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128 c Brazilian Journal of Physics, vol. 30, no. 1, Maro, 2000 Di usion-Limited Annihilation and the Reunion of Bounded Walkers M. J. de Oliveira sica Instituto de F a Universidade de S~o Paulo Caixa Postal 66318 a 05315-970 S~o Paulo, SP, Brazil Received 15 December 1999 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 Introduction 5,10,13,16-22,30,32,34-36 as well as in general dimen- sion 3,8,11,12,14,23,25,28 . Some works concentrated The study of reaction-di usion processes has been the on the case of multimolecular reactions in one dimen- subject of much interest in the last two decades 1-40 . sion 24,26,27,29,33 as well as in general dimension In a reaction-di usion system the reactants are trans- 7,31 . ported by di usion. These systems have two charac- The classical rate equation, or mean- eld equation, teristic time scales: the reaction time and the di usion for the density of particles is given by time. When the reaction time is much larger than the d = ,a k di usion time the process is called reaction-limited. In dt 1 this case most of the time the reactants are performing di usion so that the whole process is limited by reac- where a a positive constant. From this equation it fol- tion. The kinetics is dominated by di usion which im- lows that the density of particles decreases asymptoti- plies that it is well described by the laws of mass-action cally according to the power law or mean- eld equations. t,1=k,1 2 When the di usion time is much larger than the reaction time the process is called di usion-limited. In This mean- eld behavior is valid for dimensions d this case the reactions take place in a very short time so greater than a critical dimension dc given by that the whole process is limited by di usion. For low 2 dc = k , 1 3 dimensions, the process is dominated by uctuations and the kinetics is no longer described by mean- eld For d dc it is conjectured that equations. t,d=2 4 Here we are interested in the limiting case of di usion-limited processes in which the reactions take For d = dc the mean- eld result is expected to have place instantaneously. Moreover, we will consider only logarithm corrections. the case of one-species annihilation and coalescence pro- The results 3 and 4 have been conjectured by cesses. More speci cally, one considers a one-species means of scaling arguments 7,8 , exact results in one process in which particles react only when a certain dimension 5,10,13,15-17,19,20,21,27 , renormalization number k of them meet, kA ! `A, with ` k. The group calculations 12,23,25,28,31 , and probabilistic extinction of particles by reaction may be total ` = 0, approaches 3,16,17 . The main purpose of the present annihilation or partial ` 6= 0, coalescence. In any article is to derive the results 2, 3 and 4 by map- case, the density of particles t vanishes in the long- ping the reaction-di usion process, in its late stages, time limit. Such systems have been studied for the into the problem of nding the time it takes for a group case of bimolecular reactions k = 2 in one dimension of random walkers, con ned in a limited space, to meet. M.J. de Oliveira 129 II Model and mean- eld a di usion solution i ; j ! i , 1; j +1 i 6= 0 j 6= k , 1 5 i ; j ! i +1; j , 1 j 6= 0 i 6= k , 1 6 In this section we consider a mean- eld approach to the di usion-limited annihilation of particles in a lat- b extinction of particles tice. We will see that, although the approach does not i; k , 1 ! i , 1; ` i 6= 0 7 provide us with equation 1, it gives other rate equa- tions from which one derives the expected asymptotic k , 1; j ! `; j , 1 j 6= 0 8 time behavior of the density given by equation 2. When a site is occupied by k particles, k , ` of them Consider a d-dimensional hypercubic lattice in disappear instantaneously and the site becomes occu- which particles di use over the sites. To each site one pied by just ` particles. In the following we will set up associates a variable i that takes the values 0, 1, 2, ..., and solve the mean- eld equations. For convenience we k , 1 according whether the site is empty, occupied by will consider only the annihilation case, ` = 0. just one particle, two particles, ..., k , 1 particles. At Let us de ne Pn t as the probability that a given each time step, a pair of nearest-neighbor sites is cho- site has n particles at time t. According to the rules sen at random and their state changes according to the above we nd the following time evolution for this prob- following rules: ability c X d P = f k,1 P k,1;m + k ,1X k,1 P1;m , P0;m g X 9 dt 0 m=1 m=0 m=1 and d P = f k,1 P X n,1;m k,1 X k,1 + Pn+1;m , Pn;m , Pn;m g X k,1 X 10 dt n m=1 m=0 m=1 m=0 for 0 n k , 1, with the condition Pk;m = 0, where Pnm t is the probability that a nearest neighbor pair of sites have n and m particles at time t, and is a constant. These equations can be written in the form d P = P , P + P , P , P 11 dt 0 k,1 k,1;0 1 0 00 and d P = P , P dt n n,1 n,1;0 + Pn+1 , Pn , Pn;0 , Pn 12 for 0 n k , 1, with the condition Pk;m = 0. d These equations are exact but cannot be solved by the solution themselves since we need the time evolution equation Pn = t,n=k,1 15 for the two body correlation Pnm. To solve them we use for n = 1; 2; :::; k , 1. In expression 15, only the dom- a truncation scheme which consists in using the approx- inant term is presented. Terms of order smaller than imation Pn;m = PnPm on the right had site of 11 and the dominant are neglected. 12. The equations then become closed in the variables The density of particles is given by Pn d P = P , P 1 , P + P dt 0 k,1 0 0 1 13 = X nPn k,1 16 and n=1 d P = P , P 1 , P + P , P 14 and has the asymptotic behavior dt n n,1 n 0 n+1 n = t,1=k,1 17 for 0 n k , 1, with the condition Pk = 0. These equations can be solved in the long time regime with as expected. 130 c Brazilian Journal of Physics, vol. 30, no. 1, Maro, 2000 III The reunion of bounded di using in a hypercubic space of dimension D = walkers dk , 1 which we may think as the direct product of k , 1 subspaces of dimension d. The projection of the It is convenient to de ne two useful quantities. The solitary walker trajectory over each of the subspaces rst is the average time interval between two consecu- gives the trajectory of each of the k , 1 walkers. The tive reactions, denoted by . Since just after a reaction problem is then reduced to nding the time it takes for the number of particles is reduced by an amount k , `, this solitary walker to reach the origin. it follows that is related to the decreasing rate of the We will consider next a random walk on a hyper- density by cubic lattice of dimension D with periodic boundary d , dt = k , ` 1 18 conditions and N = LD distinct lattice points. The walker starts from a given point x0 6= 0 and at each The other quantity, denoted by L, is related to the mean time step the walker jumps to one of the 2D nearest distance between particles, or more precisely, it is the neighbor sites with equal probability. To calculate the size of a hypercubic cell containing k particles, on the average time to reach the origin we let the origin be an average. If the lattice is partitioned into hypercubic absorbing point. Let Px; t be the probability that the cells of linear size L with k particles each, on the aver- walker be at site x at time t. Its time evolution obeys age, it follows that L is related to the density of particles the equation by 1 = k Ld 19 d Px; t = Xfwx + Px + ; t , wxPx; tg If a relation between and L is found then this re- dt lation together with equations 18 and 19 will allow 20 us to obtain the density as a function of time. where wx is the rate of jumping to a neighboring site The relation between and L can be obtained as fol- and the summation is over the 2D nearest neighbor lows. Consider a typical hypercubic cell of linear size L sites of site x. The rate wx = , a nonzero constant, in the d-dimensional space where the particles, or ran- for x 6= 0 and w0 = 0 since the origin is an absorbing dom walkers, are di using according to the rules 5, site. 6, 7, and 8. This typical cell will have k walkers The probability that the walker be at the origin and which are assumed to be bounded to move inside this remain forever there at time t is P0; t. For a nite cell. An estimate of the quantity will be the time lattice, in any dimension, P 0; t ! 1 as t ! 1 since it will take for these k walkers to meet, starting, for x = 0 is an absorbing state. In this case, the average time to reach the origin will be nite and is given by instance, far away from each other. Consider k random walkers con ned on a region of = Z 1 d t dt P0; tdt 21 linear size L of a d-dimensional lattice. The walkers perform independent Brownian motion and eventually 0 meet. Assuming translational invariance it su ces to This formula can be understood observing that the consider the movement of the walkers relative to one of probability that the walker reach the origin between the walkers which we place at the origin. The problem t and t + t is Pt + t , Pt dP=dtt. bx; z = Z 1 Px; te,ztdt then becomes equivalent to nding the time it takes Using the Laplace transform for the other k , 1 walkers to meet at the origin. The relation between and is = Ld . P 22 Since the walkers move independently of each other 0 the kinetics can be reduced to just one random walker equation 20 becomes c b z P x; z , Px; 0 = Xfwx + Pbx + ; z , wxPbx; zg 23 d M.J. de Oliveira 131 where Px; 0 = x; x0 since at time t = 0 the walker = 2 L2 lnL D=2 34 is at position x = x0. We de ne next the Fourier transform of P x; zb = CD LD given by e Xb where the constant CD is given by D 2 35 P q; z = P x; zeiqx x 24 where the summation is over the sites of the hypercubic 1Z 1 CD = 2D q dD q 36 lattice, and q is a vector belonging to the rst Brillouin zone. From equation 23 it follows that For D 2, the dominant contribution comes from small e + iqx0 b P q; z = z e q + z + q P 0; z 25 q q. Using these results with D = dk , 1 and equations 18 and 19, and recalling that = Ld , we obtain where q = 2 X D 1 , cos qi 26 the following asymptotic behavior for the density 2 i=1 t1=k,1 d k , 1 37 Summing the right and left hand sides of equation 25 over q and taking into account that 1 ln t d=2 d = k , 1 b 1 Xe P x; z = N P q; zeiqx 27 t 2 38 q t,d=2 d k , 1 39 where the summation in q is over the rst Brillouin zone, we get IV Conclusion b z P 0; z = Gx0; z zG0; 28 We have studied the long time behavior of a one-species where 1 Gx; z = N X eiqx 29 reaction-di usion process in a hypercubic lattice where a speci ed number of particles coalesce into a smaller q z + q number of particles. The asymptotic behavior of the density of particle was obtained by mapping the pro- is the lattice Green function. b From the Laplace transform P 0; z we can obtain cess into the problem of the reunion of random walkers that are con ned to move in a limited region. the probability P 0; t and from it the average time b . However, it is possible to calculate directly from P 0; z. Indeed, from 21 it follows that References b = lim f, d z P 0; z g z!0 dz 30 1 Ya. B. Zeldovich and A. A. Ovchinnikov, Sov. Phys. JETP 47, 829 1978. For small values of z, the Green function 29 behaves 2 A. A. Ovchinnikov and Ya. B. Zeldovich, Chem. Phys. as 1 Gx; z = Nz f1 + z X eiqx g 31 28, 215 1978. 3 M. Bramson and D. Gri eath, Z. Wahrsch. verw. Ge- q q6=0 biete 53, 183 1980; Ann. Prob. 8, 183 1980. as long as N is nite. Owing to equation 28, we get 4 A. S. Mikhailov, Phys. Lett. A 85, 214, 427 1981. nally the following result for the rst-passage time 41 5 D. C. Torney and H. M. McConnell, J. Phys. 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