7023
J . Phys. Chem. 1989, 93, 7023-7026
Random Walks in Liquids Michael F. Shlesinger* Physics Division, Office of Naval Research, 800 North Quincy Street. Arlington, Virginia 2221 7-5000
and Joseph Klafter School of Chemistry, Tel-Aviu University. Ramat-Aviu. Tel-Aviv, Israel 69978 (Received: February 17, 1989)
An extension of the Montroll-Weiss continuous-time random walk is presented which can incorporate multistage transport processes. One such example would be the successive detrapping, ballistic, Brownian, trapping sequence in the motion of excess electrons in fluids as suggested by Kunhardt. In any stage, coupled space-time memories can be employed to account for dissipative processes. For example, a single-stage process with a memory reflecting Kolmogorov scaling will recover Richardson’s law of turbulent diffusion.
I. Introduction The history of random walks1 goes back at least as far as Jacob Bernoulli’s posthumously published (17 13) Ars Conjectandi (The Art of Conjecture) where he calculated the duration of a game of chance. The progress of the game could be represented as a random walk on a lattice between two absorbing barriers. Reaching a barrier meant one of the players had lost all of his money and the termination of the game. This constituted thefirst, first passage time calculation. Jacob’s nephew DanielZdeveloped a random flight picture, with collisions, for the first kinetic theory of gases. Although this work contained the seed of the modern theory of gases, it was dismissed, in part, because the existence of atoms was not yet confirmed and the notion that random events underlie physical laws seemed preposterous.’ Since the work of the Bernoulli’s, random walks have appeared in many theories of transport in fluids, be they laminar, turbulent, glassy, or polymeric. We cite a few of these examples, before we introduce our own work in the next section. Einstein: in 1905, showed that the Brownian motion of a large particle in a fluid obeyed the diffusion equation. Actually, many of the key ideas of Brownian motion were first given in the 1900 doctoral thesis of L. Batchelier5 (a Poincar6 student), who derived the then surprising result that probability radiated according to Fourier’s heat conduction law. Debye: in 1913, analyzed the rotational Brownian motion of large polar molecules of radius R in a fluid of viscosity 9 at temperature T . Random walking fluid particles hit the polar molecules, induce rotational Brownian motion, and cause the dipole correlation to decay exponentially, with time constant 4 r R 3 9 / k B T .In glassy liquids, a recent model of relaxation,’ based on an analogue of Debye’s ideas, has fractal-time random walking defects hitting frozen-in dipoles. The result is the ubiquitous stretched exponential relaxation law appearing as a probability limit distribution. de Gennes* has used a random walk model with trapping to model the flow of a liquid through a porous medium. A distribution of trapping times reflects the time spent in excursions into blind alleys. Bouchard and Georgesg have further extended this line of investigation to show (1) Todhunter, I. History of the Mathematical Theory of Probability; (Cambridge University Press: Cambridge, 1965. A Chelsea Reprint, 1949. (2) Brush, S . G. Statistical Physics and the Atomic Theory of Matter, From Boyle and Newton to Landau and Omager; Princeton University Press: Princeton, NJ, 1983. (3) Bernoulli, D. Hydrodynamica, sive de viribus et motibusfluidorum commentarii, Strasbourg, 1738. (4) Einstein, A. Ann. Phys. 1905, 17, 549. (5) Batchelier, L. Ann. Sci. de Pecole Normale Suppl. 1900, No. 101 7 , Suppl. 3. Translated to English in: Cootner, P. The Random Nature ofthe Stock Market; MIT Press: Cambridge, MA, 1964. (6) Debye, P. Polar Molecules; Dover Press: New York, 1949. (7) Shlesinger, M.; Montroll, E. W. Proc. Natl. Acad. Sci. U.S.A. 1984, 81, 1280. Klafter, J.; Shlesinger, M. F. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 848. Shlesinger, M. F. Annu. Rev. Phys. Chem. 1988, 39, 269. (8) de Gennes, P. G. J . Fluid Mech. 1983, 136, 189.
0022-3654/89/2093-7023$01.50/0
that the longitudinal diffusion of a tracer particle depends linearly on its average velocity, or its square, depending on whether the dispersion is dominated by spatial fluctuations of the velocity field or by holdup mechanisms. Weisslo has also used a random walk model with transport and trapping states for a study of flow in chromatographic columns. Zwanzig” has contributed significantly to the theory of transport in liquids through the elegant use of projection operator techniques to derive the equations of motion for the particles based on the underlying dynamics of the system. Goldsteinlz introduced correlated random walks where the walker has a tendency to continue in the direction it is moving. He was able from this starting point to derive the telegrapher’s equation. This has an advantage over the diffusion equation in that propagation occurs with a finite velocity, while in diffusion the probability is nonzero that a particle is arbitrarily far away from its origin even after an infinitesimal time. This difference can be important in modeling systems such as the spread of pollution in the oceans. Taylor” discussed correlated random walks of fluid particles which could yield a mean-square displacement ( R 2 ( t ) ) t2. More recently,14 Levy’s scale invariant random walks were used to provide equations to describe diffusion in a fully developed turbulent flow where ( R 2 ( t ) ) t 3 . The focus of this paper is to set up the equations to study multistage transport in fluids such as can occur in the motion of excess electrons in fluids where trapped, ballistic, and Brownian stages of motion can occur in sequence.
-
-
11. Random Walks with Instantaneous Jumps A very clear treatment of random walks, based on Green’s function techniques, was given by Montroll and Weiss.I5 This process, known as a continuous-time random walk (CTRW), allows a walker to wait a random time at a site before it instantaneously moves to a new site. No information about how it moves between sites is considered. The waiting time probability density function $ ( t ) to remain at a site for a time t , and the jump probability p ( R ) for the jump to go a distance R , are the only input (except for initial and boundary conditions) needed to calculate all quantities. The probability density Q(R,t) to reach a site R exactly at time t satisfies the Green’s function e q ~ a t i o n ~ ~ ~ ~ ~
(9) Bouchaud, J. P.; Georges, A. C. R. Acad. Sci., Ser. 2 1988, 307, 1431. (10) Weiss, G. H. On a Generalized Transport Equation for Chromatographic Systems, in Transport and Relaxation in Random Systems; Klafter, J., Rubin, R. J., Shlesinger, M. R., a s . ; World Scientific: Singapore, 1986. (1 1) Zwanzig, R. W. In Lectures in Theoretical Physics; Brittin, W. E., Ed.; Wiley: New York, 1961. (12) Goldstein, S. Q. J . Mech. Appl. Math. 1950, 4, 129. (13) Taylor, G. I. Proc. London Math. Sci., Ser. 2 1921, 20, 196. (14) Shlesinger, M. F.; West, B. J.; Klafter, J. Phys. Rev. Lett. 1987, 58, 1100. Klafter, J.; Blumen, A.; Shlesinger, M. F. Phys. Rev. A 1987, 35, 3081. Shlesinger, M. F.; Klafter, J.; Wong, Y. M. J . Stat. Phys. 1982, 27, 499. (15) Montroll, E. W.; Weiss, G. H. J. Math. Phys. 1965, 6, 167.
0 1989 American Chemical Society
7024 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
Shlesinger and Klafter
which takes the form in Fourier-Laplace space of
0
Note that the probability P(R,t) to be at a site R at time t differs from Q(R,t). It is given by P(R,t) = J‘Q(R,t-7) W ( T )d r
(3) 0
where W ( T )= .f:+(t) dt. The W term takes into account that the walker reached site R at the earlier time, t - T , and then waits at R without leaving for a time longer than T . The mean-square displacement of the walker is given by (when ( R ( t ) )= 0) ( R 2 ( t ) )= Z R 2 P ( R , t )= - 1 - I
.
Figure 1. Dots represent the set of points visited by a random walker. In the CTRW the walker can remain at a site for a random time. After that it instantaneously moves to a new site. Only the dots are visited.
dzP(k,s)/ak21k,o
=-LI a2Q( k , s ) / d k 2 W(s)II;=o
where -&-I
is the inverse Laplace transform.
111. Multistage Random Walks A . The Propagator. As before, let the walker remain trapped at a site for a time governed by a waiting time distribution +(t). When the walker leaves the trap, let it undergo a stochastic process for a random time until it is trapped again. We will now explicitly keep track of the particle, while in the last section only the probability of being at trapping sites was calculated. Let 4(R,t) be the probability density that the walker has a lifetime t before it is trapped again and that it experiences a total displacement of R in this time t. We can write +(R,t)in two equivalent ways
4(RJ) = p(Rlt) 4(t)
(5)
4(R,t) = 4(tlR) P ( R )
(6)
Figure 2. A three-stage random walk schematic is shown, where the dots represent trapping sites, the straight lines represent ballistic motion, and the wavy lines represent Brownian motion. We have chosen the trapping sites to be the same as in Figure 1. In the text we have calculated the probability for the random walker to be anywhere, Le., at trapping or nontrapping sites. For our treatment of turbulent diffusion the dots would be turning points and only one type of trajectory would exist. The velocity, however, would depend on the distance between turning points.
If upon detrapping the walker undergoes a sequence of n random processes governed by probability densities 4i(R,t), i = 1, ..., n, then
or where $ ( t ) is the probability density that the transport process lasts for a time t before the walker is retrapped, p ( R ) is the probability that the trajectory between two successive trap sites has a relative displacement of R , p(Rlt) is the conditional probability of having a trajectory with displacement R , given that the time spent is t , and 4(tlR) is the conditional probability density that the walk took a time t , given that it covered a displacement of R . Note that
C4(R,t)= $ ( r ) , R
since C p ( R J t )= 1 R
and
J 4 ( R , t ) dt = p ( R ) , since I @ ( t l R dt ) = 1 The equation for Q(R,t)to reach a trapping site R exactly at time t is
Q(R,t) = ~ A ‘ J ‘ Q ( R - R ’ , I - T )~ ( T - T ’ ) ~ ( R ’ , T d7’ / ) d7 + 6(R) 6 ( t )
Note that Q(R,t)satisfies the Markovian chain Green’s function equation, given in eq 7, and it is the propagator for this stochastic process. Kunhardt” has modeled the motion of excess electrons in liquids, in applied fields, and suggested that the case n = 2 is needed for a description of their motion. After detrapping, the electron moves ballistically until sufficient energy is lost to the surroundings that Brownian motion occurs. This latter process consists of many hops and collisions and continues until the electron is retrapped, and then the above sequence begins again. Our equations provide a generalization of the CTRW approach by accounting for the entire trajectory rather than just concentrating on the trapping points. B. The Probability. Remember that Q(R,t) is the probability density to become trapped at site R exactly at time t. The probability P ( R , t ) to be at site R a t time f is more complicated because not only can the walker be at site R as a trapping site, but also it may just be passing through R on its way to being trapped somewhere else. For a two-stage walk with alternating trappings and flights (trapped with waiting density + ( t ) and transporting with @ ( R , t )= p(Rlt) 4(t)),P(R,t) has two terms P(R,t) = J ‘ Q ( R , ~ - T T) ] ~ + ( dx x) di
(7)
Le., the walker reaches trapping site R - R’at time t - T , waits there for a time T - #, and then leaves following a trajectory which survives for a time T’ and undergoes a displacement of R’in the time T’. The walker now finds itself trapped at R at time t . In Fourier-Laplace space the solution to eq 7 is
(16) We use the summation symbol in a generalized sense to refer to all sites which the walker can visit. Of course, if this set is continuous then we interpret the sum as an integral.
~J‘Q(R-R’,t-T)~ t f‘4(x) ~ m ~ ( R ~ T - Tdx ~ ;I)(T’) x ) dT’ dT R‘
0
(10)
The first term is the same type as found in the CTRW where the walker reaches R at an earlier time and then waits there, while in the second term, as with the CTRW, the Q term brings the walker to R - R’exactly at time t - T and the )I term accounts for the walker being trapped at R - R’ for a time T ’ . the p term (17) Kunhardt, E. Preprint.
The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
Random Walks in Liquids accounts for the walker to go the remaining distance R’in a time T - T’, given that the lifetime of the trajectory (between trapping sites) is x, and the integral over the $ term ensures that all trajectories considered have a lifetime greater than T - T’, so that the trajectory need not become trapped a t R but may just be passing through on its way to a different trapping point. In this section, for the examples we consider, the p term will not depend on x , the lifetime of the trajectory. In the next section, on turbulence, the lifetime of a trajectory will be coupled to its velocity, so p will be an explicit function of x. Let us check the normalization of P(R,t). Since the walker must be somewhere, x R P ( R , t ) = 1 or in Fourier-Laplace space P(k=O,s) = l/s. Using the fact that ~ ~ ~ ( R I T - = T ’1,)we find by summing over all R in eq 10 and taking the Laplace transform that 1-W) + m ) ( l - d(s))
1
= (1
s
- W) $(J[
S
1
Le., normalization is satisfied. C. The Mean-Square Displacement. Let us now assume that in the transport stage there is no preferred direction so ( R ( t ) ) = 0. We now calculate, using eq 10,
$(k,s) = f(dJ(t) P(klt)) For the sake of concreteness, we choose in 3D p(Rlt) = ( 4 ~ d ( t ) ) exp(-R2/4d(t)) -~/~ The Fourier transform is given by
We choose d ( t ) = Dt”. If + ( t ) = X exp(-At) and $ ( t ) = y exp( y t ) ,then the mean-square displacement asymptotically behaves as At
with
-
Note that the normal diffusion result (Rz) t is obtained, even though the law d ( t ) = Dtn was used. This is because the exponential $ law sets a characteristic time ( l / y ) for the duration of a random flight. The d ( t ) law implies that the mean-square displacement of l / y ” is achieved in this time. Thus, the walk is composed of waiting times of mean duration 1/X and random flights of a finite mean-square displacement taking a finite amount of time. This leads to the standard Brownian motion result at long times. Note that if y 0 then the y X / ( y A) term y; Le., the diffusion is governed by the rate-limiting step. If on the 0, then y X / ( y A) A; Le., the rate-limiting other hand, X step of waiting at a site dominates the transport step in determining the diffusion constant. For a three-stage random walk exhibiting l / y + l/v)-I would overall diffusive behavior, a factor ( l / h appear in the diffusion constant. To have the stochastic process not belong to the domain of attraction of normal diffusion, one must introduce probability distributions without characteristic scales. If the duration of the flights between hopping sites had an infinite mean duration, then the d ( t ) = 1” law would be manifest, as we explore in the next
-
-+
+
+(t)
-
(0
t-1-
< < 1) (Y
-
so the first moment ( t ) = l t $ ( t ) dt = m. Then +(s) 1 - sa for small s, rather than the 1 - s ( t ) form which applies when ( t ) is finite. This leads to (R2(t))
-
(13)
fa
which reflects that the dominant motion of the walk is in the trapping stage where no characteristic time scale for release exists. IV. Turbulent Diffusion Let us now apply our methods to diffusion of a passive scalar in a turbulent flow.14 The random walker will always be in motion; Le., we can omit the trapped waiting time density $ ( t ) in the general formula in eq 7. We also assume that trajectories of length R are transversed with a velocity V. In the case of turbulence, for a walker under the influence of a coherent structure of size R its velocity Vwill be a function of R; Le., V = V(R). Larger vortices have more energy than smaller ones, so for the walker to move coherently for a longer distance it will do so with a larger velocity. We will show below according to Kolmogorov’s scaling hypothesis that V(R) = R1l3is a proper choice. To incorporate this idea, we choose
and the probability P(R,t) is given by
p(klt) = exp(-k2d(t))
-
section on turbulent diffusion. Suppose that the trapping waiting time distribution had a long tail, e.g.,
After the time t , the walker can come under the influence of another structure and change its general behavior. So we envision the walker transporting between sites which we call turning points. These are precisely the trapping points discussed in the section on multistage random walks, except now the walker does not stop at these sites; rather, it only changes its velocity and general direction. The Green’s function for this random walk is now given by 1 (15) = 1 - $(k,s)
Let us set
(R2(t))
7025
+
-
+
-
P(R,t) = EJ‘Q(R-R’,t-7) R’ 0
W(R’,7) dT
(16)
where the Q term brings the walker to R - R’at time t - T , and the W term brings the walker the remaining displacement of R’ in the remaining time T . The Wterm must take into account that the trajectory passes through R at time t but that it does not necessarily stop there. We write W a s
wheref(Y7) is the probability that the trajectory has velocity V, given that the duration of the trajectory is T . If the walker is at R - R’at time t - T and is on a trajectory with velocity V = R’/s, it will be at R at time t even if the trajectory continues for a time greater than 7 . The $ term sums over all such trajectories of duration greater than 7 . For V = R’/7, the walker covers the distance (R’~’/T)in the time 7’. For V = V ( R ) = Rz, thenf(V = R‘/717’) = ~(R-V(R)T’)with R = R’7‘/7. Kolmogorov in his famous 1941 paper (now called K41) assumed that the rate of energy transfer across a scale R, tR, was independent of R, Le., tR = t. The energy in this scale ER is proportional to V(R)*, and the time to transverse this distance t R = R/V(R). We can write t = E R / t R = V(R)3/R, or V(R)
-
t1/3R1/3
-
(18)
The Fourier transform of the energy Ek t2/3k-5/3,which is the famous Komogorov -5/3 law. thus, we set z = 1/3, choose p(R)
- R-’-*
(19)
7026
J . Phys. Chem. 1989, 93, 7026-7031
and use eq 17 to find, asymptotically at long times, that
The t 3 , result corresponds to walks with the mean time ( t ) spent in traversing a path being infinite. This case is known as Richardson's law of turbulent diffusion. The second case has ( t ) finite, but ( t Z )infinite, and the last case with the Brownian motion result occurs when ( t 2 ) is finite. This last case shows that Kolmogorov scaling ( z = 1/3) in itself is not sufficient to product turbulent diffusion. The condition @ I1 / 3 is also necessary. V. Conclusions We have generalized the Montroll-Weiss continuous-time random walk to include multistage processes. Our analysis of the random walk equations focused on the asymptotic behavior of an
unbiased walker. We investigated how, when probability distributions without characteristic scales are introduced, one can avoid the Gaussian limit. We studied cases where the walk was mostly waiting at a trapping site and where the walk was mostly transporting. The more detailed description provided by the multistage random walk will be important for first passage time calculations which are needed for a theory of reactions. The CTRW would have the walker wait at a site until it is time for an instantaneous transition to the next site, while in practice the walk could undergo a reaction on its way to the new site. The study of reactions in multiwalker systems and systems with biases fall easily into the multistage method which we have presented and will be the subject of further study. Acknowledgment. The authors thank Dr. Eric Kunhardt for suggesting the multistage random walk problem in regard to the motion of excess electrons in liquids.
Longitudinal Dielectric Relaxation Daniel Kivelson* Department of Chemistry and Biochemistry. University of California, Los Angeles, Los Angeles, California 90024
and Harold Friedman Department of Chemistry, State University of New York, Stony Brook, New York 11794 (Received: April 28, 1989; In Final Form: June IO, 1989)
Because of some apparent misunderstanding in the literature of the significanceof longitudinal dielectric relaxation, we discuss this property. We emphasize that because the ratio of the longitudinal ( T L ) to the transverse ( T ~correlation ) time is e(m)/e(O), where ~ ( 0 is ) a thermodynamic property of the system and c(m) an electronic property of the molecules, it follows that, for any dielectric material, both correlation times must contain the exact same dynamical information. Nevertheless, for nonexponential decay very rapid molecular motions will be easier to detect in longitudinal than in transverse relaxation; we show why this is so, and why oscillatory motions such as librations (torsional oscillations connected with Poley absorption) and dipolarons (inertial oscillations) are detectable in longitudinal relaxation. We relate the collective, low k, or continuum dielectric relaxation times to molecular reorientation times, but many interesting molecular phenomena depend heavily on high k properties.
1. Introduction To chemists, dielectric relaxation in liquids is of interest to the extent that it yields information concerning the dynamical behavior of individual molecules. The principal use of dielectric phenomena has been directed toward studies of molecular rotation, but there has also been much interest of late in understanding the effect of the fluctuating dielectric environment upon the course of chemical reactions. The treatment of dielectric relaxation and its relationship to molecular phenomena is a subtle subject, one that has been fraught with uncertainty, error, and controversy. In this article we comment on one aspect of the problem, the interrelationship of the transverse or Debye (TD) and the longi) relaxation times, both of which are matudinal ( T ~ dielectric croscopic or collective quantities, and the molecular quantities to which they are connected. That at least these two relaxation times are relevant to the theory of dielectric relaxation in isotropic, nonionic liquids has been indicated by numerous recent studies.'-9a The transverse (1) (2) (3) (4)
Berne, B. J. J. Chem. Phys. 1975, 62, 1154. Hubbard, J. B.; Onsager, L. J. Chem. Phys. 1977, 67, 4850. Hubbard, J. B. J. Chem. Phys. 1978,68, 1649. Fulton, R. L. J. Chem. Phys. 1975,63, 77; Mol. Phys. 1975,29,405.
0022-3654/89/2093-7026$01.50/0
time TD can be associated with the relaxation of the displacement vector D after a jump in the electric field E, while the longitudinal time T L is associated with the relaxation of E after a jump in D.Z6*'O Alternatively, rD can be associated with [ ~ ( u-)e(-)], where ~ ( o ) is the dielectric permittivity, while T~ can be associated with the To formulate the problem on the functional [c(m)-' molecular level, we use the fact that D and E are vector fields which are the transverse and longitudinal parts, respectively, of the polarization P. Here longitudinal and transverse means parallel or perpendicular to the wave vector k, respectively. We are interested in the source of the anisotropy which makes TD differ from T L in isotropic media, and how, in molecular fluids, these two times are connected to the molecular motions. e(u)-1].499ab
(5) Frohlich, H. Theory of Dielectrics, 2nd ed.; Oxford University Press: London, 1958. (6) Friedman, H. L. J. Chem. SOC.,Faraday Trans. 2 1983, 79, 1465. (7) Pollack, E. L.; Alder, B. J. Phys. Reu. Lett. 1981, 46, 950. (8) Impey, R. W.; Madden, P. A,; MacDonald, I. R.Mol. Phys. 1982,46, 513. (9) Madden, P.; Kivelson, D. Adu. Chem. Phys. 1984, LVI, 467-566; (a) p 485-490; (b) p 522; (c) pp 490-501; (d) pp 473, 478; (e) pp 520-524; (f) 523; (9) pp 524-538; (h) pp 479-485; 6 ) pp 490-493. (10) Friedman, H. L.; Newton, M. D. Faraday Discuss. Chem. Soc. 1983, 74, 73.
0 1989 American Chemical Society
