Di usion-Limited Annihilation and the Reunion of Bounded Walkers

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                                                             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
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