[page 46, §1]
[46.1.1] An introduction to fractional derivatives would be incomplete without an introduction to applications. [46.1.2] In the past fractional calculus has been used predominantly as a convenient calculational tool [76, 89, 26]. [46.1.3] A well known example is Riesz’ interpolation method for solving the wave equation [20]. [46.1.4] In recent times, however, fractional differential equations appear as ‘‘generalizations’’ of more or less fundamental equations of physics [18, 3, 90, 12, 104, 43, 60, 46, 54, 52, 56, 55, 58, 23, 91, 102, 129, 119]. [46.1.5] The idea is that physical phenomena can be described by fractional differential equations. [46.1.6] This practice raises at least two fundamental questions:
[46.1.7] Are mathematical models with fractional derivatives consistent with the fundamental laws and fundamental symmetries of nature ?
[46.1.8] How can the fractional order
[46.1.9] Both questions will be addressed here. [46.1.10] The answer to the first question is provided by the theory of fractional time evolutions [43, 47], the answer to the second question by anomalous subdiffusion [60, 46].
[46.2.1] Fractional derivatives are nonlocal operators. [46.2.2] Nevertheless, numerous authors have proposed fractional differential equations involving fractional spatial derivatives. [46.2.3] Particularly popular are fractional powers of the Laplace operator due to the well known work of Riesz, Feller and Bochner [97, 13, 27]. The nonlocality of fractional spatial derivatives raises serious (largely) unresolved physical problems.
[46.3.1] As an illustration of the problem with spatial
fractional derivatives consider the one
dimensional potential equation for functions
(2.136) |
on the open interval
[page 47, §0] specification
(2.137) |
and the perturbed boundary specification
(2.138) |
with
[47.1.1] Consider now a fractional generalization of (2.136) that arises for example as the stationary limit of (Bochner-Levy) fractional diffusion equations with a fractional Laplace operator [13]. [47.1.2] Such a onedimensional fractional Laplace equation reads
(2.139) |
where
[47.2.1] Locality in space is a basic and firmly established principle of physics (see e.g. [115, 35]). [47.2.2] Of course, one could argue that relativistic effects are negligible, and that fractional spatial derivatives might arise as an approximate phenomenological model describing an underlying physical reality that obeys spatial locality. [47.2.3] However, spatial fractional derivatives imply not only action at a distance. [47.2.4] As seen above, they imply also that the exterior domain cannot be decoupled from the interior by conventional walls or boundary conditions. [47.2.5] This has far reaching consequences for theory and experiment. [47.2.6] In theory it invalidates all arguments based on surface to volume ratios becoming negligible in the large volume limit. [47.2.7] This includes many concepts and results in thermodynamics and statistical physics that depend on the lower dimensionality of the boundary. [47.2.8] Experimentally it becomes difficult to isolate a system from its environment. [47.2.9] Fractional diffusion would never come to rest inside a vessel with thin rigid walls unless the equilibrium concentration prevails also outside the vessel. [47.2.10] A fractionally viscous fluid at rest inside a container with thin rigid walls would have to start to move when the same fluid starts flowing outside the vessel. [47.2.11] It seems therefore difficult to reconcile nonlocality in space with theory and experiment.
[page 48, §1]
[48.1.1] Nonlocality in time, unlike space, does not violate basic principles of physics, as long as it respects causality [49, 48, 43, 47, 54]. [48.1.2] In fact, causal nonlocality in time is a common nonequilibrium phenomenon known as history dependence, hysteresis and memory. [48.2.1] Theoretical physics postulates time translation invariance as a fundamental symmetry of nature. [48.2.2] As a consequence energy conservation is fundamental, and the infinitesimal generator of time translations is a first order time derivative. [48.2.3] Replacing integer order time derivatives with fractional time derivatives raises at least three basic questions:
[48.2.4] What replaces time translations as the physical time evolution ?
[48.2.5] Is the nonlocality of fractional time derivatives consistent with the laws of nature ?
[48.2.6] Is the asymmetry of fractional time derivatives consistent with the laws of nature ?
[48.2.7] These questions as well as ergodicity breaking, stationarity, long time limits and temporal coarse grainig were discussed first within ergodic theory [49, 48, 47] and later from a general perspective in [54].
[48.3.1] The third question requires special remarks because irreversibility is a longstanding and controversial subject [71]. [48.3.2] The problem of irreversibility may be formulated briefly in two ways.
[48.4.1] Assume that time is reversible. Explain how and why time irreversible equations arise in physics.
[48.5.1] Assume that time is irreversible. Explain how and why time reversible equations arise in physics.
[48.6.1] While the normal problem has occupied physicists and mathematicians for more than a century, the reversed problem was apparently first formulated in [59]. [48.6.2] Surprisingly, the reversed irreversibility problem has a clear and quantitiative solution within the theory of fractional time. [48.6.3] The solution is based on the simple postulate that every time evolution of a physical system is irreversible. [48.6.4] It is not possible to repeat an experiment in the past [59]. [48.6.5] This empiricial fact seems to reflect a fundamental law of nature that rivals the law of energy conservation.
[48.7.1] The mathematical concepts corresponding to irreversible time evolutions are operator semigroups and abstract Cauchy problems [15, 93]. [48.7.2] The following brief introduction to fractional time evolutions (sections 2.3.3.2–2.3.3.8) is in large parts identical to the brief exposition in [59]. [48.7.3] For more details see [54].
[page 49, §1]
[49.1.1] A physical time evolution
(2.140) | |||
(2.141) |
for all
[49.2.1] The linear operator
(2.142) |
with domain
(2.143) |
is called the infinitesimal generator of the semigroup.
[49.2.2] Here
[49.3.1] Physical time evolution is continuous. [49.3.2] This requirement is represented mathematically by the assumption that
(2.144) |
holds for all
[49.4.1] Homogeneity of time means two different requirements: [49.4.2] Firstly, it requires that observations are independent of a particular instant or position in
[page 50, §0] time. [50.0.1] Secondly, it requires arbitrary divisibility of time durations and self-consistency for the transition between time scales.
[50.1.1] Independence of physical processes from their position on the time axis requires that physical experiments are reproducible if they are ceteris paribus shifted in time. [50.1.2] The first requirement, that the start of an experiment can be shifted, is expressed mathematically as the requirement of invariance under time translations. [50.1.3] As a consequence one demands commutativity of the time evolution with time translations in the form
(2.145) |
for all
(2.146) |
[50.1.5] Note that
[50.2.1] The second requirement of homogeneity is
homogeneous divisibility.
[50.2.2] The semigroup property (2.140) implies that for
(2.147) |
holds.
[50.2.3] Homogeneous divisibility of a physical time evolution
requires that there exist rescaling factors
(2.148) |
exists und defines a time evolution
[50.3.1] Causality of the physical time evolution requires
that the values of the image function
[50.4.1] The requirement (2.145)
of homogeneity implies that the operators
[page 51, §0]
exists a finite Borel measure
(2.149) |
holds [128],[114, p.26].
[51.0.1] Applying this theorem to physical time evolution operators
(2.150) |
with
[51.1.1] The requirement of causality implies that the support
[51.2.1] The convolution semigroups with support in the positive
half axis
(2.151) | |||
(2.152) |
holds for all
[51.3.1] The characterization is given by the following
theorem [10, p.68].
[51.3.2] There exists a one-to-one mapping between the
convolution semigroups
(2.153) |
with
[51.4.1] The requirement of homogeneous divisibility further
restricts the set of admissible Bernstein functions.
[51.4.2] It leaves only those measures
(2.154) |
[51.4.3] Such limit measures
[51.5.1] The remaining measures define the class of fractional
time evolutions
[page 52, §1] [52.1.1] These remaining fractional measures have a density and they can be written as [43, 48, 49, 47, 54]
(2.155) |
where
(2.156) |
allowing to identify
(2.157) |
[52.2.1] The infinitesimal generators of the fractional
semigroups
(2.158) |
are fractional time derivatives of Marchaud-Hadamard type [98, 51]. [52.2.2] This fundamental and general result provides the basis for generalizing physical equations of motion by replacing the integer order time derivative with a fractional time derivative as the generator of time evolution [43, 54].
[52.3.1] For
[52.4.1] Homogeneous divisibility formalizes the fact that a verbal statement in the present tense presupposes always a certain time scale for the duration of an
[page 53, §0] instant. [53.0.1] In this sense the present should not be thought of as a point, but as a short time interval [59, 54, 48].
[53.1.1] Fractional time evolutions seem to be related
to the subjective human experience of time.
[53.1.2] In physics the time duration is measured
by comparison with a periodic reference (clock)
process.
[53.1.3] Contrary to this, the subjective human experience of
time amounts to the comparison with an hour glass,
i.e. with a nonperiodic reference.
[53.1.4] It seems that a time duration is experienced as ‘‘long’’
if it is comparable to the time interval that has
passed since birth.
[53.1.5] This phenomenon seems to be reflected in fractional stationary
states defined as solutions of the stationarity condition
[53.2.1] Finally, also the special case
[53.3.1] Consider now the second basic question of Section 2.3.1:
How can the fractional order
[53.4.1] The term fractional diffusion can refer either to diffusion with
a fractional Laplace operator or to diffusion equations with
a fractional time derivative.
[53.4.2] Fractional diffusion (or Fokker-Planck) equations with a
fractional Laplacian may be called Bochner-Levy diffusion.
[53.4.3] The identification of the fractional order
[page 54, §0] with Bochner-Levy diffusion [65]9 (This is a footnote:) 9Also, Herrmann Weyl, who pioneered fractional as well as functional calculus and worked on the foundations of physics, seems not to have applied fractional derivatives to problems in physics.. [54.0.1] A possible reason might be the unresolved problem of locality discussed above. [54.0.2] Bochner himself writes ‘‘Whether this (equation) might have physical interpretation, is not known to us’’ [13, p.370].
[54.1.1] Diffusion equations with a fractional time derivative will be called Montroll-Weiss diffusion although fractional time derivatives do not appear in the original paper [87] and the connection was not discovered until 30 years later [60, 46]. [54.1.2] As shown in Section 2.3.3, the locality problem does not arise. [54.1.3] Montroll-Weiss diffusion is expected to be consistent with all fundamental laws of physics. [54.1.4] The fact that the relation between Montroll-Weiss theory and fractional time derivatives was first established in [60, 46] seems to be widely unknown at present, perhaps because this fact is never mentioned in widely read reviews [82] and popular introductions to the subject [112]10 (This is a footnote:) 10Note that, contrary to [112, p.51], fractional derivatives are never mentioned in [6]. .
[54.2.1] There exist several versions of diffusion equations
with fractional time derivatives, and they differ
physically or mathematically
from each other [127, 104, 54, 82, 130].
[54.2.2] Of interest here will be
the fractional diffusion equation
for
(2.159) |
with a fractional time derivative of order
[54.3.1] An alternative to eq. (2.159), introduced in [54, 53], is
(2.160) |
with a Riemann-Liouville fractional
time derivative
[54.4.1] Before discussing how
[page 55, §0]
[55.1.1] In the table
(2.161) |
was used for the
[page 56, §1]
[56.1.1] The results in the table show that the normal diffusion
(
[56.2.1] The fractional diffusion equation (2.159)
can be related rigorously to the microscopic model of
Montroll-Weiss continuous time random walks (CTRW’s)
[87, 64] in the same way as ordinary
diffusion is related to random walks [27].
[56.2.2] The fractional order
[56.3.1] The basic integral equation for separable continuous time random walks describes a random walker in continuous time without correlation between its spatial and temporal behaviour. [56.3.2] It reads [87, 88, 118, 39, 64]
(2.162) |
where
[56.4.1] The fractional master equation introduced in [60]
with inital condition
(2.163) |
with fractional transition rates
(2.164) |
[page 57, §0]
for the Fourier transformed
transition rates
(2.165) |
for the waiting time density, where
(2.166) |
for
(2.167) |
for
[57.1.1] It had been observed already in the early 1970’s that continuous time random walks are equivalent to generalized master equations [66, 9]. [57.1.2] Similarly, the Fourier-Laplace formula
(2.168) |
for the solution of CTRW’s with algbraic tails of the form (2.167) was well known (see [117, eq.(21), p.402] [110, eq.(23), p.505] [67, eq.(29), p.3083]). [57.1.3] Comparison with row 2 of the table makes the connection between the fractional diffusion equation (2.159) and the CTRW-equation (2.162) evident. [57.1.4] However, this connection with fractional calculus was not made before the appearance of [60, 46]. [57.1.5] In particular, there is no mention of fractional derivatives or fractional calculus in [6].
[57.2.1] The rigorous relation between fractional diffusion and CTRW’s, established in [60, 46] and elaborated in [50, 54, 53, 57], has become a fruitful starting point for subsequent investigations, particularly into fractional Fokker-Planck equations with drift [19, 83, 81, 111, 80, 100, 33, 51, 61, 82, 130, 112].
Acknowledgement: The author thanks Th. Müller and S. Candelaresi for reading the manuscript.