3 Applications

[111.2.1] To simplify the notation T¯α⁢t¯ will be denoted as Tα⁢t in the following. [111.2.2] A first application of fractional time evolutions Tα⁢t concerns the important notion of stationarity. [111.2.3] This amounts to setting the left and right hand sides in eq. (2) to zero. [111.2.4] Surprisingly, the importance of the condition "dα⁢f/d⁢tα"=0 for the infinitesimal generators of fractional dynamics has rarely been noticed. [111.2.5] Stationary states f⁢s may be defined more generally as states that are invariant under the time evolution after a sufficient amount of time has elapsed during which all the transients have had time to decay.

[111.3.1] The function f⁢s=f0 where f0 is a constant is asymptotically and strictly stationary under the fractional time evolutions Tα⁢t. [111.3.2] This follows readily by insertion into the definition, and by noting that hα⁢x is a probability density.

[111.4.1] In addition to the conventional constants there exists a second class of stationary states given by

[page 112, §0]    where f0 and γ are constants. [112.0.1] To see this one evaluates

where relations (170) and (172) were used to rewrite the H-function in eq. (69). [112.0.2] Using the integral (178), the reduction formulae (167) and (169), and property (171) one finds

[112.0.3] An application of the series expansion (181) gives

[112.0.4] For s/t→∞ only the k=0 term in the series contributes and this shows that Tα⁢t⁢f⁢s=f⁢s in the limit. [112.0.5] These considerations show that fractional time evolutions have the usual constants as strict stationary states, but admit also algebraic behaviour as a novel type of stationary states.

[112.1.1] To elucidate the significance of the new type of stationary states it is useful to consider the infinitesimal form, Aα⁢f=0, of the stationarity condition. [112.1.2] The nature of the limit s/t→∞ suggests that their appearance might be related to the initial conditions. [112.1.3] To incorporate initial conditions into the infinitesimal generator it is necessary to consider a Riemann-Liouville representation of the fractional time derivative.

[112.2.1] The Riemann-Liouville algorithm for fractional differentiation is based on integer order derivatives of fractional integrals.

[113.1.1] The following generalized definition, based on differentiating fractional integrals, seems to be new.

[113.1.3] The Riemann-Liouville fractional derivative Da±α:=Da±α,0 corresponds to a>-∞ and type β=0. [113.1.4] Fractional derivatives of type β=1 are discussed in Chapter I  and were employed in [4]. [113.1.5] It seems however that fractional derivatives of general type 0<β<1 have not been considered previously. [113.1.6] A relation between fractional derivatives of the same order but different types is given in Chapter IX. [113.1.7] For subsequent calculations it is useful to record the Laplace-Transformation

where the inital value Da+1-β⁢α-1,0⁢f⁢0+ is the Riemann-Liouville derivative for t→0+. [113.1.8] Note that fractional derivatives of type β=1 involve nonfractional initial values.

[113.2.1] It is now possible to discuss the infinitesimal form of fractional stationarity where the generator Aα for initial conditions of type 0≤β≤1 is represented by D0+α,β. [113.2.2] The fractional differential equation

for f with initial condition

[page 114, §0]    defines fractional stationarity of order α and type β. [114.0.1] Of course, for α=1 this definition reduces to the conventional definition of stationarity. [114.0.2] Equation (107) is solved by

[114.0.3] This may be seen by inserting f⁢t into the definition

and using the basic fractional integral

(derived in eq. (1.30) in Chapter I). [114.0.4] Note that the fractional integral

remains conserved and constant for all t while the function itself varies. [114.0.5] In particular limt→0⁡f⁢t=∞ and limt→∞⁡f⁢t=0. [114.0.6] For β=1 and for α=1 one recovers f⁢t=f0 as usual.

[114.1.1] The new types of stationary states for which a fractional integral rather than the function itself is constant were first discussed in [6, 9]. [114.1.2] It seems to me that the lack of knowledge about fractional stationarity is partially responsible for the difficulty of deciding which type of fractional derivative should be used when generalizing traditional equations of motion.

[114.2.1] Another simple instance of a fractional differential equation is the equation

with C∈R a constant, and with initial condition

as before. [114.2.2] Laplace transformation using eq. (106) gives

and thence

[page 115, §0]    [115.0.1] For β=1 this reduces to

[115.1.1] Consider the fractional Cauchy problem

for f with initial condition

where C is a (‘‘fractional relaxation’’) constant. [115.1.2] Laplace Transformation gives

[115.1.3] To invert the Laplace transform rewrite this equation as

with

[115.1.4] Inverting the series term by term using L⁢xα-1/Γ⁢α=u-α yields the result

[115.1.5] The solution may be written as

using the generalized Mittag-Leffler function defined by

[page 116, §0]    for all a>0,b∈C. [116.0.1] This function is an entire function of order 1/a [38]. [116.0.2] Moreover it is completely monotone if and only if 0<a≤1 and b≥a [39].

[116.1.1] For C=0 the result reduces to eq. (109) because Ea,b⁢0=1/Γ⁢b. [116.1.2] Of special interest is again the case β=1. [116.1.3] It has the well known solution

where Eα⁢x=Eα,1⁢x denotes the ordinary Mittag-Leffler function.

[116.2.1] Consider the fractional partial differential equation for f:Rd×R+→R

with Laplacian Δ and fractional ‘‘diffusion’’ constant C. [116.2.2] The function f⁢r,t is assumed to obey the initial condition

where δ⁢r is the Dirac measure at the origin. [116.2.3] Fourier Transformation, defined as

and Laplace transformation of eq. (127) now yields

[116.2.4] Using the result (124) for the inverse Laplace transform of (120) gives

[116.2.5] Setting q=0 shows that the solution of (127) cannot be a probability density except for β=1. [116.2.6] For β≠1 the spatial integral is time dependent, and f would need to be divided by t1-β⁢α-1 to admit a probabilistic interpretation.

[116.3.1] To invert eq. (130) completely it seems advantageous to first invert the Fourier transform and then the Laplace transform. [116.3.2] The Fourier transform may be inverted by noting the formula [40]

[page 117, §0]    which leads to

with r=r. [117.0.1] To invert the Laplace transform one uses again the relation (78) with the Mellin transform defined in eq. (79). [117.0.2] Setting A=r/C, λ=α/2, ν=d-2/2 and μ=β⁢α-1+α⁢d-2/4 and using the general relation

leads to

[117.0.3] The Mellin transform of the Bessel function reads [32]

[117.0.4] Inserting this, using eq.(78), and restoring the original variables then yields

for the Mellin transform of f. [117.0.5] Comparing this with the Mellin transform of the H-function in eq. (175) allows to identify the H-function parameters as m=0,n=2,p=2,q=1, A1=A2=1/α, a1=1-d/2-β-1⁢1-1/α, a2=1-β⁢1-1/α, b1=0 and B1=1 if α⁢d/2+β-1⁢α-1>0. [117.0.6] Then the result becomes

[page 118, §0]    [118.0.1] This may be simplified using eqs.(170), (171) and (172) to become finally

[118.0.2] The result reduces to the known result [15, 8] for β=1. [118.0.3] In that case f⁢r,t is also a probability density. [118.0.4] For β≠1 the function f⁢r,t does not have a probabilistic interpretation because its normalization decays as t1-β⁢α-1.

[118.1.1] The fractional diffusion eq. (127) of type β=1 has a probabilistic interpretation as noted after eq. (131). [118.1.2] f⁢r,t may be viewed as the probability density for a random walker or diffusing object to be at position r at time t under the condition that it started from the origin r=0 at time t=0. This probabilistic interpretation is very helpful for understanding the meaning of the fractional time derivative appearing in eq. (127). [118.1.3] Rewriting equation (127) in integral form it becomes

where the initial condition has been incorporated. [118.1.4] This integral equation is very reminiscent of the integral equation for continuous time random walks [41, 42].

[118.2.1] In a continuous time random walk one imagines a random walker that starts at r=0 at time t=0 and proceeds by successive random jumps [43, 44, 45, 46, 47, 48]. [118.2.2] The probability density for a time interval of length t between two consecutive jumps is denoted ψ⁢t and the probability density of a displacement by a vector r in a single jump is denoted p⁢r. [118.2.3] Then the integral equation of continuous time random walk theory reads

where Φ⁢t is the probability that the walker survives at the origin for a time of length t. [118.2.4] Here the walker is assumed to be prepared in its initial position from which it develops according to ψ⁢t. [118.2.5] In general the first step needs special consideration [49, 49, 45]. [118.2.6] The survival probablity Φ⁢t is related to the waiting

[page 119, §0]    time density through

[119.1.1] The formal similarity between eqs. (141) and (140) suggests that there exists a relation between them. [119.1.2] To establish the relation note that eq. (130) for β=1 gives the solution of eq. (127) in Fourier-Laplace space as

[119.1.3] The Fourier-Laplace solution of eq.(141) is [44, 50, 51, 46]

[119.1.4] Equating these two equations yields

[119.1.5] Because the left hand side does not depend on u and the right hand side is independent of q they must both equal a common constant τ0α. [119.1.6] It follows that

identifying the constant C⁢τ0α as the mean square displacement of a single jump. [119.1.7] For the waiting time density one finds

which may be inverted in the same way as eq. (120) to give

where Ea,b⁢x is again the Mittag-Leffler function defined in eq. (40).

[119.2.1] For α=1 the waiting time density becomes exponential

[page 120, §0]    [120.0.1] For 0<α<1 characteristic differences arise from the asymptotic behaviour for t→0 and t→∞. [120.0.2] The asymptotic behaviour of ψ⁢t for t→0 is obtained by noting that Eα,α⁢0=1, and hence

for t→0. [120.0.3] For α<1 the waiting time density is singular at the origin implying a statistical abundance of short intervals between jumps compared to the exponential case α=1. [120.0.4] For large t→∞ recall the asymptotic series expansion [52]

valid for arg⁡-z<1-a/2⁢π and z→∞. [120.0.5] It follows that Ea,a⁢-x∼x-2 for x→∞ and hence

for t→∞. [120.0.6] This shows that fractional diffusion is equivalent to a continuous time random walk whose waiting time density is a generalized Mittag-Leffler function. [120.0.7] The waiting time density has a long time tail of the form usually assumed in the general theory [53, 49, 54, 46] and exhibits a power law divergence at the origin. [120.0.8] The exponent of both power laws is given by the order of the fractional derivative.