Complexity and Generalized Exponential Relaxation

Complexity and Generalized Exponential Relaxation: Memory versus Renewal 1

Paolo Grigolini1,2,3

Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427, USA 2 Istituto dei Processi Chimico Fisici del CNR, Area della Ricerca di Pisa, Via G. Moruzzi, 56124, Pisa, Italy and 3 Dipartimento di Fisica ”E.Fermi” - Universit´ a di Pisa, Largo Pontecorvo, 3 56127 PISA (Dated: August 26, 2007) We illustrate two distinct approaches to the Mittag-Le✏er relaxation, as a mathematical expression suitable for the interpretation of real data produced by complex systems, and especially those of physiological interest. The first approach is based on interpreting the fluctuation-dissipation process under study as obtained via Subordination to the Ordinary Fluctuation-Dissipation (SOFD) process. The second approach rests on the Generalized Langevin Equation (GLE). We prove that in the real cases of truncated time series the two theories generate a survival probability with the form of a stretched exponential, and that this property makes it hard to assess if a given time series obeys the GLE or the SOFD prescription. Some conjectures are made on the possibility of distinguishing the GLE from the SOFD predictions through the analysis of a single time series. PACS numbers: 05.40.Fb, 05.40.-a, 02.50.-r,,82.20.Uv

I.

INTRODUCTION

In the last few years there has been an increasing interest for the relaxation processes described by the Mittag-Le✏er (ML) function [Metzler & Klafter, 2000a; Mainardi,1996]. The reason of the interest is that the ML function interpolates between a stretched exponential relaxation at short times and an inverse power law relaxation at long times. Pottier [Pottier, 2003] derived this form of relaxation from the Generalized Langevin Equation (GLE) [Mori, 1965; Kubo, 1966]. Other authors have derived the same kind of anomalous regression to equilibrium using the theoretical perspective of Continuous Time Random Walk (CTRW) [Barkai & Silbey, 2000]. The two theoretical approaches are quite di↵erent, insofar as the former has a Hamiltonian foundation [Mori, 1965; Kubo, 1966], whereas the compatibility of the second approach with a Hamiltonian foundation is questionable. In fact, according to some authors [Kenkre & Knox, 1974; Budini, 2005] the formal equivalence between the non-Markov master equation generated by the CTRW and the one produced by the Zwanzig contraction approach [Zwanzig, 1961] corresponds to a genuinely physical equivalence. Other authors [Bologna et al., 2003; Cakir et al., 2007 b; Balescu, 2007] studied the time convoluted form shared by CTRW and the Zwanzig projection method [Kenkre & Knox, 1974; Budini, 2005], and reached the opposite conclusion that formal equivalence does not ensure physical equivalence. The discussion of the physical di↵erence between the CTRW and the Zwanzig projection method is not merely academic. In fact, in the CTRW there are genuine events, while the derivation of a generalized master equation from a Liouville statistical equation, by means of the Zwanzig projection method, does not generate any evident genuine event. The equivalence between these two pictures, once firmly established, would make subjective the claims of event detection.

In the specific case discussed in this paper, about GLE and CTRW yielding ML function, there is no formal equivalence between the two prescriptions, and the physical equivalence is limited only to the first moment regression to equilibrium. As far as the second moment is concerned, the two prescriptions are known [Pottier, 2003] to yield di↵erent results. However, as we shall see, the finite size of the data has the e↵ect of annihilating this important di↵erence, thereby weakening any claim about the event renewal character.

The current interest for the adoption of GLE as a proper theoretical tool for the study of physiological processes [Moksin et al., 2005; Yulmetyev et al., 2005] and the recent discovery of renewal events in the Electroencephalogram (EEG) data [Buiatti et al., 2007; Kalashian et al., 2007; Bianco et al., 2007] makes it important to design new techniques of analysis of EEG data for the purpose of establishing event-dependent conditions where the equivalence between CTRW and GLE method is violated.

The main purpose of this paper is to analyze the intriguing issue of the equivalence between GLE and CTRW from the point of view of time series, having in mind those of physiological nature, so as to help the researchers of this field to answer the important question: Are there renewal events in physiological processes? We shall show that renewal events may emerge also from the GLE theory, more as an artifact of the detection method than as a genuinely physiological property. This has the e↵ect of making harder the search for genuinely renewal events. We shall find however a promising way to assess the renewal character of the process, which can be made compelling by its application to real data and by numerical treatment of artificial sequences.

2 II.

A DEMON WITH SOCIAL LIFE

Let us consider the ordinary fluctuation-dissipation process, namely, the process described by the Langevin equation dy = dn

y(n) + f (n),

where the friction parameter

(1)

fits the condition

> 1,

(3)

which allows us to consider it virtually equivalent to a continuous time. In the discrete time n, we adopt the Fokker-Planck representation @ p(y, n) = LF P p(y, n), @n

and that the parameter T has the specific role of establishing a connection between the time region where the inverse power law condition applies, t >> T , and the microscopic time scale t < T . Let us assume that the realization y(n) occurs at time t(n). In this case the realization y(n + 1) occurs at time t(n) + ⌧ (n). Using the CTRW prescription, we write

(4)

1 Z X

n=0

t

dt0

n (t

0

) (t

t0 )exp (nLF P ) p(y, 0),

0

(9) where n (t) is the probability that the n th realization y(n) occurs exactly at t after a sequel of fluctuations y(n0 ), with n0 < n, whose precise time of occurrence does not matter. We assume that the times ⌧ (n) are drawn from the distribution of Eq. (7), randomly, i.e., with no correlation among themselves. As a consequence the resulting process is renewal. The function (t) is defined by Z t (t) = 1 dt0 (t0 ). (10) 0

with LF P ⌘

@ @2 y + D 2, @y @y

(5)

and D=

< y 2 >eq ,

(6)

which is indeed the well known fluctuation-dissipation relation. Note that < ... > denotes average on a Gibbs ensemble and the subscript eq indicates that it is done at equilibrium. The fluctuation y(t) generated by a complex process is expected to depart from the condition of exponential relaxation produced by the ordinary fluctuation-dissipation picture. Let assume therefore that the process under study is obtained via Subordination to the Ordinary Fluctuation-Dissipation (SOFD) process. The subordination method, currently used by many investigators [Sokolov, 2000; Barkai & Silbey, 2000; Metzler & Klafter, 2000b], is adopted here as a way to generalize the ordinary fluctuation-dissipation process. Let us illustrate this generalization by imagining that the realization of the ordinary fluctuation-dissipation process is the result of the action of a Demon. This Demon has a social life, which prevents it from being always active. We assume that the Demon stops working between the realization of y(n) and the realization of y(n + 1) for a time ⌧ (n) derived randomly from a time distribution density with the form (⌧ ) = (µ

1)

Tµ 1 . (⌧ + T )µ

(7)

Note that this form fits the normalization condition Z 1 dt (t) = 1 (8) 0

Thus, Eq. (9) is the probability of observing at time t the fluctuation y, realized by the Demon at time n with probability exp(LF P n)p(y, 0). The renewal character of the process and the time convoluted structure of Eq. (9) makes it convenient to adopt the Laplace transform formalism. We shall use throughR1 out the notation fˆ(u) = 0 exp( ut)f (t)dt. We note that the renewal character of the process ensures the property ˆn (u) = ( ˆ(u))n .

(11)

Thanks to the time convolution theorem and to Eq. (9), it is straightforward to prove that the Laplace transform of p(y, t), denoted by the symbol pˆ(y, u), reads pˆ(y, u) =

1 1

ˆ(u) p(y, 0). u

1

ˆ(u)exp(LF P )

(12)

Note that to get this result we used the summation property 1 X n=

xn =

1 1

x

,

(13)

which can be easily extended to the case containing the operator exp(nLF P ) = (exp(LF P ))n . Note that the operator LF P has the eigenvalues m . Consequently, by expressing p(y, 0) as the P1 sum of the corresponding eigenstates, p(y, 0) = m=0 wm (y), with p(y, 0)  1P and wm (y)  1, we obtain exp(LF P n)p(y, 0) = 1 m=0 exp( mn )wm (y), which fulfills the condition p(y, n) < 1 almost everywhere, except the zero measure n = 0. We also have ˆ(u) < 1, except at u = 0, where, due to the normalization condition, we have ˆ(0) = 1.

3 Let us now adopt the condition of Eq. (2). The operator LF P is characterized by eigenvalues proportional to . Thus, by truncating the Taylor series expansion of exp(LF P ) at first order in LF P , we obtain pˆ(y, u) =

1 1

ˆ(u)

ˆ(u) p(y, 0), u

1 LF P

ˆ(u)

(14)

which yields pˆ(y, u) =

1 ˆ (u)LF P

u

p(y, 0),

(15)

where

and ↵

⌘



1 ↵

(2

.

µ)T ↵

(23)

It is interesting to notice that when µ < 2, we have ↵ < 1, which makes Eq. (23) depart from the conventional exponential relaxation corresponding to ↵ = 1. We also notice that when µ > 2, neglecting terms decaying to zero for u ! 0 faster than u, yields ˆ(u) = 1

h⌧ i u,

(24)

Y (0) Yˆ (u) = , u + h⌧ i

(25)

which generates ˆ (u) ⌘

u ˆ(u) . (1 ˆ(u))

(16)

The function (t), defined by means of Eq. (16), is the memory kernel of a Generalized Fokker-Planck Equation (GFPE). We note, in fact, that Eq. (15) is the Laplace transform of the following equation of motion Z t @ p(y, t) = dt0 (t0 )LF P p(y, t t0 ). (17) @t 0 Eq. (17) is a GFPE that according to Kenkre and Knox [Kenkre & Knox, 1974] can also be derived from the Liouville equation of system plus thermal bath, using the Zwanzig projection approach [Zwanzig, 1961]. We invite the interested reader to consult [Budini, 2005] for a more recent discussion of the derivation of this time convoluted master equation with a projection approach from the Liouville equation of system plus reservoir. We shall refer to this theoretical tool as GFPE1. It is interesting to notice that this process makes a form of relaxation emerge that is an attractive generalization of the conventional exponential relaxation: the MittagLe✏er relaxation [Metzler & Klafter, 2000a; Mainardi, 1996]. Let us define Y (t) ⌘< y(t) > .

(18)

From Eq. (17) we derive Yˆ (u) =

Y (0) . u + ˆ (u)

(19)

(2

µ)(T u)µ

1

.

(20)

Consequently, we obtain Yˆ (u) =

1 u + ( ↵ )↵ u1

↵

,

(21)

1

h⌧ i

(22)

t).

(26)

It is worth to recall that we are considering the condition of Eq. (2). As we have earlier shown, this condition, which is essential to derive Eq. (17), renders virtually continuous the discrete time n. On the same token we have to assume T of Eq. (7) much larger than the unit time step. As a consequence h⌧ i >> 1. Thus, in the case µ > 2, the social life of the Demon has the e↵ect of making the relaxation process much slower, but it does not a↵ord any hint on the the Demon’s social life. From, this result it is in fact impossible to get information on µ. There is no qualitative di↵erence between a Poisson, µ = 1, and a non-Poisson Demon, if µ > 2. In fact, for all values of µ > 2, the exponential condition of Eq. (26) applies, with h⌧ i =

T (µ

2)

(27)

.

The Demon’s social life with µ < 2 is more attractive. In fact, in this case we get Eq. (23), which is characterized by the important property Y (t) = exp( ( ↵,

↵ ↵ ) t),

(28)

and Y (t) /

1 , tµ 1

(29)

for 1/ ↵ Ttrunc . IV.

GENERALIZED LANGEVIN EQUATION

The GLE is a well known way to extend the fluctuation-dissipation relation to the non-Markov case [Mori, 1965; Kubo, 1966]. This equation reads Z t d y(t) = dt0 '(t0 )y(t t0 ) + F (t), (34) dt 0

where the memory kernel '(t) is related to the equilibrium correlation function of the fluctuation F (t) by means of '(t

t0 ) =

hF (t)F (t0 ieq hy 2 ieq

.

(35)

Many years ago Adelman [Adelman, 1976] noticed that the formal solution of Eq. (34) can be expressed in the form Z t y(t) = dt0 E(t t0 )F (t0 ) + E(t)y(0), (36) 0

with the function E(t) defined by means of its Laplace ˆ transform E(u) as ˆ E(u) =

1 . u + '(u) ˆ

(37)

In the ordinary case where F (t) is a noise with no correlation (white noise), the Laplace transform of the memory kernel is independent of u, '(u) ˆ = , thereby making the Laplace transform of Eq. (37) become 1/(u + ), the Laplace transform of the ordinary exponential exp( t). Thus, Eq. (36) turns into the solution of the ordinary Langevin equation. Adelman [Adelman, 1976] proved that under the assumption that F (t) is a Gaussian fluctuation, the equation of motion for the probability density function p(y, t) becomes  ˙ ⌦ ↵ @ E(t) @ p(y, t) = y + y 2 eq p(y, t). (38) @t E(t) @y

We shall refer to this equation as GFPE2, and we invite the reader to compare it to GFPE1, defined by Eq. (17). We note that both equations rest on the ordinary fluctuation-dissipation process described by the FokkerPlanck operator of Eq. (5). In the former case, GFPE1, the generalization is done through the time convolution of LF P p(y, t) with (t), whereas in the latter, the constant dissipation parameter is replaced by the time dependent dissipation rate (t) defined by (t) ⌘

˙ E(t) . E(t)

(39)

5 It is interesting to note that there is a case where the Adelman’s generalized exponential of Eq. (37) becomes identical to the ML generalized exponential. According to Weiss [Weiss, 1999] the memory kernel generated by a non-ohmic bath can be expressed as Z !D '(t) = C ⇢(!)cos(!t)d!, (40) 0

where C is a suitable constant factor, ⇢(!) is the frequency distribution and, finally, !D is the Debye frequency. Weiss discusses the general case 1

⇢(!) / !

1

(42)

.

We see that the super-Ohmic case generates the anomalous relaxation process described by ˆ E(u) =

1 u + ( ↵ )↵ u1

↵

(43)

with ↵

⌘ k 1/↵

hy(t)i = E↵ ( ( hy(0)i

(44)

↵ ↵ t) ) ,

GFPE1 and GFPE2 generate ⌦ 2 ↵ ⌦ 2↵ n y (t) = y eq 1 [E↵ ( (

(49)

o

(50)

↵ ↵ t) )]} ,

(51)

↵ 2 ↵ t) )]

and

(41)

,

with ranging from = 0 to = 2. The cases > 1 and ! < 1 are denoted as super-Ohmic and sub-Ohmic, respectively. It is shown [Pottier, 2003] that '(u) ˆ = ku

straightforward exercise to prove however that the two pictures yield di↵erent results for the higher moments. Thus, while both of them yield

⌦

↵ ⌦ ↵ y 2 (t) = y 2 eq {1

[E↵ ( 2(

respectively. Thus, in principle, it should be possible to assess if a given complex time series is generated by SOFD or by GLE. However, the problem is harder than expected. In fact, it has been recently remarked [Bianco et al., 2007; Failla et al., 2004] that a finite time series is characterized by the truncation e↵ects described in Section III, thereby justifying why the events produced by EEG’s dynamics are satisfactorily described by stretched exponential functions. If the truncation time is comparable to 1/ ↵ , see Eq. (33), the ML power law does not show up and only the stretched exponential is visible. In this case, there is apparently a complete equivalence between GLE and SOFD.

and ↵=2

,

(45)

yielding ↵ < 1, and consequently the ML relaxation. Of course, the ordinary exponential relaxation is generated by the Ohmic bath, with = 1. V.

TRUNCATION-INDUCED EQUIVALENCE BETWEEN GLE AND SOFD

Let us consider an out of equilibrium initial condition with < y(0) >6= 0.

(46)

Under the condition (t) =

(t)

(47)

the GLE of Eq. (34) yields the same relaxation process as the GFP1, Eq. (17), described by Z t d < y(t) >= dt0 (t t0 ) < y(t0 ) > . (48) dt 0 The case µ < 2, according to Eq. (22) yields the same stretched exponential parameter ↵ as the super-Ohmic bath, as prescribed by Eq. (45), and consequently a relaxation process described by the ML function. It is a

VI.

CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH WORK

There are good reasons to believe that EEG are renewal processes with µ < 2 [Buiatti et al., 2007]. However, this does not rule out the possibility that the GLE approach a↵ords a picture as satisfactory as the SOFD theoretical perspective. Let us consider the case where the two methods yield < y(t) > / < y(0) >= E↵ (

↵ t),

(52)

with ↵ < 1.

(53)

Let us make also the assumption that the inverse power law of the ML function is not visible, and that only the stretched exponential shows up. The experimental detection of the parameter ↵ of the stretched exponential allows us to determine the power index µ µ=1+↵

(54)

only in the case where the SOFD theoretical perspective applies. In this case in fact, as explained in Section II, the parameter ↵ is generated by the renewal Demon’s activities. If the EEG theory applies, the parameter ↵ does not have anything to do with Demon’s social life: it is

6 rather the manifestation of long-time memory produced by the super-Ohmic bath. Let us look therefore for a property that is peculiar manifestation of the Demon’s action. Let us focus our attention on the time series y(t). Let us fix a threshold R and let us monitor the times ti at which |y(ti )| = R.

(55)

Let us evaluate the distribution of the times ⌧i = ti+1

ti .

(56)

Let us call this distribution d(⌧ ). Let us define the corresponding survival probability Z 1 D(t) = d⌧ d(⌧ ). (57) t

On the basis of the work of Ref. ⌦ ↵ [Cakir, 2007a] we expect that for R2 of the order of y 2 eq , both theories yield D(t) /

1 tµR

1

,

(58)

with µR = 2

↵ , 2

(59)

which is in general di↵erent from µ = 1 + ↵, although sharing with it the property µR < 2. The analysis of Ref. [Cakir et al., 2007a] proves that the prediction of Eq. (59), determined essentially by the origin re-crossing, is characterized by renewal aging, regardless of what is the source of the scaling ↵/2, either the renewal Demon or the trajectory long-time memory. Therefore this result is not a reliable way to determine the genuine parameter µ, if the complex process is of SOFD ⌦ type. ↵ We think that the case R2 >> y 2 eq is much more promising. In fact in this case, in the natural time scale, according to the Kramers theory [Kramers, 1940] the survival probability D(n) is given by D(n) = gexp( gn),

(60)

g = Aexp(

(61)

with R2 /B),

where A⇠

(62)

and ⌦ ↵ B ⇠ y 2 eq .

(63)

By applying the same approach as that used in Section II, we conclude that D(t) is a stretched exponential with ( ↵ )↵ / g = Aexp( R2 /B), a prediction that can be checked experimentally.

In conclusion, both GLE and SOFD theories generate stretched exponential relaxation. This equivalence has the e↵ect of weakening the claims that EEG dynamics is characterized by renewal and non-ergodic events [Bianco et al., 2007]. However, the adoption of the threshold method outlined in this section may a↵ord a crucial way to prove that the EEG fluctuations are really driven by the action of the renewal non-Poisson Demon described in this paper. The work of Ref. [Bianco et al., 2007] proves that a set of interacting electrodes is characterized by the presence of non-ergodic renewal events. The discovery of non-ergodic renewal events imbedded in the single electrode EEG data would confirm the finding of Ref. [Buiatti et al., 2007] and consequently prove that the single electrode inherits the complexity of the whole system. We hope that the publication of this article may stimulate some research work in this direction. Acknowledgment We acknowledge ARO and Welch for financial support through Grants W911NF-5-00059 and Grant B-1577, respectively. References Adelman, S. A. [1976] “FokkerPlanck equations for simple non-Markovian systems”, J. Chem. Phys. 64, 124-130. Balescu, R. [2007] ”V-Langevin equations, continuous time random walks and fractional di↵usion”,Chaos,Solitons & Fractals 34, 62-80. Barkai, E. & Silbey, R. J. [2000] “Fractional Kramers Equation,” J. Phys. Chem. B104, 3866-3874. Bel, G. & Barkai, E. [2006] “Random walk to a nonergodic equilibrium concept,” Phys Rev E73, 016125 (114). Bianco, S, Geneston, E., Grigolini, P & Ignaccolo, P. [2006], “Renewal Aging as Emerging Property of Phase Synchronization”, arXiv:cond-mat/0611035. Bianco, S., Ignaccolo, M, Rider M. S., Ross, M. J., Winsor, P. & Grigolini, P. [2007] ”Brain, music and nonPoisson renewal processes”, Phys. Rev. E 75, 061911 (1-10). Bologna, M., Grigolini, P., Palatella, L. & Pala, M. [2003]“Decoherence, wave function collapses and nonordinary statistical mechanics,” Chaos, Solitons & Fractals 17, 601-608. Budini, A.A. [2005] “Random Lindblad equations from complex environments,” Phys. Rev. E72, 056106 (1-11). Buiatti, M., Papo, D., Baudonniere, P.-M., van Vreeswijk, C. [2007] ”Feedback modulates the temporal scale-free dynamics of brain electrical activity in a hypothesis testing task,” Neuroscience 4146 1400-1412. Cakir, R., Grigolini, P. & Krokhin, A. A. [2007a] “Dynamical origin of memory and renewal” Phys. Rev. E 74, 021108 (1-6). Cakir, R., Krokhin, A. & Grigolini, P, [2007b] “From the trajectory to the density memory”, Chaos, Solitons & Fractals 34 19-32. Failla, R., Grigolini, P., Ignaccolo, M. & Schwettmann, A.[2004] “Random growth of interfaces as a subordinated process” Phys. Rev. E 70, 010101(R) (1-4).

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