[111.2.1] To simplify the notation

[111.2.6] An observable or state

(98) |

holds for

[111.3.1] The function

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

(99) |

[page 112, §0]
where

(100) |

where relations (170) and (172) were used to
rewrite the

(101) |

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

(102) |

[112.0.4] For

[112.1.1] To elucidate the significance of the new type of stationary states
it is useful to consider the infinitesimal form,

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

[112.2.2] The right-sided
Riemann-Liouville fractional integral of order

(103) |

[page 113, §0]
for

(104) |

for

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

[113.1.2] The (right-/left-sided) fractional derivative of order

(105) |

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

[113.1.3] The Riemann-Liouville fractional derivative

(106) |

where the inital value

[113.2.1] It is now possible to discuss the infinitesimal form
of fractional stationarity where the generator

(107) |

for

(108) |

[page 114, §0]
defines fractional stationarity of order

(109) |

[114.0.3] This may be seen by inserting

(110) |

and using the basic fractional integral

(111) |

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

(112) |

remains conserved and constant for all

[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

(113) |

with

(114) |

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

(115) |

and thence

(116) |

[page 115, §0]
[115.0.1] For

(117) |

[115.1.1] Consider the fractional Cauchy problem

(118) |

for

(119) |

where

(120) |

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

(121) |

with

(122) |

[115.1.4] Inverting the series term by term using

(123) |

[115.1.5] The solution may be written as

(124) |

using the generalized Mittag-Leffler function defined by

(125) |

[page 116, §0]
for all

[116.1.1] For

(126) |

where

[116.2.1] Consider the fractional partial differential equation
for

(127) |

with Laplacian

(128) |

where

(129) |

and Laplace transformation of eq. (127) now yields

(130) |

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

(131) |

[116.2.5] Setting

[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]

(132) |

[page 117, §0] which leads to

(133) |

with

(134) |

leads to

(135) |

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

(136) |

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

(137) |

for the Mellin transform of

(138) |

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

(139) |

[118.0.2] The result reduces to the known result [15, 8]
for

[118.1.1] The fractional diffusion eq. (127) of type

(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

(141) |

where

[page 119, §0] time density through

(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

(143) |

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

(144) |

[119.1.4] Equating these two equations yields

(145) |

[119.1.5] Because the left hand side does not depend on

(146) |

identifying the constant

(147) |

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

(148) |

where

[119.2.1] For

(149) |

[page 120, §0]
[120.0.1] For

(150) |

for

(151) |

valid for

(152) |

for