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

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19 Diﬀusion on Fractals

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

19.1 Introduction: What a Fractal is

In recent years, there has been a growing interest in the physics and physi-

cal chemistry of fractals. Fractal systems show new physical properties and

anomalous behaviour. This is particularly interesting in the ﬁeld of transport

processes. The name fractal was coined in 1975 by Benoit B. Mandelbrot, the

founder of modern fractal science. It is derived from the Latin verb frangere:

to break (fractum: broken), cf. words like fraction, fracture, fragment . . .

A fractal is deﬁned as a mathematical point set or a physical (chemical,

biological...)systemwithafractionalgeometrical(spatial)dimension(more

exactly: dimensionality). This dimension is called the fractal dimension or

Mandelbrot dimension

d. Its rigorous and general mathematical foundation is

given by the Hausdorﬀ dimension d

, see [1]. There also exist typical fractals

which (by chance) carry integer dimensions, e.g., certain Viscek fractals or

the Wiener trajectory of a Brownian particle during normal diﬀusion (

d =2),

see Sect. 19.4. This fact indicates diﬃculties to give a suitable deﬁnition of

the fundamental concept “fractal” which is not too narrow and not too wide

– these diﬃculties being shared by deﬁnitions of many other basic notions.

Objects embedded in our common (Euclidean) space are characterized by

(topological) dimensions d = 1, 2, or 3. In order to understand the notion of

a fractional dimension attributed to a physical system let us consider some

speciﬁc fractal structures. A good introductory example is a solid surface with

defects like steps, dislocations, cavities, with regard to its capacity to adsorb

say atoms (Fig. 19.1). With increasing number of defects, the adsorptive

capacity becomes larger and larger and approaches that of a sponge, to which

can be assigned a dimension d = 3. Therefore, it makes sense to introduce an

eﬀective dimension

d regarding adsorption, where 2 <

d<3. The topological

dimension of the surface is now d = 2 as before: One indeed deals with

a surface, even if curvature and many twists are present. The embedding

dimension (or Euclidean dimension) of the system is d

=3.

The surface of a real solid is a typical stochastic fractal. In principle, one

has to distinguish two types of fractals:

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

Fig. 19.1. A real solid surface shows a larger capacity to adsorb atoms than an

ideal ﬂat surface.

– regular, deterministic or ideal fractals, model fractals; these fractals are

constructed as mathematical objects according to a well-deﬁned (unique)

prescription or algorithm;

– irregular, random or stochastic (or statistical) or real fractals, they are

objects of nature or in the laboratory, yet they can be constructed theo-

retically as well.

The fractal systems of interest in physical science can be divided into three

classes, typical representatives of which are:

1. Natural objects (systems with mass) like

solid surfaces, enzyme surfaces, irregular porous solids, e. g. catalysts,

polymer systems (see Chap. 10), percolation networks (see Chap. 22),

irregular particle aggregates, and many others.

2. Particle trajectories of classical Brownian motion (Wiener process), par-

ticle trajectories of anomalous diﬀusion e. g. on fractal systems (the so-

called fractional Brownian motion [2]) or due to time-correlations in non-

Markovian stochastic processes, quantum mechanical state vectors (wave

function) of disordered electronic systems.

3. Sets in state diagrams, in particular boundaries of attractors and attrac-

tor basins of chaotic systems, i. e. classical dissipative dynamical systems

with deterministic chaos, sets of complex zeros of partition functions in

the frame of statistical thermodynamics (Lee-Yang theorem).

In order to deﬁne and investigate the basic properties of these systems, it

is convenient to consider ideal deterministic fractals. A typical example is the

elementary Sierpi´nski fractal (Sierpi´nski network, lattice, triangle or gasket);

its construction is indicated in Fig. 19.2.

With growing number n of construction steps the structure becomes

ﬁner and ﬁner, nevertheless one has nothing else than a surface with holes,

i. e. an object with the geometrical (topological) dimension d =2.Only

in the limit n →∞is a fractal generated; its dimension (see below) is

d =ln3/ ln 2 = 1.5849 ... < 2. The Sierpi´nski fractal is a prototype of a

19 Diﬀusion on Fractals 795

Fig. 19.2. Construction of a Sierpi´nski gasket; n: number of construction steps.

Sketched are prefractals; the proper fractal is created in the limit n →∞.

model for the physics of fractals and renders good services in the exploration

of the new and sometimes strange phenomena occurring in fractal systems.

Ideal fractals are characterized by two basic, closely related properties:

– Self-similarity at all length scales, dilatation invariance (structural invari-

ance against the dilatation group), i.e., existence of a hierarchical struc-

ture.

– The lack of any characteristic length (including such important character-

istic lengths of real physical systems as atomic diameter, lattice constant,

mean free paths, and correlation lengths).

Stochastic fractals are said to possess “stochastic self-similarity”, de-

ﬁned below. The best-known representative is the percolation cluster (see

Chap. 22), generated by randomly breaking either bonds or sites from, say,

a square lattice. For small break-oﬀ probability just some bonds are missing,

but there are still many connected paths that lead through the lattice. On the

other hand, for large break-oﬀ probability the lattice breaks into disjointed,

irregular sublattices. At the critical break-oﬀ probability where connected

paths through the lattice cease to exist the system becomes a stochastic frac-

tal, i.e., average properties attain fractal dimensions.

By deﬁnition, ideal fractals have an inﬁnitely ﬁne structure and a spatially

inﬁnite extension. When such a fractal is used to model a real physical system,

it is necessary to introduce two cut-oﬀ lengths: a lower cut-oﬀ given e. g. by

the lattice constant or atomic diameters, and an upper cut-oﬀ due to the

ﬁnite size of the system considered. At length scales between these two cut-oﬀ

lengths, the fractal is a good model for a disordered system, e. g. a percolation

labyrinth or a porous solid with stochastic distributions of pore locations and

pore diameters. In Sect. 10.5.3 of Chap. 10 even grains of sand are identiﬁed

as particles exhibiting surface fractality between certain cut-oﬀ lengths. The

essential property of (stochastic) self-similarity remains still valid over some

length scales.

With this in mind we can now deﬁne the notion of a fractal dimension. The

non-integer dimension of a fractal system is caused by the imperfect ﬁlling of

space; there are typically holes at all orders of magnitude (see Fig. 19.2). For

a rigorous and quantitative deﬁnition of the fractal dimension, it is useful to

assign a mass to the fractal (say, with mass density 1). The total mass M of

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

Fig. 19.3. Another construction of the Sierpi´nski gasket a) The initiator: L = L

M = M

. b) The generator: L

=2L

, M

=3M

.c)n =2:L

, M

a common object in an Euclidean space of dimension d =1,2,or3(say,a

given straight line, a square, or a cube) scales with the length (edge length)

L according to

M ∼ L

. (19.1)

The factor of proportionality plays no role; important is only the algebraic

functional law between M and L which is determined by the dimensionality

of the system. Generally one deﬁnes

extensive property ∼ length

dimension

. (19.2)

For an ideal fractal, the dimension

d is deﬁned by

M ∼ L

. (19.3)

However, it is now important to take the limit n →∞in the deﬁnition of

the fractal.

As an example, let us again consider the Sierpi´nski gasket, now con-

structed with another algorithm, as shown in Fig. 19.3. After step n one

has a prefractal with

Equation (19.3) yields

d = lim

n→∞

ln M

ln L

= lim

n→∞

ln(3

)

ln(2

)

= lim

n→∞

ln 3

+lnM

ln 2

+lnL

= lim

n→∞

n ln 3

n ln 2

ln 3

ln 2

< 2.

(19.4)

The quantities L

, M

, and the factor of proportionality no longer play any

role.

For (nearly) all ideal fractals constructed in this manner one can easily

ﬁnd the fractal dimension

d by the rule

19 Diﬀusion on Fractals 797

d =

ln(scaling factor of mass)

ln(scaling factor of length)

. (19.5)

A stochastic fractal is given a dimension

d by the relation

M∼L

; (19.6)

the averaging ···is to be performed over many samples of the fractal con-

sidered. Often, of course, one is confronted with the task of determining the

fractal dimension of a natural structure where no generating algorithm is

known. Methods of measuring the fractal dimension of a given object are

reviewed in [3]. The basic idea behind these methods is the assignment of a

mass function M(L)toadeﬁnitevolumeV ∼ L

in the embedding space and

then to measure this mass for a suitably chosen sequence of test volumes. This

yields a sequence of data M(L

) from which one obtains the fractal dimension

via (19.5).

Clearly, a fractal deﬁned in this manner is not only invariant with re-

gard to dilatations of space. It is also irrelevant where in space the structure

is studied: All triangles in the Sierpi´nsky gasket are treated in the same

way in all consecutive iteration steps. However, fractal systems may be more

complicated. A multifractal is, crudely spoken, a fractal system with a space-

dependent fractal dimension. Typical multifractals are aggregates of (col-

loidal) particles. Such structures are not covered in this brief introduction to

diﬀusion on fractals. More important for our purposes is another variant of

fractals, viz. self-aﬃne fractals. Isotropic fractals as, e.g., the Sierpi´nsky gas-

ket remain invariant under rescaling all the directions of the embedding space

by the same factor. Another important generalization is the introduction of

anisotropic scaling where each direction of the embedding space is scaled by

a diﬀerent amount. Structures that are invariant under such rescaling are

self-aﬃne fractals. A common example (in this case, of a stochastic fractal) is

the trajectory of a random walk where the time direction is scaled diﬀerently,

as discussed below. More about the fundamentals and applications of fractal

science can be found in excellent books [1, 3–5]; a short review of selected

applications is given in [6].

19.2 Anomalous Diﬀusion: Phenomenology

Diﬀusion is the fundamental process of mass transport in a tremendous va-

riety of systems. Since also fractal structures are not uncommon (of course,

with the limiting upper and lower cut-oﬀs, resp., in real systems) it is rather

important to investigate diﬀusion processes on fractal systems. Our topic is

the overall behaviour of diﬀusing particles, i.e., the statistics of the many

steps of one, several, or many particles, not the elementary process of one

step.

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

Let us ﬁrst recall the basic facts of normal diﬀusion in common (i. e.

Euclidean) systems (see also Chaps. 1, 10, 18, 23 and [7, 8]). Our starting

point is the linear diﬀusion equation of Fick for the particle number den-

sity n(x, y, z, t)=n(r,t) in a one-component system

∂

∂t

n = D∆n, (19.7)

where D is the diﬀusion coeﬃcient and ∆ = ∂

/∂x

+ ∂

/∂y

+ ∂

/∂z

the

Laplacian. This equation results from the existence of a diﬀusive current

j = −D∇n (19.8)

due to the random motion of particles (Fick’s law) and the continuity equa-

tion

∂

∂t

n = −∇j, (19.9)

resulting from the conservation of the number of particles.

Let N be the total number of particles in the system, then

P (r,t)=

n(r,t)

(19.10)

appears as the spatial probability density to ﬁnd one particle at the place r

at time t.

The diﬀusion equation

∂

∂t

P = D∆P (19.11)

with the initial condition P (r,t

)=δ(r − r

) (corresponding to particles

localized at the point r

at time t

) has as solution the Gaussian distribution

P (r,t|r

(4πD (t − t

))

d/2

exp



−

(r − r

)

4D (t − t

)



. (19.12)

This is the Green’s function (or propagator) of the diﬀusion equation. (R. P.

Feynman has ﬁrst coined the name propagator, but for the quantum mechan-

ical transition probability amplitude.) The evaluation for the case of normal

diﬀusion of, e.g., guest molecules in zeolites is given in Chap. 23.

The basic quantity P (r,t|r

) is the conditional spatial probability den-

sity to ﬁnd one particle at r,t, given the initial position r

.Inphysical

terms it is simply the spatial transition probability density from r

r,t for one particle. We remark that the propagator is not a self-correlation

function in the proper sense, though it may be represented in terms of the self-

correlation function of δ(r − r(t)) where r(t) denotes the position of a given

particle at time t. Hence in M¨oßbauer and neutron scattering spectroscopy

(see Chaps. 2 and 3) both terms are often used simultaneously.

With the variables r

=0,t

= 0 the propagator

19 Diﬀusion on Fractals 799

Fig. 19.4. Normal diﬀusion. In

equal time intervals ∆t the mean

square displacement increases by

equal amounts.

P (0,t|0, 0) ∼ t

−d/2

(19.13)

represents the probability density to ﬁnd a particle at the (arbitrary) origin

at time t, if it was at the origin at time 0.

The mean square displacement, with (19.12) and r

= t

= 0, becomes



(t)





P (r,t|0, 0)d

=2dDt (d =1, 2, 3, 4,...).

(19.14)

The proportionality between



(t)



and t for all integer dimensions d reﬂects

the universality of normal diﬀusion (see Fig. 19.4).

The diﬀusion equation (19.11) allows for the factorization of the solution

P :

P (x, y, z, t)=P (x, t)P(y, t)P(z, t). (19.15)

This means that there is no correlation between the particle motions in x-,

y-, and z-direction, respectively. Likewise, for the normal diﬀusive motion of

one particle, there is no correlation of the individual particle displacements

in time, which is reﬂected in the Gaussian distribution of the probability

density and therefore also in a linear relation between r

 and t as shown

in Fig. 19.4. Rigorously speaking, the universality of normal diﬀusion results

from the central limit theorem which asserts that independent, identically

distributed increments of a random variable (here the particle position) with

ﬁnite mean and variance always lead to a Gaussian distribution.

The anomalous diﬀusion on fractal systems, e. g. on the network of a

Sierpi´nski gasket, is caused by the self-similarity at all length scales [5]. A

diﬀusing particle has to circumvent all holes resulting in an increasing eﬀect

of hindrance. Experiments show, however, that it is not only the diﬀusion

coeﬃcient D which is diminished; even the functional law (19.14) is changed,

leading to the relation



(t)



∼ t

,κ<1. (19.16)

Thetimeexponentκ shows no universality:

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

Fig. 19.5. Mean square displacement for anomalous diﬀusion; κ =1/2; cf.

Fig. 19.4. The diﬀusion law (19.16) acts as an asymptotic law; therefore, the curve

is not valid during a time interval ∆t

immediately after the starting time t

the process. This time interval covers say 3 ...10 elementary steps of the walking

particle.

1. κ is a speciﬁc quantity for every fractal type, κ depends on the fractal

dimension

2. The anomalous diﬀusion becomes a stochastic process with correlation in

time, its increments are no longer independent; see below and [9].

3. It follows from the time hindrance as schematically shown in Fig. 19.5

that the initial time t

and hence the time interval ∆t is no longer ar-

bitrary, contrary to the situation in normal diﬀusion (Fig. 19.4). This is

important for the correct theoretical description of experiments on anom-

alous diﬀusion, e. g. by use of the pulsed-ﬁeld-gradient NMR technique

(cf. Fig. 19.5, Chap. 10 and [10,11]).

It can easily be understood by means of the following simple calculation that

independent time increments are incompatible with anomalous diﬀusion.

Following the representation of Fig. 19.6 we decompose the transition

from, say, the origin, to a position r into two consecutive transitions ﬁrst

toapointr

at some time t

and by another displacement r

= r − r

Therefore

r(t)=r

+ r

, r

= r

), r

= r

)

with t = t

+ t

,t,t

are ﬁxed times.

Then the mean square displacement is given by



(t)





+ r

)











+2r

· r

. (19.17)

With (19.14) for normal diﬀusion, (19.17) yields



(t)



=2dD(t

+ t

)+2r

· r

.

19 Diﬀusion on Fractals 801

Fig. 19.6. Displacements for the ex-

planation of anticorrelation during

anomalous diﬀusion.

On the other hand,



(t)



=2dDt which – since t = t

– is in agreement

with the absence of correlations (i.e., K(t

) ≡r

· r

 = 0) between the

increments r

and r

, respectively.

In contrast, for anomalous diﬀusion with (19.16) one obtains

= t

+ t

+2K(t

), (19.18)

which proves that the correlation function K(t

) is non-zero and, for κ<1,

negative:

K(t

) < 0forκ<1. (19.19)

Equation (19.19) quantiﬁes the phenomenon of “anticorrelation” and also

provides a physical interpretation (Fig. 19.6): The diﬀusing particle is walking

preferentially in the “backward” direction with an (averaged) angle larger

than π/2 between the last two steps. Thus the diﬀusion is slowed down and

one speaks of subdiﬀusive behaviour. The quantity

=2/κ (19.20)

is called the walk dimension as it describes the distribution of trajectories

the random walker may take. One has

κ =

≤ 1, 2 ≤ d

≤ 6. (19.21)

The limiting case d

= 2 corresponds to normal diﬀusion: κ =1.

Closely related to the anomalous walk dimension in a fractal is the obser-

vation of an anomalous decay

P (0,t|0, 0) ∼ t

−

d/2

. (19.22)

of the return probability (19.13). It is remarkable that

d =

d. In fact, (19.22)

deﬁnes a new dynamical dimension, the so-called spectral dimension

d,gov-

erning the dynamics of fractal systems to a large extent (though one should

be aware of exceptions [20]). Often one has

d ≈ 4/3 [21–23].