Diffusion-limited annihilation and the reunion of bounded walkers

Abstract

We study the long time behavior of a one-species reaction-di usion process kA !`A where k particles coalesce into`particles. The asymptotic time behavior of the density of particles is derived by mapping the reaction-di usion process into the problem of the reunion of k random walkers bounded to move in a limited region. I IntroductionThe study of reaction-di usion processes has been the subject of much i n terest in the last two decades 1-40 . In a reaction-di usion system the reactants are transported by di usion. These systems have t wo c haracteristic time scales: the reaction time and the di usion time. When the reaction time is much larger than the di usion time the process is called reaction-limited. In this case most of the time the reactants are performing di usion so that the whole process is limited by reaction. The kinetics is dominated by di usion which implies that it is well described by the laws of mass-action or mean-eld equations.When the di usion time is much larger than the reaction time the process is called di usion-limited. In this case the reactions take place in a very short time so that the whole process is limited by di usion. For low dimensions, the process is dominated by uctuations and the kinetics is no longer described by mean-eld equations.Here we are interested in the limiting case of di usion-limited processes in which the reactions take place instantaneously. Moreover, we will consider only the case of one-species annihilation and coalescence processes. More speci cally, one considers a one-species process in which particles react only when a certain number k of them meet, kA !`A, with` k . The extinction of particles by reaction may be total `= 0 , annihilation or partial `6 = 0, coalescence. In any case, the density of particles t v anishes in the longtime limit. Such systems have been studied for the case of bimolecular reactions k = 2 in one dimension 5,10,13,[16][17][18][19][20][21][22]30,32,[34][35][36] as well as in general dimension 3, 8,11,12,14,23,25,28 The results 3 and 4 have been conjectured by means of scaling arguments 7,8 , exact results in one dimension 5,10,13,15-17,19,20,21,27 , renormalization group calculations 12,23,25,28,31 , and probabilistic approaches 3,16,17 . The main purpose of the present article is to derive the results 2, 3 and 4 by mapping the reaction-di usion process, in its late stages, into the problem of nding the time it takes for a group of random walkers, con ned in a limited space, to meet.

Tagged particle dynamics in one dimensional ${A+ A \to kA}$ models with the particles biased to diffuse towards their nearest neighbour

“…The simplest form of particle reaction is when a certain number l of the particles meet: lA → kA with k < l. It is well known that annihilating random walkers with l = 2 and k = 0 corresponds to the Ising-Glauber kinetics while the coalescing case with l = 2 and k = 1 describes the dynamics of the q state Potts model with q → ∞, both at zero temperature and in one dimension [5]. Such systems have been studied in one dimension [6][7][8][9][10][11][12][13] as well as in higher dimensions [14][15][16][17]. Depending on the initial condition, whether one starts with even or odd number of particles, the steady state will contain no particles or one particle respectively.…”

A+A→∅model with a bias towards nearest neighbor

“…Diffusion controlled annihilation problems have received lots of attention over the years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. These are non-equilibrium systems of diffusing particles, which undergo reactions such as pairwise annihilation.…”

Effect of bias in a reaction-diffusion system in two dimensions

“…It is to be noted that in Eqs. (12) and (13) we have used the exact form of ρ(t) for L → ∞. Finally, the persistence probability P (t) is studied; for small ε ′ it shows a fast decay and goes to zero.…”