[111.2.1] To simplify the notation will be denoted as
in the following.
[111.2.2] A first application of fractional time evolutions
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 "
"
for the infinitesimal generators of fractional
dynamics has rarely been noticed.
[111.2.5] Stationary states
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.
[111.2.6] An observable or state is called stationary
or asymptotically invariant under the time evolution
if
![]() |
(98) |
holds for .
[111.2.7] It is called
stationary in the strict sense, or strictly invariant
under
, if condition (98)
holds for all
and
.
[111.3.1] The function where
is a constant
is asymptotically and strictly stationary under
the fractional time evolutions
.
[111.3.2] This follows readily by insertion into the definition,
and by noting that
is a probability density.
[111.4.1] In addition to the conventional constants there exists a second class of stationary states given by
![]() |
(99) |
[page 112, §0]
where and
are constants.
[112.0.1] To see this one evaluates
![]() |
(100) |
where relations (170) and (172) were used to
rewrite the -function in eq. (69).
[112.0.2] Using the integral (178), the reduction formulae
(167) and (169), and property (171)
one finds
![]() |
![]() |
(101) |
[112.0.3] An application of the series expansion (181) gives
![]() |
(102) |
[112.0.4] For only the
term in the series contributes
and this shows that
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, ,
of the stationarity condition.
[112.1.2] The nature of the limit
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.
[112.2.2] The right-sided
Riemann-Liouville fractional integral of order
of a locally integrable function
is defined as
![]() |
(103) |
[page 113, §0]
for , the left-sided Riemann-Liouville fractional integral
is defined as
![]() |
(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 and type
with respect to
is defined by
![]() |
(105) |
for functions for which the expression on the right hand side exists.
[113.1.3] The Riemann-Liouville fractional derivative
corresponds to
and type
.
[113.1.4] Fractional derivatives of type
are discussed
in Chapter I and were employed in [4].
[113.1.5] It seems however that fractional derivatives of general type
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
![]() |
(106) |
where the inital value is
the Riemann-Liouville derivative for
.
[113.1.8] Note that fractional derivatives of type
involve nonfractional
initial values.
[113.2.1] It is now possible to discuss the infinitesimal form
of fractional stationarity where the generator
for initial conditions of type
is represented by
.
[113.2.2] The fractional differential equation
![]() |
(107) |
for with initial condition
![]() |
(108) |
[page 114, §0]
defines fractional stationarity of order and
type
.
[114.0.1] Of course, for
this definition reduces to the
conventional definition of stationarity.
[114.0.2] Equation (107) is solved by
![]() |
(109) |
[114.0.3] This may be seen by inserting into the definition
![]() |
(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 while the function
itself varies.
[114.0.5] In particular
and
.
[114.0.6] For
and for
one recovers
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
![]() |
(113) |
with a constant, and with initial condition
![]() |
(114) |
as before. [114.2.2] Laplace transformation using eq. (106) gives
![]() |
(115) |
and thence
![]() |
(116) |
[page 115, §0]
[115.0.1] For this reduces to
![]() |
(117) |
[115.1.1] Consider the fractional Cauchy problem
![]() |
(118) |
for with initial condition
![]() |
(119) |
where is a (‘‘fractional relaxation’’) constant.
[115.1.2] Laplace Transformation gives
![]() |
(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
yields the result
![]() |
(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 .
[116.0.1] This function is an entire function of order
[38].
[116.0.2] Moreover it is completely monotone if and only if
and
[39].
[116.1.1] For the result reduces to eq. (109) because
.
[116.1.2] Of special interest is again the case
.
[116.1.3] It has the well known solution
![]() |
(126) |
where denotes the ordinary Mittag-Leffler
function.
[116.2.1] Consider the fractional partial differential equation
for
![]() |
(127) |
with Laplacian and fractional ‘‘diffusion’’
constant
.
[116.2.2] The function
is assumed to obey the initial condition
![]() |
(128) |
where is the Dirac measure at the origin.
[116.2.3] Fourier Transformation, defined as
![]() |
(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 shows that the solution of (127) cannot be a
probability density except for
.
[116.2.6] For
the spatial integral is time dependent, and
would need to be divided by
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]
![]() |
(132) |
[page 117, §0] which leads to
![]() |
(133) |
with .
[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
,
,
and
and using the general
relation
![]() |
(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 .
[117.0.5] Comparing this with the Mellin transform of the
-function
in eq. (175) allows to identify the
-function
parameters as
,
,
,
,
,
,
,
and
if
.
[117.0.6] Then the result becomes
![]() |
(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 .
[118.0.3] In that case
is also a probability density.
[118.0.4] For
the function
does not have
a probabilistic interpretation because its normalization
decays as
.
[118.1.1] The fractional diffusion eq. (127) of type
has a probabilistic interpretation as noted after eq.
(131).
[118.1.2]
may be viewed as the probability density
for a random walker or diffusing object to be at position
at time
under the condition that it started
from the origin
at time
.
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
![]() |
(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
at time
and proceeds by successive random
jumps [43, 44, 45, 46, 47, 48].
[118.2.2] The probability density for a time interval of
length
between two consecutive jumps is
denoted
and the probability density of a
displacement by a vector
in a single
jump is denoted
.
[118.2.3] Then the integral equation of continuous time
random walk theory reads
![]() |
(141) |
where is the probability that the walker
survives at the origin for a time of length
.
[118.2.4] Here the walker is assumed to be prepared in its
initial position from which it develops according
to
.
[118.2.5] In general the first step needs special consideration
[49, 49, 45].
[118.2.6] The survival probablity
is related to the waiting
[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 gives the solution of eq. (127) in
Fourier-Laplace space as
![]() |
(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 and
the right hand side is independent of
they
must both equal a common constant
.
[119.1.6] It follows that
![]() |
(146) |
identifying the constant as the mean square displacement
of a single jump.
[119.1.7] For the waiting time density one finds
![]() |
(147) |
which may be inverted in the same way as eq. (120) to give
![]() |
(148) |
where is again the Mittag-Leffler function defined
in eq. (40).
[119.2.1] For the waiting time density becomes exponential
![]() |
(149) |
[page 120, §0]
[120.0.1] For characteristic differences arise from
the asymptotic behaviour for
and
.
[120.0.2] The asymptotic behaviour of
for
is obtained by
noting that
, and hence
![]() |
(150) |
for .
[120.0.3] For
the waiting time density is singular at the
origin implying a statistical abundance of short intervals
between jumps compared to the exponential case
.
[120.0.4] For large
recall the asymptotic series expansion
[52]
![]() |
(151) |
valid for and
.
[120.0.5] It follows that
for
and
hence
![]() |
(152) |
for .
[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.