Tiger and Rabbits: a single trap and many random walkers

Transcription

1 Physica A 266 (1999) Tiger and Rabbits: a single trap and many random walkers Haim Taitelbaum a;, Zbigniew Koza b, Tomer Yanir a, George H. Weiss c a Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel b Institute of Theoretical Physics, University of Wroc law, Wroc law, Poland c Center for Information Technology, National Institutes of Health, Bethesda, MD 20892, USA Abstract We study a one-dimensional system with a single trap (Tiger) initially located at the origin, and many random-walkers (Rabbits) initially uniformly distributed throughout the innite or the semi-innite space. For a mobile imperfect trap, we study the spatiotemporal properties of the system, such as the trapping rate, the particle distribution and the segregation around the trap, all as a function of the diusivities of both the trap and the walkers. For a static trap, we present results of various measures of segregation, in particular on a few types of disordered chains, such as random local bias elds (the Sinai model) and random transition rates. c 1999 Elsevier Science B.V. All rights reserved. 1. Introduction The standard reaction rate theory for diusion-limited elementary reactions of the type A + B CorA+B B is based on many simplifying assumptions. In Smoluchowski s approach [1] one assumes the B species to be so dilute that the system can be considered as consisting of only a single particle B surrounded by a swarm of diusing particles A. Moreover, the B is assumed to be an immobile sphere acting on the surrounding, freely diusing point-like particles A as a perfect trap (i.e., upon collision of an A with the trap reaction is certain to occur). In recent years it was demonstrated [2 5] that in restricted (i.e., low-dimensional or disordered) geometries, Smoluchowski s theory must be modied to account for the self-segregation Corresponding author /99/$ - see front matter c 1999 Elsevier Science B.V. All rights reserved. PII: S (98)

2 H. Taitelbaum et al. / Physica A 266 (1999) of the reactants. Physical examples of such systems are provided by exciton trapping, quenching or fusion, electron hole and soliton antisoliton recombination, phonon up-conversion and free-radical scavenging. Examples of reactions in conned geometries include quasi-one-dimensional crystals grown inside pores and microcapillaries, polymer chains in dilute blends, catalytic surface reactions, and heterofusion in ultrathin molecular wires, laments and pores [6,7]. In this paper, we investigate a one-dimensional system in which there is a single diusing imperfect trap T ( Tiger ) surrounded by many Brownian particles A ( Rabbits ) initially uniformly lling up the whole available space. Dening the trap absorptivity V as a parameter ranging between perfect trapping and no trapping, we also examine the role played by another of Smoluchowski s assumptions that reaction at the trap is inevitable. In Section 2 we study the spatio-temporal properties of this system, such as the time evolution of the mean concentration of A s at any point x, the local and total rate of trapping, which we denote by R(x; t) and R(t), respectively. In Section 3 we discuss various measures of segregation around the trap. Finally, in Section 4, we examine the segregation around a static trap in the presence of various types of disorder. 2. Mobile trap spatiotemporal properties Consider a one-dimensional system. At time t = 0 a single trap T is placed at x =0, and a swarm of diusing A particles is uniformly distributed on both sides of the trap. The initial concentration of particles A will be denoted by 0, and the diusion constants of the A s and the trap are D A and D T, respectively. Upon contact with the trap T, particles A may react with it and be removed from the system. The motion of the trap is independent of the locations and velocities of particles A. To investigate the evolution of this system, we introduce the conditional concentration of particles A, f(x; t y), dened as the expected number of A s that at time t can be found at site x provided that at this time the mobile trap is located at y. The equation for f(x; t y) takes 2 f = f T V(x y)f ; where V is the trapping rate constant, ranging from 0 (no trapping) to (perfect trapping). The initial condition, f(x; t =0 y)= 0 (y) ; (2) corresponds to the uniform distribution of A s, and the trap located at the origin. Once we have computed f(x; t y), the expected concentration a(x; t) of particles A at (x; t) can be calculated from a(x; t)= f(x; t y)dy: (3)

3 282 H. Taitelbaum et al. / Physica A 266 (1999) Note that although we study a one-dimensional system, our mathematical treatment is carried out in a two-dimensional space, with x and y treated as independent variables and the trapping occurring along the line y = x. We adopt a convention that x always denotes positions of particles A, and y refers to the position of the trap T. In the long-time limit the solutions of (1) and (3) become independent of V and converge to those obtained in the limit V unless V =0. In this limit it is possible to derive in a closed form the long-time asymptotics of f(x; t y) for any values of V, D T and D A. To this end notice that the ratio a(x; t)= 0 is dimensionless and D T =D A, x= D T t, and V t=d T are the only mutually independent dimensionless combinations of x, t, D T, D A and V. Therefore a(x; t) must be of the form a(x; t)= 0 F(D T =D A ;x= D T t;v t=d T ) (4) for which the limits V and t, x= t = const are equivalent. Therefore, to investigate the long-time behavior of the system described by (1) with any nonzero V it suces to concentrate on the limiting case V of the perfect trap. If V, upon contact with the trap, particles A inevitably react. Therefore solving (1) in this limit is reduced to solving the standard 2 f = f 2 (5) with the boundary condition x = y f(x; t y)= 0 (6) imposed by the instantaneous reaction at x = y One-sided initial distribution Because the reaction renders the solutions of (5) non-analytic along the line y = x, it is convenient to solve (5) and (6) with the initial state composed of particles A uniformly distributed only on a half-line x 0, f 1 (x; t =0 y)= 0 (y)h(x) ; (7) where H(x) denotes the Heaviside step function, and we added the index 1 to f to distinguish it from the solution obtained for the full initial state (2), which will be given below. Eqs. (6) and (7) imply x6y f 1 (x; t y)=0; so that (3) can be rewritten as a 1 (x; t)= x f 1 (x; t y)dy; (9) where a 1 (x; t) denotes the concentration of A s at (x; t) for the one-sided initial distribution (7). (8)

4 H. Taitelbaum et al. / Physica A 266 (1999) The solution of (5) (8) is then given by where and f 1 (x; t y)=(x; y; t) (c 1 x +(1 c 1 )y; (1 + c 1 )x c 1 y; t) ; (10) (x; y; t) c 1 D T D A D T + D A : ( ) ( ) 0 y 2 x 4 D T t exp erfc 4D T t 4DA t For D T =0, D A 0, Eq. (10) reduces to ( ) x f 1 (x; t y)= 0 (y) erf H(x) ; (13) 4DA t and for D A =0, D T 0, it becomes f 1 (x; t y)= [ ( ) ( )] 0H(x) y 2 (2x y) 2 exp exp : (14) 4D T t 4D T t 4D T t One can now proceed to the calculation of a 1 (x; t), by inserting the results for f 1 (x; t y) into (9). However, due to the form of the upper limit in (9), a 1 (x; t) can be investigated analytically only when D T = D A, which implies c 1 = 0. In this limit it is also possible to calculate the mean local rate of trapping, R 1 (x; t), which is proportional to the average number of particles A that are being trapped at (x; t). Detailed results can be found in [8] Two-sided initial distribution We can now generalize our results for the initial condition (2) made up of particles A uniformly distributed on both sides of the trap simply by using the principle of superposition, which leads to (11) (12) f(x; t y)=f 1 (x; t y)+f 1 ( x; t y) : (15) This implies that for any values of D A and D T a(x; t)=a 1 (x; t)+a 1 ( x; t) : (16) In Fig. 1 we show the analytical result of a(x; t) for D A =D T = 1, compared with numerical simulations performed in the method described in detail in [8]. An interesting feature of the system is the dependence of a(x; t) at the origin (x = 0) on the values of the diusion constants D T and D A. We nd that [ 1 a(0;t)=2a 1 (0;t)= ( )] D T + D A D T D A arctan D T D A 2 ; (17) D T D A which means that in the long-time limit, or, equivalently, in the limit of perfect trap, the concentration of particles A at the origin is independent of time. It can be seen in

5 284 H. Taitelbaum et al. / Physica A 266 (1999) Fig. 1. The average concentration of particles A for D A =D T =1, 0 =0:8 and L = The solid lines were computed from (16), (9) and (10). Note the constant value of a(x; t) atx =0. As t, it becomes the value of a(x; t) for any x. Fig. 1 that a(x; t)=a(0;t) 1ast, for any x, namely that the mean concentration of particles A at any point x asymptotically goes to a constant value. Another interesting feature of a(0;t) (or a 1 (0;t)) given by (17) is that it drops to 0 only if either D T or D A goes to 0. Moreover, it actually depends only on the ratio D A =D T of the diusion constants, attains the maximal value a 1 (0;t)= 0 ( 1 4 1=2) 0:09 0 for D T = D A, and is not sensitive to interchanging the values of D T and D A. In Fig. 2 we present the semi-log plot of a 1 (0;t)= 0, computed from (17), as a function of D A =D T. 3. Segregation at a single trap A number of recent studies [9 19] analyze the problem of segregation at a single trap, which is the depletion of the particles in the neighborhood of the trap induced by the trapping events. The depletion zone can be characterized by the distance from the trap to a point at which the concentration prole of the diusing particles, a(x; t), reaches an arbitrary fraction (0 1) of its bulk value (which will be referred to as the -distance). This distance, x (t), is dened through the equation a(x (t);t)= 0 ; where 0 is the initial constant concentration of the A particles. Another, more complicated measure of segregation is the average distance between the trap and the nearest unreacted particle (nearest-neighbor distance), which is dened as L(t) = 0 Lf(L; t)dl; (18) (19)

6 H. Taitelbaum et al. / Physica A 266 (1999) Fig. 2. The average relative concentration of particles A at the origin, a 1 (0;t)= 0, as a function of D A =D T, plotted using (17). The maximal value is 1=4 1=2 0:09. where f(l; t) is the probability density function for the distance, L, of the nearest particle to the trap at time t. In low dimensions, these properties have been found to dier signicantly from the classical three-dimensional results. For a static tiger and mobile rabbits in one dimension, the -distance is easily shown to increase asymptotically as t 1=2, which follows directly from the diusion mechanism that controls the reaction, but the nearest-neighbor distance increases asymptotically as t 1=4 [9]. In the opposite case, in which a single mobile trap diuses into a sea of static A particles (The Target problem [20]), the nearest-neighbor distance scales as t 1=2 [10], as opposed to the above-mentioned t 1=4 result. The intermediate, general case of a one-dimensional system in which both the trap T and particles A are mobile is conceptually a simple extension of the Smoluchowski model. However, its rigorous mathematical treatment is very dicult, due to the many-body nature of this problem. Schoonover et al. [11] conjectured that asymptotically with where L(t) t ; = 1 arctan( 1+2D) ; (20) (21) D D T =D A : The form of Eq. (21) has been suggested based on an heuristic analogy with an exponent obtained in a related problem of the survival probability of a single particle A surrounded by two traps T [21]. It is exact for D = 0 and D, yielding =1=4 (22)

7 286 H. Taitelbaum et al. / Physica A 266 (1999) and =1=2, respectively. For other values of D a qualitative agreement with computer simulations has been obtained, but only for a particular initial condition in which particles A were placed on only one side of the trap T. We have performed rened and extensive simulations of the generalized Smoluchowski model in one dimension, carried out with better statistics, much longer times and much larger systems [19], for two dierent initial conditions: the one-side initial condition where particles A are placed on one side of the trap, and the two-side initial state in which A s are distributed on both sides of the trap. We have found that when both the trap and particles A are mobile, the value of is not only non-universal, but also depends on the initial conditions. For the initial condition studied in [11], i.e. in the one-side case, assumes values close or even a little larger than those predicted in Eq. (21). However, in the two-side case the values of become signicantly smaller than those predicted in Eq. (21). The results of our simulations are presented in Fig. 3(a). The estimated asymptotic values of for the one-side and the two-side cases will be denoted as 1 and 2, respectively. The asymptotic value conjectured in [11] (Eq. (21)) will be denoted as. For D = 0 both 1 and 2 go to 1=4, and we obtained 1 =0:252 and 2 =0:253; for D both 1 and 2 converge to 1=2 and we obtained 1 =0:496 and 2 =0:492. In Fig. 3(b) we plot the dierence 1 2 as a function of D=(D + 1). It shows that this dierence goes to 0 only as D 0orD. We have shown that the conjectured form (21) can be used only when particles A are distributed on one side of the trap; in the case where they can be found on both sides, Eq. (21) overestimates the value of by up to 15%, with the error diminishing as D 0orD. We suggest that the sensitivity of to the initial conditions is related to the fact that in each case the major contribution to the ensemble average of the nearest-neighbor distance, L(t), comes from entirely dierent realizations of the system. In the one-side case, the most important contribution comes from systems in which the trap moves in the direction opposite to the location of A s; however, in the two-side case this kind of motion would make the trap diuse deep into a region densely occupied by A s, and so the corresponding nearest-neighbor distance would be very small. The above heuristic argument can be easily extended and used to prove that in general 1 2. It remains a challenge to nd a rigorous relation between 1 and 2 for 0 D. 4. Segregation in the presence of disorder The results described so far pertain to diusion of non-interacting particles in a translationally invariant space. It is therefore of interest to investigate the behavior of the two measures of segregation at a static trap in disordered media, in which diusive properties are anomalous [22,23]. Recent related studies include diusion in a fractal medium [12], long-tailed continuous-time random walk [13,14], Levy processes [17] and a uniform bias eld [15,18].

8 H. Taitelbaum et al. / Physica A 266 (1999) Fig. 3. (a) Comparison of the values of estimated from simulations, 1 for the one-side and 2 for the two-side initial conditions, respectively, with the values predicted in Eq. (21), denoted as. (b) The dierence between 1 and 2 as a function of D=(D + 1). The parameters are t =10 5, L 7000, N = We therefore study the segregation at the single trap on two dierent types of disordered chains. The rst is a chain with local random bias elds (the so-called Sinai model), and the second is the case of random transition rates which represents a weaker type of disorder. These are simplied models that mimic eects of external potentials or internal interactions on the kinetics of the depletion zone. Our study is based on extensive numerical simulations and scaling arguments [15,16]. We start with the case of local random bias elds, which is an example of the so-called Sinai model [24]. In this model, each site i along the linear chain has an associated transition probability of moving to the right P i = 1 2 (1 + E i), where each E i is assigned a value of ±E, (0 E 1), with equal probability 1=2. Sinai proved that

9 288 H. Taitelbaum et al. / Physica A 266 (1999) the rms displacement of a particle diusing in this system increases asymptotically as ln 2 t, which represents a remarkable slowing down as compared to the standard diusion process. This is due to the diculty in moving against local elds induced by stretches of bias with the same sign. The Sinai model has been suggested as being relevant to various physical phenomena. We have studied the segregation on the Sinai chain by extensive numerical simulations [16]. We have found that the average prole P(x; t) is a scaling function of x=ln 2 t, which implies that the -distance scales as ln 2 t, following the Sinai type of diusion of the bulk. The statistical properties of the average nearest-neighbor distance L(t) have been studied using an independent set of Monte-Carlo simulations [16]. We have found that the nearest-neighbor distance exhibits the same asymptotic dependence on time as does the -distance, namely both scale as ln 2 t. This is surprising since it diers from the result in regular diusion where the two measures have dierent scaling behavior as a function of time. We suggest that this has to do with the relation between the nearest-neighbor distance measure and diusive properties of tagged, hard-core particles. Keeping track of the nearest amongst indistinguishable particles, is equivalent to following a tagged particle which cannot pass its successors along the chain. It is well known that the eect of the hard-core interaction in ordinary diusion is to change the asymptotic rms displacement from a t 1=2 to a t 1=4 behavior [25,26], due to mutual interactions between the particles. Therefore, although the t 1=4 result for the nearest-neighbor distance in regular diusion has been established for non-interacting particles, it basically reects a measure which is related to diusion subject to hard-core interaction. In the Sinai model, one can argue that the localization induced by the random elds is so strong that hard-core eects are also negligible. Indeed, Koscielny-Bunde et al. [27] examined in detail the eect of hard-core interaction on the diusion properties of the Sinai model, and found that the leading asymptotic behavior of the rms displacement is the same as for non-interacting particles. Hence, we expect that in those systems where hard-core interaction changes the asymptotic rms displacement of diusion, the behavior of the nearest-neighbor distance and the -distance should be dierent. In order to test this prediction we study the segregation at the single static trap for a diusion process with random transition rates W which are chosen from a slowly decaying distribution having the power-law form P(W ) W ; (06 1). The asymptotic rms displacement of Brownian particles in this system is known to be proportional to t 1=dw, with d w =(2 )=(1 ) [22,23]. Using extensive numerical simulations [16], we found that the average prole near the trap at the origin scales asymptotically like x=t 1=dw, with a corresponding -distance behavior of t 1=dw,as expected. The average nearest-neighbor distance for this case has been studied using Monte-Carlo simulations [16]. Koscielny-Bunde et al. [27] showed that the eect of the hard-core interaction on diusion in a system with such a long-tailed distribution of transition rates is to change the form of the asymptotic rms displacement from a proportionality to t (1 )=(2 ) to a proportionality to t (1 )=(4 3). Indeed, when we t our results for the nearest-neighbor distance to this form, the data clearly agrees

10 H. Taitelbaum et al. / Physica A 266 (1999) Table 1 A summary of the segregation measures at a single trap in one dimension for the cases of regular diusion, random bias elds (Sinai model) and random transition rates Case x (t) L(t) No disorder t 1=2 t 1=4 Random bias elds (Sinai model) ln 2 t ln 2 t Random transition rates t (1 )=(2 ) t (1 )=(4 3) with this prediction [16]. Note that = 0 corresponds to regular diusion, without disorder. Our conclusions about the segregation are summarized in Table 1. The two measures of segregation, the -distance and the nearest-neighbor distance, can increase asymptotically either with the same or with dierent time dependence. This depends on the eect of hard-core interaction on diusion in these systems. If such an interaction changes the leading asymptotic behavior of the rms displacement in that system, the nearest-neighbor distance measure also changes in a corresponding manner. The -distance, however, always scales the same as the standard rms displacement in the system. Acknowledgements Support by the Israel Science Foundation (HT) and the Polish KBN grant no. 2 P03B (ZK) is gratefully acknowledged. References [1] M. von Smoluchowski, Z. Phys. Chem. 92 (1917) 129. [2] A.A. Ovchinnikov, Y.B. Zeldowich, Chem. Phys. 28 (1978) 215. [3] D. Toussaint, F. Wilczek, J. Chem. Phys. 78 (1983) [4] A. Blumen, J. Klafter, G. Zumofen, in: I. Zschokke (Ed.), Optical Spectroscopy of Glasses, Reidel, Dordrecht, 1986, p [5] M. Bramson, J.L. Lebowitz, J. Stat. Phys. 65 (1991) 941. [6] R. Kopelman, Science 241 (1988) [7] S.J. Parus, R. Kopelman, Phys. Rev. B 39 (1989) 889. [8] Z. Koza, H. Taitelbaum, Phys. Rev. E 57 (1998) 237. [9] G.H. Weiss, R. Kopelman, S. Havlin, Phys. Rev. A 39 (1989) 466. [10] D. Ben-Avraham, G.H. Weiss, Phys. Rev. A 39 (1989) [11] R. Schoonover, D. Ben-Avraham, S. Havlin, R. Kopelman, G.H. Weiss, Physica A 171 (1991) 232. [12] S. Havlin, R. Kopelman, R. Schoonover, G.H. Weiss, Phys. Rev. A 43 (1991) [13] G.H. Weiss, S. Havlin, J. Stat. Phys. 63 (1991) [14] G.H. Weiss, J. Masoliver, Physica A 174 (1991) 209. [15] H. Taitelbaum, G.H. Weiss, Mat. Res. Soc. Symp. Proc. 290 (1993) 351. [16] H. Taitelbaum, G.H. Weiss, Phys. Rev. E 50 (1994) [17] G. Zumofen, J. Klafter, Phys. Rev. E 51 (1995) [18] C.A. Condat, G. Sibona, C.E. Budde, Phys. Rev. E 51 (1995) [19] Z. Koza, T. Yanir, H. Taitelbaum, Phys. Rev. E 58 (1998) 6821.

11 290 H. Taitelbaum et al. / Physica A 266 (1999) [20] A. Blumen, G. Zumofen, J. Klafter, Phys. Rev. B 30 (1984) [21] D. Ben-Avraham, J. Chem. Phys. 88 (1988) 941. [22] S. Havlin, D. Ben-Avraham, Adv. Phys. 36 (1987) 695. [23] A. Bunde, S. Havlin, Fractals and Disordered Systems, Springer, Berlin, [24] Ya.G. Sinai, Theory Prob. Appl. 27 (1982) 256. [25] S. Alexander, P. Pincus, Phys. Rev. B 18 (1978) [26] T.E. Harris, J. Appl. Prob. 2 (1965) 323. [27] E. Koscielny-Bunde, A. Bunde, S. Havlin, H.E. Stanley, Phys. Rev. A 37 (1988) 1821.