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3 Applications

3.1 Fractional Invariance and Stationarity

[111.2.1] To simplify the notation T¯αt¯ will be denoted as Tαt in the following. [111.2.2] A first application of fractional time evolutions Tαt 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 "dαf/dtα"=0 for the infinitesimal generators of fractional dynamics has rarely been noticed. [111.2.5] Stationary states fs 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.

Definition 3.1

[111.2.6] An observable or state ft is called stationary or asymptotically invariant under the time evolution Tαt if

 Tα⁢t⁢f⁢s=f⁢s (98)

holds for s/t. [111.2.7] It is called stationary in the strict sense, or strictly invariant under Tαt, if condition (98) holds for all t0 and sR.

[111.3.1] The function fs=f0 where f0 is a constant is asymptotically and strictly stationary under the fractional time evolutions Tαt. [111.3.2] This follows readily by insertion into the definition, and by noting that hαx is a probability density.

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

 f⁢s=f0⁢sγ-1    for ⁢s>00    for ⁢s≤0 (99)

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

 Tα(t)f(s)=∫0∞f(s-x)1thα(xt)dx=f0∫0s(s-x)γ-11α⁢tH1101(xt|1-1/α,1/α0,1)dx (100)

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

 Tα⁢t⁢f⁢s =f0sγ-1Γ(γ)H1101((st)α|1,11-γ,α). (101)

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

 Tα⁢t⁢f⁢s=f0⁢sγ-1⁢Γ⁢γ⁢∑k=0∞-1k⁢t/sk⁢αk!⁢Γ⁢γ-k⁢α. (102)

[112.0.4] For s/t only the k=0 term in the series contributes and this shows that Tαtfs=fs 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, Aαf=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.

Definition 3.2 (Riemann-Liouville fractional integral)

[112.2.2] The right-sided Riemann-Liouville fractional integral of order α>0,αR of a locally integrable function f is defined as

 Ia+α⁢f⁢x=1Γ⁢α⁢∫axx-yα-1⁢f⁢y⁢d⁢y (103)

[page 113, §0]    for x>a, the left-sided Riemann-Liouville fractional integral is defined as

 Ia-α⁢f⁢x=1Γ⁢α⁢∫xay-xα-1⁢f⁢y⁢d⁢y (104)

for x<a.

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

Definition 3.3 (Fractional derivatives)

[113.1.2]  The (right-/left-sided) fractional derivative of order 0<α<1 and type 0β1 with respect to x is defined by

 Da±α,β⁢f⁢x=±Ia±β⁢1-α⁢dd⁢x⁢Ia±1-β⁢1-α⁢f⁢x (105)

for functions for which the expression on the right hand side exists.

[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 . [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

 L⁢Da+α,β⁢f⁢x⁢u=uα⁢L⁢f⁢x⁢u-uβ⁢α-1⁢Da+1-β⁢α-1,0⁢f⁢0+ (106)

where the inital value Da+1-βα-1,0f0+ is the Riemann-Liouville derivative for t0+. [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 0β1 is represented by D0+α,β. [113.2.2] The fractional differential equation

 D0+α,β⁢f⁢t=0 (107)

for f with initial condition

 I0+1-β⁢1-α⁢f⁢0+=f0 (108)

[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

 f⁢t=f0⁢t1-β⁢α-1Γ⁢1-β⁢α-1+1. (109)

[114.0.3] This may be seen by inserting ft into the definition

 D0+α,β⁢f⁢x=I0+β⁢1-α⁢dd⁢x⁢I0+1-β⁢1-α⁢f⁢x (110)

and using the basic fractional integral

 Ia+α⁢x-aβ=Γ⁢β+1Γ⁢α+β+1⁢x-aα+β (111)

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

 I0+1-β⁢1-α⁢f⁢t=f0 (112)

remains conserved and constant for all t while the function itself varies. [114.0.5] In particular limt0ft= and limtft=0. [114.0.6] For β=1 and for α=1 one recovers ft=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

 D0+α,β⁢f⁢t=C (113)

with CR a constant, and with initial condition

 I0+1-β⁢1-α⁢f⁢0+=f0 (114)

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

 f⁢u=Cuα+1+f0uα+β⁢1-α (115)

and thence

 f⁢t=C⁢tαΓ⁢α+1+f0⁢t1-β⁢α-1Γ⁢1-β⁢1-α+1. (116)

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

 f⁢t=C⁢tαΓ⁢α+1+f0. (117)

3.2 Generalized Fractional Relaxation

[115.1.1] Consider the fractional Cauchy problem

 D0+α,β⁢f⁢t=-C⁢f⁢t (118)

for f with initial condition

 I0+1-β⁢1-α⁢f⁢0+=f0 (119)

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

 f⁢u=uβ⁢α-1⁢f0C+uα. (120)

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

 f⁢u=uα-γC+uα=u-γ⁢1C⁢u-α+1=∑k=0∞-Ck⁢u-α⁢k-γ (121)

with

 γ=α+β⁢1-α. (122)

[115.1.4] Inverting the series term by term using Lxα-1/Γα=u-α yields the result

 f⁢t=tγ-1⁢∑k=0∞-C⁢tαkΓ⁢α⁢k+γ. (123)

[115.1.5] The solution may be written as

 f⁢t=f0⁢t1-β⁢α-1⁢Eα,α+β⁢1-α⁢-C⁢tα (124)

using the generalized Mittag-Leffler function defined by

 Ea,b⁢x=∑k=0∞xkΓ⁢a⁢k+b (125)

[page 116, §0]    for all a>0,bC. [116.0.1] This function is an entire function of order 1/a . [116.0.2] Moreover it is completely monotone if and only if 0<a1 and ba .

[116.1.1] For C=0 the result reduces to eq. (109) because Ea,b0=1/Γb. [116.1.2] Of special interest is again the case β=1. [116.1.3] It has the well known solution

 f⁢t=f0⁢Eα⁢-C⁢tα (126)

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

3.3 Generalized Fractional Diffusion

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

 D0+α,β⁢f⁢r,t=C⁢Δ⁢f⁢r,t (127)

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

 I0+1-β⁢1-α⁢f⁢r,0+=f0⁢r=f0⁢δ⁢r (128)

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

 F⁢f⁢r⁢q=∫Rdei⁢q⋅r⁢f⁢r⁢d⁢r, (129)

and Laplace transformation of eq. (127) now yields

 f⁢q,u=uβ⁢α-1⁢f0C⁢q2+uα. (130)

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

 f⁢q,t=f0⁢t1-β⁢α-1⁢Eα,α+β⁢1-α⁢-C⁢q2⁢tα. (131)

[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 β1 the spatial integral is time dependent, and f would need to be divided by t1-βα-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 

 2⁢π-d/2⁢∫ei⁢q⋅r⁢rm1-d/2⁢Kd-2/2⁢m⁢r⁢d⁢r=1q2+m2 (132)

[page 117, §0]    which leads to

 f⁢r,u=f0⁢2⁢π⁢C-d/2⁢rC1-d/2⁢uβ⁢α-1+α⁢d-2/4⁢Kd-2/2⁢r⁢uα/2C (133)

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=βα-1+αd-2/4 and using the general relation

 M{xqg(bxp)}(s)=1pb-s+q/pg(s+qp)(b,p>0) (134)

 M⁢f⁢r,u⁢s=f0λ⁢2⁢π⁢C-d/2⁢A1-d/2⁢A-s+μ/λ⁢M⁢Kν⁢u⁢s+μ/λ. (135)

[117.0.3] The Mellin transform of the Bessel function reads 

 M⁢Kν⁢x⁢s=2s-2⁢Γ⁢s+ν2⁢Γ⁢s-ν2. (136)

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

 M⁢f⁢r,t⁢s=f0α⁢r2⁢πd/2⁢r2⁢C2⁢1-β⁢1-1/α⁢r2⁢C2⁢s/α⁢Γ⁢d2+β-1⁢1-1α-sα⁢Γ⁢1+β-1⁢1-1α-sαΓ⁢1-s (137)

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-11-1/α, a2=1-β1-1/α, b1=0 and B1=1 if αd/2+β-1α-1>0. [117.0.6] Then the result becomes

 f(r,t)=f0α⁢r2⁢πd/2(r2⁢C)2⁢1-β⁢1-1/αH2102((2⁢Cr)2/αt|1-d2+1-β⁢1-1α,1α,1-β⁢1-1α,1α0,1). (138)

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

 f(r,t)=f0⁢t1-β⁢α-1r2⁢πd/2H1220(r24⁢C⁢tα|1+1-β⁢α-1,αd/2,1,1,1). (139)

[118.0.2] The result reduces to the known result [15, 8] for β=1. [118.0.3] In that case fr,t is also a probability density. [118.0.4] For β1 the function fr,t does not have a probabilistic interpretation because its normalization decays as t1-βα-1.

3.4 Relation with Continuous Time Random Walk

[118.1.1] The fractional diffusion eq. (127) of type β=1 has a probabilistic interpretation as noted after eq. (131). [118.1.2] fr,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

 f⁢r,t=δr⁢0+CΓ⁢α⁢∫0tt-sα-1⁢Δ⁢f⁢r,t⁢d⁢s (140)

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 ψt and the probability density of a displacement by a vector r in a single jump is denoted pr. [118.2.3] Then the integral equation of continuous time random walk theory reads

 f⁢r,t=δr⁢0⁢Φ⁢t+∫0tψ⁢t-s⁢∫Rdp⁢r-r′⁢f⁢r,t⁢d⁢r′⁢d⁢s (141)

where Φt 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 ψt. [118.2.5] In general the first step needs special consideration [49, 49, 45]. [118.2.6] The survival probablity Φt is related to the waiting

[page 119, §0]    time density through

 Φ⁢t=1-∫0tψ⁢t′⁢d⁢t′. (142)

[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

 f⁢q,u=uα-1C⁢q2+uα. (143)

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

 f⁢q,u=1u⁢1-ψ⁢u1-ψ⁢u⁢p⁢q. (144)

[119.1.4] Equating these two equations yields

 1-p⁢qC⁢q2=1-ψ⁢uuα⁢ψ⁢u. (145)

[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

 p⁢q=1-C⁢τ0α⁢q2 (146)

identifying the constant Cτ0α as the mean square displacement of a single jump. [119.1.7] For the waiting time density one finds

 ψ⁢u=11+τ0α⁢uα, (147)

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

 ψ⁢t;α,τ0=1τ0⁢tτ0α-1⁢Eα,α⁢-tατ0α (148)

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

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

 ψ⁢t;1,τ0=1τ0⁢e-t/τ0. (149)

[page 120, §0]    [120.0.1] For 0<α<1 characteristic differences arise from the asymptotic behaviour for t0 and t. [120.0.2] The asymptotic behaviour of ψt for t0 is obtained by noting that Eα,α0=1, and hence

 ψ⁢t∼tα-1 (150)

for t0. [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 

 Ea,b⁢z=-∑n=1Nz-nΓ⁢b-a⁢n+O⁢zN (151)

valid for arg-z<1-a/2π and z. [120.0.5] It follows that Ea,a-xx-2 for x and hence

 ψ⁢t∼t-1-α (152)

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.