3 Applications

3.1 Fractional Invariance and Stationarity

[111.2.1] To simplify the notation ${\overline{\mbox{\rm T}}}_{\alpha}({\overline{t}})$ will be denoted as $\mbox{\rm T}_{\alpha}(t)$ in the following. [111.2.2] A first application of fractional time evolutions $T_{\alpha}(t)$ 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 " $\mbox{\rm d}^{\alpha}f/\mbox{\rm d}t^{\alpha}$ " $=0$ 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.

Definition 3.1

[111.2.6] An observable or state $f(t)$ is called stationary or asymptotically invariant under the time evolution $\mbox{\rm T}_{\alpha}(t)$ if

$\mbox{\rm T}_{\alpha}(t)f(s)=f(s)$

(98)

holds for $s/t\to\infty$ . [111.2.7] It is called stationary in the strict sense, or strictly invariant under $\mbox{\rm T}_{\alpha}(t)$ , if condition (98) holds for all $t\geq 0$ and $s\in\mathbb{R}$ .

[111.3.1] The function $f(s)=f_{0}$ where $f_{0}$ is a constant is asymptotically and strictly stationary under the fractional time evolutions $\mbox{\rm T}_{\alpha}(t)$ . [111.3.2] This follows readily by insertion into the definition, and by noting that $h_{\alpha}(x)$ is a probability density.

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

$f(s)=\begin{cases}f_{0}s^{{\gamma-1}}&\text{\ \ \ \ for }s>0\\ 0&\text{\ \ \ \ for }s\leq 0\end{cases}$

(99)

[page 112, §0] where $f_{0}$ and $\gamma$ are constants. [112.0.1] To see this one evaluates

$\mbox{\rm T}_{\alpha}(t)f(s)=\int _{0}^{\infty}f(s-x)\frac{1}{t}h_{\alpha}\left(\frac{x}{t}\right)\; dx=f_{0}\int _{0}^{s}(s-x)^{{\gamma-1}}\frac{1}{\alpha t}H^{{01}}_{{11}}\left(\frac{x}{t}\left|\begin{array}[]{l}{(1-1/\alpha,1/\alpha)}\\ {(0,1)}\end{array}\right.\right)\mbox{\rm d}x$

(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

$\displaystyle\mbox{\rm T}_{\alpha}(t)f(s)$

$\displaystyle=f_{0}s^{{\gamma-1}}\Gamma(\gamma)H^{{01}}_{{11}}\left(\left(\frac{s}{t}\right)^{\alpha}\left|\begin{array}[]{l}{(1,1)}\\ {(1-\gamma,\alpha)}\end{array}\right.\right).$

(101)

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

$\displaystyle\mbox{\rm T}_{\alpha}(t)f(s)=f_{0}s^{{\gamma-1}}\Gamma(\gamma)\sum _{{k=0}}^{\infty}\frac{(-1)^{k}(t/s)^{{k\alpha}}}{k!\Gamma(\gamma-k\alpha)}.$

(102)

[112.0.4] For $s/t\to\infty$ only the $k=0$ term in the series contributes and this shows that $\mbox{\rm T}_{\alpha}(t)f(s)=f(s)$ 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, $\mbox{\rm A}_{\alpha}f=0$ , of the stationarity condition. [112.1.2] The nature of the limit $s/t\to\infty$ 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 $\alpha>0,\alpha\in\mathbb{R}$ of a locally integrable function $f$ is defined as

$(\mbox{\rm I}^{{\alpha}}_{{a+}}f)(x)=\frac{1}{\Gamma(\alpha)}\int _{a}^{x}(x-y)^{{\alpha-1}}f(y)\;\mbox{\rm d}y$

(103)

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

$(\mbox{\rm I}^{{\alpha}}_{{a-}}f)(x)=\frac{1}{\Gamma(\alpha)}\int _{x}^{a}(y-x)^{{\alpha-1}}f(y)\;\mbox{\rm d}y$

(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 $0<\alpha<1$ and type $0\leq\beta\leq 1$ with respect to $x$ is defined by

$\mbox{\rm D}^{{\alpha,\beta}}_{{a\pm}}f(x)=\left(\pm\mbox{\rm I}^{{\beta(1-\alpha)}}_{{a\pm}}\frac{\mbox{\rm d}}{\mbox{\rm d}x}(\mbox{\rm I}^{{(1-\beta)(1-\alpha)}}_{{a\pm}}f)\right)(x)$

(105)

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

[113.1.3] The Riemann-Liouville fractional derivative $\mbox{\rm D}^{{\alpha}}_{{a\pm}}:=\mbox{\rm D}^{{\alpha,0}}_{{a\pm}}$ corresponds to $a>-\infty$ and type $\beta=0$ . [113.1.4] Fractional derivatives of type $\beta=1$ are discussed in Chapter I and were employed in [4]. [113.1.5] It seems however that fractional derivatives of general type $0<\beta<1$ 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

$\displaystyle{\mathcal{L}}\left\{\mbox{\rm D}^{{\alpha,\beta}}_{{a+}}f(x)\right\}(u)=u^{\alpha}{\mathcal{L}}\left\{ f(x)\right\}(u)-u^{{\beta(\alpha-1)}}(\mbox{\rm D}^{{(1-\beta)(\alpha-1),0}}_{{a+}}f)(0+)$

(106)

where the inital value $(\mbox{\rm D}^{{(1-\beta)(\alpha-1),0}}_{{a+}}f)(0+)$ is the Riemann-Liouville derivative for $t\to 0+$ . [113.1.8] Note that fractional derivatives of type $\beta=1$ involve nonfractional initial values.

[113.2.1] It is now possible to discuss the infinitesimal form of fractional stationarity where the generator $\mbox{\rm A}_{\alpha}$ for initial conditions of type $0\leq\beta\leq 1$ is represented by $\mbox{\rm D}^{{\alpha,\beta}}_{{0+}}$ . [113.2.2] The fractional differential equation

$\mbox{\rm D}^{{\alpha,\beta}}_{{0+}}f(t)=0$

(107)

for $f$ with initial condition

$\mbox{\rm I}^{{(1-\beta)(1-\alpha)}}_{{0+}}f(0+)=f_{0}$

(108)

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

$f(t)=\frac{f_{0}\; t^{{(1-\beta)(\alpha-1)}}}{\Gamma((1-\beta)(\alpha-1)+1)}.$

(109)

[114.0.3] This may be seen by inserting $f(t)$ into the definition

$\mbox{\rm D}^{{\alpha,\beta}}_{{0+}}f(x)=\left(\mbox{\rm I}^{{\beta(1-\alpha)}}_{{0+}}\frac{\mbox{\rm d}}{\mbox{\rm d}x}(\mbox{\rm I}^{{(1-\beta)(1-\alpha)}}_{{0+}}f)\right)(x)$

(110)

and using the basic fractional integral

$\mbox{\rm I}^{{\alpha}}_{{a+}}(x-a)^{\beta}=\frac{\Gamma(\beta+1)}{\Gamma(\alpha+\beta+1)}(x-a)^{{\alpha+\beta}}$

(111)

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

$\mbox{\rm I}^{{(1-\beta)(1-\alpha)}}_{{0+}}f(t)=f_{0}$

(112)

remains conserved and constant for all $t$ while the function itself varies. [114.0.5] In particular $\lim _{{t\to 0}}f(t)=\infty$ and $\lim _{{t\to\infty}}f(t)=0$ . [114.0.6] For $\beta=1$ and for $\alpha=1$ one recovers $f(t)=f_{0}$ 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

$\mbox{\rm D}^{{\alpha,\beta}}_{{0+}}f(t)=C$

(113)

with $C\in\mathbb{R}$ a constant, and with initial condition

$\mbox{\rm I}^{{(1-\beta)(1-\alpha)}}_{{0+}}f(0+)=f_{0}$

(114)

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

$f(u)=\frac{C}{u^{{\alpha+1}}}+\frac{f_{0}}{u^{{\alpha+\beta(1-\alpha)}}}$

(115)

and thence

$f(t)=\frac{C\; t^{\alpha}}{\Gamma(\alpha+1)}+\frac{f_{0}\; t^{{(1-\beta)(\alpha-1)}}}{\Gamma((1-\beta)(1-\alpha)+1)}.$

(116)

[page 115, §0] [115.0.1] For $\beta=1$ this reduces to

$f(t)=\frac{C\; t^{\alpha}}{\Gamma(\alpha+1)}+f_{0}.$

(117)

3.2 Generalized Fractional Relaxation

[115.1.1] Consider the fractional Cauchy problem

$\mbox{\rm D}^{{\alpha,\beta}}_{{0+}}f(t)=-C\; f(t)$

(118)

for $f$ with initial condition

$\mbox{\rm I}^{{(1-\beta)(1-\alpha)}}_{{0+}}f(0+)=f_{0}$

(119)

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

$f(u)=\frac{u^{{\beta(\alpha-1)}}\; f_{0}}{C+u^{\alpha}}.$

(120)

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

$f(u)=\frac{u^{{\alpha-\gamma}}}{C+u^{\alpha}}=u^{{-\gamma}}\frac{1}{Cu^{{-\alpha}}+1}=\sum _{{k=0}}^{\infty}(-C)^{k}u^{{-\alpha k-\gamma}}$

(121)

with

$\gamma=\alpha+\beta(1-\alpha).$

(122)

[115.1.4] Inverting the series term by term using ${\mathcal{L}}\left\{ x^{{\alpha-1}}/\Gamma(\alpha)\right\}=u^{{-\alpha}}$ yields the result

$f(t)=t^{{\gamma-1}}\sum _{{k=0}}^{\infty}\frac{(-Ct^{\alpha})^{k}}{\Gamma(\alpha k+\gamma)}~.$

(123)

[115.1.5] The solution may be written as

$f(t)=f_{0}\; t^{{(1-\beta)(\alpha-1)}}E_{{\alpha,\alpha+\beta(1-\alpha)}}(-Ct^{\alpha})$

(124)

using the generalized Mittag-Leffler function defined by

$E_{{a,b}}(x)=\sum _{{k=0}}^{\infty}\frac{x^{k}}{\Gamma(ak+b)}$

(125)

[page 116, §0] for all $a>0,b\in\mathbb{C}$ . [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 $0<a\leq 1$ and $b\geq a$ [39].

[116.1.1] For $C=0$ the result reduces to eq. (109) because $E_{{a,b}}(0)=1/\Gamma(b)$ . [116.1.2] Of special interest is again the case $\beta=1$ . [116.1.3] It has the well known solution

$f(t)=f_{0}\; E_{\alpha}(-Ct^{\alpha})$

(126)

where $E_{\alpha}(x)=E_{{\alpha,1}}(x)$ denotes the ordinary Mittag-Leffler function.

3.3 Generalized Fractional Diffusion

[116.2.1] Consider the fractional partial differential equation for $f:\mathbb{R}^{d}\times\mathbb{R}_{+}\to\mathbb{R}$

$\mbox{\rm D}^{{\alpha,\beta}}_{{0+}}f(\boldsymbol{r},t)=C\;\Delta f(\boldsymbol{r},t)$

(127)

with Laplacian $\Delta$ and fractional ‘‘diffusion’’ constant $C$ . [116.2.2] The function $f(\boldsymbol{r},t)$ is assumed to obey the initial condition

$\mbox{\rm I}^{{(1-\beta)(1-\alpha)}}_{{0+}}f(\boldsymbol{r},0+)=f_{{0\boldsymbol{r}}}=f_{0}\delta(\boldsymbol{r})$

(128)

where $\delta(\boldsymbol{r})$ is the Dirac measure at the origin. [116.2.3] Fourier Transformation, defined as

${\mathcal{F}}\left\{ f(\boldsymbol{r})\right\}(\boldsymbol{q})=\int _{{\mathbb{R}^{d}}}e^{{i\boldsymbol{q}\cdot\boldsymbol{r}}}f(\boldsymbol{r})\mbox{\rm d}\boldsymbol{r},$

(129)

and Laplace transformation of eq. (127) now yields

$f(\boldsymbol{q},u)=\frac{u^{{\beta(\alpha-1)}}\; f_{0}}{C\boldsymbol{q}^{2}+u^{\alpha}}.$

(130)

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

$f(\boldsymbol{q},t)=f_{0}\; t^{{(1-\beta)(\alpha-1)}}E_{{\alpha,\alpha+\beta(1-\alpha)}}(-C\boldsymbol{q}^{2}t^{\alpha}).$

(131)

[116.2.5] Setting $\boldsymbol{q}=0$ shows that the solution of (127) cannot be a probability density except for $\beta=1$ . [116.2.6] For $\beta\neq 1$ the spatial integral is time dependent, and $f$ would need to be divided by $t^{{(1-\beta)(\alpha-1)}}$ 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]

$(2\pi)^{{-d/2}}\int e^{{i\boldsymbol{q}\cdot\boldsymbol{r}}}\left(\frac{|\boldsymbol{r}|}{m}\right)^{{1-(d/2)}}K_{{(d-2)/2}}\left(m|\boldsymbol{r}|\right)\;\mbox{\rm d}\boldsymbol{r}=\frac{1}{\boldsymbol{q}^{2}+m^{2}}$

(132)

[page 117, §0] which leads to

$f(\boldsymbol{r},u)=f_{0}(2\pi C)^{{-d/2}}\left(\frac{r}{\sqrt{C}}\right)^{{1-(d/2)}}u^{{\beta(\alpha-1)+\alpha(d-2)/4}}K_{{(d-2)/2}}\left(\frac{ru^{{\alpha/2}}}{\sqrt{C}}\right)$

(133)

with $r=|\boldsymbol{r}|$ . [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 $A=r/\sqrt{C}$ , $\lambda=\alpha/2$ , $\nu=(d-2)/2$ and $\mu=\beta(\alpha-1)+\alpha(d-2)/4$ and using the general relation

${\mathcal{M}}\left\{ x^{q}g(bx^{p})\right\}(s)=\frac{1}{p}b^{{-(s+q)/p}}g\left(\frac{s+q}{p}\right)\qquad(b,p>0)$

(134)

leads to

${\mathcal{M}}\left\{ f(r,u)\right\}(s)=\frac{f_{0}}{\lambda}(2\pi C)^{{-d/2}}A^{{1-(d/2)}}A^{{-(s+\mu)/\lambda}}{\mathcal{M}}\left\{ K_{\nu}(u)\right\}\left((s+\mu)/\lambda\right).$

(135)

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

${\mathcal{M}}\left\{ K_{\nu}(x)\right\}(s)=2^{{s-2}}\Gamma\left(\frac{s+\nu}{2}\right)\Gamma\left(\frac{s-\nu}{2}\right).$

(136)

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

${\mathcal{M}}\left\{ f(r,t)\right\}(s)=\frac{f_{0}}{\alpha(r^{2}\pi)^{{d/2}}}\left(\frac{r}{2\sqrt{C}}\right)^{{2(1-\beta)(1-(1/\alpha))}}\left(\frac{r}{2\sqrt{C}}\right)^{{2s/\alpha}}\frac{\Gamma\left(\frac{d}{2}+(\beta-1)(1-\frac{1}{\alpha})-\frac{s}{\alpha}\right)\Gamma\left(1+(\beta-1)(1-\frac{1}{\alpha})-\frac{s}{\alpha}\right)}{\Gamma(1-s)}$

(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 $m=0$ , $n=2$ , $p=2$ , $q=1$ , $A_{1}=A_{2}=1/\alpha$ , $a_{1}=1-(d/2)-(\beta-1)(1-(1/\alpha))$ , $a_{2}=(1-\beta)(1-(1/\alpha))$ , $b_{1}=0$ and $B_{1}=1$ if $(\alpha d/2)+(\beta-1)(\alpha-1)>0$ . [117.0.6] Then the result becomes

${f(r,t)}=\frac{f_{0}}{\alpha(r^{2}\pi)^{{d/2}}}{\left(\frac{r}{2\sqrt{C}}\right)^{{2(1-\beta)(1-(1/\alpha))}}}H^{{02}}_{{21}}\left(\left(\frac{2\sqrt{C}}{r}\right)^{{2/\alpha}}t\left|\begin{array}[]{l}{(1-\frac{d}{2}+(1-\beta)(1-\frac{1}{\alpha}),\frac{1}{\alpha}),((1-\beta)(1-\frac{1}{\alpha}),\frac{1}{\alpha})}\\ {(0,1)}\end{array}\right.\right).$

(138)

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

$f(r,t)=\frac{f_{0}\; t^{{(1-\beta)(\alpha-1)}}}{(r^{2}\pi)^{{d/2}}}H^{{20}}_{{12}}\left(\frac{r^{2}}{4Ct^{\alpha}}\left|\begin{array}[]{l}{(1+(1-\beta)(\alpha-1),\alpha)}\\ {(d/2,1),(1,1)}\end{array}\right.\right).$

(139)

[118.0.2] The result reduces to the known result [15, 8] for $\beta=1$ . [118.0.3] In that case $f(\boldsymbol{r},t)$ is also a probability density. [118.0.4] For $\beta\neq 1$ the function $f(\boldsymbol{r},t)$ does not have a probabilistic interpretation because its normalization decays as $t^{{(1-\beta)(\alpha-1)}}$ .

3.4 Relation with Continuous Time Random Walk

[118.1.1] The fractional diffusion eq. (127) of type $\beta=1$ has a probabilistic interpretation as noted after eq. (131). [118.1.2] $f(\boldsymbol{r},t)$ may be viewed as the probability density for a random walker or diffusing object to be at position $\boldsymbol{r}$ at time $t$ under the condition that it started from the origin $\boldsymbol{r}=\boldsymbol{0}$ 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

$f(\boldsymbol{r},t)=\delta _{{\boldsymbol{r}\boldsymbol{0}}}+\frac{C}{\Gamma(\alpha)}\int _{0}^{t}(t-s)^{{\alpha-1}}\Delta f(\boldsymbol{r},t)\;\mbox{\rm d}s$

(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 $\boldsymbol{r}=\boldsymbol{0}$ 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 $\psi(t)$ and the probability density of a displacement by a vector $\boldsymbol{r}$ in a single jump is denoted $p(\boldsymbol{r})$ . [118.2.3] Then the integral equation of continuous time random walk theory reads

$f(\boldsymbol{r},t)=\delta _{{\boldsymbol{r}\boldsymbol{0}}}\Phi(t)+\int _{0}^{t}\psi(t-s)\int _{{\mathbb{R}^{d}}}p(\boldsymbol{r}-\boldsymbol{r}^{{\prime}})f(\boldsymbol{r},t)\;\mbox{\rm d}\boldsymbol{r}^{{\prime}}\;\mbox{\rm d}s$

(141)

where $\Phi(t)$ 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 $\psi(t)$ . [118.2.5] In general the first step needs special consideration [49, 49, 45]. [118.2.6] The survival probablity $\Phi(t)$ is related to the waiting

[page 119, §0] time density through

$\Phi(t)=1-\int _{0}^{t}\psi(t^{{\prime}})\;\mbox{\rm d}t^{{\prime}}.$

(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 $\beta=1$ gives the solution of eq. (127) in Fourier-Laplace space as

$f(\boldsymbol{q},u)=\frac{u^{{\alpha-1}}}{C\boldsymbol{q}^{2}+u^{\alpha}}.$

(143)

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

$f(\boldsymbol{q},u)=\frac{1}{u}\frac{1-\psi(u)}{1-\psi(u)p(\boldsymbol{q})}.$

(144)

[119.1.4] Equating these two equations yields

$\frac{1-p(\boldsymbol{q})}{C\boldsymbol{q}^{2}}=\frac{1-\psi(u)}{u^{\alpha}\psi(u)}.$

(145)

[119.1.5] Because the left hand side does not depend on $u$ and the right hand side is independent of $\boldsymbol{q}$ they must both equal a common constant $\tau _{0}^{\alpha}$ . [119.1.6] It follows that

$p(\boldsymbol{q})=1-C\tau _{0}^{\alpha}\boldsymbol{q}^{2}$

(146)

identifying the constant $C\tau _{0}^{\alpha}$ as the mean square displacement of a single jump. [119.1.7] For the waiting time density one finds

$\psi(u)=\frac{1}{1+\tau _{0}^{\alpha}u^{\alpha}},$

(147)

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

$\psi(t;\alpha,\tau _{0})=\frac{1}{\tau _{0}}\left(\frac{t}{\tau _{0}}\right)^{{\alpha-1}}E_{{\alpha,\alpha}}\left(-\frac{t^{\alpha}}{\tau _{0}^{\alpha}}\right)$

(148)

where $E_{{a,b}}(x)$ is again the Mittag-Leffler function defined in eq. (40).

[119.2.1] For $\alpha=1$ the waiting time density becomes exponential

$\psi(t;1,\tau _{0})=\frac{1}{\tau _{0}}e^{{-t/\tau _{0}}}.$

(149)

[page 120, §0] [120.0.1] For $0<\alpha<1$ characteristic differences arise from the asymptotic behaviour for $t\to 0$ and $t\to\infty$ . [120.0.2] The asymptotic behaviour of $\psi(t)$ for $t\to 0$ is obtained by noting that $E_{{\alpha,\alpha}}(0)=1$ , and hence

$\psi(t)\sim t^{{\alpha-1}}$

(150)

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

$E_{{a,b}}(z)=-\sum _{{n=1}}^{N}\frac{z^{{-n}}}{\Gamma(b-an)}+O(|z|^{N})$

(151)

valid for $|\arg(-z)|<(1-(a/2))\pi$ and $z\to\infty$ . [120.0.5] It follows that $E_{{a,a}}(-x)\sim x^{{-2}}$ for $x\to\infty$ and hence

$\psi(t)\sim t^{{-1-\alpha}}$

(152)

for $t\to\infty$ . [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.

Author:	R. Hilfer
originally published in:	Applications of Fractional Calculus in Physics, World Scientific, Singapore, page 87-130 (2000), ISBN 981-02-3457-0
originally published:	March, 2000