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2 Foundations

2.1 Basic Desiderata for Time Evolutions

[91.2.1] The following basic requirements define a time evolution in this chapter.

1. 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.

2. Continuity
[92.0.3] The time evolution is assumed to be strongly continuous in by demanding

 (9)

for all .

3. 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.

4. Causality
[92.0.6] The time evolution operator should be causal in the sense that the function should depend only on values of for .

5. 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.

2.2 Evolutions, Convolutions and Averages

[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].

Theorem 2.1

[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]

Definition 2.1 (Averaging)

[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

Theorem 2.2

[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)
Proof.

[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.

Definition 2.2 (Convolution semigroup)

[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

Corollary 2.1

[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 .

Proof.

[97.1.5] Follows from Theorem 2.2 and the observation that would violate the causality condition. ∎

[97.2.1] Equation (29) establishes the basic convolution structure of the assertion in eq. (5). [97.2.2] It remains to investigate the requirement that should arise from a coarse graining procedure, and to establish the nature of the kernel in eq. (5).

2.3 Time Averaging and Coarse Graining

[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)

gives with eqs. (37) and (38)

 (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]

Proposition 2.1

[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.

Definition 2.3 (Coarse Graining)

[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 .

2.4 Coarse Graining Limits and Stable Averages

[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.

Proposition 2.2

[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.

Theorem 2.3 (Coarse Graining Limit)

[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)
Proof.

[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

Definition 2.4 (Stable Averages)

[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]

Definition 2.5

[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.

Theorem 2.4

[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 .

Proof.

[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)

[page 107, §0]    of a function . [107.0.1] Using the Mellin transform [32]

 (80)

valid for and it follows that

 (81)

[107.0.2] The general relation then implies

 (82)

 (83)

by identification with eq. (153) below. [107.0.3] Restoring a shift yields the result of eq. (69). ∎

[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

Corollary 2.2

[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).

Proof.

[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). ∎

2.5 Macroscopic Time Evolutions

[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.

Theorem 2.5 (Universality Classes of Time Evolutions)

[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.

Theorem 2.6 (Macroscopic Time Evolution)

[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.

Proof.

[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]

2.6 Infinitesimal Generators

[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

Theorem 2.7

[110.2.3] The infinitesimal generator of the macroscopic time evolutions is related to the infinitesimal generator of through

 (97)
Proof.

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.