[91.2.1] The following basic requirements define a time evolution in this chapter.
Semigroup
[91.2.2] A time evolution is
a pair
where
is a semigroup of operators
mapping the Banach space
of functions
on
to itself.
[91.2.3] The argument
of
represents a time
duration, the argument
of
a
time instant.
[91.2.4] The index
indicates the units (or scale) of time.
[91.2.5] Below,
will again be frequently suppressed to simplify the notation.
[91.2.6] The elements
represent observables or
[page 92, §0]
the state of a physical system as function of the time coordinate
.
[92.0.1] The semigroup conditions require
![]() |
![]() |
(7) | |
![]() |
![]() |
(8) |
for ,
and
.
[92.0.2] The first condition may be viewed as representing the unlimited
divisibility of time.
Continuity
[92.0.3] The time evolution is assumed to be strongly continuous
in by demanding
![]() |
(9) |
for all .
Homogeneity
[92.0.4] The homogeneity of the time coordinate
requires commutativity with translations
![]() |
(10) |
for all and
.
[92.0.5] This postulate allows to shift the origin of time
and it reflects the basic symmetry of time translation
invariance.
Causality
[92.0.6] The time evolution operator should be causal in the sense that
the function should depend only on values
of
for
.
Coarse Graining
[92.0.7] A time evolution operator should be obtainable from
a coarse graining procedure.
[92.0.8] A precise definition of coarse graining is given in
Definition 2.3 below.
[92.0.9] The idea is to combine a time average
in the limit
with a rescaling of
and
.
[92.0.10] While the first four requirements are conventional the fifth requires comment. [92.0.11] Averages over long intervals may themselves be timedependent on much longer time scales. [92.0.12] An example would be the position of an atom in a glass. [92.0.13] On short time scales the position fluctuates rapidly around a well defined average position. [92.0.14] On long time scales the structural relaxation processes in the glass can change this average position. [92.0.15] The purpose of any coarse graining procedure is to connect microscopic to macroscopic scales. [92.0.16] Of course, what is microscopic
[page 93, §0] depends on the physical situation. [93.0.1] Any microscopic time evolution may itself be viewed as macroscopic from the perspective of an underlying more microscopic theory. [93.0.2] Therefore it seems physically necessary and natural to demand that a time evolution should generally be obtainable from a coarse graining procedure.
[93.1.1] There is a close connection and mathematical similarity
between the simplest time evolution and
the operator
of time averaging defined as
the mathematical mean
![]() |
(11) |
where is the length of the averaging interval.
[93.1.2] Rewriting this formally as
![]() |
(12) |
exhibits the relation between and
.
[93.1.3] It shows also that
commutes with translations
(see eq. (10)).
[93.2.1] A second even more suggestive relationship between
and
arises because
both operators can be written as convolutions.
[93.2.2] The operator
may be written as
![]() |
(13) |
where the kernel
![]() |
(14) |
is the characteristic function of the unit interval.
[93.2.3] The Laplace convolution in the last line requires .
[93.2.4] The translations
on the other hand may be
[page 94, §0] written as
![]() |
(15) |
where again is required for the Laplace convolution
in the last equation.
[94.0.1] The similarity between eqs. (15) and (13)
suggests to view the time translations
as a degenerate
form of averaging
over a single point.
[94.0.2] The operators
and
are both convolution operators.
[94.0.3] By Lebesgues theorem
so that
in analogy with eq. (8) which holds for
.
[94.0.4] However, while the translations
fulfill eq. (7)
and form a convolution semigroup whose kernel
is the Dirac measure at 1,
the averaging operators
do not form a semigroup as will be
seen below.
[94.1.1] The appearance of convolutions and convolution semigroups
is not accidental.
[94.1.2] Convolution operators arise quite generally from the symmetry
requirement of eq. (10) above.
[94.1.3] Let denote the Lebesgue spaces of
-th power integrable functions, and let
denote the Schwartz space of test functions for tempered
distributions [27].
[94.1.4] It is well established that all bounded linear operators
on
commuting with translations (i.e.
fulfilling eq. (10)) are of convolution
type [27].
[94.1.5] Suppose the operator ,
is linear, bounded and commutes with translations.
[94.1.6] Then there exists a unique tempered distribution
such that
for all
.
[94.2.1] For the tempered distributions in this theorem
are finite Borel measures.
[94.2.2] If the measure is bounded and positive this means that
the operator B can be viewed as a weighted averaging
operator.
[94.2.3] In the following the case
will be of interest.
[94.2.4] A positive bounded measure
on
is uniquely determined by its distribution function
defined by
![]() |
(16) |
[94.2.5] The tilde will again be omitted to simplify the notation.
[94.2.6] Physically a weighted average represents
the measurement of a signal
using an apparatus
with response characterized by
and resolution
.
[94.2.7] Note that the resolution (length of averaging interval) is a duration
and cannot be negative.
[page 95, §1]
[95.1.1] Let be a (probability) distribution function on
, and
.
[95.1.2] The weighted (time) average of a function
on
is
defined as the convolution
![]() |
(17) |
whenever it exists.
[95.1.3] The average is called causal if the support of is in
.
[95.1.4] It is called degenerate if the support of
consists of a single point.
[95.2.1] The weight function or kernel corresponding to a
distribution
is defined as
whenever
it exists.
[95.3.1] The averaging operator in eq. (11)
corresponds to a measure with distribution function
![]() |
(18) |
while the time translation corresponds to the (Dirac) measure
concentrated at
with distribution function
![]() |
(19) |
[95.3.2] Both averages are causal, and the latter is degenerate.
[95.4.1] Repeated averaging leads to convolutions.
[95.4.2] The convolution of two distributions
on
is
defined through
![]() |
(20) |
[95.4.3] The Fourier transform of a distribution is defined by
![]() |
(21) |
where the last equation holds when the distribution admits a
weight function.
[95.4.4] A sequence of distributions is said to
converge weakly to a limit
,
[page 96, §0] written as
![]() |
(22) |
if
![]() |
(23) |
holds for all bounded continuous functions .
[96.1.1] The operators and
above have positive
kernels, and preserve positivity in the sense that
implies
.
[96.1.2] For such operators one has
[96.1.3] Let T be a bounded operator on ,
that is translation invariant in the sense that
![]() |
(24) |
for all and
, and
such that
and
almost everywhere
implies
almost everywhere.
[96.1.4] Then there exists a uniquely determined bounded measure
on
with mass
such that
![]() |
(25) |
[96.1.5] For the proof see [28]. ∎
[96.1.6] The preceding theorem suggests to represent those time evolutions that fulfill the requirements 1.– 4. of the last section in terms of convolution semigroups of measures.
[96.1.7] A family of positive bounded measures on
with the properties that
![]() |
![]() |
(26) | |
![]() |
![]() |
(27) | |
![]() |
![]() |
(28) |
is called a convolution semigroup of measures on .
[page 97, §1]
[97.1.1] Here is the Dirac measure at
and the limit is
the weak limit.
[97.1.2] The desired characterization of time evolutions now becomes
[97.1.3] Let be a strongly continuous time evolution
fulfilling the conditions of homogeneity and causality,
and being such that
and
almost everywhere implies
almost everywhere.
[97.1.4] Then
corresponds uniquely to a convolution
semigroup of measures
through
![]() |
(29) |
with for all
.
[97.1.5] Follows from Theorem 2.2 and
the observation that
would violate the causality condition.
∎
[97.3.1] The purpose of this section is to motivate the definition of coarse graining. [97.3.2] A first possible candidate for a coarse grained macroscopic time evolution could be obtained by simply rescaling the time in a microscopic time evolution as
![]() |
(30) |
where would be macroscopic times.
[97.3.3] However, apart from special cases,
the limit will in general not exist.
[97.3.4] Consider for example a sinusoidal
oscillating
around a constant.
[97.3.5] Also, the infinite translation
is
not an average, and this conflicts with the requirement
above, that coarse graining should be a smoothing operation.
[97.4.1] A second highly popular candidate for coarse graining
is therefore the averaging operator .
[97.4.2] If the limit
exists and
is
integrable in the finite interval
then
the average
![]() |
(31) |
is a number independent of the instant .
[97.4.3] Thus, if one wants to study the macroscopic time dependence of
,
it is necessary to consider a scaling limit in
[page 98, §0]
which also .
[98.0.1] If the scaling limit
is performed such that
is constant, then
![]() |
(32) |
becomes again an averaging operator over the infinitely
rescaled observable.
[98.0.2] Now still does not qualify as a coarse grained
time evolution because
as will be
shown next.
[98.1.1] Consider again the operator defined
in eq. (11).
[98.1.2] It follows that
![]() |
(33) |
and
![]() |
(34) |
[98.1.3] Thus twofold averaging may be written as
![]() |
(35) |
where
![]() |
(36) |
is the new kernel.
[98.1.4] It follows that , and hence
the averaging operators
do not form a semigroup.
[98.2.1] Although the iterated average is again
a convolution operator with support
compared
to
for
.
[98.2.2] Similarly,
has support
.
[98.2.3] This suggests to investigate the iterated average
in a scaling limit
.
[98.2.4] The limit
smoothes the function by
enlarging the
[page 99, §0]
averaging window to ,
and the limit
shifts the origin to infinity.
[99.0.1] The result may be viewed as a coarse grained time evolution
in the sense of a time evolution on time scales
"longer than infinitely long".
[99.0.2] c (This is a footnote:) c
The scaling limit was called "ultralong time limit" in [10]
It is therefore necessary to rescale
.
[99.0.3] If the rescaling factor is called
one is
interested in the limit
with
fixed, and
with
and fixed
![]() |
(37) |
whenever this limit exists.
[99.0.4] Here denotes the macroscopic time.
[99.1.1] To evaluate the limit note first that eq. (11) implies
![]() |
(38) |
where denotes the rescaled observable
with a rescaling factor
.
[99.1.2] The
-th iterated average may now be calculated by
Laplace transformation with respect to
.
[99.1.3] Note that
![]() |
(39) |
for all ,
where
is the generalized Mittag-Leffler function defined as
![]() |
(40) |
for all and
.
[99.1.4] Using the general relation
![]() |
(41) |
![]() |
(42) |
where is the Laplace transform of
.
[99.1.5] Noting that
it becomes apparent that a
limit
will exist if the rescaling factors are
[page 100, §0]
chosen as .
[100.0.1] With the choice
and
one finds
for the first factor
![]() |
(43) |
[100.0.2] Concerning the second factor assume that for each the limit
![]() |
(44) |
exists and defines a function .
[100.0.3] Then
![]() |
(45) |
and it follows that
![]() |
(46) |
[100.0.4] With Laplace inversion yields
![]() |
(47) |
[100.0.5] Using eq. (12) the result (47) may be expressed symbolically as
![]() |
(48) |
with .
[100.0.6] This expresses the macroscopic or coarse grained time evolution
as the scaling limit of a
microscopic time evolution
.
[100.0.7] Note that there is some freedom in the choice of the
rescaling factors
expressed by the prefactor
.
[100.0.8] This freedom reflects the freedom to choose the time units
for the coarse grained time evolution.
[100.1.1] The coarse grained time evolution
is again a translation.
[100.1.2] The coarse grained observable
corresponds
to a microscopic average by virtue
of the following result [29].
[page 101, §1]
[101.1.1] If is bounded from below and one of the limits
![]() |
or
![]() |
exists then the other limit exists and
![]() |
(49) |
[101.1.2] Comparison of the last relation with eq. (44)
shows that is a microscopic average of
.
[101.1.3] While
is a microscopic time coordinate, the time coordinate
of
is macroscopic.
[101.2.1] The preceding considerations justify to view the time evolution
as a coarse grained time evolution.
[101.2.2] Every observation or measurement of a physical quantity
requires a minimum duration
determined by the
temporal resolution of the measurement apparatus.
[101.2.3] The value
at the time instant
is always an average
over this minimum time interval.
[101.2.4] The averaging operator
with kernel
defined in equation (11) represents an
idealized averaging apparatus that can be switched on
and off instantaneously, and does not otherwise influence
the measurement.
[101.2.5] In practice one is usually confronted with finite startup
and shutdown times and a nonideal response of the apparatus.
[101.2.6] These imperfections are taken into account by using
a weighted average with a weight function or kernel
that differs from
.
[101.2.7] The weight function reflects conditions of the measurement, as well as
properties of the apparatus and its interaction with the system.
[101.2.8] It is therefore of interest to consider causal averaging operators
defined in eq. (17)
with general weight functions.
[101.2.9] A general coarse graining procedure is then
obtained from iterating these weighted averages.
[101.2.10] Let be a probability distribution on
, and
,
a sequence of rescaling factors.
A coarse graining limit is defined as
![]() |
(50) |
[page 102, §0]
whenever the limit exists.
[102.0.1] The coarse graining limit is called causal if
is causal, i.e. if
.
[102.1.1] The purpose of this section is to investigate the coarse graining procedure introduced in Definition 2.3. [102.1.2] Because the coarse graining procedure is defined as a limit it is useful to recall the following well known result for limits of distribution functions [30]. [102.1.3] For the convenience of the reader its proof is reproduced in the appendix.
[102.1.4] Let be a weakly convergent sequence of distribution functions.
[102.1.5] If
, where
is nondegenerate
then for any choice of
and
there exist
and
such that
![]() |
(51) |
[102.2.1] The basic result for coarse graining limits can now be formulated.
[102.2.2] Let be such that the limit
defines the Fourier transform of a function
.
[102.2.3] Then the coarse graining limit exists and defines
a convolution operator
![]() |
(52) |
if and only if for any there are constants
and
such that the distribution function
obeys the
relation
![]() |
(53) |
[102.2.4] In the previous section the coarse graining limit was evaluated
for the distribution from eq. (18) and
the corresponding
was found in eq. (47)
to be degenerate.
[102.2.5] A degenerate distribution
trivially obeys eq. (53).
[102.2.6] Assume therefore from now on that neither
nor
are degenerate.
[102.3.1] Employing equation (17) in the form
![]() |
(54) |
[page 103, §0]
one computes the Fourier transformation of
with respect to
![]() |
(55) |
[103.0.1] By assumption
has a limit whenever
with
.
[103.0.2] Thus the coarse graining limit exists and is a convolution
operator whenever
converges
to
as
.
[103.0.3] Following [30] it will be shown that this is true
if and only if the characterization (53)
and
with
apply.
[103.0.4] To see that
![]() |
(56) |
holds, assume the contrary.
Then there is a subsequence converging to a finite
limit.
[103.0.5] Thus
![]() |
(57) |
so that
![]() |
(58) |
for all .
[103.0.6] As
this leads to
for all
and hence
must be degenerate contrary to assumption.
[103.1.1] Next, it will be shown that
![]() |
(59) |
[103.1.2] From eq. (56) it follows that
and therefore
![]() |
(60) |
and
![]() |
(61) |
Substituting by
in
eq. (60) and by
in
eq. (61) shows that
![]() |
(62) |
[page 104, §0]
[104.0.1] If then there exists
a subsequence of either
or
converging to a constant
.
[104.0.2] Therefore eq. (62) implies
which upon iteration yields
![]() |
(63) |
[104.0.3] Taking the limit then gives
implying that
is degenerate contrary to assumption.
[104.1.1] Now let be two constants.
[104.1.2] Because of (56) and (59)
it is possible to choose for each
and
sufficiently large
an index
such that
![]() |
(64) |
[104.1.3] Consider the identity
![]() |
(65) |
By hypothesis the distribution functions corresponding to
converge to
as
.
[104.1.4] Hence each factor on the right hand side converges and
their product converges to
.
[104.1.5] It follows that the distribution function on the
left hand side must also converge.
[104.1.6] By Proposition 2.2 there must exist
and
such that the left hand side differs from
only as
.
[104.2.1] Finally the converse direction that the coarse graining
limit exists for is seen to follow from
eq. (53).
[104.2.2] This concludes the proof of the theorem.
∎
[104.3.1] The theorem shows that the coarse graining limit, if it
exists, is again a macroscopic weighted average .
[104.3.2] The condition (53) says that this macroscopic average
has a kernel that is stable under convolutions, and this motivates the
[104.3.3] A weighted averaging operator is called stable
if for any
there are constants
and
such that
![]() |
(66) |
holds.
[104.4.1] This nomenclature emphasizes the close relation with the limit theorems of probability theory [30, 31]. [104.4.2] The next theorem provides the explicit form for distribution functions satisfying eq. (66). [104.4.3] The proof uses Bernsteins theorem and hence requires the concept of complete monotonicity.
[page 105, §1]
[105.1.1] A -function
is called
completely monotone if
![]() |
(67) |
for all integers .
[105.2.1] Bernsteins theorem [31, p. 439] states that a function
is completely monotone if and only if it is the the
Laplace transform ()
![]() |
(68) |
of a distribution or of a density
.
[105.3.1] In the next theorem the explicit form of stable averaging kernels
is found to be a special case of the general -function.
[105.3.2] Because the
-function will reappear in other results
its general definition and properties
are presented separately in Section 4.
[105.3.3] A causal average is stable if and only if its weight function is of the form
![]() |
(69) |
where ,
and
are constants
and
.
[105.3.4] Let without loss of generality.
[105.3.5] The condition (66) together with
defines one sided
stable distribution functions [31].
[105.3.6] To derive the form (69) it suffices to consider
condition (66) with
.
[105.3.7] Assume thence that for any
there exists
such that
![]() |
(70) |
where the convolution is now a Laplace convolution
because of the condition .
[105.3.8] Laplace tranformation yields
![]() |
(71) |
[105.3.9] Iterating this equation (with ) shows that
there is an
-dependent constant
such that
![]() |
(72) |
[page 106, §0] and hence
![]() |
(73) |
[106.0.1] Thus satisfies the functional equation
![]() |
(74) |
whose solution is with some real constant
written as
with hindsight.
[106.0.2] Inserting
into eq.(72) and substituting
the function
gives
![]() |
(75) |
[106.0.3] Taking logarithms and substituting this becomes
![]() |
(76) |
[106.0.4] The solution to this functional equation is .
[106.0.5] Substituting back one finds
and therefore
is of the general form
with
.
[106.0.6] Now
is also a distribution function. Its normalization
requires
and this restricts
to
.
[106.0.7] Moreover, by Bernsteins theorem
must be completely
monotone.
[106.0.8] A completely monotone function is positive, decreasing
and convex.
[106.0.9] Therefore the power in the exponent
must have a negative prefactor, and
the exponent is restricted to the range
.
[106.0.10] Summarizing, the Laplace transform
of a distribution
satisfying (70) is of the form
![]() |
(77) |
with and
.
[106.0.11] Checking that
does indeed satisfy
eq. (70) yields
as the relation between the constants.
[106.0.12] For the proof of the general case of eq. (66)
see Refs. [30, 31].
[106.1.1] To invert the Laplace transform it is convenient to use the relation
![]() |
(78) |
between the Laplace transform and the Mellin transform
![]() |
(79) |
[107.0.4] Note that is the standardized form
used in eq. (5).
[107.0.5] It remains to investigate the sequence of rescaling factors
.
[107.0.6] For these one finds
[107.0.7] If the coarse graining limit exists and is nondegenerate
then the sequence of rescaling factors has the
form
![]() |
(84) |
where and
is slowly varying,
i.e.
for all
(see
Chapter IX, Section 2.3).
[33][107.0.8] Let .
[107.0.9] Then for all
and any fixed
![]() |
(85) |
[107.0.10] On the other hand
![]() |
(86) |
where the remainder tends uniformly to zero on every
finite interval.
[107.0.11] Suppose that the sequence is unbounded
so that there is a subsequence with
.
[107.0.12] Setting
in eq. (86) and using
eq. (85) gives
[page 108, §0]
which cannot be satisfied
because
.
[108.0.1] Hence
is bounded.
[108.0.2] Now the limit
in eqs. (85) and (86) gives
![]() |
(87) |
[108.0.3] This requires that
![]() |
(88) |
implying eq. (84) by virtue of the Characterization Theorem 2.2 in Chapter IX. [108.0.4] (For more information on slow and regular variation see Chapter IX and references therein). ∎
[108.1.1] The preceding results show that a coarse graining limit is
characterized by the quantities .
[108.1.2] These quantities are determined by the coarsening weight
.
[108.1.3] The following result, whose proof can be found in
[33, p. 85], gives their relation with the
coarsening weight.
[108.1.4] In order that a causal coarse graining limit based on
gives rise to a macroscopic average with
it is necessary and sufficient that
behaves as
![]() |
(89) |
in a neighbourhood of , and that
is slowly
varying for
.
[108.1.5] In case
the rescaling factors can be chosen as
![]() |
(90) |
while the case reduces to the degenerate case
.
[108.2.1] The preceding theorem characterizes the domain of attraction of a universality class of time evolutions. [108.2.2] Summarizing the results gives a characterization of macroscopic time evolutions arising from coarse graining limits.
[108.2.3] Let be such that the limit
defines the Fourier transform of a function
.
[108.2.4] If
is a causal average whose coarse graining limit
exists with
as
[page 109, §0] in the preceding theorem then
![]() |
(91) |
defines a family of one parameter semigroups with
parameter
indexed by
.
[109.0.1] Here
denotes the translation
semigroup, and
is a constant.
[109.0.2] Noting that and
combining Theorems 2.3 and 2.4 gives
![]() |
(92) |
where ,
and
are the constants from
theorem 2.4 and the last equality defines the operators
with
and
.
[109.0.3] Fourier transformation then yields
![]() |
(93) |
and the semigroup property (7) follows from
![]() |
(94) |
by Fourier inversion. [109.0.4] Condition (8) is checked similarly. ∎
[109.0.5] The family of semigroups indexed by
that can arise from coarse graining limits are called
macroscopic time evolutions.
[109.0.6] These semigroups are also holomorphic, strongly continuous
and equibounded (see Chapter III).
[109.1.1] From a physical point of view this
result emphasizes the different role
played by and
.
[109.1.2] While
is the
macroscopic time coordinate whose values are
, the duration
is positive.
[109.1.3] If the dimension of a microscopic time duration
is [s], then
the dimension of the macroscopic time duration
is [s
].
[page 110, §1]
[110.1.1] The importance of the semigroups for
theoretical physics as universal attractors of coarse
grained macroscopic time evolutions seems not to have
been noticed thus far.
[110.1.2] This is the more surprising as their mathematical
importance for harmonic analysis and probability
theory has long been recognized [31, 34, 35, 28].
[110.1.3] The infinitesimal generators are known
to be fractional derivatives [31, 35, 36, 37].
[110.1.4] The infinitesimal generators are defined as
![]() |
(95) |
[110.1.5] For more details on semigroups and their infinitesimal generators see Chapter III.
[110.2.1] Formally one calculates by applying direct and
inverse Laplace transformation
with
in eq. (91) and using eq. (77)
![]() |
(96) |
[110.2.2] The result can indeed be made rigorous and one has
[110.2.3] The infinitesimal generator of the macroscopic
time evolutions
is related to the infinitesimal
generator
of
through
![]() |
(97) |
See Chapter III. ∎
[page 111, §1]
[111.1.1] The theorem shows that fractional derivatives
of Marchaud type arise as the infinitesimal
generators of coarse grained time evolutions in physics.
[111.1.2] The order of the derivative lies between zero and unity,
and it is determined by the decay of the averaging kernel.
[111.1.3] The order
gives a quantitative measure for the decay of the
averaging kernel.
[111.1.4] The case
indicates that memory effects and history
dependence may become important.