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
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 t≥0 and s∈R.
[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
fs=f0sγ-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-11αtH1101(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
Tαtfs | =f0sγ-1Γ(γ)H1101((st)α|1,11-γ,α). | | (101) |
[112.0.3] An application of the series expansion (181)
gives
Tαtfs=f0sγ-1Γγ∑k=0∞-1kt/skαk!Γγ-kα. | | (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,α∈R
of a locally integrable function f is defined as
Ia+αfx=1Γα∫axx-yα-1fydy | | (103) |
[page 113, §0]
for x>a, the left-sided Riemann-Liouville fractional integral
is defined as
Ia-αfx=1Γα∫xay-xα-1fydy | | (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
0≤β≤1 with respect to x is defined by
Da±α,βfx=±Ia±β1-αddxIa±1-β1-αfx | | (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 [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
LDa+α,βfxu=uαLfxu-uβα-1Da+1-βα-1,0f0+ | | (106) |
where the inital value Da+1-βα-1,0f0+ 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
ft=f0t1-βα-1Γ1-βα-1+1. | | (109) |
[114.0.3] This may be seen by inserting ft into the definition
D0+α,βfx=I0+β1-αddxI0+1-β1-αfx | | (110) |
and using the basic fractional integral
Ia+αx-aβ=Γβ+1Γα+β+1x-aα+β | | (111) |
(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→0ft=∞ and
limt→∞ft=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
with C∈R a constant, and with initial condition
as before.
[114.2.2] Laplace transformation using eq. (106) gives
fu=Cuα+1+f0uα+β1-α | | (115) |
and thence
ft=CtαΓα+1+f0t1-βα-1Γ1-β1-α+1. | | (116) |
[page 115, §0]
[115.0.1] For β=1 this reduces to
3.2 Generalized Fractional Relaxation
[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
fu=uα-γC+uα=u-γ1Cu-α+1=∑k=0∞-Cku-αk-γ | | (121) |
with
[115.1.4] Inverting the series term by term using
Lxα-1/Γα=u-α
yields the result
ft=tγ-1∑k=0∞-CtαkΓαk+γ. | | (123) |
[115.1.5] The solution may be written as
ft=f0t1-βα-1Eα,α+β1-α-Ctα | | (124) |
using the generalized Mittag-Leffler function defined by
Ea,bx=∑k=0∞xkΓak+b | | (125) |
[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,b0=1/Γb.
[116.1.2] Of special interest is again the case β=1.
[116.1.3] It has the well known solution
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,βfr,t=CΔfr,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-β1-αfr,0+=f0r=f0δr | | (128) |
where δr is the Dirac measure at the origin.
[116.2.3] Fourier Transformation, defined as
Ffrq=∫Rdeiq⋅rfrdr, | | (129) |
and Laplace transformation of eq. (127) now yields
fq,u=uβα-1f0Cq2+uα. | | (130) |
[116.2.4] Using the result (124) for the inverse Laplace transform
of (120) gives
fq,t=f0t1-βα-1Eα,α+β1-α-Cq2tα. | | (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 β≠1 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
[40]
2π-d/2∫eiq⋅rrm1-d/2Kd-2/2mrdr=1q2+m2 | | (132) |
[page 117, §0]
which leads to
fr,u=f02πC-d/2rC1-d/2uβα-1+αd-2/4Kd-2/2ruα/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) |
leads to
Mfr,us=f0λ2πC-d/2A1-d/2A-s+μ/λMKνus+μ/λ. | | (135) |
[117.0.3] The Mellin transform of the Bessel function reads [32]
MKνxs=2s-2Γs+ν2Γs-ν2. | | (136) |
[117.0.4] Inserting this, using eq.(78), and restoring the
original variables then yields
Mfr,ts=f0αr2πd/2r2C21-β1-1/αr2C2s/αΓd2+β-11-1α-sαΓ1+β-11-1α-sαΓ1-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)=f0αr2πd/2(r2C)21-β1-1/αH2102((2Cr)2/αt|1-d2+1-β1-1α,1α,1-β1-1α,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)=f0t1-βα-1r2πd/2H1220(r24Ctα|1+1-βα-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 β≠1 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
fr,t=δr0+CΓα∫0tt-sα-1Δfr,tds | | (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
fr,t=δr0Φt+∫0tψt-s∫Rdpr-r′fr,tdr′ds | | (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
[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]
fq,u=1u1-ψu1-ψupq. | | (144) |
[119.1.4] Equating these two equations yields
1-pqCq2=1-ψuuαψu. | | (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
identifying the constant Cτ0α 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
ψt;α,τ0=1τ0tτ0α-1Eα,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
ψt;1,τ0=1τ0e-t/τ0. | | (149) |
[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 ψt for t→0 is obtained by
noting that Eα,α0=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]
Ea,bz=-∑n=1Nz-nΓb-an+OzN | | (151) |
valid for arg-z<1-a/2π 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.