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2 Relation between fractional and fractal walks

[1.3.3.1] Let us start by recalling briefly the general theory of continuous time random walks [5, 7, 8]. [1.3.3.2] The basic equation of motion is the continous time random walk (CTRW) integral equation [16]

p⁢r→,t=δr→⁢0⁢Φ⁢t+∫0tψ⁢t-t′⁢∑r→′λ⁢r→-r→′⁢p⁢r→′,t′⁢d⁢t′(2.1)

[page 2, §0]    describing a random walk in continous time without correlation between its spatial and temporal behaviour. [2.1.0.1] Here, as in (1.2), p⁢r→,t denotes the probability density to find the diffusing entity at the position r→∈Rd at time t if it started from the origin r→=0 at time t=0. [2.1.0.2] λ⁢r→ is the probability for a displacement r→ in each single step, and ψ⁢t is the waiting time distribution giving the probability density for the time interval t between two consecutive steps. [2.1.0.3] The transition probabilities obey ∑r→λ⁢r→=1. [2.1.0.4] The function Φ⁢t is the survival probability at the initial position which is related to the waiting time distribution through

Φ⁢t=1-∫0tψ⁢t′⁢d⁢t′.(2.2)

[2.1.0.5] The objective of this paper which was defined in the introduction is to show that the fractional master equation (1.2) is a special case of the CTRW-equation (2.1), and to find the appropriate waiting time density.

[2.1.1.1] The translation invariant form of the transition probabilities in (2.1) allows a solution through Fourier-Laplace transformation. [2.1.1.2] Let

ψ⁢u=L⁢ψ⁢t⁢u=∫0∞e-u⁢t⁢ψ⁢t⁢d⁢t(2.3)

denote the Laplace transform of ψ⁢t and

λ⁢q→=F⁢λ⁢r→⁢q→=∑r→ei⁢q→⋅r→⁢λ⁢r→(2.4)

the Fourier transform of λ⁢r→, which is also called the structure function of the random walk [5]. [2.1.1.3] Then the Fourier Laplace transform p⁢q→,u of the solution to (2.1) is given as [5, 7, 8, 16]

p⁢q→,u=1u⁢1-ψ⁢u1-ψ⁢u⁢λ⁢k=Φ⁢u1-ψ⁢u⁢λ⁢q→(2.5)

where Φ⁢u is the Laplace transform of the survival probability.

[2.1.2.1] Similarly the fractional master equation (1.2) can be solved in Fourier-Laplace space with the result

p⁢q→,u=uω-1uω-w⁢q→(2.6)

where w⁢q→ is the Fourier transform of the kernel w⁢r→ in (1.2). [2.1.2.2] Eliminating p⁢q→,u between (2.5) and (2.6) gives the result

1-ψ⁢uuω⁢ψ⁢u=λ⁢q→-1w⁢q→=C(2.7)

where C is a constant. [2.1.2.3] The last equality obtains because the left hand side of the first equality is q→-independent while the right hand side is independent of u.

[2.1.3.1] From (2.7) it is seen that the fractional master equation characterized by the kernel w⁢r→ and the order ω corresponds to a special class of space time decoupled continuous time random walks characterized by λ⁢r→ and ψ⁢t. [2.1.3.2] This correspondence is given precisely as

ψ⁢u=11+C⁢uω(2.8)

and

λ⁢q→=1+C⁢w⁢q→(2.9)

with the same constant C appearing in both equations. [2.2.0.1] Not unexpectedly the correspondence defines the waiting time distribution uniquely up to a constant while the structure function is related to the Fourier transform of the transition rates.

[2.2.1.1] To invert the Laplace transformation in (2.8) and exhibit the form of the waiting time density ψ⁢t in the time domain it is convenient to introduce the Mellin transformation

f⁢s=M⁢f⁢x⁢s=∫0∞xs-1⁢f⁢x⁢d⁢x(2.10)

for a function f⁢x. [2.2.1.2] The Mellin transformed waiting time density is obtained as

ψ⁢s=M⁢ψ⁢t⁢s=1ω⁢C1/ω⁢1C1/ω-s⁢Γ⁢1ω-sω⁢Γ⁢1-1ω+sωΓ⁢1-s(2.11)

where Γ⁢x denotes the Gamma function. [2.2.1.3] To obtain (2.11) from (2.8) the relation between Laplace and Mellin transforms

M⁢L⁢f⁢t⁢u⁢s=Γ⁢s⁢M⁢f⁢t⁢1-s,(2.12)

the special result

M⁢11+x⁢s=Γ⁢s⁢Γ⁢1-s(2.13)

and the general relation

M⁢f⁢a⁢xb⁢s=1b⁢a-s/b⁢M⁢f⁢x⁢s/b(2.14)

valid for a,b>0 have been employed. [2.2.1.4] Using the definition of the general H-function given in the appendix one obtains the result

ψ(t)=ψ(t;ω,C)=1ω⁢C1/ωH1211(tC1/ω|1-1ω,1ω1-1ω,1ω0,1)(2.15)

which may be rewritten as

ψ(t;ω,C)=1tH1211(tC1/ω|1,11,11,ω)(2.16)

with the help of general relations for H-functions [24]. [2.2.1.5] The dependence on the parameters ω and C has been indicated explicitly. [2.2.1.6] From the series expansion of H-functions given in the appendix one finds

[page 3, §0]

ψ⁢t;ω,C=tω-1C⁢∑k=0∞1Γ⁢ω⁢k+ω⁢-tωCk(2.17)

showing that ψ⁢t behaves as

ψ⁢t∝t-1+ω(2.18)

for small t→0. [3.1.0.1] Because 0<ω≤1 the waiting time density is singular at the origin. [3.1.0.2] The series representation (2.17) shows that the waiting time density is a natural generalization of an exponential waiting time density to which it reduces for ω=1, i.e. ψ⁢t;1,C=1/C⁢exp⁡t/C. [3.1.0.3] The series in (2.17) is recognized as the generalized Mittag-Leffler function Eω,ω⁢x [25], and ψ⁢t may thus be written alternatively as

ψ⁢t;ω,C=tω-1C⁢Eω,ω⁢-tωC.(2.19)

[3.1.0.4] Of course the result (2.17) can also be obtained more directly, but we have presented here a method using Mellin transforms because it remains applicable in cases where a direct inversion fails [23]. [3.1.0.5] The asymptotic expansion of the Mittag-Leffler function for large argument [25] yields

ψ⁢t∝t-1-ω(2.20)

for large t→∞ and 0<ω<1. [3.1.0.6] This result shows that the waiting time distribution has an algebraic tail of the kind usually considered in the theory of random walks [12, 13, 14, 15, 16].