Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

Подождите немного. Документ загружается.

802 Uwe Renner, Gunter M. Sch¨utz, and G¨unter Vojta

19.3 Stochastic Theory of Diﬀusion on Fractals

The aim of a more detailed stochastic theory of dynamic processes in fractal

systems is twofold,

– to furnish methods for the analytical or numerical calculation of dynami-

cal dimensions, mean square displacements, reaction velocities and other

quantities,

– to provide a deeper understanding of dynamic processes in fractals.

Given the scale invariance both of the fractal and of the trajectory of the dif-

fusing particle (see below) the natural choice of approach is the use of scaling

relations, and, for more detailed study, the renormalization group formalism.

The central idea behind all scaling approaches and renormalization schemes

is the study of quantities of interest under a repeated rescaling of space and

time. Renormalization can be implemented in many diﬀerent ways which we

cannot review here in this brief survey in any detail. We only point out some

of the relevant literature for further reading.

A main problem is the evaluation of the propagator P (r,t|r

) which no

longer satisﬁes the simple equation (19.11), and of quantities derived from it,

in particular, the mean square displacement (19.16) and the return probabil-

ity (19.22). There exist several analytical methods [5,7,9] (see also Chap. 18)

which make use of either

– the simple random walk (RW) of one diﬀusing particle. This particle

jumps in ﬁxed time steps to neighbouring lattice sites and thus accom-

plishes a Markovian process,

– or the continuous time random walk (CTRW), the diﬀusing particle jumps

at arbitrary times to neighbouring sites and generally performs a non-

Markovian process (technically a semi-Markovian process) which shows

some type of memory.

For simulation purposes the most transparent way to study the proper-

ties of diﬀusion in fractals is the simple (discrete-time) random walk. This

is a legitimate approach to obtaining more general results because usually

power law relations such as those discussed above are universal reﬂecting the

properties of large classes of systems rather than the properties of only the

speciﬁc model under investigation. In the basic relations (19.16), (19.22) the

parameter t is then understood as a large integer number, representing the

number of steps a random walker has taken.

There are two related formalisms for the analytical investigation of

CTRW, namely the master equation method [5, 7] and the propagator for-

malism [7, 9].

The basic quantity of the propagator formalism is the waiting time distri-

bution (probability density) w(t) (another widely used symbol is ψ(t)). The

19 Diﬀusion on Fractals 803

waiting time is deﬁned as the time which a particle spends at a deﬁnite site

until the next jump. For the special case of an exponential distribution

w(t)=αe

−αt

, (19.23)

where 1/α = t is the mean waiting time, the total process (in time) becomes

a Poisson process which is Markovian.

Generally, the propagator P (r,t|r

) is given by the summation of wait-

ing time distributions up to the ﬁnite time t, more precisely by the convolution

product (power)

P (t)=w(t)

∗k

,k=1, 2,.... (19.24)

After Laplace transformation in time a convenient formalism emerges [9]. A

dynamic real-space renormalization procedure then yields the walk dimension

(19.20) as well as P (r,t|r

) [9, 18, 21]. The walk dimension, e. g. of

the Sierpi´nski gasket, is d

=ln5/ ln 2 > 2; its spectral dimension is

d =

ln 9/ ln 5 < 2. These quantities may also be derived using renormalization

within the master equation approach [22]. Using renormalization techniques

the spectral dimensions for a variety of lattices have been calculated [23].

The CTRW formalism can be generalized by including a probability dis-

tribution of random ﬂying times of the diﬀusing particle considered; this sup-

plies a generalized non-Markovian master equation and the equivalent general

propagator formalism [19]. Other theoretical possibilities are furnished by the

fractional calculus (with the application of diﬀerential quotients and integrals

of fractional order) [9,24]. The diﬀerential equations of diﬀusion, i .e . the ex-

tensions of the Fick equation (19.7) or (19.11), are more complicated and no

longer universal; their analytical form depends on the fractal type.

19.4 Anomalous Diﬀusion: Dynamical Dimensions

Anomalous diﬀusion can be conveniently described by use of dynamical di-

mensions which only in part imply a geometrical meaning. Simple scaling

laws already give valuable insights into their signiﬁcance and interrelations,

without the necessity of employing the full renormalization group machinery.

The return probability for normal diﬀusion is connected with the propa-

gator formula (19.13). Generalizing (19.13) we have introduced the spectral

dimension

d (19.22) (another common symbol: d

) of anomalous diﬀusion.

Another deﬁnition of the spectral dimension is possible on the basis of the

Debye theory of atomic vibrations in a crystalline monatomic solid. The vi-

brational spectrum (hence the name “spectral dimension”) is determined by

the vibrational density of states

!(ω) ∼ ω

d−1

for normal solids (d =2, 3, 4,...)

!(ω) ∼ ω

d−1

for fractals, (19.25)

804 Uwe Renner, Gunter M. Sch¨utz, and G¨unter Vojta

i. e. for prefractals whose vertices (lattice sites) are occupied by vibrating

atoms. Both deﬁnitions (19.22) and (19.25) yield the same spectral dimension

[5].

A second important dynamical dimension is the walk dimension d

(19.20)

for the mean square displacement. Since the walk dimension describes the

typical region explored by the random walker after some time t one deﬁnes

a diﬀusion length l ≡ l(t)bymeansof



(t)



= l

. (19.26)

Then (19.20) can be given the form

t ∼ l

. (19.27)

The time t is proportional to the contour length of the trajectory of the

diﬀusing particle. In lattice language this corresponds to the number of steps

the random walker has taken. Considering the trajectory itself as a stochastic

fractal, the contour length is also proportional to the mass of the fractal and

(19.27) becomes a special case of the general formula (19.2).

Therefore, in most cases the walk dimension d

is identical to the geo-

metrical fractal dimension

d which can be assigned to the single-particle

trajectory of the diﬀusion process considered as Wiener process or general-

ized Wiener process. The Wiener theory of normal diﬀusion describes the

usual Brownian motion discovered nearly 200 years ago. The correspond-

ing trajectory is schematically represented in Fig. 19.7. It is the simplest

possible rigorous mathematical theory of a (continuous) Markovian process

(i. e. a process without memory) governing normal diﬀusion. Nevertheless,

the particle trajectory is a very complicated curve, ﬁlling a (twisted) plane

essentially densely and being characterized by a dimension

d =2=d

.This

curve is everywhere continuous, but nowhere diﬀerentiable. These are typi-

cal features of a fractal which in this case is a self-aﬃne structure. This can

be seen from (19.27): The particle trajectory remains (on average) invariant

under the anisotropic scaling r → br, t → b

t in the embedding (r,t) space.

This dynamical scale invariance with d

= 2 shows up also in the propaga-

tor (19.13). For hindered diﬀusion one gets

d = d

> 2, that means that

the particle trajectory is even more densely packed. There are, however, also

exceptions of the rule d

d [12].

Since the trajectory of a Brownian particle itself deﬁnes a fractal the

theory of diﬀusion on fractals has to deal with understanding the impact of

an underlying fractal space on the statistical properties of curves which are

already (self-aﬃne, dynamical) fractals in normal (Euclidean) space. As a

general relation between the three dimensions

d, d

one has

κ =

. (19.28)

19 Diﬀusion on Fractals 805

Fig. 19.7. Wiener trajectory

of normal diﬀusion, coarse

grained.

Here

d denotes the fractal dimension of the system considered (not of the

particle trajectory). For a common (Euclidean) system one has

d =

d = d,

= 2. A fractal system is characterized by κ<1, i. e.

A proof of the basic relation (19.28) can be given by the following scaling

argument. It is a typical example of the methodology of fractal science. With

the deﬁnition (19.26) of the diﬀusion length l one has

l = l(t) ∼ t

1/d

. (19.29)

The volume visited by a particle on the average up to the time t is given by

V (t) ∼ l

∼ t

d/d

. (19.30)

The probability (density) to ﬁnd the particle at time t at any deﬁnite place

within the fractal, e. g. at the origin, hence follows as

P (0,t) ∼

V (t)

∼ t

−

d/d

. (19.31)

On the other hand, according to (19.22) one has

P (0,t) ∼ t

−

d/2

. (19.32)

The comparison of (19.31) and (19.32) yields (19.28). This proof is essentially

rigorous. Similar scaling arguments may also be employed for other transport

processes, e.g., electronic transport [5,13].

An important quantity intimately connected with the return probability

is the number S(t)ofdiﬀerent locations in space that the diﬀusing particle

has visited after time t. Here a “location” means some small volume element

of the order of the particle size. One may therefore think of the (continuous)

space as being divided in a lattice with a lattice constant equal to the particle

diameter and view the random motion as a hopping process between these

sites as explained in detail in Chap. 18. In d-dimensional Euclidean space

806 Uwe Renner, Gunter M. Sch¨utz, and G¨unter Vojta

(19.30) reads V (t) ∼ t

d/2

. On the other hand, the number of jumps N (t)

made after time t is proportional to t. Therefore the mean time spent on

each site is given by the ratio

V (t)/N (t) ∼ t

1−d/2

In more than two dimensions this mean time decreases with increasing time.

This implies that the probability that a random walker ever returns to a given

site tends to zero as t becomes large and one speaks of a non-recurrent random

walk. Hence the number S(t) of diﬀerent visited sites is just proportional to

N(t) and therefore increases linearly in time. On the other hand, for d<2the

mean time spent at each site diverges and the return probability approaches

1; the random walk is recurrent. In this case S(t) ∼ t

d/2

is proportional to

the explored volume V (t). In the borderline case of two dimensions there is

a logarithmic correction to the asymptotic law S ∼ t. For recurrent walks

on fractal objects these considerations apply accordingly. Using (19.30) one

ﬁnds

S(t) ∼ t

d/d

(19.33)

which may be written in terms of the spectral dimension as S(t) ∼ t

d/2

.For

d>2 the random walk on the fractal is non-recurrent. In this case S(t) ∼ t.

19.5 Anomalous Diﬀusion and Chemical Kinetics

An important ﬁeld of application of diﬀusion on fractals are diﬀusion-limited

reactions, e.g., on the fractal surface of catalysts. Usually chemical reactions

are studied in terms of parameters which concern the electronic structure of

the molecules or atoms involved in that reaction, the environment in which

the reactions take place seems irrelevant. This treatment then leads to rate

equations with reaction constants that depend on those microscopic molecular

parameters. Probably the simplest example is the bimolecular reaction

A+A

−→ C

where A are the reactants and C is an inert reaction product. This is a

reaction where two A particles are “annihilated” when they meet inside some

eﬀective reaction volume, with a reaction constant k. In the absence of a back

reaction C → 2A all A-particles will eventually disappear. We point out that

such a reaction mechanism need not necessarily be of chemical nature where

new chemical bonds are formed between reactants. Also physical processes

such as laser-induced exciton hopping can be described in terms of such

processes [31]. In this case the “particles” are excitons moving e.g. along a

polymer chain. The reaction corresponds to the decay upon encounter under

the emission of light (playing the part of the inert reaction product).

19 Diﬀusion on Fractals 807

Since two A-particles are necessary for an annihilation to take place the

rate equation describing the temporal change of the concentration C of A-

particle simply reads

˙c = −kc

. (19.34)

This is the well-known law of mass action. Equation (19.34) is easy to inte-

grate, with the result

c(t)=

1+c

for an initial concentration c

of A-particles. At long times the concentration

decays algebraically

c =1/(kt) (19.35)

and is quite remarkably independent of the initial concentration. The same

reasoning may be applied to a reaction of the type A + A → A+C or even to

mixed reactions where either both particles disappear or only one, with some

probability. Various models for these systems can be shown to be exactly

equivalent by a mathematical transformation of the observables [26, 29, 32].

The assumptions regarding the spatial distribution of A-particles which

lead to (19.34) are two-fold: Firstly, one assumes the reaction to take place in

a homogeneous environment so that for a homogeneous initial distribution of

A-particles the particle concentration would remain space-independent for all

times. Secondly – and this is more subtle – one assumes that at each instant of

time the probability of ﬁnding a second A-particle inside the reaction volume

of the ﬁrst is proportional to the overall probability of ﬁnding particles, i.e.,

to the concentration. This is a mean-ﬁeld assumption which would be valid

if correlations between particle positions could be ignored.

Of course, the validity of such an assumption is somewhat questionable,

particularly if one thinks of reacting particles diﬀusing e.g. on the surface of

a catalyst. If a reaction has just taken place the reaction volume is empty

rather than ﬁlled with a probability proportional to c. Only after surround-

ing particles have had suﬃcient time to diﬀuse and thus restore an uncor-

related ﬁlling of space, the mean-ﬁeld assumption will again be valid and

further annihilations can take place according to (19.34). Hence the mean-

ﬁeld treatment of an intrinsically diﬀusion-limited reaction is justiﬁed only if

the reacting particles are stirred suﬃciently quickly (so that the reaction is

not diﬀusion-limited anymore) or if diﬀusion is suﬃciently eﬃcient to main-

tain an essentially uncorrelated state of the system at all times.

It was already realized by Smoluchowski [25] that in low-dimensional re-

action media the mean ﬁeld law (19.34) is actually not valid. Here diﬀusive

mixing is ineﬀective and large correlations persist for all times. Instead of

assuming that there is a second reacting particle with a probability propor-

tional to c one has to consider the ﬂux of particles from the environment

into the reaction volume of the ﬁrst particle. This approach leads to an eﬀec-

tive reaction “constant” k(t) determined not only by microscopic molecular

parameters but also by the process of diﬀusion as described by Fick’s law

808 Uwe Renner, Gunter M. Sch¨utz, and G¨unter Vojta

(19.8). In one space dimension calculating the ﬂux leads to a decaying reac-

tion parameter k(t) ∼ 1/

√

t. Inserting this into (19.34) leads to an anomalous

decay

c ∼ 1/

√

t (19.36)

of the particle density. This demonstrates that the eﬀect of correlations is

particularly pronounced in low dimensions which has been veriﬁed experi-

mentally in the last two decades (for a review see [31]). Exact results derived

from lattice models (reviewed in [28]) show that the time-dependent concen-

tration becomes independent of the initial concentration, as in mean ﬁeld

theory. Exactly solvable lattice models allow not only for the explicit calcu-

lation of the concentration. Also density proﬁles for spatially inhomogeneous

initial distributions and correlations can be calculated. Thus one has a very

detailed knowledge of the properties of the reaction A + A → Candsome

other simple reactions which goes far beyond the ﬂuctuation-corrected mean-

ﬁeld theory of Smoluchowski.

According to Smoluchowski’s theory the validity of the asymptotic mean

ﬁeld law depends on the dimensionality of the reaction medium because of the

properties of the particle ﬂux. In two dimensions one has a logarithmic correc-

tion to (19.35), whereas in dimensions higher than two mean ﬁeld theory re-

mains valid. For other reactions, e.g., the bimolecular Langmuir-Hinshelwood

reaction A + B → C, the critical dimension above which mean-ﬁeld theory is

valid is equal to four for initially equal concentrations of randomly distributed

A and B particles. Below four dimensions one has c ∼ t

−d/4

. Hence already

in ordinary three-dimensional reaction environments mean-ﬁeld theory fails,

in this case due to concentration ﬂuctuations inside pure A and B domains

which build up as the reaction goes on. This is conﬁrmed by renormalization

group studies [30].

The anomalous behaviour of diﬀusion-limited reactions in low-dimensional

systems naturally leads to the question what happens in a fractal where the

dimension is not integer and, moreover, the environment is not homogeneous

anymore so that the ﬂux cannot be calculated simply using Fick’s law. It

comes as no surprise that as a result of anomalous diﬀusion, chemical re-

actions in fractal media (heterogeneous reactions, heterogeneous catalysis)

show strange behaviour [2, 14–17]. Also in the presence of disorder, diﬀu-

sion can become anomalous (Chap. 18 and 22) and the decay law (19.36)

changes [26,27].

Rather than calculating the ﬂux it is convenient to consider S(t), the

number of diﬀerent sites visited, since the change of this quantity in time is

proportional to the eﬀective reaction constant,

k ∼

dS(t)

. (19.37)

The quantity S(t) has been discussed in Sect. 19.4 in some detail. Its connec-

tion to the ﬂux can be understood qualitatively by noting that the particle

19 Diﬀusion on Fractals 809

ﬂux entering the Smoluchowski approach is proportional to the probability

ﬂux of ﬁnding a particle. This probability ﬂux, however, is proportional to the

change in the number of sites visited because of the recurrence properties of

the underlying random walk. For a fractal one may then use (19.33) together

with (19.34) to obtain an eﬀective rate equation for the diﬀusion-limited an-

nihilation process on a fractal. One then ﬁnds for long times

c ∼ t

−

d/d

(19.38)

for spectral dimensions below 2. The mechanism for this fractal correction to

mean ﬁeld behaviour is the same as in a low-dimensional Euclidean medium.

As time passes particles tend to be more isolated than they would be in

an uncorrelated random ensemble, i. e. particle positions become increasingly

anticorrelated. Diﬀusion under recurrence conditions is too ineﬃcient to wash

out those anticorrelations since particles are not suﬃciently likely to explore

a large volume in between reactions.

The decrease of the reaction rate has a striking eﬀect when comparing to

originally identical reaction vessels, one of which is stirred very quickly (thus

destroying anticorrelations) after some time. From this point on the reac-

tion in the undisturbed vessel will proceed more slowly. In a rate-controlled

reaction such stirring would have no eﬀect on the reaction velocity. Similar

considerations may be adapted to the heteronuclear Langmuir-Hinshelwood

reaction A + B → C.

19.6 Conclusion

The importance and usefulness of the fractal concept can be summarized as

follows:

1. Fractals possess mathematically strange properties and imply very inter-

esting novel physical phenomena.

2. Fractals are suitable to model complicated real systems.

3. Fractals can characterize disordered systems by only a few structural

quantities, viz. geometric fractal dimension and dynamical dimensions

(dynamical exponents).

4. Fractals allow for the application of rigorous scaling and renormalization

methods without approximations.

For the study of random walks (which are stochastic fractals by them-

selves, even without underlying fractal medium) not only the fractal dimen-

sion

d of the medium is relevant. The distribution of possible pathways gives

rise to an independent exponent, the walk exponent d

, describing the sta-

tistical properties of trajectories of diﬀusing particles. In the presence of

interactions, e.g. chemical reactions, correlations between particle positions

810 Uwe Renner, Gunter M. Sch¨utz, and G¨unter Vojta

are a new determining factor for the behaviour of diﬀusion-limited chem-

ical reactions. However, for some simple paradigmatic processes the origin

of these correlations (in fact, anticorrelations) can be traced back to the

statistical properties of single random walks. Anomalous decay laws result

from ineﬃcient diﬀusive mixing in recurrent random walks and hence can

be found in fractals as well as in low-dimensional Euclidean media. An en-

tirely unexplored ﬁeld are nonequilibrium interacting particle systems un-

dergoing biased diﬀusion on a fractal under the action of a driving force.

One-dimensional (Euclidean) lattice models for such systems are reasonably

well-understood [28, 33]. However, even basic questions such as the behav-

iour of the stationary particle current as a function of particle density are

unanswered for fractals.

Notation

c particle concentration

d integer (Euclidean) dimension

(Euclidean) embedding dimension

d geometric fractal dimension, Mandelbrot dimension

(also Hausdorﬀ dimension)

d spectral dimension

walk dimension

D diﬀusion coeﬃcient

h heterogeneity exponent

k chemical rate constant

l diﬀusion length

n = n(r,t) particle number density

N total number of particles in a system

P (r,t)=P(r,t|r

) propagator

r reaction order

r =(x, y, z) coordinate vector

S(t) number of diﬀerent sites visited by a diﬀusing particle

on average up to time t

V (t) volume visited by a diﬀusing particle on average up

to time t

w(t) waiting time distribution

ε reaction eﬃciency

κ diﬀusion exponent

!(ω) vibrational density of states

References

1. B.B. Mandelbrot: The Fractal Geometry of Nature (Freeman, New York 1983)

2. I.M. Sokolov, J. Klafter, A. Blumen: Phys. Today 55, 48 (2002)

19 Diﬀusion on Fractals 811

3. Fractals in Science, ed by A. Bunde, S. Havlin (Springer, Berlin Heidelberg

New York 1995)

4. M. Schroeder: Fractals, Chaos, Power Laws (Freeman, New York 1991)

5. Fractals and Disordered Systems, 2nd edn, ed by A. Bunde, S. Havlin (Springer,

Berlin Heidelberg New York 1996)

6. H.Spindler,G.Vojta:Z.Chem.28, 421 (1988)

7. J.W.Haus,K.W.Kehr:Phys.Rep.150, 263 (1987)

8. J. K¨arger, D.M. Ruthven: Diﬀusion in Zeolites and other Microporous Solids

(Wiley, New York 1992)

9. U. Renner: Stochastische Theorie von Transportprozessen in ungeordneten und

fraktalen Systemen. PhD thesis, Universit¨at Leipzig (1993)

10. J. K¨arger, G. Vojta: Chem. Phys. Lett. 141, 411 (1987)

11. J. K¨arger, H. Pfeifer, G. Vojta, Phys. Rev. A 37, 4514 (1988)

12. Ph. de Forcrand, J. Pasche, D. Petritis: J. Phys. A: Math. Gen. 21, 3771 (1988)

13. U. Renner, G. Vojta. In: Synergie, Syntropie, Nichtlineare Systeme,vol1(Wis-

senschaftszentrum Leipzig, 1995) p 35

14. P. Pfeifer: Chimia 39, 120 (1985)

15. A. Blumen, G.H. K¨ohler: Proc. R. Soc. London, Ser. A 423, 189 (1989)

16. R. Kopelman: Science 241, 1620 (1988)

17. G. Vojta: Materials Science Forum 62, 743 (1990)

18. C. Van den Broeck: Phys. Rev. Lett. 62, 1421 (1989)

19. S. Richter, G. Vojta: Physica A 188, 631 (1992)

20. A. Vojta, G. Vojta, M. Vojta, T. Vojta: J. Phys.: Condens. Matter 7,L67

(1995)

21. S. Alexander, R. Orbach: J. Phys. (Paris) Lett. 43, L625 (1982)

22. R. Rammal, G. Toulouse: J. Phys. (Paris) Lett. 44, L13 (1983)

23. R.A. Guyer: Phys. Rev. A 29, 2751 (1984); 30, 1112 (1984)

24. H.E. Roman, P.H. Alemany: J. Phys. A: Gen. Math. 27, 3407 (1994)

25. M. von Smoluchowski: Z. Phys. Chem. 92, 129 (1917)

26. G.M. Sch¨utz: Z. Phys. B 104, 583 (1997)

27. G.M. Sch¨utz, K. Mussawisade: Phys. Rev. E 57, 2563 (1998)

28. G.M. Sch¨utz. In: Phase Transitions and Critical Phenomena, vol 19, ed by C.

Domb, J. Lebowitz (Academic Press, London 2000) pp 1–251

29. M. Henkel, E. Orlandini, G.M. Sch¨utz: J. Phys. A 28, 6335 (1995)

30. B.P. Lee: J. Phys. A 27, 2633 (1994)

31.

Nonequilibrium Statistical Mechanics in One Dimension,edbyV.Privman

(Cambridge University Press, Cambridge U.K. 1997)

32. H. Simon: J. Phys. A 28, 6585 (1995)

33. T.M. Liggett: Stochastic Interacting Systems: Contact, Voter and Exclusion

Processes (Springer, Berlin Heidelberg New York 2000)