2.3 Physical Introduction to Fractional Derivatives

[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:

1. [48.2.4] What replaces time translations as the physical time evolution ?

2. [48.2.5] Is the nonlocality of fractional time derivatives consistent with the laws of nature ?

3. [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.

[48.6.1] 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 T⁢Δ⁢t:0≤Δ⁢t<∞ is defined as a one-parameter family (with time parameter Δ⁢t) of bounded linear time evolution operators T⁢Δ⁢t on a Banach space B. [49.1.2] The parameter Δ⁢t represents time durations. The one-parameter family fulfills the conditions

for all Δ⁢t1,Δ⁢t2≥0, t0∈R and f∈B. [49.1.3] The elements f∈B represent time dependent physical observables, i.e. functions on the time axis R. [49.1.4] Note that the argument Δ⁢t≥0 of T⁢Δ⁢t has the meaning of a time duration, while t∈R in f⁢t means a time instant. [49.1.5] Equations (2.140) and (2.141) define a semigroup. [49.1.6] The inverse elements T⁢-Δ⁢t are absent. [49.1.7] This reflects the fundamental difference between past and future.

[49.2.1] The linear operator A defined as

with domain

is called the infinitesimal generator of the semigroup. [49.2.2] Here s-lim⁢f=g is the strong limit and means lim⁡f-g=0 in the norm of B as usual.

[49.3.1] Physical time evolution is continuous. [49.3.2] This requirement is represented mathematically by the assumption that

holds for all f∈B, where s-lim is again the strong limit. [49.3.3] Semigroups of operators satisfying this condition are called strongly continuous or C0-semigroups [15, 93]. [49.3.4] Strong continuity is weaker than uniform continuity and has become recognized as an important continuity concept that covers most applications [2].

[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

for all Δ⁢t≥0 und t0,τ∈R. [50.1.4] Here the translation operator T⁢t is defined by

[50.1.5] Note that τ∈R is a time shift, not a duration. [50.1.6] It can also be negative. Physical experiments in the past have the same outcome as in the present or in the future. [50.1.7] Outcomes of past experiments can be studied in the present with the help of documents (e.g. a video recording), irrespective of the fact that the experiment cannot be repeated in the past.

[50.2.1] The second requirement of homogeneity is homogeneous divisibility. [50.2.2] The semigroup property (2.140) implies that for Δ⁢t>0

holds. [50.2.3] Homogeneous divisibility of a physical time evolution requires that there exist rescaling factors Dn for Δ⁢t such that with Δ⁢t=Δ⁢t¯/Dn the limit

exists und defines a time evolution T¯⁢Δ⁢t¯. [50.2.4] The limit n→∞ corresponds to two simultaneous limits n→∞,Δ⁢t→0, and it corresponds to the passage from a microscopic time scale Δ⁢t to a macroscopic time scale Δ⁢t¯.

[50.3.1] Causality of the physical time evolution requires that the values of the image function g⁢t=T⁢Δ⁢t⁢f⁢t depend only upon values f⁢s of the original function with time instants s<t.

[50.4.1] The requirement (2.145) of homogeneity implies that the operators T⁢Δ⁢t are convolution operators [128, 114]. Let T be a bounded linear operator on L1⁢R that commutes with time translations, i.e. that fulfills eq. (2.145). [50.4.2] Then there

[page 51, §0]    exists a finite Borel measure μ such that

holds [128],[114, p.26]. [51.0.1] Applying this theorem to physical time evolution operators T⁢Δ⁢t yields a convolution semigroup μΔ⁢t of measures T⁢Δ⁢t⁢f⁢t=μΔ⁢t*f⁢t

with Δ⁢t1,Δ⁢t2≥0. [51.0.2] For Δ⁢t=0 the measure μ0 is the Dirac-measure concentrated at 0.

[51.1.1] The requirement of causality implies that the support supp⁢μΔ⁢t⊂R+=0,∞ of the semigroup is contained in the positive half axis.

[51.2.1] The convolution semigroups with support in the positive half axis 0,∞ can be characterized completely by Bernstein functions [10]. [51.2.2] An arbitrarily often differentiable function b:0,∞→R with continuous extension to 0,∞ is called Bernstein function if for all x∈0,∞

holds for all n∈N. [51.2.3] Bernstein functions are positive, monotonously increasing and concave.

[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 μt:t≥0 with support on 0,∞ and the set of Bernstein functions b:0,∞→R [10]. [51.3.3] This mapping is given by

with Δ⁢t>0 and u>0.

[51.4.1] The requirement of homogeneous divisibility further restricts the set of admissible Bernstein functions. [51.4.2] It leaves only those measures μ that can appear as limits

[51.4.3] Such limit measures μ¯ exist if and only if b⁢x=xα with 0<α≤1 and Dn∼n1/α holds [32, 11, 54].

[51.5.1] The remaining measures define the class of fractional time evolutions Tα⁢Δ⁢t that depend only on one parameter, the fractional order α.

[page 52, §1]    [52.1.1] These remaining fractional measures have a density and they can be written as [43, 48, 49, 47, 54]

where Δ⁢t≥0 and 0<α≤1. [52.1.2] The density functions hα⁢x are the one-sided stable probability densities [43, 49, 48, 47, 54]. [52.1.3] They have a Mellin transform [131, 103, 45]

allowing to identify

in terms of H-functions [30, 103, 45, 95].

[52.2.1] The infinitesimal generators of the fractional semigroups Tα⁢Δ⁢t

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 α=1 one finds h1⁢x=δ⁢x-1 from eq. (2.158), and the fractional semigroup Tα=1⁢Δ⁢t reduces to the conventional translation semigroup T1⁢Δ⁢t⁢f⁢t0=f⁢t0-Δ⁢t. [52.3.2] The special case α=1 occurs more frequently in the limit (2.154) than the cases α<1 in the sense that it has a larger domain of attraction. [52.3.3] The fact that the semigroup T1⁢Δ⁢t can often be extended to a group on all of R provides an explanation for the seemingly fundamental reversibility of mechanical laws and equations. [52.3.4] This solves the "reversed irreversibility problem".

[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 Tα⁢Δ⁢t⁢f⁢t=f⁢t. Fractional stationarity requires a generalization of concepts such as ‘‘stationarity’’ or ‘‘equilibrium’’. [53.1.6] This outlook could be of interest for nonequilibrium and biological systems [43, 49, 48, 47, 54].

[53.2.1] Finally, also the special case α→0 challenges philosophical remarks [59]. [53.2.2] In the limit α→0 the time evolution operator degenerates into the identity. [53.2.3] This could be expressed verbally by saying that for α=0 ‘‘becoming’’ and ‘‘being’’ coincide. [53.2.4] In this sense the paradoxical limit α→0 is reminiscent of the eternity concept known from philosophy.

[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 α in Bochner-Levy diffusion equations has been known for more than five decades [13, 26, 14]. [53.4.4] For a lucid account see also [27]. [53.4.5] The fractional order α in this case is the index of the underlying stable process [13, 27]. [53.4.6] With few exceptions [77] these developments in the nation of mathematics did, for many years, not find much attention or application in the nation of physics although eminent mathematical physicists such as Mark Kac were thoroughly familiar

[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 f:Rd×R+→R

with a fractional time derivative of order α and type 1. [54.2.3] The Laplace operator is Δ and the fractional diffusion constant is C. [54.2.4] The function f⁢r,t is assumed to obey the initial condition f⁢r,0+=f0⁢δ⁢r. [54.2.5] Equation (2.159) was introduced in integral form in [104], but the connection with [87] was not given.

[54.3.1] An alternative to eq. (2.159), introduced in [54, 53], is

with a Riemann-Liouville fractional time derivative D0+α of type 0. [54.3.2] This equation does not describe diffusion of Montroll-Weiss type [53]. [54.3.3] It has therefore been called ‘‘inconsistent’’ in [81, p.3566]. As emphasized in [53] the choice of D0+α in (2.159) is physically and mathematically consistent, but corresponds to a modified initial condition, namely I0+1-α⁢f⁢r,0+=f0⁢δ⁢r. [54.3.4] Similarly, fractional diffusion equations with time derivative D0+α,β of order α and type β have been investigated in [54]. [54.3.5] For α=1 they all reduce to the diffusion equation.

[54.4.1] Before discussing how α arises from an underlying continuous time random walk it is of interest to give an overall comparison of ordinary diffusion with

[page 55, §0]    α=1 and fractional diffusion of the form (2.159) with α≠1. [55.0.1] This is conveniently done using the following table published in [46]. [55.0.2] The first column gives the results for α=1, the second for 0<α<1 and the third for the limit α→0. [55.0.3] The first row compares the infinitesimal generators of time evolution Aα. [55.0.4] The second row gives the fundamental solution f⁢k,u in Fourier-Laplace space. [55.0.5] The third row gives f⁢k,t and the fourth f⁢r,t. [55.0.6] In the fifth and sixth row the asymptotic behaviour is collected for r2/tα→0 and r2/tα→∞.

[55.1.1] In the table Eα,β⁢x denotes the generalized Mittag-Leffler function from eq. (2.130), Kν⁢x is the modified Bessel function [1], d>2, cα=2-α⁢αα/2-α and the shorthand

was used for the H-function H1220. [55.1.2] For information on H-functions see [30, 79, 95, 54].

[page 56, §1]    [56.1.1] The results in the table show that the normal diffusion (α=1) is slowed down for 0<α<1 and comes to a complete halt for α→0. [56.1.2] For more discussion of the solution see [46].

[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 α can be identified and has a physical meaning related to waiting times in the Montroll-Weiss model. [56.2.3] The relation between fractional time derivatives and CTRW’s was first exposed in [60, 46]. [56.2.4] The relation was established in two steps. First, it was shown in [60] that Montroll-Weiss continuous time random walks with a Mittag-Leffler waiting time density are rigorously equivalent to a fractional master equation. [56.2.5] Then, in [46] this underlying random walk model was connected to the fractional diffusion equation (2.159) in the usual asymptotic sense [109] of long times and large distances¹¹ (This is a footnote:) ¹¹This is emphasized in eqs. (1.8) and (2.1) in [46] that are, of course, asymptotic.. [56.2.6] For additional results see also [50, 54, 53, 57]

[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]

where f⁢r,t denotes the probability density to find the walker at position r∈Rd after time t if it started from r=0 at time t=0. [56.3.3] The function λ⁢r is the probability for a displacement by r in each step, and ψ⁢t gives the probability density of waiting time intervals between steps. [56.3.4] The transition probabilities obey ∑rλ⁢r=1, and Φ⁢t=1-∫0tψ⁢t′⁢d⁢t′ is the survival probability at the initial site.

[56.4.1] The fractional master equation introduced in [60] with inital condition f⁢r,0=δr,0 reads

with fractional transition rates w⁢r obeying ∑rw⁢r=0. [56.4.2] Note, that eq. (2.162) contains a free function ψ⁢t that has no counterpart in eq. (2.163). [56.4.3] The rigorous relation between eq. (2.162) and eq. (2.163), first established in [60], is given by the relation

[page 57, §0]    for the Fourier transformed transition rates w⁢r and probabilities λ⁢r, and the choice

for the waiting time density, where τ>0 is a characteristic time constant. [57.0.1] With Eα,α⁢0=1 it follows that

for t→0. [57.0.2] From Eα,α⁢x∼x-2 for x→∞ one finds

for t→∞. [57.0.3] For α=1 the waiting time density becomes the exponential distribution, and for α→0 it approaches 1/t.

[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

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.