1 Introduction
[page 89, §1]
[89.1.1] A large number of problems in theoretical physics,
including Schrödingers, Maxwell and Newtons
equations, can be formulated as initial value problems
for dynamical evolution equations of the form
where t∈R denotes time and B is an
operator on a Banach space.
[89.1.2] Depending on the initial data f0=f0 describing
the state or observable of the system at time t=0 the problem
is to find the state or observable ft of the system at later
times t>0.
[89.2.1] Many authors, mostly driven by the needs of applied problems,
have considered generalizations of equation (1)
of the form
in which the first order time derivative d/dt is replaced
with a certain fractional time derivative ‘‘dα/dtα’’
of order α>0
(see e.g.
[1]–[24]
and the Chapters IV–VIII in this book).
[89.2.2] A number of fundamental questions are raised by such a replacement.
[89.2.3] In order to appreciate these it is useful to recall that
the appearance of d/dt in eq. (1) reflects
not only a basic symmetry of nature but also the basic
principle of locality.
[89.2.4] Of course, the symmetry in question is time translation invariance.
[89.2.5] Remember that
ddsfs=limt→0fs-fs-tt=-limt→0Ttfs-fst | | (3) |
[page 90, §0]
identifies -d/dt as the infinitesimal generator of
time translations
defined by
[90.0.1] Equation (2)
abandons Tt as the
general time evolution, and this raises the question
what replaces eq. (4), and
how a fractional derivative can
arise as the generator of a physical time evolution.
[90.0.2] Most workers in fractional calculus have avoided these questions,
and my purpose in this chapter is to review and
discuss an answer provided recently in
[6, 7, 8, 9, 10, 11].
[90.1.1] Derivatives of fractional order 0<α≤1 were found to
emerge quite generally as the infinitesimal generators
of coarse grained macroscopic time evolutions given by
[6, 7, 8, 9, 10, 11]
Tαtft0=∫0∞Tsft0hαstdst | | (5) |
where t≥0 and 0<α≤1.
[90.1.2] Explicit expressions for the kernels hαx
for all 0<α≤1 are given in eq. (69) below.
[90.1.3] It is the main objective of this chapter to
show that (in a certain sense) all macroscopic
time evolutions have the form of eq. (5),
and that fractional time derivatives
are their infinitesimal generators.
[90.2.1] Given the great difference between Tαt in
eq. (5)
and T1t=Tt in eq. (4) it
becomes clear that basic issues, such as irreversibility,
translation symmetry, or the meaning of stationarity
are inevitably involved when proposing fractional dynamics.
[90.2.2] Let me therefore advance the basic postulate that all time
evolutions of physical systems are irreversible.
[90.2.3] Obviously this law of irreversibility must
be considered to be an empirical law of nature
equal in rank to the law of energy conservation.
[90.2.4] Reversible behaviour is an idealization.
[90.2.5] Its validity or applicability in physical experiments
depends on the degree to which the system can be isolated
(or decoupled) from its past history and its environment.
[90.2.6] According to this view the irreversible flow of time
is more fundamental than the time reversal symmetry of Newtons
or other equations.
[90.2.7] My starting point is therefore that for a general time
evolution operator Tt the evolution parameter t
is not a time instant (which could be positive
or negative), but a duration, which cannot be negative.
[90.3.1] An immediate consequence of the postulated
law of irreversibility is that the classical irreversibility
problem of theoretical physics becomes reversed.
[page 91, §0]
[91.0.1] Now the theoretical task is not to explain how irreversibility
arises from reversible evolution equations, but how seemingly
reversible equations arise as idealizations from
an underlying irreversible time evolution.
[91.0.2] A possible explanation is provided by the present
theory based on eq. (5).
[91.0.3] It turns out that the case α=1 in eq. (5)
is of predominant mathematical and physical importance, because it
is in a quantifiable sense a strong universal attractor.
[91.0.4] In this case the kernel h1x becomes
h1x=limα→1-hαx=δx-1, | | (6) |
and the time evolution T1t=Tt in (5)
reduces to a simple translation as in eq.(4).
[91.0.5] T1t with t≥0 is a
representation of the time semigroup (R+,+).
[91.0.6] It can be extended to one of the full group (R,+).
[91.0.7] This is not possible for Tα with 0<α<1.
[91.0.8] The physical interpretation of α is seen from
supphα=R+ for α≠1 and supph1=1
for α=1.
[91.0.9] Hence the parameter α classifies and quantifies
the influence of the past history.
[91.0.10] Small values of α correspond to a strong influence of
the past history.
[91.0.11] For α=1
the influence of
the past history is minimal in the sense that
it enters only through the present state.
[91.1.1] The basic result in eq. (5) was
given in [6] and subsequently rationalized
within ergodic theory by investigating the
recurrence properties of induced automorphisms
on subsets of measure zero [9, 10, 11].
[91.1.2] In these investigations the existence of a recurrent
subset of measure zero had to be assumed.
[91.1.3] Such an assumption becomes plausible from observations
in low dimensional chaotic systems
(see e.g. [25, 26] and Chapter V).
[91.1.4] A rigorous proof for any given dynamical
system, however, appears difficult, and it is therefore
of interest to rederive the emergence of Tαt
from a different, and more general, approach.