Diffusion on two-dimensional percolation clusters with multifractal jump probabilities

By means of Monte Carlo simulations we studied the properties of diffusion limited recombination reactions (DLRR's) and random walks on two dimensional incipient percolation clusters with multifractal jump probabilities. We claim that, for these kind of geometric and energetic heterogeneous substrata, the long time behavior of the particle density in a DLRR is determined by a random walk exponent. It is also suggested that the exploration of a random walk is compact. It is considered a general case of intersection ind euclidean dimension of a random fractal of dimension DF and a multifractal distribution of probabilities of dimensionsDq (q real), where the two dimensional incipient percolation clusters with multifractal jump probabilities are particular examples. We argue that the object formed by this intersection is a multifractal of dimensionsD'q=Dq+DF-d, for a finite interval ofq.


I. Introduction
Recently, fractals and multifractals [1 a] which appear in many physical applications have attracted a growing interest (Refs. 1 5 and references cited therein).For example, it is known [-6 and references therein] that the surface of most solids at the molecular scale is fractal.Consequently, many physico-chemical properties and processes related to such systems have to be carefully examinated.In this context, diffusion and reaction studies on disordered systems have recently been reported [-7-123.Major interest arises because a random walk on fractal media exhibits anomalous long-time behavior which implies that the reaction order of a diffusion limited recombination reaction (henceforth DLRR) is also anomalous in a low concentration regime [9][10][11][12] and this fact has been experimentally verified [9,11].* Researcher of the Consejo Nacional de Investigaciones Cientificas y T6cnicas (CONICET), Argentina On the other hand, the study of the properties of random walks on square lattices with multifractal distribution of jumping probabilities [13], which constitutes an interesting open problem, has recently been initiated.
The purpose of this work is to study, by means of the Monte Carlo simulations, the behavior of random walkers and DLRR's on two dimensional percolation clusters with multifractal jump probabilities.These substrata can be considered as the "intersection" of two dimensional incipient percolation clusters (i.e.geometric heterogeneous objects) and planar multifractal distributions of jumping probabilities (i.e.energetic heterogeneous objects), where the concept of "intersection" has to be interpreted in a restricted manner as it is discussed in Sect.II.As it is shown in Sect.III, these substrata are a new kind of multifractals, called percolation multifractals (PM), which have a non trivial spatial fractal dimension.The PM's combine both geometric and energetic heterogeneities which are strongly complex and to the best of our knowledge, this is the first model introduced in order to study the diffusion on substrata with these properties.Explicitly, our study is mainly focused on the computation of random walk exponents and the exponent related to the time behaviour of the particle density in a DLRR and to analyse the relation between them.Our hope is that this investigation could help in the comprehension of complex recombination reactions which occur on fractal catalysts with geometric and energetic heterogeneities.
It is also presented a conjecture about the set of dimensions of the objects formed by the intersection of a random geometric fractal and a multifractal distribution of probabilities.
The paper is organized as follows: In Sect.II it is defined the PM model.In Sect.III the theoretical arguments to obtain the conjecture about the intersection of fractals and multifractals are discussed.In Sect.IV it is presented the theory about random walkers and DLRR's on multifractal structures.In Sect.V the Monte Carlo simulation is described.Finally in Sect.VI the discussion of the results and the conclusion are stated.

II. The PM model
Let us start with the definition of planar multifractals which have recently been proposed by Meakin [-1 e, 13] and will be used in this work.The multifractal distribution of probabilities on square lattices of size L x L, with L = 2", are constructed as follows.Four normalized probabilities Pz (i = 1 .... ,4), are selected.In the first step, these probabilities are randomly assigned to the four quadrants of linear size ll=L/2, of the lattice (see Fig. 1).
In the second step, each quadrant is divided into four smaller quadrants of linear size 12 =L/22 (in the m-th step, lm=L/2m), and the probability associated to each quadrant prior to the division is multiplied by /)1, P2, P3 and P4 in random order.This procedure is continued and after n-generations each lattice site B is associated to a probability #B of the form #, = eSl pS2 pS~ pS~, (1) with $1 + S: + $3 + $4 = n.In the limit n-~ oo a multifractal distribution of probabilities on the two dimensional space is obtained (see below).Also, one can assume that #,=exp(-E,/kT), (2) where EB is the activation energy of diffusion at the site B, k the Boltzmann constant and T the temperature.Equation ( 2) is consistent with the jumping probability of a random walk defined in Sect.IV (see (29)).
A multifractal is characterized by an infinite set of dimensions Dq (q real) [-3, 5].For the planar multifractals used in this work, these dimensions can be defined by 22m E #q~(lm/L)(q-1)Dq (3) S=I where the sum runs over all the quadrants S of linear size Ira, and/~s is the probability (or measure) associated to the S th quadrant.This measure is given by #s = ~ #B, B where the sum runs over all the sites of the S th quadrant.This implies that in the limit lm/L--*O, the number % of measures behaving as (for more details see [5]) where e=d[(q-1)Dql/dq and f(cQ=qc~-(q-1)Dq.This means that the system consists of pieces with fractal dimension f(~) and power law singularities e. Explicitly, in our case, from relation (3) one obtains Dq = in (P~ + P~ + P~ + P~) The percolation model has been extensively studied (see for example [-14-16] and references therein) in the field of geometric phase transitions, and its definition and properties will neither be presented nor discussed here.But let us recall that the incipient percolation cluster in two dimensions is a geometric fractal structure with fractal dimension Dv=91/48 [17].Let us stress that, in general, for a geometric fractal of dimension DF one can assume that #c = const if the site C belongs to a fractal, and/~c--0 otherwise.Then using the appropriate version of relation ( 3) one obtains Dq = D v for all q.In this sense a geometric fractal is a "trivial" multifractal.
Let us finally define the PM model as the intersection of an incipient percolation cluster on a square lattice with a planar multifractal of probabilities #B defined above.Note that both, the percolation clusters and the multifractals are confined in 2 D planes and the intersection considered here is restricted to the case where the angle 5 between these planes is zero (cases with 6 4=0 which cause the formation of 1 D objects are not considered).So, in the PM model the probability/~ associated to a site B is given by C#~, if the site B belongs #~ = to the incipient percolation cluster (7) 0, otherwise where C is the normalization constant obtained by demanding that the following equation holds B where the sum runs over all B sites of the square lattice (note that ~/~B = 1).

B
Let us stress that the explicit value of C does not affect at all the jumping probability of random walkers (see ( 29)), but we impose the condition (8) in order to deal with normalized probability distributions as used in multifractals (as it will be shown in Sect.III, the PM is a multifractal in the limit of very large square lattices).
Due to the intersection, the PM is a structure which combines both, the geometric (fractal) heterogeneity of the percolation cluster and the energetic heterogeneity (see (2)) of the planar multifractal.It can also be thought of as a dilute planar multifractal.

III. The intersection of fractals and multifractals
Let us consider a general case (in which the PM is a special one) of intersection, in a space of d euclidean dimensions, of a geometric random fractal structure of dimension DF with a multifractal (normalized) probability distribution, both with overall linear size R and a short cut off r0, where the restricted interpretation of the concept of intersection has to be remembered.
At the scale r, with r o__<r~R (strictly speaking in the limit fiR+O), the fractal is composed by (r/R) -DF boxes of linear size r and the multifractal has boxes of measure ~~(r/R)=.(10) The intersection of two fractal objects of dimensions Dr and f respectively, gives a new ffactal of dimension f', where [18,193 f, ~f+Dv-d, if f+Dv-d>O Before the intersection, the measures are normalized The intersection is defined following the same procedure as in the PM.That is, at the short cut off ro the measure #B becomes #~ = Co #B if the box B belongs to the fractal; and #~ = 0, otherwise, where C o is the normalization constant.Consequently, at the scale r > r o one has with the same C value for all c(, and the measures #' remain normalized Assuming f+ Dr-d > O, one obtains from ( 9) and ( 11)
As usual, small values of f correspond to large values of lql, so (20) is only valid for qmin < q < q .... where the values qmin and qmax depend on each particular intersection.In summary, we claim that in d euclidean dimensions the equation (20) holds for the intersection of a random geometric fractal of dimension D e with a multifractaI probability distribution of dimensions Dq, at least for a finite interval of q.In this sense, the intersection gives a new rnultifractal structure.Let us note that for both, the case q=0 (D O is the fractal dimension of the support of the measures) and the case of a trivial multifractal (that is Dq= D, for all q), the Eq. ( 20) is verified because it corresponds to the intersection of two fractal objects [18,19].
For the special case of PM's, D'q can be defined through (3) but replacing #s by #} (#s = ~ #B, where the B sum runs over all sites of the S th quadrant).Then is the so-called information dimension [3].On the other hand, for a planar multifractal one has (see ( 6)) q--*X q ln2 i=1 Finally, Eq. ( 20) implies where a' and a are obtained from ( 23) and ( 26) respectively.For q = 2 one has Di = D2 + ~-2, ( where D~ and D2 are obtained from ( 21) and ( 6) with q = 2, respectively.As we will see below (in Sect.VI) the Monte Carlo results suggest that ( 27) and (28) are fulfilled.

IV. Random walkers and recombination reactions on multifractal structures
Meakin [13] has studied the properties of random walkers on planar multifractals.According to this work, we have assumed that the probability PBc of a random walk at the site B with measure/.t~to jump into a randomly chosen nearest-neighbour site C with measure #b is In the computation of the time t only the jumping attempts to sites with no null measures are considered (see (7)).The average number SN of distinct sites visited by a random walker after N steps is expected to behave, for large N, as where ~1(0<~<1) is the random walk exponent I/ (for geometric fractals, q =g/2, for g<2, where ~ is the spectral dimension related to the density of states for scalar harmonic excitations of the fractal [20,21]).
The visitative efficiency e of a random walker is defined through the derivative of SN with respect to time e=dSN/dt.
(31) For DLRR's between A particles; where the products are removed from the substratum; the reaction rate may be written as where p is the density of A particles.This relation is only rigorous for the two body approximations (i.e. using the concept of relative diffusion between two walkers [9][10][11]) and for the substrata used in this work, the validity of the above mentioned relation is the first hypothesis.Note that t is proportional to the number of jumping attempts whereas N is the number of successful jumping events.Then, one could assume (the second hypothesis) that in average for large N. Let us stress that in a multifractal the jumping probability strongly depends on the spatial region of the substratum, then the relation (34) is not trivial at all (note that for some substrats, N,-~ t ~, with fl=# 1 [22]).From (30), ( 31), ( 33) and (34) one obtains in the low density regime (i.e. in the limit t ~ oo) In summary, the conjecture is that in multifractals the diffusion limited recombination reaction exponent tl (35) is the same as the random walk exponent 11 (30).
On the other hand, the behaviour for large N of the average square distance R~ from the origin of the walk is characterized by the random walk exponent v.That is R~.,~N 2~. (36) Furthermore, as Do is the fractal dimension of the support of the measure, one can see that the number of points within a region of radius RN behaves, for large N, as R~v ~ Then from (36) one gets where ZN is the number of accessible sites for a random walker after N steps.Now assuming that the random walk explores all the accessible space [21,23] (i.e.SN ~ ZN), one obtains from (30) and (37) that where the case q = 1 means that the number SN of distinct sites visited by a random walker cannot exceed the number of N steps.This compact exploration assumption holds in fractals [21,23] and it will be analysed for the multifractals used in this work.

V. The Monte Carlo simulation
Monte Carlo simulations are performed using L x L quare lattices with L = 28= 256.The planar multifractals are generated by using where Q(0< Q < 1) is a free parameter [13].For obtaining PM's only clusters which percolate in both directions of the square lattice (i.e. they have both their width and their length equal to L) at the critical probability Pc = 0.5927 (see for example [24,25]) are selected.Free (periodic) boundary conditions for the random walkers (DLRR's, respectively) are used.
For a simple random walk (using the jumping probability (29), SN and RN (see ( 30) and ( 36)) are computed as a function of N and these results are averaged over a large number of walks (see Figs. 3 and 4).
To simulate the DLRR's the sites with no null measures are covered by A particles at random.After that the diffusion starts.The probability of a randomly selected A particle to jump into a randomly chosen nearest-neighbour site with a no null measure is defined as for random walkers (see (29)).When two A particles are at the same substratum site as a consequence of the jumps, both particles are removed from the substratum (successful recombination event).The time t is defined as t=0.05M, where M is the number of jumping attempts per particle (i.e. the number of successful jumping events plus the failed ones).The density p of A particles against t (see (35)) is obtained and averaged over many simulation reactions (see Figs. 3 and 4).In Fig. 2 we have a plot of yl versus ln2-" and ln-ln plot of Y2 versus 2 -m for PM's with Q=0.5.For this case, and using ( 6) and ( 26

VI. Discussion of the results and conclusion
Comparing both Yl and Y2 with the two straight lines of slopes 1.54 and D~-1 (with D~= 1.30) respectively, in Fig. 2, one can see that the Monte Carlo results are in agreement with ( 27) and (28).Then the relation ( 20) is verified for PM's in the limit q ~ 1 and for q = 2.

VI.2. The Exponent r I
Figures 3 and 4 show plots of SN versus N and p versus t obtained for PM's with Q=0.75 and Q=0.50, respectively.The slopes of these curves strongly suggest that the random walk and the DLRR exponents t/ (see (30) and ( 35)) are the same.This conclusion agrees with the results obtained working with other PM's as well as with planar multifractals (see Table 1).
Based on all these evidences, we claim that for the multifractals studied in this work, the random walk exponent tl (30) and the diffusion limited recombination reaction exponent rl (35) are the same.Then, the simulation of DLRR's is an alternative method to obtain the random walk exponent q.Furthermore, in DLRR's t/ is related to the reaction order X= 1 + l/t/ [-9-12, 26], which can be experimentally determined.
Table 1.The exponents q and v.The error bars are of about __+5% (a), + 10% (b), and they do not take into account any possible corrections due to finite size effects.For PM substrata, the ~/values were obtained considering both, random walk and DLRR simulation results.For planar multifractals the r/values correspond to DLRR simulations only [26].The square L x L lattice size is L=256.~ and ~' are the values previously published by Meakin [13] for random walks, assuming that SN behaves as N r or (N/In N) r respectively; and on square lattices of size L = 1024.Note the remarkable agreement with the results obtained from DLRR's even when the used lattice sizes were different  1.Let us note that: i) on an incipient percolation cluster the visitation is compact [-21, 23]; ii)the fractal dimension Do of PM substrata is the same as that of the incipient percolation cluster; (DF) and iii)the obtained exponents (see (Table 1) for MP's (with Q < 1) are smaller than the v value (v = 0.352) for the percolation cluster.This implies that for the number N of steps, the distance R N travelled by the random walk on a PM structure is smaller than that over a percolation cluster.Therefore, one expects that the exploration would also be compact on PM substrata.Moreover, the results shown in Table 1 agree,, within the error bars, with ~ = vD0, and Meakin [133 has shown that the visitation is compact on planar multifractals.Summing up, from these evidences it seems that the compact exploration assumption holds for the multifractals studied in this work.
We would like to express our gratitude to Hans Herrmann who has introduced us in the fascinating world of multifractals.H.O.M. would like to thank Professor Abdus Salam, The International Atomic Energy Agency and UNESCO for kind hospitality at the International Centre for Theoretical Physics, Trieste, where the manuscript was prepared.H.O.M. wishes to acknowledge discussions with Garani Ananthakrishna and Indrani Bose.
Fig. I a, b.An example of construction of a planar multifractal.a first step; b second step
Figures 3 and 4 also show plots of R 2 versus N. The v exponents obtained fi'om these figures and for the case Q =0.25 are presented in Table