FRACTAL GEOMETRY

Definitions One of the common features of fractal objects is that they are self-similar (scale invariant). We call embedding dimension the Euclidean dimension d of the space the fractal can be embedded in. In addition, d has to be the smallest such dimension. In general, we call a physical object fractal, if measuring its volume, surface or length with d, d − 1 etc. dimensional hyperballs it is not possible to obtain a well converging finite measure for these quantities when changing l over several orders of magnitude. It is possible to construct mathematical objects which satisfy the criterion of self-similarity exactly, and their measured volume depends on l even if l or (l/L) becomes smaller than any finite value. Growing and division fractals Mathematical and physical fractals D will be called fractal dimension. In the case of growing fractals, where there exists a smallest typical size a, one cuts out d-dimensional regions of linear size L from the object and the volume, V (L), of the fractal within these regions is considered. When determining V (L), the structure is covered by balls or boxes of unit volume (l = a = 1 is usually assumed), therefore V (L) = N(L), where N(L) is the number of such balls or boxes. For fractals having fixed L and details on very small length scale D is defined through the scaling of N(l) as a function of decreasing l, where N(l) is the number of d dimensional balls of diameter l needed to cover the structure. Correspondingly,

N(L) ∼ L^{D}  (2.2)

and

D = lim_{L→∞} \frac{ln N(L)}{ln(L)}  (2.3)

for the growing case, where l = 1. Here, as well as in the following expressions the symbol ∼ means that the proportionality factor, not written out in (2.2), is independent of l. For fractals having a finite size and infinitely small ramifications we have

N(l) ∼ l^{−D}  (2.4)

with

D = lim_{l→0} \frac{ln N(l)}{ln(1/l)}.  (2.5)

for example

N(L) = 5^{k}  with  L = 3^{k},  (2.6)

where k is the number of iterations completed and D = ln 5/ ln 3 = 1.465... The finite size (L < ∞) of physical fractals makes it possible to treat all of the physically relevant cases using a dimensionless quantity

ϵ = \frac{l}{L}  (2.7)

Therefore, (2.2) and (2.4) in terms of ϵ are recast into the same form

N(ϵ) ∼ ϵ^{−D},  (2.8)

where ϵ ≪ 1, N(ϵ) is the number of d dimensional balls of radius ϵL needed to cover the fractal, and D is the same fractal dimension as in (2.2) and (2.4). The original intention of Hausdorff was to define a measure being independent of the resolution of the measurement, ϵ. Hausdorff dimension

M(ϵ, D_{h}) = inf_{all coverings} ∑_{i} r_{i}^{D_{h}}  for  r_{i} < ϵ

M(D_{h}) = lim_{ϵ→0} M(ϵ, D_{h})

D_{H} = sup(D_{h}, for which M(D_{h}) > 0)

Dimension of the set of rational numbers on [0,1]. Enumerable set. Cover all of the points up to the k-th point with intervals of length (1/2)^{k}.

M(k, D_{h}) = k(1/2)^{k}D_{h}

goes to zero if k → ∞ for any D_{h} > 0, thus D_{H} = 0. D_{box} = 1!! USEFUL RULES

– projecting a D < d_{p} dimensional fractal onto a d_{p} dimensional surface results in D_{p} = D. For D ≥ d_{p}, however, D_{p} = d_{p}.

– Cutting out a d − m dimensional slice (cross-section) of a D dimensional fractal embedded into a d dimensional space usually leads to a D − m dimensional object.

– Consider two sets A and B with D_{A} and D_{B}, respectively. Multiplying them results in D = D_{A} + D_{B}.

– The union of two fractal sets A and B with D_{A} > D_{B} has the dimension

D = D_{A}.

– The fractal dimension of the intersection of two fractals with D_{A} and D_{B} is given by D_{A∩B} = D_{A} + D_{B} − d. To see this, consider a box of linear size L within the overlapping region of two growing stochastic fractals. The density of A and B particles is respectively proportional to L^{D_{A}}/L^{d} and L^{D_{B}}/L^{d}. The number of overlapping sites N ∼ L^{D_{A∩B}} is proportional to these densities and to the volume of the box which leads to the above given relation.

– The distribution of empty regions (holes) in a fractal of dimension D scales as a function of their linear size with an exponent −D − 1. E.g, in 1d

n(ϵ, Δϵ) ∼ ϵ^{−D−1}Δϵ,

where n(ϵ, Δϵ) is the number of gaps (empty regions) of length between ϵ and ϵ−Δϵ. This can be seen by noting that the total length covered with intervals ϵ is

L(ϵ) ∼ ϵ^{−D+1}.

The increase of the uncovered part when ϵ is decreased to ϵ − Δϵ is

\frac{dL(ϵ)}{dϵ}Δϵ ∼ ϵ^{−D}Δϵ.

This comes from the gaps of length between ϵ and ϵ − Δϵ, because they will not be covered any more. Thus,

ϵ^{−D}Δϵ ∼ n(ϵ, Δϵ)ϵ

which is equivalent to the statement.

2.3. TYPES OF FRACTALS 2.3.1. Deterministic and random fractals

Fractals constructed by deterministic iteration. n copies each reduced by a factor 1/r. k steps, k → ∞.

D = \frac{ln n}{ln r}  (2.10)

which is an exact expression for D. Similarly for growing. Non-uniform fractals: As before, the fractal is produced by dividing an original object into parts being reduced versions of it, but this time the factors r_{i} > 1, (i = 1, 2, ...n) can not be all identical. Let N_{i}(ϵ) denote the number of balls of radius ϵ needed to cover the ith part. The number of balls needed to cover the whole fractal is

N(ϵ) = ∑_{i=1}^{n} N_{i}(ϵ).  (2.11)

Since the fractal is self-similar,

N_{i}(ϵ/r_{i}) = N(ϵ)  (2.12)

Then substitution of (2.4) and (2.12) into (2.11) leads to

∑_{i=1}^{n} (\frac{1}{r_{i}} )^{D} = 1  (2.13)

Stochastic fractals by iteration Random fractals (no iteration). They can be best characterized by the so called density-density or pair correlation function

c(r⃗) = \frac{1}{V} ∑_{r⃗′} ρ(r⃗ + r⃗′) ρ(r⃗′)  (2.14)

which is the expectation value of the event that two points separated by r⃗ belong to the structure. For growing fractals the volume of the object is V = N, where N is the number of particles in the cluster, and (2.14) gives the probability of finding a particle at the position r⃗ + r⃗′, if there is one at r⃗′. In (2.14) ρ is the local density, i.e., ρ(r⃗) = 1 if the point r⃗ belongs to the object, otherwise it is equal to zero. Ordinary fractals are typically isotropic ( c(r⃗) = c(r). An object is non-trivially scale-invariant if its correlation function is unchanged up to a constant under rescaling of lengths by an arbitrary factor b:

c(br) ∼ b^{−α}c(r)  (2.15)

with α a non-integer number larger than zero and less than d. The only function which satisfies (2.15) is

c(r) ∼ r^{−α}  (2.16)

To relate D and α we calculate (for a growing fractal) the number of particles N(L) within a sphere of radius L

N(L) ∼ ∫_{0}^{L} c(r)d^{d}r ∼ L^{d−α},  (2.17)

where the summation in (2.14) has been replaced by integration. Thus

D = d − α  (2.18)

which is a result widely used for the determination of D from the density correlations within a random fractal.

EXAMPLES Example 2.1. One of the simplest and best known fractals is the so called triadic Cantor set. (2.10) gives D = ln 2/ ln 3 = 0.6309.... In general, Cantor sets with various n and r can be constructed. Two sets with n = 2, r = 4 and n = 3, r = 9 have the same fractal dimension D = 1/2, but different overall appearance. Example 2.2. Let us consider again the unit interval, and replace it with a curve consisting of 5 intervals of unit length. For the so called Peano curve D = ln 9/ ln 3 = 2, and in the limit of k → ∞ it establishes a continuous correspondence between the straight line and the plane. Example 2.3. The Sierpinski gasket. Loops on all levels. D = ln 3/ ln 2 ≃ 1.585. Example 2.4. Julia sets are derived from

z′ = f(z) = z^{2} − μ,  (2.19)

where z and μ are complex numbers. The set of z values (points in the plane) which is invariant under the transformation (2.19) can be called self-squared, and for a fixed value of μ this set is in most of the cases a fractal. Mandelbrot studied the convergence properties of the recursion

z_{k+1} = z_{k}^{2} − μ  (2.20)

corresponding to (2.19), as a function of μ. He found that the region of μ values for which the iterates of z_{0} = 0 under (2.20) fail to converge to ∞ is bounded by a fractal curve. Moreover, marking the points in the μ plane with colours depending on the number of iterations k needed for z_{k} > 2 one obtains an extremely complex, beautiful picture. Example 2.5. Non-Uniform tree. To obtain the fractal dimension of this tree-like object we first note that reducing the configuration generated in the kth step by a factor 2^{k} one obtains a structure that would have been generated by the division technique with 1/r_{1} = 1/r_{2} = 1/2 and 1/r_{3} = 1/r_{4} = b/2. Now we use (2.13) and calculate D from the equation

2 (\frac{1}{2})^{D} + 2 (\frac{b}{2})^{D} = 1  (2.22)

which can be solved numerically for any given b. For some of the b values the implicit equation (2.22) can be inverted, e.g., if b = 1/2, D = 1 − ln(\sqrt{3} − 1)/ ln 2 ≃ 1.45. Example 2.6. For random walks < R^{2}(t) >∼ t independently of d. Since (N(R) ∼ t), the above expression is equivalent to

N(R) ∼< R^{2}(t) >  (2.23)

and comparing (2.23) with (2.2) we find D = 2 < d if d > 2. Example 2.7. Levy flight. All directions are chosen with the same probability, and the probability of making a jump longer than Δx_{0} is P r(Δx > Δx_{0}) = Δx_{0}^{−D_{L}}, except that P r(Δx > Δx_{0}) = 1 when Δx_{0} < 1. All jumps are independent, and the resulting trajectory has a fractal dimension D = D_{L}. 2.3.2. Self-affine fractals They are invariant under dilation transformation only if the lengths are rescaled by direction dependent factors.

If for a single-valued, nowhere-differentiable function F (x) has the property

F (x) ≃ b^{−H}F (bx)  (2.24)

it is self-affine, where H > 0 is some exponent. Invariant under rescaling: shrinking along the x axis by a factor 1/b, followed by rescaling of values of the function by a different factor equal to b^{−H}. For random functions the above considerations are valid in a stochastic sense (expressed by using the sign ≃). Equivalent definition:

< |F (x + Δx) − F (x)| >∼ Δx^{H}.

Deterministic model for a self-affine function defined on the unit interval. Local and global dimensions. First one should make a choice for the unit of both F and t. Then a quantity t_{c} can be defined through

|F (t + t_{c}) − F (t)| ∼ |t_{c}|.  (2.29)

This t_{c} can be called the crossover scale for the given process. t_{c} depends on the units, therefore, the position of the crossover is in general not intrinsic.

EXAMPLES Example 2.8. X_{H}(t) denotes the distance of a particle from the origin, randomly walking on a straight line, as a function of the time t. The distances are measured from the starting point, the direction of the jumps is chosen randomly (but not necessarily independently), and

⟨X_{H}^{2}(t)⟩ ∼ t^{2H}.  (2.31)

A random walk with 0 < H ≠ 1/2 < 1 is called fractional Brownian motion. It is well known that for the ordinary Brownian motion, when the jumps are independent and their distances have a Gaussian distribution, H = 1/2 (i.e., D_{B} = 2−H = 1.5) and X_{1/2}(t) satisfies (2.24) stochastically. Similarly, X_{H}(t) and b^{−H}X_{H}(bt) can be shown to be identical in distribution. Since 0 < H < 1, it follows from (2.31) that the global dimension corresponding to the behaviour for t_{c} ≫ 1 is 1, because X_{H}(t)/t → 0 as t → ∞. The local dimension: During a time interval δt, |max [X_{H}(t)] − min [X_{H}(t)]| is of the order of (δt)^{H}. Covering the part of X_{H}(t) on the interval δt by squares of side δt requires on the order of (δt)^{H}/δt = δt^{H−1} squares. Therefore, covering X_{H}(t) on the interval [0, 1] requires

N(δt) ∼ \frac{δt^{H−1}}{δt} = δt^{−(2−H)}  (2.32)

which according to the definition (2.5) leads to D_{B} = 2 − H

The points at which X_{H}(t) = 0 form the zero set of the fractional Brownian motion. It is a random Cantor set of fractal dimension D = 1 − H. 0 < H < 1/2 is antipersistent, the case H > 1/2 is persistent, meaning that its jumps have positive correlation with the direction of the previous jump. To see this we set X_{H}(0) = 0 and define the past increment as −X_{H}(−t) and the future increment as X_{H}(t). Then

⟨−X_{H}(−t)X_{H}(t)⟩ =
= 2^{−1}{⟨[X_{H}(t) − X_{H}(−t)]^{2}⟩ − 2⟨[X_{H}(t)]^{2}⟩} =
= 2^{−1}(2t)^{2H} − t^{2H}.  (2.33)

Dividing by ⟨X_{H}^{2}(t)⟩ = t^{2H}, one obtains the correlation of increments which is independent of t; it is equal to 2^{2H−1} − 1 vanishing, as expected, for H = 1/2. Fourier spectrum: the coefficients of the series, A(f), are independent Gaussian random variables scaling as

|A(f)| ∼ f^{−H− \frac{1}{2}} .  (2.34)

Brownian surfaces. Example 2.9 In 1872 Weierstrass-Mandelbrot function:

C(t) = ∑_{n=−∞}^{∞} \frac{1 − cos(b^{n}t)}{b^{(2−D)n}},  (2.35)

In the range of parameter values

1 < D < 2,  b > 1  (2.36)

C(t) is continuous but dC(t)/dt diverges everywhere. The selfaffinity of C(t) for b > 0 can be easily shown by a formal replacement of n by n + 1 in (2.35) leading to the scaling relation C(t) = b^{−(2−D)}C(bt) which is equivalent to the definition (2.24) of self-affine functions. Example 2.10 Multiplying two cantor sets. Let us assume that in the kth step the sides of the rectangles are l_{1}^{(k)} = l_{1}^{k} and l_{2}^{(k)} = l_{2}^{k}, where l_{1} < l_{2}. Thus, D_{1} = ln 2/ ln(1/l_{1}) and D_{2} = ln 2/ ln(1/l_{2}), respectively, and we obtain for the local fractal dimension D_{L} = ln 2/ ln(1/l_{1}) + ln 2/ ln(1/l_{2}). 2.3.3. Fat fractals Until now D < d.

In general, the resolution dependent volume of an object can be written in the form

V (l) = V (0) + f(l),  (2.37)

where V (0) = V_{0} is the volume in the limit l → 0. For thin (ordinary) fractals V (0)=0, and f(l) ∼ l^{d−D} with D < d. For fat fractals V (0) > 0, but f(l) follows a power law with an exponent.

V (l) ≃ V (0) + Al^{β}  (2.38)

β = lim_{l→0} \frac{ln[N(l)l^{d}] − V (0)}{ln l},  (2.39)

where N(l) is the number of d-dimensional balls needed to cover the structure. By definition β > 0, and it is equal to ∞ for non-fractal sets. Relation to Peano curve, objects like blood vessels

EXAMPLES Example 2.11 Modified Cantor set. To “fatten” the thin fractal delete instead the central \frac{1}{3}, then \frac{1}{9}, then \frac{1}{27}, etc., of each remaining interval. If the sum of deleted intervals does not go to 1, the remaining set has a dimension D=d=1. In general: Suppose that we cut out a piece of volume v_{1} from the unit hypercube of dimension d. Next, from each of these cubes a similar piece of relative volume v_{2} is cut out, and this process is repeated infinitely many times. The total volume remaining after the kth iteration is

V_{k} = (1 − v_{1})(1 − v_{2})...(1 − v_{k}) = ∏_{0}^{k}(1 − v_{k}).  (2.44)

V_{k} decreases as k → ∞ to a limiting value V. For v_{k} fixed one has V = 0, however, if ∑_{0}^{∞} v_{k} < ∞, the limiting volume is positive, ∏_{0}^{∞}(1 − v_{k}) > 0.

FRACTAL MEASURES

Scalar (field-like) quantities in the presence of a fractal object. Consider an isolated, charged object. For a branching ramified fractal, the electric field (the derivative of the potential) at the surface has infinitely many singularities (divergence and approach to zero) with different non-integer exponents. 3.1. MULTIFRACTALITY Deterministic multiplicative process in 2d. In the first step the starting object (square) is replaced by its three smaller copies with corresponding reduction factors 1/r_{1}, 1/r_{2} and 1/r_{3} with weight factor (probability) denoted by P_{1}, P_{2} and P_{3}. In the next step (k = 2) the same procedure is repeated for each of the squares. Fractal measure at k → ∞. If P_{1} > P_{2} and and k ≫ 1, P_{1}^{k} ≫ P_{2}^{k}. Weights and sizes of all magnitude. Imagine that a d dimensional hypercubic lattice with a lattice constant l is put on the fractal, and denote by p_{i} the stationary probability associated with the ith box of volume l^{d}, where

p_{i} = ∫_{ith box} dμ(x⃗)  (3.1)

with ∑_{i} p_{i} = 1. We denote ϵ = l/L, and ϵ → 0 (corresponding to either l → 0 or in the growing case to L → ∞) will be studied.

p(ϵ) ∼ ϵ^{α},  (3.2)

where ϵ ≪ 1 and α can take on a range of values. There are usually many boxes with the same index α.

N_{α}(ϵ) ∼ ϵ^{−f(α)},  (3.3)

where, in view of (2.4), f(α) is the fractal dimension of the subset of boxes characterized by the exponent α. The exponent α can take on values from the interval [α_{∞}, α_{−∞}], and f(α) is usually a single humped function with a maximum

max_{α} f(α) = D.  (3.4)

The f(α) spectrum of an ordinary uniform fractal is a single point on the f − α plane. In other words, a measure is multifractal if the histogram N(p, ϵ) of the weights p(ϵ) tends to assume a single non-trivial function in the limit of ϵ → 0 after rescaling. Rescaling here means that we plot − ln N(p, ϵ)/ ln ϵ versus ln p(ϵ)/ ln ϵ. 3.2. RELATIONS AMONG THE EXPONENTS The qth moment of the box probability distribution is

χ_{q}(ϵ) ≡ ∑_{i} p_{i}^{q}  (3.5)

for −∞ < q < ∞. For q = 0 (3.5) gives N(ϵ), the number of boxes of size ϵ needed to cover the fractal support (the region, where p_{i} ≠ 0). Therefore,

χ_{0}(ϵ) = N(ϵ) ∼ ϵ^{−D},  (3.6)

where D is the dimension of the support. Since the distribution is normalized, χ_{1}(ϵ) = 1. The scaling of χ_{q}(ϵ) for ϵ → 0 generally depends on q in a non-trivial way

χ_{q}(ϵ) ∼ ϵ^{(q−1)D_{q}},  (3.7)

The factor (q − 1) was chosen to satisfy the relation χ_{1}(ϵ) = 1 automatically. Correspondingly, for all q we have D_{q} > 0. It can be shown that the D_{q} values monotonically decrease with growing q. From the comparison of (3.6) with (3.7) it follows that D_{0} = D. In the simple case of uniform fractals with a uniform distribution all D_{q}-s are equal to D. When ϵ → 0 the dominant contribution to the sum (3.5) comes from a subset of boxes forming a fractal with a fractal dimension f_{q}

N_{q}(ϵ) ∼ ϵ^{−f_{q}},  (3.8)

where N_{q}(ϵ) is the number of boxes giving the essential contribution in (3.5). In addition, all of these boxes have the same p_{i} = p_{q}.

p_{q} ∼ ϵ^{α_{q}}  (3.9)

for ϵ → 0. The f_{q} and α_{q} spectra defined by (3.8) and (3.9) provide an alternative description of a fractal measure with regard to (3.2) and (3.3). Why f_{q} and α_{q}? Even if f_{q} and qα_{q} smoothly changing functions, because ϵ → 0 the function N_{q}p_{q}^{q} = e^{−f_{q}+qα_{q}} ha a very sharp maximum ϵ → 0 around the minimum of e^{−f_{q}+qα_{q}}. For all other values of α ≠ α_{q} the contribution of the corresponding boxes is negligible. For a simple fractal f_{q} = α_{q} = D.

Relation between D_{q} and the spectra f_{q} and α_{q} since

χ_{q}(ϵ) ≃ N_{q}(ϵ)p_{q}^{q},  (3.10)

and the essential contribution to χ_{q}(ϵ) comes from boxes with p_{q}. Inserting (3.7), (3.8) and (3.9) into (3.10) we get

(q − 1)D_{q} = qα_{q} − f_{q}.  (3.11)

Consistent with D = D_{0} = f_{0}. Since f is finite we have D_{±∞} = α_{±∞}. Now it is possible to relate the generalized dimensions to the f(α) spectrum. First we express χ_{q}(ϵ) for ϵ → 0 through f(α) using (3.2), (3.3) and (3.5)

χ_{q}(ϵ) ∼ ∫_{α_{∞}}^{α_{−∞}} ϵ^{qα′−f(α′)}dα′,  (3.12)

where α is a quasi-continuous variable. For ϵ ≪ 1 the integral will be dominated by the value of α which minimizes the exponent. This leads to the condition

\frac{df(α)}{dα}|_{α_{q}} = q,  (3.13)

where α_{q} is the value of α for which qα − f(α) is minimal. We see that (3.12) is consistent with (3.11) if, in addition to (3.13),

f_{q} = f(α_{q}).  (3.14)

Thus, knowing f(α) we can find α_{q} from (3.13) and then f_{q} from (3.14) (Halsey et al 1986). Alternatively, if D_{q} is given, one can calculate α_{q} from the relation

α_{q} = \frac{d}{dq} [(q − 1)D_{q}]  (3.15)

which can be obtained from (3.11) using (3.13) and (3.14). If we know α_{q} and D_{q}, f_{q} can be determined from (3.11) and finally, f(α) from (3.14). Therefore, the spectra D_{q} and f(α) represent equivalent descriptions of multifractals, since they are Legendre transforms of each other. For the evaluation of experimental data usually first the set of p_{i} values is determined.

D_{q} = lim_{ϵ→0} [\frac{1}{q − 1} \frac{ln ∑_{i} p_{i}^{q}}{ln ϵ} ]  (3.16)

with the help of the corresponding log-log plot. Finally, the resulting plot of D_{q} versus q is numerically derived to obtain f(α). For the q = 1 case we have from equation (3.11) and its derivative taken at q = 1

D_{1} = α_{1} = f_{1}.  (3.17)

Furthermore, inserting (3.5) into (3.7) and taking the q → 1 limit (using l’Hospital’s rule) one gets

D_{1} = lim_{q→1} \frac{ln ∑ p_{i}^{q}}{(q − 1) ln ϵ}

lim_{q→1} \frac{d/dq(ln ∑ p_{i}^{q})}{d/dq[(q − 1) ln ϵ]}

using d/dq [ln h(q)] = 1/h(q) d/dq[h(q)] and dp^{q}/dq = p^{q} ln p, after taking the limit we get

− ∑_{i} p_{i} ln p_{i} ∼ D_{1} ln(1/ϵ).  (3.18)

The left hand side of (3.18) is a familiar expression from information theory and corresponds to the amount of information associated with the distribution of p_{i} values. D_{1} is called information dimension. A distribution with D_{1} < D is necessarily a fractal measure. 3.3. FRACTAL MEASURES CONSTRUCTED BY RE-CURSION In the first step (k = 1) the ith part obtained from the seed object by reduction using a rescaling factor (1/r_{j}) < 1 will be given a probability P_{j}, with ∑_{j}^{n} P_{j} = 1, where n is the number of pieces the starting configuration is substituted with. At the next stage (k = 2) each part is divided into n parts with probabilities reduced further by P_{j} and size rescaled by 1/r_{j}. The multifractal is obtained in the k → ∞ limit. n parts each being a rescaled version of the whole support by a factor 1/r_{j}. The total measure associated with the jth such part is P_{j}. Therefore,

χ_{q,j}(ϵ) = ∑_{i} p_{j,i}^{q} = P_{j}^{q}χ_{q}(ϵr_{j}),  (3.19)

where χ_{q,j}(ϵ) is the quantity defined by (3.5) evaluated for the jth part and p_{j,i} is the probability of the ith box in the jth part. For the whole system

χ_{q}(ϵ) = ∑_{j=1}^{n} χ_{q,j}(ϵ).  (3.20)

Using (3.7) and (3.19) in (3.20) we get

∑_{j=1}^{n} P_{j}^{q}r_{j}^{(q−1)D_{q}} = 1  (3.21)

which is an implicit equation for the generalized dimensions D_{q}. Special cases. For general q, but identical rescaling factors r_{j} ≡ r equation (3.21) can be solved for D_{q}

D_{q} = \frac{1}{q − 1} \frac{ln ( ∑_{j=1}^{n} P_{j}^{q})}{ln(1/r)}.  (3.22)

Multifractal properties are trivially lost, if all P_{j}-s and r_{j}-s are equal. Furthermore, distributing the measure on the support with a constant density, i.e. choosing

P_{j} = \frac{r_{j}^{−d}}{∑_{j=1}^{n}(1/r_{j})}  (3.23)

results in a non-trivial D_{q} spectrum (multifractal geometry.

EXAMPLES

Example 3.1. Hierarchical network of resistors. At the kth step of the iteration procedure each bond of length l = 1 is replaced with a configuration having a linear size equal to r = 3 and made of n = 6 bonds of unit length. This growing fractal in the k → ∞ limit has a fractal dimension D = ln 6/ ln 3. The bonds have the same resistivity and the voltage between the two terminal bonds is equal to V_{0} = 1. Then, in the unit cell there will be two bonds with voltage drop V_{1} = 1/3 and four others with V_{2} = 1/6. The sum of all voltage drops is not equal to 1, therefore, the distribution is not normalized. (It can be normalized by simply choosing V_{0} = 3/4.)

∑_{i} v_{i}^{q} ∼ L^{−(q−1)D_{q}},  (3.34)

where v_{i} is the voltage drop across the ith bond and L = (l/ϵ) = 3^{k} is the linear size of the network. The sum of the v_{i}^{q}-s after completing the kth recursion is

∑_{m} N_{m}v_{m}^{q} = (2V_{1}^{q} + 4V_{2}^{q})^{k}.  (3.35)

In the above expression

N_{m} = 2^{k+m}\binom{k}{m}  (3.36)

is the number of bonds with the voltage drop

v_{m} = V_{1}^{k−m}V_{2}^{m}  (3.37)

with 0 ≤ m ≤ k. Substituting (3.35) into (3.34) and using the actual voltage values we get

D_{q} = \frac{1}{q − 1} \frac{ln[2 (\frac{1}{3})^{q} + 4 (\frac{1}{6})^{q}]}{ln(1/3)}  (3.38)

LOCAL GROWTH MODELS

Particles, clusters, aggregates. Lattice or off-lattice. First or further neighbours. Stochasticity. In principle all stochastic models can lead to all possible configurations. Typical config. 5.1. SPREADING PERCOLATION Seed particle(s). Neighbouring sites are considered live. Next, one of these is chosen randomly and (i) filled with a particle with a probability p or (ii) killed with a probability 1 − p. Generate r with a uniform distribution on [0, 1] and fill the site if r < p. New neighbours become living sites. Spreading phenomena, e.g., epidemics, chemical reactions, flame propagation, etc. relation to ”equilibrium” percolation. When p is increased, at p = p_{c} a transition. Appearance of a connected infinite cluster having a density for 0 < p − p_{c} ≪ 1

P (p) ∼ (p − p_{c})^{β},  (5.3)

where p_{c} is called percolation threshold. correlation length of he infinite cluster at p_{c}

ξ ∼ |p − p_{c}|^{−ν_{p}}  (5.4)

where at ξ the power law decay of the pair correlation function c(r) (see (2.14)) crosses over into a constant behaviour. As before, because of scaling we can assume that for ξ ≫ a and r ≫ a (a is the lattice spacing) the only relevant length is ξ

c(r) ∼ P (p)f(r/ξ),  (5.5)

where the scaling function f(x) approaches a constant for x → ∞. For x ≪ 1 we expect that c(r) is independent of ξ which can be satisfied only if f(x) ∼ x^{−β/ν_{p}} for x ≪ 1, since

p − p_{c} ∼ ξ^{−1/ν_{p}}  and  P (p) ∼ ξ^{−β/ν_{p}}

From here c(r) ∼ r^{−β/ν_{p}} for r ≪ ξ. Thus, the fractal dimension at p_{c} (ξ → ∞)

D = d − \frac{β}{ν_{p}} .  (5.6)

For example, in two dimensions ν_{p} = \frac{4}{3} and β = \frac{5}{36} gives for the fractal dimension of the infinite cluster at p_{c} D = \frac{91}{48} ≃ 1.896. 5.2. INVASION PERCOLATION Fluid in porous media.

i) A random number (on the unit interval) is assigned to each site. ii) Starts with a seed particle or a surface and goes on by subsequent occupation of one of the perimeter sites. iii) The perimeter site with the smallest random number r is occupied. Leads to fractal (first spanning) cluster with D similar to standard percolation. In this 5.4. RANDOM WALKS Trails of walks on a lattice. Mean squared end-to-end distance ⟨R_{e}^{2}(t)⟩ = R_{0}^{2}(t),

R_{0}(t) ∼ t^{ν}  (5.20)

the quantity D = 1/ν is the fractal dimension of the chain.

5.4.1. Self-intersecting random walks

End-to-end vector at time t is R⃗_{e}(t) = ∑_{i=1}^{t} a⃗_{i}, where a⃗_{i} is a vector of length a equal to the lattice spacing and it has z possible orientation with z being the coordination number. Then

⟨R_{e}^{2}(t)⟩ = ∑_{i,j}⟨a⃗_{i}a⃗_{j}⟩ = a^{2}t + 2 ∑_{i>j}⟨a⃗_{i}a⃗_{j}⟩ ∼ t,  (5.21)

since due to the independence of the orientations of a⃗_{i}-s the cross products vanish. For a walk starting at t = 0 it is also straightforward to show that the probability of finding the walker at a distance R P (R, t) ∼ e^{−R^{2}/Dt}, where D = a^{2}/Δt with Δt = 1 in our case. True self-avoiding walks (TSAW). In particular, the probability p_{ij} of jumping from site i to the neighbouring site j is in the model equal to

p_{ij} = \frac{e^{−g_{ij}n_{j}}}{∑_{k=1}^{z} e^{−g_{ik}n_{k}}} ,  (5.27)

where g_{ij} is a parameter controlling the degree of inhibition, n_{k} is the number of times the site k has been visited before and z is the lattice coordination number. The strongest inhibition is provided by the limit when all g_{ij} → ∞, but TSAW can cross itself even in this limit. This occurs when there is no previously unvisited neighbour, and is made possible by the normalization included in (5.27). Changes the scaling only in 1d.

5.4.2. Growing self-avoiding walks Polymer chains In the case of GSAW the walker chooses an unoccupied neighbouring site randomly, with a probability p = 1/n, where n is the number of free neighbours. If n = 0, the walk terminates; Indefinitely growing self-avoiding walk (IGSAW). The walker recognizes possible traps and avoids them. To generate such a walk in two dimensions one needs local information on the walk only. In order to avoid trapping, he can use the knowledge of the local configuration and the winding number. This number is equal to n_{l} − n_{r}, where n_{l} and n_{r} denote respectively the number of left and right turns made previously by the walker. The value

D = 1.75 = \frac{7}{4}  (5.33)

can be shown to be exact by establishing a correspondence between IGSAW and walks along the external perimeter or “hull” of an infinite percolation cluster. 5.4.3. Walks on fractals The mean squared distance from the origin is expected to scale (time) as

R_{0}^{2}(t) ∼ t^{2/d_{w}},  (5.36)

where the exponent d_{w} depends on the structure of the fractal in a non-trivial way. In general d_{w} > 2. Since t ∼ R_{0}^{d_{w}}, d_{w} is a fractal dimension-like quantity. However, d_{w} is typically not the fractal dimension of the trail, but describes the fractal scaling of mass (number of steps) within a region of radius R_{0}. In the case of fractal substrates returns to already visited sites play a relevant role in all d. This is indicated by the fact that generally d_{w} > D. For example, d_{w} = ln(d + 3)/ ln 2 > ln(d + 1)/ ln 2 = D for the d-dimensional versions of Sierpinski gaskets, or d_{w} = D + 1 for the loopless fivefold fractal.

DIFFUSION-LIMITED GROWTH

Growth processes governed by non-local fields (e.g. temperature distribution). Typically: Laplace equation with moving boundary conditions (diffusion-limited growth). Complex, branching patterns, selforganization.

6.1. DIFFUSION-LIMITED AGGREGATION (DLA) Electrolite with positive metallic ions in a small concentration, and a negative electrode. (DLA) was introduced by Witten and Sander (1981). Rules: One puts a seed particle at the origin of a lattice. Another particle is launched far from the origin and is allowed to walk at random (diffuse) until it arrives at a site adjacent to the seed particle. Then it is stopped, and another particle is launched which stops when adjacent to the two occupied sites, and so forth.

A typical cluster. Fluctuations and instability. Tricks: i) Surrounding radius, ii) killing radius, iii) bookkeeping of the distances 6.1.1. Fractal dimension

N(R) ∼ R^{D}  (6.1)

results in D ≃ 1.7, or calculate the density-density correlation function c(r) (2.14).

c(r) ∼ r^{−α}  (6.2)

leading to α ≃ 0.29 in d = 2. In good agreement with (2.18) ( D = d − α ≃ 1.71). Higher d:

d − 1 < D < d.  (6.4)

Mean-field prediction D = (d^{2} + 1)/(d + 1).

Table 6.1. The fractal dimension (D) of DLA clusters grown on 2 ≤ d ≤ 6 dimensional hypercubic lattices.

d  D  \frac{(d^{2}+1)}{(d+1)}
2  1.71 ± 0.02  1.667 3  2.53 ± 0.06  2.500 4  3.31 ± 0.10  3.400 5  4.20 ± 0.16  4.333 6  5.3  5.286

Lower bound: trail of a randomly walking particle is a fractal of dimension 2. The fractal dimension of the intersection of two fractals of dimensions D and 2 embedded into a d dimensional space is

D_{∩} = D + 2 − d.  (6.5)

If the cluster was ”transparent” for the diffusing particle, i.e., D_{∩} <, the particle would not stick to the aggregate or would stick with the same probability everywhere (leading to a homogeneous structure). Therefore, D_{∩} <, and D > d − 2. Variants, universality.

i) DLA with sticking probability less than 1. ii) Off-lattice DLA. Anisotropy is different! 6.1.2. Anisotropy in DLA Because of growth, radial direction is preferred. To see this one calculates the tangential correlation function c_{R}(θ) defined in d = 2

c_{R}(θ) = \frac{1}{N} ∑_{θ′} ρ_{R}(θ + θ′) ρ_{R}(θ′),  (6.9)

where N is the number of particles in the aggregate, ρ_{R}(θ) = k if there are k particles in a box of size RΔθΔR at the point (R, θ) and ρ_{R}(θ) = 0 otherwise. The summation in (6.9) is taken over θ′ values gradually increased by a fixed small Δθ′ from θ′ = 0 to θ′ = π. For θ ≪ 1 numerically, c_{R}(θ) ∼ θ^{−α_{⊥}} with α_{⊥} ≃ 0.41. Different from α ≃ 0.29 (overall). Since α_{⊥} > α we assume that

c(r, φ) ≃ ρ_{0}r^{−α_{∥}} cos^{2} φ+
ρ_{1}r^{−α_{⊥}} sin^{2} φ,  (6.10)

where ρ_{0} and ρ_{1} are constants and α_{∥} describes the decay of c(r, φ) in the radial direction.

N(a) ∼ ∫_{0}^{a} rdr ∫_{0}^{2π} c(r, φ)dφ ∼
∼ πa^{2−α_{∥}}(ρ_{0} + ρ_{1}a^{α_{∥}−α_{⊥}}),  (6.11)

where N(a) is the number of particles within a circle of radius a. The fractal dimension is given by

D = lim_{a→∞} \frac{ln N(a)}{ln a} = lim_{a→∞} 2 − α_{∥} + D_{1}(a) = 2 − α_{∥}  (6.12)

with D_{1} = [ln π(ρ_{0}+ρ_{1}a^{α_{∥}−α_{⊥}})]/ ln a representing a slowly decaying correction to scaling (agreement with observation). Direction dependent sticking probability. i) the particle sticks with probability 1 if the left or right nearest neighbour sites to its actual position are occupied, ii) otherwise, it only has a probability p_{s} < 1 of sticking. The characteristic lengths of the aggregate X and Y in the easy (x) and hard (y) direction of growth increase as X ∼ N^{\frac{2}{3}} and Y ∼ N^{\frac{1}{3}} in the limit N → ∞. Since X × Y ∼ N, the area covered by the cluster grows linearly with N. 6.1.3. Theoretical approaches to DLA Mean field (fluctuations are averaged out) result:

D = \frac{d^{2} + 1}{d + 1}  (6.17)

which gives values surprisingly close to the simulation results. Most successful: Investigation of the growth probability scaling near the tips based on the electrostatic analogy. The probability of finding a diffusing particle in a certain site r⃗ outside the cluster satisfies the Laplace equation ∇^{2}p(r⃗) = 0 with the boundary condition p = 0 on sites adjacent to the cluster. The flux of particles onto the cluster at a point r⃗_{0} on its surface is proportional to ∇p(r⃗_{0}).

Electrostatic analog: a charged conductor having the shape of a cluster. The local electric field E⃗ = −∇ϕ is the analog of the flux of particles dR/dt = (dN/dt)/(dN/dR) onto that point. We assume that the tip area can be represented by a cone of exterior half angle φ. From conformal transformation it is known that the potential around such a tip is

ϕ(r, ϑ) = C r^{π/(2φ)} cos(πϑ/2φ),  (6.24)

where C is a normalization factor. The flux of particles onto the cone edge at a distance λ from its tip is

∇ϕ(λ, φ) = \frac{C π}{2φ} λ^{π/(2φ)−1}.  (6.25)

One obtains dN/dt by integrating the above expression from λ = 0 to some large cutoff at λ ∼ R and dR/dt by integrating up to a small cutoff λ ∼ a, where a = 1 is the size of diffusing particles or the lattice spacing. We find from (6.25) that dN/dt ≃ C R^{π/(2φ)} and dR/dt ≃ C which leads to

R^{D−1} ≃ \frac{dN}{dR} = \frac{dN/dt}{dR/dt} ≃ R^{π/(2φ)}.  (6.26)

Thus,

D = 1 + \frac{π}{2φ} = \frac{3π − θ}{2π − θ}  (6.27)

Good agreement, however, θ is input parameter. θ = \frac{π}{2} gives D = \frac{5}{3}. For θ_{eff} ≃ 0.59π ≃ 106^{∘} one gets D ≃ 1.71.

6.1.4. Multifractal scaling The growth is governed by the distribution of p(r⃗_{j}), where p(r⃗_{j}) is the probability that the next growth event takes place at the site being at r⃗_{j}. Multifractal. Growth velocity distribution. Covering with boxes. Then,

D_{q} = lim_{(l/L)→0} \frac{1}{q − 1} \frac{ln ∑_{i}[p_{i}(l/L)]^{q}}{ln(l/L)} ,  (6.30)

The f(α) Legendre transformation. Bell shaped? Then there is an α_{min} (or p_{max}). Writing (6.27) in the form D = 1 + x and (6.25) as ∇ϕ ∼ l^{x−1}, we see that the integral of ∇ϕ ∼ p_{i} (total probability in a box of size l at the tip) scales as l^{x} ∼ l^{α_{min}}. Thus,

D = 1 + α_{min}.  (6.31)

The ”measured” value α_{min} ≃ 0.7 is in good accord with this (D ≃ 1.7) (smallest singularity means fastest growth) Dependence of α_{max}, or the minimum growth probability p_{min} on the cluster diameter L?

i) For finite α_{max} (fjords)

p_{min} ∼ L^{−α_{max}}.  (11.1)

ii) For ”channels”

p_{min} ≃ e^{−aL},  (11.2)

where a is some constant. iii) Another possibility is

p_{min} ≃ e^{−b(ln L)^{c}},  (11.3)

with b and c ≃ 2 constants. i) Multiplicative screening, ”regular” f(α). Let us now discuss the ii) Exponential screening: phase transition (convexity criterion). Exponentially small p means arbitrarily large α. The shape of f(α) must be either a) If we require f(α) > 0 then after some α_{0} it remains constant as α → ∞ (left-sided f(α)) (nonanalytic), or b) f(α) becomes negative for large α values, still ending in a straight line. The notion phase transition is motivated by the formal analogy with thermodynamics: χ_{q}, can be rewritten in the form

Z(β, L) = ∑_{i} p_{i}^{β} ∼ L^{−F (β),}  (11.5)

where Z → χ, β → q and F → (q − 1)D_{q} play respectively the role of the partition function, the temperature and the free energy. Non-analycity in D_{q} results in the same in f(α).

6.2. DIFFUSION-LIMITED DEPOSITION

Fractal dimension is the same as for DLA. Analogy with a cut at large radius. α_{∥} and α_{⊥}. Ensemble of aggregates. Cluster size distribution function, n_{s}(N) (normalized). Scaling

n_{s}(N) ∼ s^{−τ} f (\frac{s^{σ}}{N} ) ,  (6.44)

where f(x) is a cutoff function with f(x) ≃ 1 for x ≪ 1 and f(x) ≃ 0 for x ≫ 1 and s ≫ 1 and N ≫ 1, N is the number of particles per system size (L).

\begin{matrix}N = ∑_{s=1}^{∞} sn_{s}(N) & ∼ ∫ ^{∞} s^{1−τ} f(s^{σ}/N)ds ≃ \\ ≃ σ^{−1}N^{(2−τ)/σ} ∫ ^{∞} z^{(2−τ−σ)/σ}f(z)dz ≃ \\ ≃ N^{(2−τ)/σ} \end{matrix}  (6.45)

scaling law

τ = 2 − σ  (6.46)

τ < 2 must hold to have a diverging sum in Eq. (6.45) if N goes to infinity. The main contribution comes from the s ≫ 1 clusters.

Deterministic model of deposition The largest clusters in the k-th stage contain s_{max}(k) = ∑_{j}^{k} 5^{j−1} = 5^{k}/4 particles and the number of these clusters per nucleation site is 3^{−k}. The cluster size distribution can be written

n_{s}(N) ∼ \frac{1}{3^{l}5^{l}} θ (1 − \frac{s}{(5^{k}/4)}) ,  (6.53)

where θ is the step function and we took into account that the normalized number of clusters of size 5^{l} (l < k) is 2/3^{l+1} and that these delta function-like peaks are separated from each other by a distance proportional to 5^{l}. Since s_{l} ∼ 5^{l} and N = (5/3)^{k} one recovers the scaling form (6.44) with τ = 1 + ln 3/ ln 5 and σ = 1 − ln 3/ ln 5, because,

3^{−l}5^{−l} ∼ s_{l}^{− ln 3/ ln 5}s_{l}^{−1};

\frac{s^{σ}}{N} = \frac{s}{N^{1/σ}} ∼ \frac{s}{(5^{k}/3^{k})^{1/σ}} = \frac{s}{5^{k}}

and k ln 5 = k(ln 5 − ln 3)/σ, so that the scaling law τ = 2 − σ is satisfied. Since D = ln 5/ ln 3, the scaling relation

τ = 1 + \frac{d_{surface}}{D}

is also fulfilled in the deterministic (and the stochastic) model. τ through D!.

6.3. DIELECTRIC BREAKDOWN MODEL

i) ϕ = ϕ_{0} = 0 in the conducting phase, where ϕ is the electric potential satisfying Laplace’s equation

∇^{2}ϕ = 0,  (6.54)

and ii) the growth velocity is stochastically proportional to some power η of the local electric field E⃗ = −∇ϕ. The actual model in two dimensions (6.54)

ϕ_{i,j} = \frac{1}{4}(ϕ_{i−1,j} + ϕ_{i+1,j} + ϕ_{i,j−1} + ϕ_{i,j+1}),  (6.55)

where ϕ_{i,j} is the value of ϕ in the grid site i, j.

ϕ_{i,j} = 0  for sites belonging to the cluster,  (6.56)

and ϕ_{i,j} = −1 for sites on a large circle. Motion of the interface is represented by an expression for the growth probability at the site i_{a}, j_{a} adjacent to the cluster

p_{i_{a},j_{a}} = C∇ϕ_{i_{a},j_{a}}^{η} = −Cϕ_{i_{a},j_{a}}^{η},  (6.57)

normalization 1/C = ∑ ϕ_{i,j}^{η} summation running over all nearest neighbours. D depends on η!

GROWING SELF-AFFINE SURFACES

Compact object with rough surface. Typical (if the development of the interface is marginally stable and the fluctuations are relevant, leads to self-affine fractals ). Here d is the substrate dimension.

Aggregation models:

1. Eden model: At each step a single perimeter site is filled with probability 1/N_{p}, where N_{p} is the total number of perimeter sites. Slight asphericity of huge single-seed Eden clusters. For deterministic trivial.

2. Ballistic deposition: Raining sticky particles. Surface is defined by ignoring overhangs

3. Simple random deposition does not lead to fractal. Heights are distributed according to a Poisson distr. Since neighbouring columns are independent, the surface is not continuous. For large N Poisson is like Gaussian, width grows as \sqrt{N/L}. t = N/L =< h > 4. However, simple deposition with surface relaxation is already non-trivial! (Edwards-Wilkinson)

5. Solid-on-solid models: height difference between two neighbours cannot be larger than 1. For the above non-trivial cases simulations show:

DYNAMIC SCALING The root mean-square of the height fluctuations w(L, t) of the surface h(r, t) is

w(L, t) = [⟨h^{2}(r, t)⟩_{r} − ⟨h(r, t)⟩_{r}^{2}]^{1/2},  (12.1)

The only scale in the problem is the linear size of the substrate L, thus, w depends only on some power of L and a ratio of the form t/L^{z}.

w(L, t) = L^{α}f(t/L^{α/β}),

where α/β = z is the dynamic scaling exponent. In the limit x ≪ 1, the width depends only on t and this implies that for small x the scaling function f(x) has to be of the form f(x) ∼ x^{β}. Saturation at large times means that f(x) goes to a constant. Thus,

f(x) ∼ \{ \begin{matrix}x^{β}, & if x ≪ 1 \\ Const, & if x ≫ 1. \end{matrix}  (12.2)

An alternative: height-height correlation (or height difference) function c(r, t)

c(r, t) = ⟨|h(r⃗′, t′) − h(r⃗′ + r⃗, t′ + t)|⟩_{r⃗′,t′},  (12.3)

c(r, t) (isotropic case) scales the same way as the width,

c(r, t) ∼ r^{α}f (\frac{t}{r^{z}} )  (12.4)

with an f(x) analogous to (12.2), however, in this case L is replaced by r. Correspondingly, for long times and for r ≪ L the correlation function c(r, 0) behaves as

c(r, 0) ∼ r^{α},  (12.5)

and for fixed r and short times

c(0, t) ∼ t^{β}.  (12.6)

CONTINUUM EQUATION APPROACH General form (x for simplicity → x):

\frac{∂h(x, t)}{∂t} = G(h(x, t), x, t) + η(x, t)

⟨η(x, t)⟩ = 0

⟨η(x, t)η(x′, t′)⟩ = 2Cδ(x − x′)δ(t − t′)

Symmetry principles (constraints for G): Invariances (equation should be invariant after certain transformations):

i) Translation in time. (t → t + δ_{t}). t^{2} allowed in the equation? No, because it becomes t^{2} + 2tδ_{t} + δ_{t}^{2}. No explicit t dependence is allowed. Note: ∂h(x, t)/∂t is OK, ∂h(x, t)/∂(t + δ_{t}) = ∂h(x, t)/∂t.

ii) Translation along growth direction (independence of where is h = 0). No explicit h dependence in G. However, ∇^{n}h possible.

iii) Translation along thew substrate. No explicite x dependence (again, ∂^{n}/∂x^{n} is OK). iv) Rotation and inversion (left and right): odd derivatives excluded.

v) Up/down symmetry. Rules out even powers (as (∇h)^{2}). Valid for equilibrium only! In non-equilibrium growth leads to selected direction.

Linear theory On large scale ∇h is small (self-affinity). Higher order terms neglected (Edwards-Wilkinson eq.)

\frac{∂h(x, t)}{∂t} = ν∇^{2}h + η(x, t)

Solving by scaling approach: x⃗ → x⃗′ = x⃗/b and h → h′ = h/b^{α} leaves the interface statistically unchanged. Time is rescaled too: t → t′ = t/b^{z}. Next:

⟨η(bx⃗, b^{z}t)η(bx⃗′, b^{z}t′)⟩ =
= 2Cb^{−(d+z)}δ(x⃗ − x⃗′)δ(t − t′)

Here we used

δ^{d}(ax⃗) = a^{−d}δ^{d}(x⃗)

a general property of the delta function. If ηη scales like above, then we assume η(bx⃗, b^{z}t) → b^{−(d+z)/2}η(x⃗, t) and

b^{α−z} \frac{∂h(x, t)}{∂t} = νb^{α−2}∇^{2}h + b^{−(d+z)/2}η

\frac{∂h(x, t)}{∂t} = νb^{z−2}∇^{2}h + b^{−(d+z)/2−α}η

for the above terms to be independent of b

α = \frac{2 − d}{2},  β = \frac{2 − d}{4},  z = 2.

d = 2 critical dimension.

Nonlinear case: The Kardar-Parisi-Zhang (KPZ) equation Growth perpendicular to the surface with a given velocity.

\frac{∂h}{∂t} = ν∇^{2}h + λ/2(∇h)^{2} + v + η(x⃗, t).  (12.13)

First term of RHS: surface tension, regularization. Smoothing effect. The second term is the lowest-order non-vanishing term in a gradient expansion. When the surface locally advances with normal velocity v, the increment projected onto the h axis is δh = [(vδt)^{2} + (vδt∇h)^{2}]^{1/2} which leads for small gradient

∂h/∂t = v[1 + (∇h)^{2}]^{1/2} ≃ v + (v/2)(∇h)^{2} + ...,  (7.25)

Also global smoothing, but combined with local sharpening. Noise is the only source for roughness. Let us use Eq. (7.24) to obtain a scaling relation among the exponents α, β and z. Here the simple scaling argument (beacuse of non-linearity) does not work. We use the simpler form

∂h̃/∂t ≃ (∇h̃)^{2}  (7.27)

which has a solution of the form

h(x, t) = t^{β}g (\frac{x}{t^{1/z}} )

After substitution

t^{β−1}g(y) − t^{β}g′(y)t^{−1/z−1} ≃ t^{2β}[g′(y)]^{2}t^{−2/z}

second term on LHS diverges more slowly. Other two balance if

β − 1 = 2β − 2/z
−z = βz − 2
α + z = 2 !

(in addition to α/β = z. Thus, there is one independent exponent). For d = 1 (growth from a line) the stationary surface is like a random walk (H = 1/2), thus

α = \frac{1}{2},  β = \frac{1}{3},  z = \frac{3}{2}.

Other equations Surface diffusion

\frac{∂h}{∂t} = −K∇^{4}h + η(x⃗, t),  (12.19)
∂h/∂t = −∇j(x, t); j(x, t) = −∇μ(x, t)
μ(x, t) ∼ −∇^{2}h(x, t)

j - current, μ - chemical potential, −∇^{2}h - appr. the inverse of the local radius of curvature with negative sign (large μ or chemical potential at ”tips” and small in ”valleys”). Scaling similar for the EW eq. (also linear).

\frac{∂h(x, t)}{∂t} = Kb^{z−4}∇^{4}h + b^{(z−d−2α)/2}η

α = \frac{4 − d}{2},  β = \frac{4 − d}{8},  z = 4.

α > 1! Non-linear terms are important.

All these agree with simulations, disagree with experiments! Bacteria colonies, wetting and fire fronts (giving α between 0.6 and 0.75), ion sputtering/etching, molecular beam epitaxy GROWTH IN A RANDOM MEDIUM Inhomogeneity represented by quenched noise: η(h, x⃗), zero mean, correlations:

⟨η(h, x⃗)η(h′, x⃗′)⟩ = δ^{d}(x⃗ − x⃗′)Δ(h − h′)

Δ(h−h′) is a coarse-grained function, is of order 1 for some distance δh and then goes to zero. ”Deterministic” growth. Pinning, depinning transition.

\frac{∂h}{∂t} = F + ν∇^{2}h + λ/2(∇h)^{2} + v + η(h(x⃗, t), x⃗).

F_{c}. ξ ∼ (F − F_{c})^{ν}, correlation length of depinned regions. If η is small enough then the interface encounters points where it becomes pinned. If the density of such pinning sites is high enough the propagation of the whole surface can be blocked.

Model of Directed Percolation Depinning blocked sites, regions under overhangs are filled in. Surface stops if a spanning directed line of blocking sites appears. Such a directed line which is like a spanning path in a directed percolation cluster is characterised by two correlation lengths, being parallel (ξ_{∥}) and perpendicular (ξ_{⊥}) to the interface. At threshold density of the pinning sites p_{c} as

ξ_{∥} ∼ |p − p_{c}|^{−ν_{∥}}, ξ_{⊥} ∼ |p − p_{c}|^{−ν_{⊥}}

with

ν_{∥} = 1.733...  and  ν_{⊥} = 1.097...

Complete blocking appears when ξ_{∥} becomes equal to the system size L. The width w of such a blocked surface is in the order of ξ_{⊥}, hence

L^{H} ∼ w ≈ ξ_{⊥} ∼ |p − p_{c}|^{−ν_{⊥}} ∼ ξ_{∥}^{ν_{⊥}/ν_{∥}} ≈ L^{ν_{⊥}/ν_{∥}},

Thus,

H = \frac{ν_{⊥}}{ν_{∥}} = 0.633...

in accord with the numerical simulation. In the moving, but noise dominated regime the interface consists of both blocked and freely moving segments. Self-Organized Pinning Fill the site with the smallest pre-assigned random number on a tilted square lattice and eliminate overhangs. No parameter is tuned (self-organization). Analogous to invasion percolation. Avalanches. DPD exponent. Heuristic scaling theory for the moving phase Method: magnitude of the terms should be the same. Consider the KPZ quenched eq. with the nonlinear and he noise terms only.

\frac{Δh}{Δt} ≃ \frac{(Δh)^{2}}{(Δx)^{2}} ≃ (Δx)^{−d/2}

Δh = Δx^{\frac{1}{2} (2−d/2)} = Δx^{(4−d)/4}

ΔhΔx^{d/2} = Δt = Δh^{1+ \frac{d}{2} \frac{4}{4−d}} = Δh^{\frac{4+d}{4−d}}

α = \frac{4 − d}{4}, β = \frac{4 − d}{4 + d}

For example, for d = 1 α = 0, 75 and β = 0, 6 in accord with some experiments and simulations.

12.6. MULTIAFFINITY The multiscaling properties of the self-affine function h(x) of a single variable can be investigated by calculating the qth order heightheight correlation function which we define as

c_{q}(x) = \frac{1}{N} ∑_{i=1}^{N} |h(x_{i}) − h(x_{i} + x)|^{q},  (12.35)

where N ≫ 1 is the number of points over which the average is taken. Multiscaling if

c_{q}(x) ∼ x^{qH_{q}}  (12.36)

where qH_{q} is the natural choice (instead of (q − 1)H_{q} as for fractal measures). E.g., velocity correlations in fully turbulent flows. It is not consistent with the expression h(x) ≃ λ^{−H}h(λx). Local scaling depends on x. Multiaffine function is like the absolute value of the derivative of a fractal measure (multifractal). Could be obtained by taking a multifractal and integrating with randomly selected signs. Exactly solvable deterministic model. Asymmetrical z made of four intervals. Depending on b_{1} very different structures can be generated, b_{2} = 0.5. The H_{q} spectrum can be calculated. Intervals of length 4^{−k} matter. Denoting with N(Δh) the number of boxes in which |h(x) − h(x + Δx)| = Δh, we have N(b_{1}^{n}b_{2}^{k−n}) = 2^{k}C_{k}^{n}, where n = 0, ..., k. Thus c_{q}(Δx) = ∑_{n=0}^{k} 2^{−k}C_{k}^{n}b_{1}^{nq}b_{2}^{(k−n)q} with Δx = 4^{−k}. Since this expression for c_{q} can be written as c_{q}(Δx) = [(b_{1}^{q} + b_{2}^{q})/2]^{k} we have

H_{q} = \frac{ln[(b_{1}^{q} + b_{2}^{q})/2]}{q ln(1/4)}.  (12.44)

In the present approach the roughness exponent introduced earlier is H = H_{1}. If b_{1} ≃ 0.72, then b_{1}^{3} ≃ 0.375, with b_{2}^{3} = 0.125 we get that H_{3} = 1/3. This is the same value as H_{3} (exact result) for turbulent velocity distribution. Surface growth/erosion dominated by turbulence (e.g., rain) may lead to such surfaces.

PATTERN FORMATION

Surface tension, in contrast cluster growth models. Anisotropy, fluctuations and driving force. Source of many types of patterns. Morphological diagram. 9.1. EQUATIONS Dimensionless field-like variable u(x⃗, t), which may denote temperature (solidification), pressure (viscous fingering) electric potential (electrodeposition, dielectric breakdown) or concentration (isothermal solidification, electrodeposition).

∇^{2}u(x⃗, t) = 0  (9.1)

with appropriate boundary conditions for slow growth (compared to relaxation time of the field). For regular growth, the continuity eq. is.

C∇^{2}u(x⃗, t) = \frac{∂u}{∂t}  (9.1a)

The normal velocity

v_{n} = −Cn̂ · ∇u,  (9.2)

In (9.2) it is assumed that ∇u = 0 in the growing phase. The growth is induced by

u_{∞} = Const.  (9.3)

different from the equilibrium value of u at the interface Γ. u_{eq} = 0 on the surface. u on the growing interface

u_{Γ} = −d_{0}κ − βv_{n}^{η},  (9.4)

where the capillary length d_{0} is proportional to the surface tension, κ denotes the local curvature of the interface (with κ > 0 for a sphere), β is the kinetic coefficient and η is an exponent depending on the physical process considered. The local curvature can be calculated using the expression κ = 1/R_{1} +...+1/R_{d−1}, where R_{i} are the local principal radii. With β = 0, (9.4) reduces to the Gibbs-Thomson relation which is valid assuming local thermodynamic equilibrium. Qualitative picture.

Mullins-Sekerka instability. Linear stability analysis (neglect higher order terms). Full analysis has to be nonlinear. Growth in z direction. New variable (comoving frame) z − vt → z For this z (9.1a) changes to:

∇^{2}u + \frac{v}{C} \frac{∂u}{∂z} = 0  (9.I)

(\frac{∂u}{∂t} = \frac{∂u}{∂z} \frac{∂z}{∂t}) Planar solution for z > 0 is

u = e^{− \frac{v}{C} z} − 1

for u → −1 if z → ∞. Next: Assume a given form, substitute, keep linear terms, satisfy boundary conditions, and see how the surface progresses. Quasistationary approximation (perturbations move slower than relaxation of the field takes place). Assume that the interface is perturbed along the x direction (and not in y) as (in other words, z of the interface is):

h(x, t) = ĥ_{k}e^{ikx+ω_{k}t} = ĥ_{k} exp[1]

ĥ_{k} ≪ 1, then the solution can be written as

u(x, t) ≃ e^{− \frac{v}{C} z} − 1 + û_{k}e^{ikx−qz+ω_{k}t} =
= u_{o} + û_{k}exp[2]  (9.II)

û_{k} ∼ ĥ_{k} and q is he positive solution of

−(v/C)q = q^{2} − k^{2} = 0

since u_{0} satisfies (9.I) and from inserting (9.II) into (9.I) we see that

−û_{k}k^{2}exp[1] + û_{k}q^{2}exp[1] − û_{k}vqexp[1]/C = 0

Now we evaluate (9.II) at h(x, t), AND linearize, AND impose u_{Γ} = −d_{0}κ, where κ ≃ −∂^{2}h/∂x^{2}

e^{− \frac{v}{C} ĥ_{k}exp[1]} − 1 + û_{k}e^{ikx−qĥ_{k}exp[1]} = −d_{0}κ

since e^{a} ≃ 1 + a for a ≪ 1,

−\frac{v}{C} ĥ_{k}exp[1] + û_{k}exp[1](1 − qĥ_{k}exp[1]) =

= −d_{0}k^{2}ĥ_{k}exp[1]

further linearizing, simplifying

−\frac{v}{C} ĥ_{k} + û_{k} = −d_{0}k^{2}ĥ_{k}

other boundary condition is v_{n} = −Cn̂ · ∇u, here we assume that ∇u ≃ ∂u/∂z, thus,

ω_{k}ĥ_{k}exp[1] = v(1 − \frac{v}{C} z − 1)+

+Cqû_{k}exp[1](1 − qĥ_{k}exp[1])

ω_{k}ĥ_{k} = −\frac{v^{2}}{C} ĥ_{k} + Cqû_{k}

de

û_{k}/ĥ_{k} = v/C − d_{0}k^{2}
ω_{k} = −v^{2}/C + qv − Cd_{0}qk^{2}

for v/kC ≪ 1, we have k ≃ q, and v^{2}/C ≪ vq, thus

ω_{k} ≃ kv (1 − \frac{d_{0}C}{v} k^{2})  (9.5)

which is the desired dispersion relation. For ω > 0 the perturbations grow exponentially in time (instability), while they quickly die out if ω < 0. The upper cutoff is due to the surface tension represented by d_{0}. If d_{0} = 0, the growth rate increases indefinitely. The fastest growth occurs for

λ_{c} = 2π/k_{c} = 2π\sqrt{3Cd_{0}/v}  (9.6)

since the maximum of (9.5) is at the positive root, k_{c}, of v−3d_{0}Ck^{2}. k_{c} is expected to be close to the characteristic wavelength. Competition of the stabilizing effect of the surface tension and the destabilizing effect of the instability.

Cascade of tip-splittings! (Tips ”fatten” and break into two). Morphological phase diagrams Role of anisotropy and randomness

9.2. MODELS RELATED TO DIFFUSION-LIMITED AGGRE-GATION The probability of finding the particle at x⃗

u(x⃗, t + 1) = \frac{1}{z} ∑_{a⃗} u(x⃗ + a⃗, t),  (9.7)

where a⃗ runs over the z neighbours of site x⃗. On a square lattice, after deducing from both sides u(x⃗, t)

u(i, j, t + 1) − u(i, j, t) =

= \frac{1}{4}[u(i − 1, j, t) − 2u(i, j, t) + u(i + 1, j, t)+

+u(i, j − 1, t) − 2u(i, j, t) + u(i, j + 1, t)],  (9.7)

we recover the finite difference version of the diffusion equation. Similarly, the probability that the perimeter site x⃗ gains a particle at time t + 1

v_{n}(x⃗, t + 1) = \frac{1}{z} ∑_{a⃗} u(x⃗ + a⃗, t).  (9.8)

(9.8) is equivalent to the discretized version of the boundary condition (9.2) (u = 0 on next-to-surface sites). However, the boundary condition (9.4) is not satisfied in DLA. 9.2.1. Effects of surface tension Sticking probability p_{s}(κ) depending on the local surface curvature.

A plausible choice is

p_{s}(κ) = Aκ + B,  (9.12)

where A and B are constants. κ for an aggregate? E.g., on the square lattice one can use:

κ = \frac{6}{πR} (\frac{N_{R}}{R^{2}} − \frac{π}{2} )

where N_{R} is the number of particles in the aggregate around the given surface site within a radius R. For flat N_{R} = πR^{2}/2, for κ = 1/R we get N_{R} ≃ 4πR^{2}/6, consistent. The analogy with the boundary condition (9.4) is provided by the fact that the sticking probability is proportional to the local growth velocity. If p_{s} calculated from (9.12) is less than 0 it is set to a small threshold value, and if p_{s} > 1, it is assumed to be equal to 1. Leads to patterns like viscous fingers. Parallel fingers. Radial geometry.

9.2.2. Noise reduction in DLA

Decrease the fluctuations in DLA. A perimeter site is filled only when it has been contacted (by random walks) m times. The scores associated with all of the other surface sites remain at their values before this event. In the limit m → ∞, the method yields the solution of the lattice version of the noiseless Laplace equation (9.1) with the boundary conditions (9.2) and (9.3). Morphological changes random fractal → dendritic → needle.

9.3. GENERALIZATIONS OF THE DIELECTRIC BREAK-DOWN MODEL Snowflakes. They (a) are quasi two-dimensional, (b) have sixfold symmetry and (c) can have a large number of entirely different shapes. A seed particle placed on a triangular lattice. At each time step u is calculated by solving the lattice version of the Laplace equation (9.1). At the surface u depends on κ (calculated as for DLA with surface tension). The gradients at the surface are normalized onto the unit interval and those perimeter sites for which the gradient is larger than q are filled. To approximate the boundary condition (9.2) one assumes that q varies in time as

q(t) = a + b (t mod [c])  (9.13)

which is a piecewise function depending on the parameters a, b and c. If the local gradient is small, its value exceeds q(t) less frequently and this results in a slower growth.

Leads to very realistic patterns if the surface at given times is plotted, for a(t) and b(t) changing randomly .

EXPERIMENTS

Different experiments, same equations. Applications 10.1. VISCOUS FINGERING Less viscous fluid is injected into a more viscous one. Navier-Stokes equation. In a porous medium viscous forces dominate the flow. 10.1.1. The Hele-Shaw cell The two plates are close, horizontal, and the flow in the x, y plane has a velocity profile v(z) = [v_{x}^{2}(z)+v_{y}^{2}(z)]^{1/2} which is approximately parabolic in the direction z perpendicular to the plates

v(z) = a(b^{2}/4 − z^{2}).  (10.1)

Furthermore, we assume that v_{z} = 0 and ∂v_{x}/∂x = ∂v_{y}/∂y = 0. For the average velocity one has

⟨v⟩ = \frac{1}{b} ∫_{−b/2}^{b/2} v(z)dz = \frac{ab^{2}}{6} .  (10.2)

Thus,

\frac{∂^{2}v}{∂z^{2}} = −2a  and  a = −\frac{6⟨v⟩}{b^{2}}

The Navier-Stokes equation has the form

m\frac{dv⃗}{dt} = −∇p + μ∇^{2}v⃗

\frac{∂p}{∂x} + \frac{∂p}{∂y} = μ (\frac{∂^{2}v_{x}(z)}{∂z^{2}} + \frac{∂^{2}v_{y}(z)}{∂z^{2}}) ,  (10.3)

where μ is the viscosity of the fluid, ρ is its density, and p denotes the pressure. For small b he inertial term is negligible (a grows quickly). We get

⟨v⃗⟩ = −\frac{b^{2}}{12μ}∇p,  (10.4)

For incompressible fluids ∇^{2}p = 0. The boundary condition (9.2) is essentially equivalent to (10.4) evaluated at the interface. Single finger in a long cell. radial fractal-like patterns. DLA-like fingers in cells with porous (random) surface, internal part. Anisotropy in viscous fingering Solidification (with roughness: fingering-like) Electrodepisition, chemical dissolution, dielectric breakdown. Field: Pressure, temperature, concentration, electric field.