasser and V.PrivmanAB model,in1d.Regarding the spatial domain structure,there is only a qualita-tive agreement in the literature,8,9including phenomenological considerations and numerical simulations.3.Exactly solvable‘sticky’two-species annihilation modelA third variant of the AB model in1d has been introduced recently by Kra-pivsky.50,51,68In this system same-species particles interact by sticking together. Same-species particle clusters coalesce on encounters—no hard-core interaction is present—and they form a larger cluster with size given by the total number of par-ticles in the merging clusters.When two clusters of different-species particles meet AB-annihilation takes place.The resulting cluster has particle number given by the particle-number difference of the reacting clusters,while the species is that of the‘parent’cluster with the larger number of particles.Numerical simulations and scaling69suggest d c=2.Synchronous-updating,cellular-automaton-like,‘sticky-particle’lattice dynam-ics models have been considered recently.Exact results have been obtained for ‘sticky’diffusion-limited single-species,22and single-and two-species10,11reactions. They have been treated by a novel approach which yields a unified exact solution of the coagulation and annihilation single-species reactions and the‘sticky’two-species reaction defined above.10The approach combines and extends several techniques de-veloped in charge-coagulation model studies70-72and earlier single-species reaction studies21,22of cellular-automaton-type models.A combinatorial argument is then utilized to relate the‘cluster history’at time t to the set of compatible initial config-urations(at t=0).51Exact solvability for the two-species reaction is based on the linearity of certain dynamical evolution equations,supplemented by a mathematical development yielding a closed form for a non-trivial double-sum.10 For the single-species reactions the expected power-law decay with exponent 1is not affected4here by the hopping-rate bias as it is for the‘driven’hard-core AB model.For unequal initial concentration of the two species,the asymptotic approach to the constant concentration difference is t−3/2.This clearly contrasts with the expo-nential dependence of the‘standard’AB model.Thus,for the multiple-occupancy, sticky-particle‘bosonic’model the effects of the hopping bias are not fundamentallyExact solutions of low-dimensional reaction-diffusion systems5 important.In fact,in the large-time asymptotic regime the changes due to bias can be fully absorbed in the diffusion constant value.This behavior is quite different from that of the single-occupancy,hard-core‘fermionic’models.To summarize,we have reviewed some common diffusion-limited reactions with emphasis on exact results and our recent exact solution of the‘sticky’two-species model.Many open problems challenge researchers in thisfield.The few exact results available yield insight on the physical problem and provide a guide for approximate analytical and numerical work.We have emphasized the differences between the three two-species reactions considered which actually belong to different universality classes.References1.T.Liggett,Interacting Particle Systems,(Springer-Verlag,New York,1985).2.V.Privman,Trends Stat.Phys.1,89(1994).3.Nonequilibrium Statistical Mechanics in One Dimension,ed.V.Privman(CambridgeUniversity Press,Cambridge),to be published.4. 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