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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 ({Tτ(t):0t<},(Bτ,)) where Tτt=Ttτ is a semigroup of operators Tt:0t< mapping the Banach space (Bτ(R),) of functions fτs=fsτ on R to itself. [91.2.3] The argument t0 of Tτt represents a time duration, the argument sR of fτs a time instant. [91.2.4] The index τ>0 indicates the units (or scale) of time. [91.2.5] Below, τ will again be frequently suppressed to simplify the notation. [91.2.6] The elements fτs=fsτBτ represent observables or

[page 92, §0]

the state of a physical system as function of the time coordinate sR. [92.0.1] The semigroup conditions require

 Tτ⁢t1⁢Tτ⁢t2⁢fτ⁢t0 =Tτ⁢t1+t2⁢fτ⁢t0 (7) Tτ⁢0⁢fτ⁢t0 =fτ⁢t0 (8)

for t1,t2>0, t0R and fτBτ. [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 t by demanding

 limt→0⁡T⁢t⁢f-f=0 (9)

for all fB.

3. Homogeneity
[92.0.4] The homogeneity of the time coordinate requires commutativity with translations

 T⁢t1⁢T⁢t2⁢f⁢t0=T⁢t2⁢T⁢t1⁢f⁢t0 (10)

for all t2>0 and t0,t1R. [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 gt0=Ttft0 should depend only on values of fs for s<t0.

5. Coarse Graining
[92.0.7] A time evolution operator Tt 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 1ts-tsftdt in the limit t,s with a rescaling of s and t.

[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 Tt=Tt and the operator Mt of time averaging defined as the mathematical mean

 M⁢t⁢f⁢s=1t⁢∫s-tsf⁢y⁢d⁢y, (11)

where t>0 is the length of the averaging interval. [93.1.2] Rewriting this formally as

 M⁢t⁢f⁢s=1t⁢∫0tf⁢s-y⁢d⁢y=1t⁢∫0tT⁢y⁢f⁢s⁢d⁢y (12)

exhibits the relation between Mt and Tt. [93.1.3] It shows also that Mt commutes with translations (see eq. (10)).

[93.2.1] A second even more suggestive relationship between Mt and Tt arises because both operators can be written as convolutions. [93.2.2] The operator Mt may be written as

 M⁢t⁢f⁢s=1t⁢∫0tf⁢s-y⁢d⁢y=∫-∞∞f⁢s-y⁢1t⁢χ0,1⁢yt⁢d⁢y=∫0sf⁢s-y⁢1t⁢χ0,1⁢yt⁢d⁢y, (13)

where the kernel

 χ0,1⁢x=1    for ⁢x∈0,10    for ⁢x∉0,1 (14)

is the characteristic function of the unit interval. [93.2.3] The Laplace convolution in the last line requires t<s. [93.2.4] The translations Tt on the other hand may be

[page 94, §0]    written as

 T⁢t⁢f⁢s=f⁢s-t=∫-∞∞f⁢s-y⁢1t⁢δ⁢yt-1⁢d⁢y=∫0sf⁢s-y⁢1t⁢δ⁢yt-1⁢d⁢y (15)

where again 0<t<s 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 Tt as a degenerate form of averaging f over a single point. [94.0.2] The operators Mt and Tt are both convolution operators. [94.0.3] By Lebesgues theorem limt0Mtfs=fs so that M0ft=ft in analogy with eq. (8) which holds for Tt. [94.0.4] However, while the translations Tt fulfill eq. (7) and form a convolution semigroup whose kernel is the Dirac measure at 1, the averaging operators Mt 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 LpRn denote the Lebesgue spaces of p-th power integrable functions, and let S denote the Schwartz space of test functions for tempered distributions [27]. [94.1.4] It is well established that all bounded linear operators on LpRn commuting with translations (i.e. fulfilling eq. (10)) are of convolution type [27].

Theorem 2.1

[94.1.5] Suppose the operator B:LpRnLqRn, 1p,q, is linear, bounded and commutes with translations. [94.1.6] Then there exists a unique tempered distribution g such that Bh=g*h for all hS.

[94.2.1] For p=q=1 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 n=1 will be of interest. [94.2.4] A positive bounded measure μ on R is uniquely determined by its distribution function μ~:R0,1 defined by

 μ~⁢x=μ(]-∞,x[)μ⁢R. (16)

[94.2.5] The tilde will again be omitted to simplify the notation. [94.2.6] Physically a weighted average Mt;μfs represents the measurement of a signal fs using an apparatus with response characterized by μ and resolution t>0. [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 μ be a (probability) distribution function on R, and t>0. [95.1.2] The weighted (time) average of a function f on R is defined as the convolution

 M(t;μ)f(s)=(f*μ(⋅/t))(s)=∫-∞∞f(s-s′)dμ(s′/t)=∫-∞∞T(s′)f(s)dμ(s′/t) (17)

whenever it exists. [95.1.3] The average is called causal if the support of μ is in R+. [95.1.4] It is called degenerate if the support of μ consists of a single point.

[95.2.1] The weight function or kernel mx corresponding to a distribution μx is defined as mx=dμ/dx whenever it exists.

[95.3.1] The averaging operator Mt in eq. (11) corresponds to a measure with distribution function

 μχ⁢x=0    for ⁢x≤0x    for ⁢0≤x≤11    for ⁢x≥1 (18)

while the time translation Tt corresponds to the (Dirac) measure δx-1 concentrated at 1 with distribution function

 μδ⁢x=0    for ⁢x<11    for ⁢x≥1. (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 κ of two distributions μ,ν on R is defined through

 κ⁢x=μ*ν⁢x=∫-∞∞μ⁢x-y⁢d⁢ν⁢y=∫-∞∞ν⁢x-y⁢d⁢μ⁢y. (20)

[95.4.3] The Fourier transform of a distribution is defined by

 F⁢μ⁢t⁢ω=μ^⁢ω=∫-∞∞ei⁢ω⁢t⁢d⁢μ⁢t=∫-∞∞ei⁢ω⁢t⁢m⁢t⁢d⁢t (21)

where the last equation holds when the distribution admits a weight function. [95.4.4] A sequence μnx of distributions is said to converge weakly to a limit μx,

[page 96, §0]    written as

 limn→∞⁡μn=μ, (22)

if

 limn→∞∫-∞∞f(x)dμn(x)=∫-∞∞f(x)dμ(x) (23)

holds for all bounded continuous functions f.

[96.1.1] The operators Mt and Tt above have positive kernels, and preserve positivity in the sense that f0 implies Mtf0. [96.1.2] For such operators one has

Theorem 2.2

[96.1.3] Let T be a bounded operator on LpR, 1p< that is translation invariant in the sense that

 T⁢T⁢t⁢f=T⁢t⁢T⁢f (24)

for all tR and fLpR, and such that fLpR and 0f1 almost everywhere implies 0Tf1 almost everywhere. [96.1.4] Then there exists a uniquely determined bounded measure μ on R with mass μR1 such that

 T⁢f⁢t=μ*f⁢t=∫-∞∞f⁢t-s⁢d⁢μ⁢s (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 μt:t>0 of positive bounded measures on R with the properties that

 μt⁢R ≤1    fort>0, (26) μt+s =μt*μs    fort,s>0, (27) δ =limt→0⁡μt⁢ (28)

is called a convolution semigroup of measures on R.

[page 97, §1]    [97.1.1] Here δ is the Dirac measure at 0 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 Tt be a strongly continuous time evolution fulfilling the conditions of homogeneity and causality, and being such that fLpR and 0f1 almost everywhere implies 0Tf1 almost everywhere. [97.1.4] Then Tt corresponds uniquely to a convolution semigroup of measures μt through

 T⁢t⁢f⁢s=μt*f⁢s=∫-∞∞f⁢s-s′⁢d⁢μt⁢s′ (29)

with suppμtR+ for all t0.

Proof.

[97.1.5] Follows from Theorem 2.2 and the observation that suppμtR- 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 Tt 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

 T∞⁢t¯⁢f⁢s=limτ→∞⁡Tτ⁢t¯⁢f⁢s=limτ→∞⁡T⁢τ⁢t¯⁢f⁢s=limτ→∞⁡f⁢s-τ⁢t¯ (30)

where 0<t¯< 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 ft oscillating around a constant. [97.3.5] Also, the infinite translation T 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 Mt. [97.4.2] If the limit t exists and ft is integrable in the finite interval s1,s2 then the average

 f¯=limt→∞⁡M⁢t⁢f⁢s1=limt→∞⁡M⁢t⁢f⁢s2 (31)

is a number independent of the instant si. [97.4.3] Thus, if one wants to study the macroscopic time dependence of f¯, it is necessary to consider a scaling limit in

[page 98, §0]    which also s. [98.0.1] If the scaling limit s,t is performed such that s/t=s¯ is constant, then

 limt,s→∞s=t⁢s¯⁡M⁢t⁢f⁢s=∫s¯-1s¯f∞⁢z⁢d⁢z=M⁢1⁢f∞⁢s¯ (32)

becomes again an averaging operator over the infinitely rescaled observable. [98.0.2] Now M1 still does not qualify as a coarse grained time evolution because M1M1M2 as will be shown next.

[98.1.1] Consider again the operator Mt defined in eq. (11). [98.1.2] It follows that

 M2⁢t⁢f⁢s=1t⁢χ0,1⁢⋅t*1t⁢χ0,1⁢⋅t*f⁢s (33)

and

 1t2⁢∫0xχ0,1⁢x-yt⁢χ0,1⁢yt⁢d⁢y=0    for ⁢x≤0xt2    for ⁢0≤x≤t2t-xt2    for ⁢t≤x≤2⁢t0    for ⁢x≥2⁢t. (34)

[98.1.3] Thus twofold averaging may be written as

 M2⁢t⁢f⁢s=∫0sf⁢s-y⁢1t⁢χ2⁢yt⁢d⁢y (35)

where

 χ2⁢x=x    for 0≤x≤12-x    for 1≤x≤20    otherwise (36)

is the new kernel. [98.1.4] It follows that M2tM2t, and hence the averaging operators Mt do not form a semigroup.

[98.2.1] Although M2tM2t the iterated average is again a convolution operator with support 0,2t compared to 0,t for Mt. [98.2.2] Similarly, M3t has support 0,3t. [98.2.3] This suggests to investigate the iterated average Mntfs in a scaling limit n,s. [98.2.4] The limit n smoothes the function by enlarging the

[page 99, §0]    averaging window to 0,nt, and the limit s 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 s. [99.0.3] If the rescaling factor is called σn>0 one is interested in the limit n,s with s¯=s/σn fixed, and σn with n and fixed t>0

 limn,s→∞s=σn⁢s¯⁡M⁢tn⁢f⁢s=limn→∞⁡M⁢tn⁢f⁢σn⁢s¯ (37)

whenever this limit exists. [99.0.4] Here s¯>1 denotes the macroscopic time.

[99.1.1] To evaluate the limit note first that eq. (11) implies

 M⁢t⁢f⁢σn⁢s¯=∫0s¯fσn⁢s¯-z⁢σnt⁢χ0,1⁢σn⁢zt⁢d⁢z (38)

where fτt=ftτ denotes the rescaled observable with a rescaling factor τ. [99.1.2] The n-th iterated average may now be calculated by Laplace transformation with respect to s¯. [99.1.3] Note that

 L⁢1c⁢χ0,1⁢xc⁢u=1-e-c⁢uc⁢u=E1,2⁢-c⁢u (39)

for all cR, where E1,2x is the generalized Mittag-Leffler function defined as

 Ea,b⁢x=∑k=0∞xkΓ⁢a⁢k+b (40)

for all a>0 and bC. [99.1.4] Using the general relation

 Ea,b⁢x=1Γ⁢b+x⁢Ea,a+b⁢x (41)

gives with eqs. (37) and (38)

 L⁢M⁢tn⁢f⁢σn⁢s¯⁢u¯=1-t⁢u¯σn⁢E1,3⁢-t⁢u¯σnn⁢1σn⁢L⁢f⁢s⁢u¯σn (42)

where fu¯ is the Laplace transform of fs¯. [99.1.5] Noting that E1,30=1/2 it becomes apparent that a limit n will exist if the rescaling factors are

[page 100, §0]    chosen as σnn. [100.0.1] With the choice σn=σn/2 and σ>0 one finds for the first factor

 limn→∞⁡1-2⁢t⁢u¯n⁢σ⁢E1,3⁢-2⁢t⁢u¯n⁢σn=e-t⁢u¯/σ. (43)

[100.0.2] Concerning the second factor assume that for each u¯ the limit

 limn→∞⁡2n⁢L⁢f⁢s⁢2⁢u¯n=f¯⁢u¯ (44)

exists and defines a function f¯u¯. [100.0.3] Then

 limn→∞⁡1σn⁢L⁢f⁢s¯⁢u¯σn=1σ⁢f¯⁢u¯σ, (45)

and it follows that

 limn→∞⁡L⁢M⁢tn⁢f⁢σn⁢s¯⁢u¯=e-t⁢u¯/σ⁢1σ⁢f¯⁢u¯σ. (46)

[100.0.4] With t¯=t/σ Laplace inversion yields

 limn,s→∞s=σn⁢s¯⁡M⁢tn⁢f⁢s=∫0s¯f¯⁢σ⁢s¯-σ⁢y¯⁢δ⁢y¯-t¯⁢d⁢y¯=f¯σ⁢s¯-t¯. (47)

[100.0.5] Using eq. (12) the result (47) may be expressed symbolically as

 limn,s→∞s/n=σ⁢s¯/2⁡1t⁢∫0tT⁢y⁢d⁢yn⁢f⁢s=f¯σ⁢s¯-t¯=T¯⁢t¯⁢f¯σ⁢s¯ (48)

with t¯=t/σ. [100.0.6] This expresses the macroscopic or coarse grained time evolution T¯t¯ as the scaling limit of a microscopic time evolution Tt. [100.0.7] Note that there is some freedom in the choice of the rescaling factors σn expressed by the prefactor σ. [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 T¯t¯ is again a translation. [100.1.2] The coarse grained observable f¯s¯ corresponds to a microscopic average by virtue of the following result [29].

[page 101, §1]

Proposition 2.1

[101.1.1] If fx is bounded from below and one of the limits

 limy→∞⁡1y⁢∫0yf⁢x⁢d⁢x

or

 limz→0⁡z⁢∫0∞f⁢x⁢e-z⁢x⁢d⁢x

exists then the other limit exists and

 limy→∞⁡1y⁢∫0yf⁢x⁢d⁢x=limz→0⁡z⁢L⁢f⁢x⁢z. (49)

[101.1.2] Comparison of the last relation with eq. (44) shows that f¯s¯ is a microscopic average of fs. [101.1.3] While s is a microscopic time coordinate, the time coordinate s¯ of f¯ is macroscopic.

[101.2.1] The preceding considerations justify to view the time evolution T¯t¯ as a coarse grained time evolution. [101.2.2] Every observation or measurement of a physical quantity fs requires a minimum duration t determined by the temporal resolution of the measurement apparatus. [101.2.3] The value fs at the time instant s is always an average over this minimum time interval. [101.2.4] The averaging operator Mt with kernel χ0,1 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 χ0,1. [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 Mt;μ 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 μ be a probability distribution on R, and σn>0, nN a sequence of rescaling factors. A coarse graining limit is defined as

 limn,s→∞s=σn⁢s¯⁡M⁢t;μn⁢f⁢s (50)

[page 102, §0]    whenever the limit exists. [102.0.1] The coarse graining limit is called causal if Mt;μ is causal, i.e. if suppμR+.

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 μns be a weakly convergent sequence of distribution functions. [102.1.5] If limnμns=μs, where μs is nondegenerate then for any choice of an>0 and bn there exist a>0 and b such that

 limn→∞⁡μn⁢an⁢x+bn=μ⁢a⁢x+b. (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 fs be such that the limit lima0af^aω=f¯^ω defines the Fourier transform of a function f¯s. [102.2.3] Then the coarse graining limit exists and defines a convolution operator

 limn,s→∞s=σn⁢s¯⁡M⁢t;μn⁢f⁢s=∫-∞∞f¯⁢s¯-s¯′⁢d⁢ν⁢s¯′/t;μ (52)

if and only if for any a1,a2>0 there are constants a>0 and b such that the distribution function νx=νx;μ obeys the relation

 ν⁢a1⁢x*ν⁢a2⁢x=ν⁢a⁢x+b. (53)
Proof.

[102.2.4] In the previous section the coarse graining limit was evaluated for the distribution μχ from eq. (18) and the corresponding ν was found in eq. (47) to be degenerate. [102.2.5] A degenerate distribution ν trivially obeys eq. (53). [102.2.6] Assume therefore from now on that neither μ nor ν are degenerate.

[102.3.1] Employing equation (17) in the form

 M⁢t;μ⁢f⁢σn⁢s¯=∫-∞∞f⁢σn⁢s¯-σn⁢y⁢d⁢μ⁢σn⁢y/t (54)

[page 103, §0]    one computes the Fourier transformation of Mt;μnf with respect to s¯

 F⁢M⁢t;μn⁢f⁢σn⁢s¯⁢ω¯=μ^⁢t⁢ω¯σnn⁢1σn⁢f^⁢ω¯σn. (55)

[103.0.1] By assumption f^ω¯/σn/σn has a limit whenever σn with n. [103.0.2] Thus the coarse graining limit exists and is a convolution operator whenever μ^tω¯/σnn converges to ν^ω¯ as n. [103.0.3] Following [30] it will be shown that this is true if and only if the characterization (53) and σn with n apply. [103.0.4] To see that

 limn→∞⁡σn=∞ (56)

holds, assume the contrary. Then there is a subsequence σnk converging to a finite limit. [103.0.5] Thus

 μ^⁢t⁢ω/σnknk=ν^⁢ω⁢1+o⁢1 (57)

so that

 μ^⁢ω=ν^⁢ω⁢σnk/t1/nk⁢1+o⁢1 (58)

for all ω. [103.0.6] As nk this leads to μ^ω=1 for all ω and hence μ must be degenerate contrary to assumption.

[103.1.1] Next, it will be shown that

 limn→∞⁡σn+1σn=1. (59)

[103.1.2] From eq. (56) it follows that limnμ^ω/σn=1 and therefore

 μ^⁢t⁢ω/σnn=ν^⁢ω⁢1+o⁢1 (60)

and

 μ^⁢t⁢ω/σn+1n+1=ν^⁢ω⁢1+o⁢1. (61)

Substituting ω by σnω/σn+1 in eq. (60) and by σn+1ω/σn in eq. (61) shows that

 limn→∞⁡ν^⁢σn+1⁢ω/σnν^⁢ω=limn→∞⁡ν^⁢σn⁢ω/σn+1ν^⁢ω=1. (62)

[page 104, §0]    [104.0.1] If limnσn+1/σn1 then there exists a subsequence of either σn+1/σn or σn/σn+1 converging to a constant A<1. [104.0.2] Therefore eq. (62) implies ν^ω=ν^Aω which upon iteration yields

 ν^⁢ω=ν^⁢An⁢ω. (63)

[104.0.3] Taking the limit n then gives ν^0=1 implying that ν is degenerate contrary to assumption.

[104.1.1] Now let 0<a1<a2 be two constants. [104.1.2] Because of (56) and (59) it is possible to choose for each ε>0 and sufficiently large n>n0ε an index mn such that

 0≤σmσn-a2a1<ε. (64)

[104.1.3] Consider the identity

 μ^⁢a1⁢t⁢ω¯σnn+m=μ^⁢a1⁢t⁢ω¯σnn⁢μ^⁢σmσn⁢a1⁢t⁢ω¯σmm. (65)

By hypothesis the distribution functions corresponding to μ^tω¯/σnn converge to νs¯ as n. [104.1.4] Hence each factor on the right hand side converges and their product converges to νa1s¯*νa2s¯. [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 a>0 and b such that the left hand side differs from νs¯ only as νas¯+b.

[104.2.1] Finally the converse direction that the coarse graining limit exists for μ=ν 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 Mt;ν. [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 Mt;μ is called stable if for any a1,a2>0 there are constants a>0 and bR such that

 μ⁢a1⁢x*μ⁢a2⁢x=μ⁢a⁢x+b (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 C-function f:]0,[R is called completely monotone if

 -1n⁢dn⁢fd⁢xn≥0 (67)

for all integers n0.

[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 (u>0)

 μ⁢u=L⁢μ⁢x⁢u=∫0∞e-u⁢x⁢d⁢μ⁢x=∫0∞e-u⁢x⁢m⁢x⁢d⁢x (68)

of a distribution μ or of a density m=dμ/dx.

[105.3.1] In the next theorem the explicit form of stable averaging kernels is found to be a special case of the general H-function. [105.3.2] Because the H-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

 hα(x;b,c)=1b1/αhα(x-cb1/α)=1α⁢x-cH1110(b1/αx-c|0,10,1/α) (69)

where 0<α1, b>0 and cR are constants and hαx=hαs;1,0.

Proof.

[105.3.4] Let c=0 without loss of generality. [105.3.5] The condition (66) together with suppμ[0,[ defines one sided stable distribution functions [31]. [105.3.6] To derive the form (69) it suffices to consider condition (66) with b=0. [105.3.7] Assume thence that for any a1,a2>0 there exists a>0 such that

 μ⁢a1⁢x*μ⁢a2⁢x=μ⁢a⁢x (70)

where the convolution is now a Laplace convolution because of the condition supp[0,[. [105.3.8] Laplace tranformation yields

 μ⁢u/a1⁢μ⁢u/a2=μ⁢u/a. (71)

[105.3.9] Iterating this equation (with a1=a2=1) shows that there is an n-dependent constant an such that

 μ⁢un=μ⁢u/a⁢n (72)

[page 106, §0]    and hence

 μ⁢ua⁢n⁢m=μ⁢un⁢m=μ⁢ua⁢nm=μ⁢u(a(n)a(m). (73)

[106.0.1] Thus an satisfies the functional equation

 a⁢n⁢m=a⁢n⁢a⁢m (74)

whose solution is an=n1/γ with some real constant written as 1/γ with hindsight. [106.0.2] Inserting an into eq.(72) and substituting the function gx=logμx gives

 n⁢g⁢u=g⁢u⁢n-1/γ. (75)

[106.0.3] Taking logarithms and substituting fx=loggex this becomes

 log⁡n+f⁢log⁡u=f⁢log⁡u-log⁡nγ. (76)

[106.0.4] The solution to this functional equation is fx=-γx. [106.0.5] Substituting back one finds gx=x-γ and therefore μu is of the general form μu=expu-γ with γR. [106.0.6] Now μ is also a distribution function. Its normalization requires μ(u=0)=1 and this restricts γ to γ<0. [106.0.7] Moreover, by Bernsteins theorem μu 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 -1γ<0. [106.0.10] Summarizing, the Laplace transform μu of a distribution satisfying (70) is of the form

 μ⁢u=hα⁢u;b,0=e-b⁢uα (77)

with 0<α1 and b>0. [106.0.11] Checking that hαu;b,0 does indeed satisfy eq. (70) yields a-α=a1-α+a2-α 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

 M⁢m⁢x⁢s=M⁢L⁢m⁢x⁢u⁢1-sΓ⁢1-s (78)

between the Laplace transform and the Mellin transform

 M⁢m⁢x⁢s=∫0∞xs-1⁢m⁢t⁢d⁢x (79)

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

 M⁢e-b⁢xα⁢s=Γ⁢s/αα⁢bs/α (80)

valid for α>0 and Res>0 it follows that

 M⁢hα⁢x;b,0⁢s=1α⁢b1-s/α⁢Γ⁢1-s/αΓ⁢1-s. (81)

[107.0.2] The general relation Mx-1fx-1s=Mfx1-s then implies

 M⁢x-1⁢hα⁢x-1;b,0⁢s=1α⁢bs/α⁢Γ⁢s/αΓ⁢s (82)

 x-1hα(x-1;b,0)=1αH1110(b1/αx|0,10,1/α) (83)

by identification with eq. (153) below. [107.0.3] Restoring a shift c0 yields the result of eq. (69). ∎

[107.0.4] Note that hαx=hαs;1,0 is the standardized form used in eq. (5). [107.0.5] It remains to investigate the sequence of rescaling factors σn. [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 σn of rescaling factors has the form

 σn=n1/α⁢Λ⁢n (84)

where 0<α1 and Λn is slowly varying, i.e. limnΛbn/Λn=1 for all b>0 (see Chapter IX, Section 2.3).

Proof.

[33][107.0.8] Let μ^nω=μ^ωn. [107.0.9] Then for all ω and any fixed k

 μ^n⁢ω/σn=e-b⁢ωα⁢1+o⁢1=μ^k⁢n⁢ω/σk⁢n. (85)

[107.0.10] On the other hand

 μ^k⁢n⁢ω/σk⁢n=μ^n⁢ω⁢σn/σk⁢nk=e-b⁢ωα⁢1+o⁢1 (86)

where the remainder tends uniformly to zero on every finite interval. [107.0.11] Suppose that the sequence σn/σkn is unbounded so that there is a subsequence with σknj/σnj0. [107.0.12] Setting ω=σknj/σnj in eq. (86) and using eq. (85) gives

[page 108, §0]    exp-bk=1 which cannot be satisfied because b,k>0. [108.0.1] Hence σn/σkn is bounded. [108.0.2] Now the limit n in eqs. (85) and (86) gives

 e-b⁢ωα=e-b⁢k⁢ωα⁢σn/σk⁢nα⁢1+o⁢1. (87)

[108.0.3] This requires that

 limn→∞⁡σk⁢nσn=k1/α (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 α,b,c,Λ. [108.1.2] These quantities are determined by the coarsening weight μ. [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 Mt;μ gives rise to a macroscopic average with hαx;b,c it is necessary and sufficient that μ^ω behaves as

 log⁡μ^⁢ω=i⁢c⁢ω-b⁢ωα⁢Λ⁢ω (89)

in a neighbourhood of ω=0, and that Λω is slowly varying for ω0. [108.1.5] In case 0<α1 the rescaling factors can be chosen as

 σn-1=inf⁡ω>0:ωα⁢Λ⁢ω=b/n (90)

while the case α>1 reduces to the degenerate case α=1.

[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 fs be such that the limit lima0af^aω=f¯^ω defines the Fourier transform of a function f¯s. [108.2.4] If Mt;μ is a causal average whose coarse graining limit exists with α,b,c as

[page 109, §0]    in the preceding theorem then

 limn,s→∞s=σn⁢s¯⁡M⁢t;μn⁢f⁢s=∫c¯∞f¯⁢s¯-y⁢hα⁢yt¯⁢d⁢yt¯=∫c¯∞T¯y⁢f¯⁢s¯⁢hα⁢yt¯⁢d⁢yt¯=M⁢t¯;hα⁢f¯⁢s¯-c¯=T¯α⁢t¯⁢f¯⁢s¯-c¯ (91)

defines a family of one parameter semigroups T¯αt¯ with parameter t¯=tαb indexed by α. [109.0.1] Here T¯t¯f¯s¯=f¯s¯-t¯ denotes the translation semigroup, and c¯=c/tb1/α is a constant.

Proof.

[109.0.2] Noting that supphαxR+ and combining Theorems 2.3 and 2.4 gives

 limn,s→∞s=σn⁢s¯⁡M⁢t;μn⁢f⁢s=∫c∞f¯⁢s¯-s¯′⁢1t⁢b1/α⁢hα⁢s¯′-ct⁢b1/α⁢d⁢s¯′=T¯α⁢t¯⁢f¯⁢s¯-c¯ (92)

where 0<α1, b>0 and cR are the constants from theorem 2.4 and the last equality defines the operators T¯αt¯ with t¯=tαb and c¯=c/tb1/α. [109.0.3] Fourier transformation then yields

 F⁢T¯α⁢t¯⁢f¯⁢s¯-c¯⁢ω¯=e-i⁢c⁢ω¯-t¯⁢i⁢ω¯α, (93)

and the semigroup property (7) follows from

 F⁢T¯α⁢t¯1⁢T¯α⁢t¯2⁢f¯⁢s¯-c¯⁢ω¯=e-i⁢c⁢ω¯-t¯1⁢i⁢ω¯α-t¯2⁢i⁢ω¯α=F⁢T¯α⁢t¯1+t¯2⁢f¯⁢s¯-c¯⁢ω¯ (94)

by Fourier inversion. [109.0.4] Condition (8) is checked similarly. ∎

[109.0.5] The family of semigroups T¯αt¯ indexed by α 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 s¯ and t¯. [109.1.2] While s¯ is the macroscopic time coordinate whose values are s¯R, the duration t¯>0 is positive. [109.1.3] If the dimension of a microscopic time duration t is [s], then the dimension of the macroscopic time duration t¯ is [sα].

[page 110, §1]

2.6 Infinitesimal Generators

[110.1.1] The importance of the semigroups T¯αt¯ 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

 Aα⁢f¯⁢s¯=limt¯→0⁡T¯α⁢t¯⁢f¯⁢s¯-f¯⁢s¯t¯. (95)

[110.1.5] For more details on semigroups and their infinitesimal generators see Chapter III.

[110.2.1] Formally one calculates Aα by applying direct and inverse Laplace transformation with c¯=0 in eq. (91) and using eq. (77)

 Aα⁢f¯⁢s¯=limt¯→0⁡12⁢π⁢i⁢∫η-i⁢∞η+i⁢∞es¯⁢u¯⁢e-t¯⁢u¯α-1t¯⁢f¯⁢u¯⁢d⁢u¯=12⁢π⁢i⁢∫η-i⁢∞η+i⁢∞es¯⁢u¯⁢limt¯→0⁡e-t¯⁢u¯α-1t¯⁢f¯⁢u¯⁢d⁢u¯=-12⁢π⁢i⁢∫η-i⁢∞η+i⁢∞es¯⁢u¯⁢u¯α⁢f¯⁢u¯⁢d⁢u¯. (96)

[110.2.2] The result can indeed be made rigorous and one has

Theorem 2.7

[110.2.3] The infinitesimal generator Aα of the macroscopic time evolutions T¯αt¯ is related to the infinitesimal generator A=-d/dt¯ of T¯t¯ through

 Aα⁢f¯⁢s¯=--Aα⁢f¯⁢s¯=-Dα⁢f¯⁢s¯=-1Γ⁢-α⁢∫0∞f¯⁢s¯-y-f¯⁢s¯yα+1⁢d⁢y=-1Γ⁢-α⁢∫0∞y-α-1⁢T¯y-1⁢f¯⁢s¯⁢d⁢y. (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 α of the derivative lies between zero and unity, and it is determined by the decay of the averaging kernel. [111.1.3] The order α gives a quantitative measure for the decay of the averaging kernel. [111.1.4] The case α1 indicates that memory effects and history dependence may become important.