Aging and rejuvenation with fractional derivatives

Transcription

1

2 PHYSICAL REVIEW E 70, (2004) Aging and rejuvenaion wih fracional derivaives Gerardo Aquino, Mauro Bologna, Paolo Grigolini,,2,3 and Bruce J. Wes 4 Cener for Nonlinear Science, Universiy of Norh Texas, P. O. Box 3427, Denon, Texas , USA 2 Diparimeno di Fisica dell Universià di Pisa and INFM, via Buonarroi 2, 5627 Pisa, Ialy 3 Isiuo dei Processi Chimico Fisici del CNR, Area della Ricerca di Pisa, Via G. Moruzzi, 5624 Pisa, Ialy 4 Mahemaics Division, Army Research Office, Research Triangle Park, Norh Carolina 27709, USA (Received 8 March 2004; published 0 Sepember 2004) We discuss a dynamic procedure ha makes fracional derivaives emerge in he ime asympoic limi of non-poisson processes. We find ha wo-sae flucuaions, wih an inverse power-law disribuion of waiing imes, finie firs momen, and divergen second momen, namely, wih he power index in he inerval 23, yield a generalized maser equaion equivalen o he sum of an ordinary Markov conribuion and a fracional derivaive erm. We show ha he order of he fracional derivaive depends on he age of he process under sudy. If he sysem is infiniely old, he order of he fracional derivaive, o, is given by o=3. A brand new sysem is characerized by he degree o= 2. If he sysem is prepared a ime a 0 and he observaion begins a ime =0, we derive he following scenario. For imes 0 a he sysem is saisfacorily described by he fracional derivaive wih o=3. Upon ime increase he sysem undergoes a rejuvenaion process ha in he ime limi a yields o= 2. The inermediae ime regime is probably incompaible wih a picure based on fracional derivaives, or, a leas, wih a mono-order fracional derivaive. DOI: 0.03/PhysRevE PACS number(s): Ey, Fb, Cd I. INTRODUCTION The fracional calculus has recenly received a grea deal of aenion in he physics lieraure, hrough he publicaion of books [,2] and review aricles [3,4], as well as an ever increasing number of research papers, some of which are quoed here [5 2]. The blossoming ineres in he fracional calculus is due, in par, o he fac ha i provides one of he dynamical foundaions for fracal sochasic processes [2,4]. The adopion of he fracional calculus by he physics communiy was inhibied hisorically because here was no clear experimenal evidence for is need. The disciplines of saisical physics and hermodynamics were hough o be sufficien for describing complex physical phenomena solely wih he use and modificaions of analyic funcions. This view was suppored by he successes of such physiciss as Onsager, who hrough he use of simple physical argumens was able o relae apparenly independen ranspor processes o one anoher, even hough hese processes are associaed wih quie differen physical phenomena [4]. His general argumens resed on hree assumpions: () microscopic dynamics have ime-reversal symmery; (2) flucuaions of he hea bah decay a he same rae as do macroscopic deviaions from equilibrium; and (3) physical sysems are aged. We refer o assumpion 2 as he Onsager principle and show ha i is ied up wih assumpion 3. Onsager s argumens focused on a sysem ha is in conac wih a hea bah sufficienly long ha he bah has come o hermal equilibrium and consequenly he sysem is aged. In saisical physics we know ha he bah is responsible for boh flucuaions and dissipaion, and if he flucuaions are whie he regression of perurbaions of he bah o equilibrium is insananeous. This means ha he energy absorbed from he sysem of ineres by he bah, hrough macroscopic dissipaion, is disribued over he bah degrees of freedom on a very much shorer ime scale han he relaxaion ime of he sysem. This propery is summarized in he well known flucuaion-dissipaion heorem, which has even been generalized o he case where he flucuaions in he bah do no regress insananeously [5]. The dynamics of he physical variables o which he Onsager principle apply are described by wo differen kinds of equaions: () he Langevin equaion, a sochasic differenial equaion for he dynamical variable and (2) he phase space equaion for he probabiliy densiy. Two disinc mehods have been developed o describe he phase space evoluion of he probabiliy densiy: he maser equaion inroduced by Pauli and he coninuous ime random walk (CTRW) approach of Monroll and Weiss [6]. The CTRW formalism describes a random walk in which he walker pauses afer each jump for a sojourn specified by a waiing ime disribuion funcion. I was shown by Bedeaux e al. [7] ha he Markov maser equaion is equivalen o a CTRW if he waiing ime disribuion is Poissonian. However, when he waiing ime disribuion is no exponenial, he case we consider here, he equivalence beween he wo approaches is mainained only by generalizing o he non-markov maser equaion, he so-called generalized maser equaion (GME) [8]. Recenly, Mezler [9] argued ha he GME unifies he fracional calculus and he CTRW. Allegrini e al. [3] have shown ha creaing a maser equaion compaible wih he Onsager principle requires ha he sysem be enangled wih he bah in such a way as o realize a condiion of sable hermodynamic equilibrium. This sysem-bah enanglemen is he resul of a rearrangemen process ha may ake an infiniely long ime o complee, leading o he replacemen of he GME of Kenkre e al. [8], which corresponds o he nonsaionary condiion, wih a new GME compaible wih he saionary condiion, and consequenly wih he Onsager principle. Herein, we exend ha discussion o include he connecion wih he fracional calculus in boh infiniely and no infiniely aged condiion /2004/70(3)/03605()/$ The American Physical Sociey

3 AQUINO e al. PHYSICAL REVIEW E 70, (2004) This will allow us o go beyond he discussion of Sokolov and Mezler [2], whose fracional derivaive refers o he young sae. We shall find a fracional derivaive operaor wih a differen index, corresponding o he infiniely aged condiion, and we shall show ha he condiion of Sokolov and Mezler is an aracor for all he sysems ha are parially aged and no infiniely aged. Beyond he Onsager principle We approach he subjec of fracional derivaives from a perspecive similar o ha of Sokolov and Mezler [2]. More specifically, we address he problem of he connecion beween he GME [8] and he saionary version of he CTRW [6]. The GME considered by Allegrini e al. [3] is he wo-sie version of he following equaion: d d p = Kpd, 0 where p is he m-dimensional populaion vecor of m sies, K is a ransiion marix beween he sies, and is he memory kernel. The CTRW prescripion for his process yields p = n=00 d n M n p0. Noe ha n is he probabiliy ha n jumps occurred and ha he las jump ook place a ime =, implying he renewal heory relaion n n d, 3 =0 where is he waiing ime disribuion funcion inroduced ino he CTRW and 0 =. While M is he ransiion marix connecing he sies afer one jump has occurred, he probabiliy ha no jump occurs in he ime inerval 0, is d. 4 = The waiing ime disribuion funcion and he memory kernel can be relaed o one anoher by aking he Laplace ransform of he GME () and he CTRW (2). This comparison, afer some algebra [3], yields ˆ uˆ u uk = M I, 5 ˆ u where I is he mm uni marix and he Laplace ransform of he funcion f is fˆu. Here, as in Allegrini e al. [3], we limi our discussion o he wo-sae case where and 2 M = K =, 7 hereby reducing Eq. (5) o ˆ uˆ u u =. 8 ˆ u This relaion beween he Laplace ransform of he memory kernel and he Laplace ransform of he waiing ime disribuion funcion was firs obained by Kenkre e al. [8] and is reviewed by Monroll and Wes [20]. In he case when he laice has only wo sies, a lef and a righ sie, he random walker corresponds o a dichoomous signal, wih he values =, for he lef sie, and =, for he righ sie. For he sake of simpliciy, we assume he wo saes o have he same saisical weigh. Also in he wo-sae CTRW, if we adop a discree ime represenaion, he moion of he random walker corresponds o a symbolic sequence, wih he form , which shows a significan persisence of boh saes. The waiing ime disribuion is he disribuion of he paches filled wih eiher + s or s. We assume symmery beween he wo saes and a finie firs momen of making i possible for us o define he auocorrelaion funcion for he flucuaions, = 0 2, 9 because he process is saionary in ime [2]. The problem of how o relae o his persisen flucuaion is delicae and will be discussed wih some deail in Sec. I B. Here we limi ourselves o noicing ha he Onsager principle bypasses he echnical difficulies wih he connecion beween he Langevin and he maser equaion picures, eiher ordinary or generalized. In fac, he maser equaion is a prescripion o deermine he probabiliy of occupying a given sae. In he case under sudy here, he maser equaion makes i possible o deermine p i, a ime, wih i=,2. The Langevin equaion, or in general any heoreical ool driving he moion of a variable, in he case under sudy here, affords a convenien means o deermine, and consequenly he auocorrelaion funcion of Eq. (9). In his original work Onsager considered he case of a macroscopic flucuaion ha regresses o equilibrium hrough he phenomenological equaions of moion. Here we adop an exension of he Onsager principle made by Allegrini e al. [3] o he case of wo saes, using he probabiliy of he random walker being in sae i=,2,p i, a ime, which allows us o deermine he auocorrelaion funcion of Eq. (9) wihou any need o esablish he Langevin-like picure equivalen o he GME under sudy here. In fac, we adop he following equaliy: = p p 2 p 0 p 2 0, 0 which yields an immediae connecion beween he GME of Eq. () and he auocorrelaion funcion of Eq. (9), provided

4 AGING AND REJUVENATION WITH FRACTIONAL PHYSICAL REVIEW E 70, (2004) ha he sysem is iniially in he ou of equilibrium sae corresponding o p 0 p Assuming a regression o equilibrium in such a way as o reain Eq. (0) we obain from he GME () using he coupling marix (7) d d. d = 20 Thus, he Laplace ransform of he auocorrelaion funcion can be relaed o he Laplace ransform of he memory kernel by ˆ u =. 2 u +2ˆ u I is ineresing o noice ha in he Poisson case, namely, when is an exponenial funcion of ime, he memory kernel of he GME, given by Eq. (8) urns ou o be equivalen o a Dirac funcion of ime, hereby implying ha he bah responsible for he flucuaions of he variable has a ime scale infiniely smaller han he sysem of ineres. In his specific case, wih he help of Eq. (2) we see ha he auocorrelaion funcion decays exponenially wih ime. This is a condiion behind he ordinary Onsager principle. Following he auhors of Ref. [3] we wan o go beyond he ordinary Onsager principle. The auhors of Ref. [3] sudied he case where he auocorrelaion funcion depars from he exponenial relaxaion and has he he following ime asympoic propery: 2. 3 The waiing ime disribuion corresponding o his auocorrelaion funcion has he following ime asympoic propery [3]:, 4 wih 2 o fi he saionary condiion. A firs sigh, one migh be surprised abou our decision o make hese complex processes obey he Onsager principle. Such processes have exoic hermodynamical properies, and in some cases hey are even shown o be nonergodic [22] and o produce aging effecs [23,26]. Anoher ineresing effec emerging from hese processes was described in Ref. [24]. These auhors used a fracional Fokker-Planck equaion, which is closely relaed o he GME used in his paper, and hey found ha he response of heir GME o exernal perurbaion is quie differen from he response of he corresponding CTRW, in conflic wih he fac ha heir GME is equivalen o a CTRW in he absence of perurbaion. All hese surprising properies, however, refer o he case 2, where no invarian measure exiss. The case 2, under sudy here, is in principle compaible wih he Onsager principle, and as a consequence our reques is no absurd. Neverheless, we shall see ha he Onsager principle requires ha he sysem is infiniely aged, an ideal condiion, and ha an even apparenly negligible deparure from his condiion yields a sriking effec: a rejuvenaion process. The auhors of Ref. [3] deermined ha he problem of how o make hese processes compaible wih he Onsager principle could be solved by expressing he CTRW in saionary form, resuling in he GME memory kernel ˆ u = u ˆ u 2 ˆ u + u+ˆ u, where is he average waiing ime, =0 d. 5 6 The form of he memory kernel given by Eq. (5) is consisen wih he equaion of moion for he auocorrelaion funcion (), and consequenly Eq. (5) is equivalen o ˆ u = 2 u ˆ. u B. Theoreical waiing ime disribuion 7 We have o remind he reader ha he saionary auocorrelaion funcion of is no relaed direcly o. Zumofen and Klafer [25] provided a prescripion for deriving he corresponding equilibrium auocorrelaion funcion of from. Their resul ress on he observaion ha is an experimenal funcion, evaluaed by observing he ime duraion of he wo saes. The connecion wih renewal heory is esablished by assuming ha he ime duraion of a sae is deermined by wo processes; one is he exracion of a random number from a heoreical inverse power-law disribuion *, wih he same power index, and he oher is a coin ossing procedure ha deermines he sign of his laminar region. Thus, a given experimenal sojourn ime in one of he wo saes may correspond o an arbirarily large number of drawings and coin ossings. Renewal heory is used o relae he auocorrelaion funcion o he waiing ime disribuion funcion *. In fac, from he renewal heory [2] we obain he following imporan resul: = * * d, 8 where * is he mean waiing ime of he * -disribuion densiy. I is ineresing o noice ha his equaion implies ha he second derivaive of he auocorrelaion funcion is proporional o *, d 2 d 2 = * *. 9 In Sec. II he deparure poin of our calculaions is given by he auocorrelaion funcion of Eq. (8). In his case is convenien o assign o his equilibrium auocorrelaion funcion a simple analyical form. This is done as follows. Firs of all we assign o * he following analyical form:

5 AQUINO e al. PHYSICAL REVIEW E 70, (2004) T * * = + T *, 20 which, as can be proved wih he help of Eq. (24), is compaible wih geing he ime asympoic form of Eq. (4). This makes i possible for us o wrie * as follows: * =0 * d = T * 2. 2 Wih he choice of Eq. (20) for *, he auocorrelaion funcion of Eq. (8) ges he aracive analyical form where = T * + T *, Thus, in he case 3, he auocorrelaion funcion of he flucuaions is no inegrable. Zumofen and Klafer [25], in addiion o explaining wih clear physical argumens he connecion beween and *, esablished ha he Laplace ransforms of he wo funcions are relaed o one oher by ˆ 2ˆ * u u =. 24 +ˆ u This imporan relaion allows us o esablish a connecion beween and *, which urns ou o be =2 *. 25 In conclusion, here are wo differen waiing ime disribuions wih he ime asympoic behavior of Eq. (4), he experimenal waiing ime disribuion and he heoreical waiing ime disribuion *. The heoreical disribuion is necessary o define he auocorrelaion funcion. Thus, a heoreical reamen involving he auocorrelaion funcion will force us o depend on *. In his case i is convenien o adop he analyical form of Eq. (20) for *, and Secs. II and III will res on his choice. In oher cases, Sec. IV, Appendix A, and Appendix B, he heoreical reamen will use he experimenal waiing ime disribuion. In hese cases i is convenien o adop he analyical form T = + T. 26 However, whaever choice is made, eiher he analyical form of Eq. (20) for * or he analyical form of Eq. (26) for, in boh cases, hanks o Eq. (24), he wo waiing ime disribuions mainain he same ime asympoic behavior, wih he same. So do he wo differen expression for he equilibrium auocorrelaion funcion, boh decaying as Eq. (3). For he main purpose of his paper he ime asympoic behavior is in fac he propery ha maers. C. Purpose of his paper The purpose of his paper is o prove ha a dichoomous flucuaion, wih he waiing ime disribuion of Eq. (4), can be described by a GME ha, in urn, is well represened by a fracional derivaive operaor. The fracional index corresponds o he realizaion of he Onsager condiion, as surprising as his fac migh be in he condiion 32, which is so far from he Poisson limi =. The Onsager principle is no oally foreign o his anomalous physical condiion, hanks o he fac ha 2 is compaible wih he exisence of hermodynamical equilibrium. However, he ime necessary o reach his hermodynamical condiion is infinie, and he sysem, observed a finie imes, no maer how long, shows a surprising rejuvenaion effec. Through his rejuvenaion effec, he fracional order compaible wih he Onsager principle slowly urns ino ha esablished by Sokolov and Mezler [2]. The ouline of he paper is as follows. In Sec. II, using he inverse Laplace ransform of Eq. (7) we deermine he unknown memory kernel, making i possible o discuss how o express he GME in erms of fracional derivaives. The case where 23 is compared o he recen work of Sokolov and Mezler [2]. We find ha he index of he fracional derivaive is 3, raher han 2, as prediced by Sokolov and Mezler. In Sec. III, we prove ha his difference in index is due o he fac ha we adop a saionary condiion, while Sokolov and Mezler do no. In Sec. IV, we also prove ha in he case of a finie, raher han infinie, age our GME makes a ransiion from he 3 h o he 2h order. The saionary case becomes sable only in he limiing case of infinie age. In Appendix A, he ineresed reader can find deails on how o esablish he order of he fracional operaor in he GME, in he whole range 3, when only he brand new condiion of Eq. (8) is considered. The accuracy of he analyical expressions ha we use in Sec. IV o illusrae he rejuvenaion process is discussed in Appendix B. In conclusion, on he one hand we shed ligh on he meaning of he work of Allegrini e al. [3], which is proven o be a subordinaion o a Markov maser equaion hrough he saionary disribuion of firs exi imes. On he oher hand, we exend he approach o sysems of any age and reveal he phenomenon of a coninuous ime random walk wih rejuvenaion. To accomplish his dual role we rely heavily on he resuls recenly obained by Barkai [26] and, o a lesser exen he resuls of Allegrini e al. [3]. However, his allows us o reveal a dependence of he fracional derivaive order on he aging and rejuvenaion process, which was no previously idenified. II. THE INVERSE LAPLACE TRANSFORM OF THE MEMORY KERNEL To esablish he form of he unknown memory kernel, we make a few preliminary observaions. Firs of all, we noe ha hrough Eq. (7) we esablish a direc connecion wih he auocorrelaion funcion and ha his auo-correlaion funcion is, in urn, direcly relaed o he

6 AGING AND REJUVENATION WITH FRACTIONAL PHYSICAL REVIEW E 70, (2004) waiing ime disribuion *, hrough Eq. (8). Thus, wih no loss of generaliy for he reasons illusraed in Sec. I i is convenien o refer o * raher han o and consequenly, according o he prescripions illusraed a he end of Sec. I B, o he analyical form of Eq. (20). For simpliciy, we se T * = hroughou his secion. Thus, he Laplace ransform of he auocorrelaion funcion is [27] ˆ u = u e u u E, 27 where 0, given he fac ha we are considering u 23, and E is a generalized exponenial funcion [2]. Thus, ˆ u diverges as u 0 and Eq. (2) yields ˆ 0=0. We explore he opposie limi u using Eq. (5), which yields ˆ u=/=/2 *. In he ime represenaion, he laer limi is equivalen o /2 * for 0. Therefore, we segmen he Laplace ransform of he GME memory kernel ino wo pars as follows: ˆ u = 2 * + ˆ au. 28 The firs erm models he shor-ime limi, while he second erm is responsible for he long-ime behavior. In he ime represenaion we have = 2 * + a. 29 Noe ha his division of he memory kernel ino a whienoise conribuion and a slow erm corresponds o a similar pariion made by Fuliński [28]. Thus, for he ime evoluion equaion of he auocorrelaion funcion of, we derive he following equaion: d = d * a d Using Eq. (9) and subsiuing ino i he explici expression of * as a funcion of, afer some algebra, we obain T + T T + T = T + T = 2 a 0 d. 3 The wo erms on he lef-hand side Eq. (3) are posiive. Due o he negaive sign on he righ-hand side of his equaion we conclude ha i migh well be ha a is always negaive. Le us concenrae on he case : using he auocorrelaion funcion of Eq. (22) (wih T * =) and using he change of ime variable +, we rewrie Eq. (3) in he form 20 = + a + d +2 a + 0 d FIG.. The slow componen of he memory kernel, a, as a funcion of ime. The black dos denoe he resul of he numerical inversion of he expression in he Laplace ransform resuling from Eqs. (5) and (28) for =0.5; he coninuous line is he analyical approximaion given by Eq. (36). In he limiing case we neglec he second erm on he lef-hand side of his equaion. This is a naural consequence of he assumpion ha he memory kernel mus end o zero wih a negaive ail as an inverse power law. Wih his assumpion i is sraighforward o prove ha he modulus of he firs erm becomes much larger han ha of he second erm on he lef-hand side of his equaion. The consequences of his crucial assumpion are suppored by he numerical resuls depiced in Fig.. Wih his assumpion Eq. (32) simplifies o a 20 d = + +, 33 which can be solved by means of he fracional calculus [2]. We use he Riemann-Liouville (RL) definiion of he fracional inegral: D q f = fd q0 q, 34 which is he aniderivaive of he fracional derivaive wih order q, wih q. Consequenly, for we can express Eq. (33) in erms of he RL fracional inegral D a = 2 + +, so ha invering his equaion we have he formal expression for he slow par of he memory kernel a = 2 D + + = 2 D,+. 35 We denoe he auocorrelaion funcion wih he inverse power + by he symbol,+. Carrying ou he required calculaions on he righ-hand side of Eq. (35), we obain (see, for example, Wes e al. [[2], p. 90],

7 AQUINO e al. PHYSICAL REVIEW E 70, (2004) a = ,. 36 A comparison wih numerical inversion of he kernel is shown in Fig.. For he sake of compleeness, i is worh noicing ha we can proceed in a similar way in he case also. In his case we find, for he conribuion a of he GME, he following ime asympoic behavior: a = 2 ++,. 37 III. THE EMERGENCE OF FRACTIONAL OPERATORS In his secion we show ha in he wo-sie case we are discussing he GME has he form of a ranspor equaion, wih wo erms on he righ-hand side. The firs has he form afforded by he ordinary maser equaion and consequenly saisfies he Onsager principle, giving a relaxaion dependen on he average waiing ime of he CTRW. The second erm corresponds o a fracional derivaive in ime, and exends he Onsager principle o he case of a relaxaion wih a fa ail. To obain hese resuls, we use wha we have learned in he preceding secion. Firs of all, since we are dealing wih he wo-sie case, using he form of M and K marices wih z=x= and y =, we rewrie Eq. () in he following form: d d p p 2 p d, =0 d d p 2 p p 2 d. = The memory kernel is relaed o he auocorrelaion funcion of he dichoomous variable hrough Eq. (7). Insering Eq. (7) ino he Laplace ransform of he se of he wo-sie dynamical equaions, solving he resuling se of equaions, and aking he corresponding inverse Laplace ransforms yields he soluions p = 2 p 2 0 p 0, p 2 = 2 + p 2 0 p Noe ha hese soluions can be combined o yield he generalized Onsager principle given by Eq. (0) in erms of he difference in he probabiliies. We now wan o find a formal equaion of evoluion for he probabiliies involving fracional operaors. We know from he preceding secion ha = 2 * + a 42 wih * =T * / 2=/, hanks o he fac ha we se T * =. Subsiuing he decomposiion of he memory kernel ino Eq. (38), we obain dp = p 2 p d 2 * a p 2 p d Wriing a as he derivaive of an as ye unspecified funcion f, and using he propery d f gd = f0g f gd d wih f=d/df, we obain dp = p 2 p d 2 * f0p 2 p + d f d0 p 2 p d. 45 We also found ha in he case 0 he asympoic behavior of he memory kernel is expressed by a ,, T * =. 46 Rewriing Eq. (47) as d a 2 d, 47 we idenify f wih /2 for. Choosing f0=0 and using he properies of he funcion, we assign o he ime asympoic equaion of moion he form dp = p p 2 d d 2 * + 22 d0 p 2 p d 48 or, in erms of he RL fracional inegral (34), dp = p p 2 d 2 * 22 D p p The same procedure applied o he equaion of moion for p 2 yields dp 2 d = p 2 p 2 * + 22 D p p

8 AGING AND REJUVENATION WITH FRACTIONAL PHYSICAL REVIEW E 70, (2004) The difference beween hese wo equaions yields d d p p 2 = p p 2 * 2 D p p 2, 5 clearly showing he wo kinds of conribuion o he generalized Onsager principle. The firs erm gives he relaxaion of he perurbaion away from equilibrium a he macroscopic rae required by Onsager. The second erm gives he addiional slow relaxaion in he form of he fracional inegral. IV. AGING ORDER We have o remark ha he condiion of Eq. (2) refers o he saionary condiion explicily considered by Klafer and Zumofen [29]. In his secion we prove ha here is a connecion beween a sysem s age and he order of he fracional derivaive in he relaxaion process. A sign of he dependence of he fracional derivaive order on age is given by he discrepancy beween he resuls of Sec. III and Ref. [2]. Le us compare Eq. (49) o Eq. (6) of Ref. [2]. We obain he fracional index raher han as in he work of Sokolov and Mezler. In Appendix A we prove ha our ime asympoic approach o fracional derivaives, in he nonsaionary case sudied by Sokolov and Mezler, yields he same index as hey obain [2]. Thus, he discrepancy beween our predicion and he predicion of Sokolov and Mezler depends on he fac ha we consider a condiion consisen wih he Onsager principle, whereas Sokolov and Mezler do no. Furhermore, if he sysem is no infiniely aged, a sor of rejuvenaion process is expeced o ake place ha will lead o he fracional order of Sokolov and Mezler. To suppor our remarks concerning he relaion beween aging and he order of he fracional operaor, here we discuss how o define a waiing ime disribuion of any age. The auhors of Ref. [3] have shown ha he waiing ime disribuion of Eq. (4) is obained from he following dynamic model. A paricle moves in an he inerval I 0, driven by he equaion of moion dy/d = y z, 52 wih z. When he paricle reaches he border y=,iis injeced back o an iniial condiion beween y=0 and y= wih uniform probabiliy. The age of he CTRW is deermined by he disribuion of firs exi imes. The ordinary CTRW is based on idenifying his disribuion wih. This means ha he CTRW is equivalen o assuming ha he sysem is prepared in a fla disribuion a =0, which coincides wih he beginning of he observaion process. Le us discuss now he consequence of beginning he observaion a significan ime afer he preparaion. Le us imagine ha he sysem is prepared in a fla disribuion a a ime = a 0, and ha he observaion begins a =0. This means ha he fla disribuion begins producing a sequence of ime inervals of size, according o he disribuion of Eq. (4); more precisely, he ime inerval beginning a = a and ending a = a +, he ime inerval 2 beginning a = a + and ending a = a + + 2, and so on. The waiing ime disribuion of age a, denoed by a, isdeermined by he firs of hese ime inervals overlapping wih 0. The ime lengh of ha overlap is he ime lengh whose disribuion deermines a. We make he assumpion ha he beginning of he firs ime inerval overlapping wih 0 occurs wih equal probabiliy a any poin beween = a and =0. The validiy of his assumpion is discussed in Appendix B, which esablishes ha his assumpion is very good for a 0 and a. In beween he asympoic limis he resuling predicion is no exac. However, since i yields simple analyical formulas, we adop his simplifying assumpion for any age. Thus, we have ha a + ydy a =0, 53 g a where g a is he normalizaion facor defined by a g a d, 54 0 and is he probabiliy ha no even occurs hroughou he ime inerval of lengh. Using for, according o he prescripion adoped in his paper, he analyical form of Eq. (26), i is easy o prove ha Eq. (53) can be wrien in he form a = 2 + T + T + a T 2 a + T This formula proves ha for a he index of he disribuion is, whereas for a he index becomes. This resul for he age-dependen waiing ime disribuion funcion agrees wih he predicions by Barkai [26] and by he auhors of Ref. [3]. Noice ha he formula Eq. (55) is equivalen o drawing he iniial condiion for y from an aged disribuion of his variable. Here, we are in a posiion o evaluae he waiing ime index a a generic ime, where we wrie a as a = AT, a + T eff. 56 Using Eq. (55) we arrive a he following formula for he ime dependence of he effecive power-law index: eff = ln + T + T + a. 57 ln + T Figure 2 illusraes he regression of he effecive powerlaw index o wih wo differen ages, and shows clearly ha he regression is slower for older sysems. This formula does more han explain he discrepancy beween Eq. (49) and Eq. (6) of Ref. [2]. In fac, i shows ha i is possible o build a GME ha a shor imes follows he prescripion of our GME and a long imes moves ino he basin of aracion of Sokolov and Mezler. This is cerainly he case if a

9 AQUINO e al. PHYSICAL REVIEW E 70, (2004) FIG. 2. The effecive power index eff as a funcion of ime, for =2.3. The curves refer, from op o boom, o a =00, 000, This aspec is imporan and needs a more exhausive illusraion. We noe ha he approach of Ref. [3] can be easily exended o he case where he disribuion of firs exi imes has a finie age. I is enough o follow he procedure of Ref. [3] and o replace he firs exi ime disribuion wih a raher han wih a =, as done in Ref. [3]. The resul of his procedure yields for he GME he following form for he Laplace ransform of he memory kernel: uˆ ˆ a u = a u. 58 +ˆ u 2ˆ a u I is sraighforward o prove ha for a =0 Eq. (58) reduces o ˆ uˆ u a =0u =. 59 ˆ u In fac, he general prescripion of Eq. (53) immediaely yields a = for a =0. To derive he memory kernel corresponding o he infiniely aged condiion of Eq. (5) we have o noice firs ha Eq. (53) yields, in accordance wih Refs. [26] and [3], a = = d. 60 To illusrae he change of he memory kernel wih ime, noice ha he Laplace ransform of Eq. (55) is ha is, ˆ a u = ˆ u e u a + e u a0 g a ˆ u e u a + ue u a0 ˆ a u = ug a a e uy ydy a e uy ydy, FIG. 3. The Laplace ransform of he a -old memory kernel a,ˆ a u, as a funcion of u, for =2.3. The curves refer, from op o boom, o a =0,0.,,0,. Le us subsiue Eq. (62) ino Eq. (58), from which we obain he Laplace ransform of he memory kernel for an arbirary age. In Fig. 3 we show he Laplace ransform of he memory kernel corresponding o a number of differen ages. Moving from he op o he boom curve he age increases from he brand new kernel a =0 o he infiniely aged or saionary kernel a =. V. CONCLUDING REMARKS The adopion of a wo-sae model o generae anomalous diffusion is no unusual in he random walk lieraure. For insance, Shushin [30] also generaed anomalous diffusion by means of a wo-sae model. However, o properly locae in he lieraure of anomalous diffusion he model of he presen paper, we have o poin ou ha he diffusion generaor in Ref. [30] is a wo-sae Markov equaion modulaed by imedependen ransiion parameers ha obey non-poisson saisics. The model sudied in he presen paper migh be adoped o generae anomalous diffusion as well, bu is quie differen from ha of Sushin. The exended ime of sojourn in one of he wo saes would produce uniform moion, wih no randomness involved, as discussed by he auhors of Ref. [2], which ress in fac on he same model as ha adoped here. In his paper we esablish ha an infiniely old sysem, wih he power-law index in he inerval 23, yields he fracional order =3. The predicion of Ref. [2], on he oher hand, yields he fracional order 2, corresponding o he brand new condiion. If he sysem is no infiniely aged, namely, a, he shor-ime behavior of he sysem for a is expeced o be ha of an aged sysem. A large observaion imes, a, rejuvenaion begins. This can be explained using he dynamical model of Eq. (52). In fac, aging has o do wih he slow regression o equilibrium, if i exiss, of he variable y, which mimics a real bah slowly regressing o equilibrium. The aging effecs discussed by Barkai [26] correspond o z2 and hus o 2. In his case he dynamic model under discussion does no have an invarian disribuion, and, consequenly, any observaion is

10 AGING AND REJUVENATION WITH FRACTIONAL PHYSICAL REVIEW E 70, (2004) done while he bah is drifing oward a condiion ha will never be reached. This aging effec affecs he form of he firs exi ime disribuion, whose index is raher han. However, afer he firs exi he rajecories are injeced back wih a uniform probabiliy, and hus all he ensuing jumps are deermined by he ordinary waiing ime disribuion. I would be desirable o have an equaion of moion wih a fracional operaor order ha changes as a funcion of ime from 3 o 2. However, here are echnical and concepual difficulies ha make i difficul, if no impossible, o realize his goal. In fac, according o he perspecive adoped in his paper, he order of he fracional operaor is esablished using ime asympoic argumens. Thus, if a we can associae a ime a wih he order o= 2 of he fracional derivaive. This is so because for a he calculaions would be virually equivalen o hose done in Appendix A. I is no clear how o proceed when is of he order of a, his being he firs reason why assigning a fracional derivaive order o any ime migh be difficul. There also exiss a physical reason ha migh make i impossible o move from he order 3 o 2. Physically, his exremely exended ransiion process migh involve a mixure of fracional derivaives of differen orders. I has been shown [3] ha he Lévy walk does no have well-defined scaling, due o aging effecs. Similarly, he adopion of a fracional ime derivaive, wih ime-dependen order, migh be inadequae o explore he regime of ransiion from he order 3 o he order 2. In conclusion, he fracional operaor and is order reflec a sable condiion, of a brand new or infiniely old sysem. The regime of ransiion from he dynamic o any of hese wo hermodynamic regimes, and he regime of ransiion from he earlier o he laer hermodynamic regime, is no ye a fully undersood physical condiion, an issue calling for furher invesigaion. I is ineresing o noice ha, even if we selec 2 and consequenly adop a condiion compaible wih he saionary condiion, he effecive index of he firs exi disribuion is locaed in he nonsaionary region if. This is probably he reason why he memory kernel seems o share he same properies as hose adoped in Refs. [6,0,32] o produce subdiffusion. Noice ha he bahs used by Luz [6] and Poier [32] have properies quie differen from he subordinaion perspecive of Ref. [0], even hough he relaxaion process semming from subordinaion [0] is quie similar o ha produced by he non-ohmic bahs of Luz and Poier. We hope ha he presen work migh help in undersanding he connecion beween he wo perspecives. This is anoher subjec for fuure research. ACKNOWLEDGMENTS G. A. and P. G. graefully acknowledge financial suppor from ARO hrough Gran No. DAAD APPENDIX A In his appendix we show how o obain he order of he fracional operaor in he GME for 3, using he Laplace ransform form for given by Eq. (8). We are using raher han *. Thus, according o he procedure adoped in his paper, we use he analyical form of Eq. (26). For he reader s convenience, we rewrie his expression here, T = T +. A I is imporan o sress ha in he ex we made he choice of Eq. (20) for *, in his way deermining, hrough Eq. (24), a form for ha depars from Eq. (A). However, boh choices lead o he same ime asympoic behavior of Eq. (4) for boh he waiing ime disribuions. We noe ha for u we ge a finie value ˆ = /T corresponding o 0 in space. As in Sec. II, we separae he kernel ino wo conribuions: = /T+ a, and inser he Laplace ransform of separaion ino Eq. (8), o wrie ˆ au = uˆ u + ˆ u ˆ u ˆ Du + ˆ u =. A2 ˆ u We inroduce ˆ Du as he Laplace ransform of he disribuion s derivaive = T /T+ + ; using he Laplace ransform of an inverse power law and subsiuing i ino Eq. (A2) we have ˆ 2 e u u E au = u e u u E ue u E u u e u u E, A3 where, as usual, for simpliciy we have se T=. Anicipaing he convoluion form of he soluion we cross-muliply o obain u e u u E ˆ au = e u u E ue u E u. A4 Using he relaion [2] 0 exp ud = a + a sin E ua e ua,, A5 and seing =, we consruc R = 2 sin a d a d, A6 where R is he inverse Laplace ransform of he righ side of Eq. (A4). Using he well known recurrence relaion of he funcion, we can combine erms in Eq. (A6) o obain

11 AQUINO e al. PHYSICAL REVIEW E 70, (2004) R = 2 a d = 0 + a d. A7 Le us consider firs he case 2, where, in he limi of (equivalen o T), we can wrie Eq. (A7) o a good approximaion as R = 3 a d = R. A8 0 Going back o he Laplace ransform represenaion, we obain he simplier expression 2 ˆ au u 2 = 2 e u u E ue u E u, A9 which afer a lile algebra yields for ˆ au ˆ au = 2 e u u E u u e u E u u u. A0 Thus using he inverse Laplace ransforms we obain he corresponding expression in he ime represenaion: a = A In he ime asympoic limi we ge a, A2 corresponding o a fracional operaor of index = 2. For he sake of compleeness, we also give he expression for he GME kernel in he case 2. Proceeding as was done earlier we obain sin A3 Finally, we wan o poin ou ha he expressions we are proposing refer o values of which are no ineger. We are exploring he inerval [,3]. Thus he he expressions we are proposing become quesionable for =2. To ge he proper expression for =2 we have o sudy expressions like hose of Eq. (A2) and (A9) a =2+, do a Taylor series expansion around =2, and assign o =2 he limiing values reached for 0. APPENDIX B This appendix is devoed o esablishing he accuracy of Eq. (53), and consequenly he validiy of he assumpion FIG. 4. The waiing ime disribuion a as a funcion o ime. The dos denoe he exac values, and he lines he predicion of Eq. (55) for =2.3,T=. Moving from boom o op in he righhand porion of he figure, a =0.0,, 0, 00. ha he beginning of he firs laminar region overlapping wih 0 is uniformly disribued beween = a and =0. The exac expression for a is where a =0 G + + a dxga x + x, n=20 d d 2 2 B d n n. B2 n 2 I is sraighforward o find he Laplace ransform of G. This is given by Ĝu = ˆ u n =. B3 n=0 ˆ u Thus, he Laplace ransform of Eq. (B) wih respec o a reads sa = s a es a s a e 0 s a y ydy. B4 By numerically ani Laplace ransforming Eq. (B4) we evaluae he ime dependence of he exac waiing ime disribuion of age a, Eq. (B). In Fig. 4 we compare he exac predicion, evaluaed numerically, o he heurisic expression of Eq. (53). We find ha a small and large values of a hese wo expressions coincide. In he inermediae region hey do no. Neverheless, we hink he agreemen beween he wo expressions is saisfacory enough for he purpose of his paper

12 AGING AND REJUVENATION WITH FRACTIONAL PHYSICAL REVIEW E 70, (2004) [] Applicaions of Fracional Calculus in Physics, edied by R. Hilfer (World Scienific, Singapore, 2000). [2] B. J. Wes, M. Bologna, and P. Grigolini, Physics of Fracal Operaors (Springer, New York, 2003). [3] R. Mezler and J. Klafer, Phys. Rep. 339, (2000). [4] G. M. Zaslavsky, Phys. Rep. 37, 46 (2002). [5] R. Mezler and A. Compe, J. Phys. Chem. B 04, 3858 (2000). [6] E. Luz, Phys. Rev. E 64, 0506 (200). [7] A. V. Chechkin, R. Gorenflo, and I. M. Sokolov, Phys. Rev. E 66, (2002). [8] J. Bisquer, Phys. Rev. Le. 9, (2003). [9] R. Mezler and J. Klafer, J. Phys. Chem. B (2000). [0] E. Barkai and R. J. Silbey, J. Phys. Chem. B (2000). [] A. V. Chechkin, J. Klafer, V. Yu. Gonchar, R. Mezler, and L. V. Tanaarov, Phys. Rev. E 67, 0002(R) (2003). [2] I. M. Sokolov and R. Mezler, Phys. Rev. E 67, 000(R) (2003). [3] P. Allegrini, G. Aquino, P. Grigolini, L. Palaella, and A. Rosa, Phys. Rev. E 68, (2003). [4] L. Onsager, Phys. Rev. 38, 2265 (93); 37, 405 (93). [5] R. Kubo, Can. J. Phys. 34, 274 (956). [6] E. W. Monroll and G. H. Weiss, J. Mah. Phys. 6, 67 (965). [7] D. Bedeaux, K. Lakaos, and K. Shuler, J. Mah. Phys. 2, 226 (97). [8] V. M. Kenkre, E. W. Monroll, and M. F. Shlesinger, J. Sa. Phys. 9, 45(973). [9] R. Mezler Phys. Rev. E 62, 6233 (2000). [20] E. W. Monroll and B. J. Wes, in Flucuaion Phenomena, edied by E. W. Monroll and J. L. Lebowiz (Norh-Holland, Amserdam, 979). [2] T. Geisel, J. Nierweberg, and A. Zacherl, Phys. Rev. Le. 54, 66 (985). [22] F. Bardou, J.-P. Bouchaud, A. Aspec, and C. Cohen- Tannoudji, Lévy Saisics and Laser Cooling (Cambridge Universiy Press, Cambridge, England, 2002). [23] X. Brokmann, J.-P. Hermier, G. Messin, P. Desbiollles, J.-P. Bouchaud, and M. Dahan, Phys. Rev. Le. 90, 2060 (2003). [24] I. M. Sokolov, A. Blumen and J. Klafer, Europhys. Le. 56, 75 (200). [25] G. Zumofen and J. Klafer, Phys. Rev. E 47, 85 (993). [26] E. Barkai, Phys. Rev. Le. 90, 040 (2003). [27] M. Bologna, P. Grigolini, and B. J. Wes, Chem. Phys. 284, 5 (2002). [28] A. Fuliński, Phys. Rev. E 50, 2668 (994). [29] J. Klafer and G. Zumofen, Physica A 96, 02 (993). [30] A. I. Shushin, Phys. Rev. E 64, 0508 (200). [3] P. Allegrini, J. Bellazzini, G. Bramani, M. Ignaccolo, P. Grigolini, and J. Yang, Phys. Rev. E 66, 050R (2002). [32] N. Poier, Physica A 37, 37 (2003)