FRACTIONAL KINETICS. Until about 10 years ago, expressions involving fractional

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1 FRACTIONAL Unil bou yers go, epressions involving frcionl derivives nd inegrls were prey much resriced o he relm of mhemics. Bu over he ps decde, mny physiciss hve discovered h number of sysems priculrly hose ehibiing nomlously slow diffusion, or subdiffusion re usefully described by frcionl clculus. Those sysems include chrge rnspor in morphous semiconducors, he spred of conminns in underground wer, relion in polymer sysems, nd rcer dynmics in polymer neworks nd in rrys of convecion rolls. Frcionl diffusion equions generlize Fick s second lw nd he Fokker Plnck equion by king ino ccoun memory effecs such s he sreching of polymers under eernl fields nd he occupion of deep rps by chrge crriers in morphous semiconducors. Such generlized diffusion equions llow physiciss o describe comple sysems wih nomlous behvior in much he sme wy s simpler sysems. Frcionl clculus Physiciss re ll fmilir wih he high-school clculus h inroduces sudens o derivives of ineger order n, d n y/d n. Those derivives nd heir inverse operions inegrions provide he lnguge for formuling nd nlyzing mny lws of physics. Bu physiciss generlly ren ugh bou frcionl-order derivives, which migh be formlly epressed s, for emple, d / y/d /. Is he frcionl clculus ll h difficul? In fc, he clculus of frcionl inegrls nd derivives is lmos s old s clculus iself. As erly s 695, Gofried von Leibniz, in reply o Guillume de l Hôpil, wroe, Thus i follows h d / will be equl o =d :,... from which one dy useful consequences will be drwn. Abou 3 yers hd o pss before wh is now known s frcionl clculus ws slowly cceped s prcicl insrumen in physics. Before h ccepnce, frcionl clculus hd o be more rigorously formuled. Imporn conribuions o h end sem from he work of IGOR SOKOLOV is professor of physics Humbold Universiy in Berlin, Germny. YOSSI KLAFTER is professor of chemisry Tel Aviv Universiy in Isrel. ALEX BLUMEN is professor of physics he Universiy of Freiburg in Germny. KINETICS I isn he clculus we knew: Equions buil on frcionl derivives describe he nomlously slow diffusion observed in sysems wih brod disribuion of relion imes. Igor M. Sokolov, Joseph Klfer, nd Alender Blumen Pierre-Simon Lplce, Bernhrd Riemnn, Joseph Liouville, Oliver Heviside, Arhur Erdélyi, nd mny ohers. One wy o formlly inroduce frcionl derivives proceeds from he repeed differeniion of n inegrl power: d n m m! () d n m n. ( m n)! For n rbirry power m, repeed differeniion gives d n m Gm ( ) m n, () d n Gm ( n ) wih gmm funcions replcing he fcorils. The gmm funcions llow for generlizion o n rbirry order of differeniion, d Gm ( ) m m. (3) d Gm ( ) The eension defined by equion 3 corresponds o he Riemnn Liouville derivive. I is sufficien for hndling funcions h cn be epnded in Tylor series. A second, more elegn nd generl wy o inroduce frcionl derivives uses he fc h he nh derivive is n operion inverse o n n-fold repeed inegrion. Bsic is he inegrl ideniy y y n... f ( y )dy... dy ( y) n f( y)dy. (4) ( n )! Clerly, he equliy is sisfied, nd i is no difficul o see ierively h he derivives of boh sides of he equliy re equl. A generlizion of he epression llows one o define frcionl inegrl of rbirry order vi D n n f () ( y) f( y)d y ( ). G ( ) A frcionl derivive of n rbirry order is defined hrough frcionl inegrion nd successive ordinry differeniion. For ddiionl elborion, see bo, which lso discusses he relion beween he vlue of he ine- 48 NOVEMBER PHYSICS TODAY Americn Insiue of Physics, S (5)

2 Bo. Definiions nd Emples The frcionl inegrion operor D is defined by equion 5. The -h frcionl derivive is hen d n D n d n D. The number of ddiionl differeniions n is equl o [], where [] is he whole pr of. From he bove definiion i follows h Gm ( ) D m m Gm ( ) s envisged in equion 3. Noe h here he lower limi of inegrion is zero. An ineresing consequence of he rule for differeniing powers is D. G( ) Th is, he derivive of consn vnishes only if he order of he derivive is ineger, in which cse G( ) diverges. Anoher ineresing resul holds for he derivive of he eponenil funcion: g(, ) D e e, G( ) where g(, ) is he incomplee g funcion. The prcicl use of frcionl clculus is underlined by he fc h, under Lplce rnsform, he operor D hs he simple form { D f( )} u { f( )}. The resul for he differeniion of n eponenil my seem disppoining. Bu if one chooses he lower limi of inegrion o be Fin equion 5, he resuling Weyl derivive sisfies F D e e. Moreover, he Weyl definiion reproduces he fmilir properies of Fourier-rnsformed inegrls nd derivives: { FD f ( )} ( iw) f( w). We see h here re severl wys o inerpre d /d, ll of which coincide wih he usul differeniion if is n ineger. Th my hve been one of he resons for he le ccepnce of frcionl clculus s ool o describe physicl phenomen. In fc, he freedom of definiion is n dvnge h llows one o ke ddiionl physicl informion (such s wheher force cing on sysem is lwys pplied or is urned on specific ime) direcly ino ccoun. grion limi nd definiions for frcionl differeniion bsed on he Lplce rnsform (when ) nd he Fourier rnsform (when F). The ble bove presens (for ) severl emples of semi-inegrls nd semi-derivives, operions for which /. In his ricle, we re generlly concerned wih frcionl ime derivives, nd we se, in effec choosing s he beginning of he sysem s ime evoluion. In priculr, we noe h he operor d D d D, (6) wih < <, plys cenrl role in generlized diffusion equions., Emples of Frcionl Clculus wih / D Semi-inegrl / d f() d 3 / / C=+++ /p =++ p =+ p / 4 /3=++ p Gm ( ) Gm ( 3/) / m / ep( ) erf( =++ ) f () =+++ p/ [ln(4 ) ] Funcion f () C, ny consn / =++ =++ m, m > ep( ) ln Semi-derivive / d D f() d f () From Fick o frcionl diffusion The foundions of kineics were esblished more hn 5 yers fer he prophecy of Leibniz, wihou he use of frcionl clculus. In 855, he young Adolf Fick, phologis he Universiy of Zürich, wroe work eniled Über Diffusion ( On Diffusion ). The work ws published in Poggendorf s Annlen der Physik, he Physicl Review Leers of h ime. Fick sred by observing h diffusion in wer confined by membrnes is no only one of he bsic fcors of orgnic life, bu i is lso n eremely ineresing physicl process nd, s such, should rc much more enion from physiciss hn i hs so fr. Diffusion processes such s hose considered by Fick, nd processes h re described by frcionl clculus, coninue o fscine physiciss nd ohers. Fick ws n eperimenl physiologis, bu his work on diffusion ws heoreicl, nd his pproch would ody be clled phenomenologicl liner-response heory pplied o diffusion. In brief, he resul of diffusion is known o be he equilibrion of concenrions. Thus, pricle curren hs o flow gins he concenrion grdien. In nlogy wih Ohm s lw for elecric curren, or wih Fourier s lw for he flow, Fick ssumed h he curren j is proporionl o he concenrion grdien, so h jr (, ) k c( r, ), (7) n equion now known s Fick s firs lw. Here, k is he diffusion coefficien wih dimension of L /T nd c is he concenrion. If, in ddiion, pricles re neiher creed nor desroyed, hen, ccording o he coninuiy equion, ]c( r, ) jr (, ). (8) ] Combining Fick s firs lw wih he coninuiy equion gives Fick s second lw, lso known s he diffusion equion: ]c( r, ) k c( r, ), (9) ] / / C/=+++ p =+ p / =+++ /p Gm ( ) Gm ( /) m / / =+++ p ep( ) erf( =++ ) ln(4 ) / =+++ p hp:// NOVEMBER PHYSICS TODAY 49

3 P (,) which is closed equion for he emporl evoluion of he concenrion. Fick s phenomenology missed he probbilisic poin of view cenrl o sisicl mechnics. I ws Alber Einsein who, 5 yers fer On Diffusion, firs derived he diffusion equion from he posules of moleculr heory, in which pricles move independenly under he influence of herml giion. In his picure, he concenrion of pricles c(r, ) some poin r is proporionl o he probbiliy P(r, ) of finding pricle here. Thus, he diffusion equion holds when probbiliies re subsiued for concenrions. If, for emple, pricle is iniilly plced he origin of coordines in d-dimensionl spce, hen is evoluion, ccording o equion 9, is given by r P( r, ) ep. () (4 pk) d/ 4k The men squred displcemen of he pricle is hus r () r P ( r,)d 3 r dk. () P (,) FIGURE. CONTINUOUS-TIME RANDOM WALKS (CTRWs) do no cover ground s quickly s simple rndom wlks. The blck lines indice specific relizion of simple rndom wlk (lef) nd CTRW (righ). Noe h CTRW seps occur very irregulrly; mos of he ime he wlker doesn move ll. As consequence, he men squre displcemen in CTRWs grows considerbly slower hn in simple rndom wlks. The blue nd he red curves indice he ypicl behvior of he displcemen: / } / for he simple rndom wlk nd / } /4 for CTRW wih power-lw prmeer (see e) /. The filled yellow curves show he probbiliy disribuions P(,) =. For regulr diffusion, which corresponds o he simple rndom wlk, he disribuion is Gussin. For diffusion governed by CTRWs, he disribuion sisfies he frcionl diffusion equion. Is chrcerisic enlike form displys cusp. Noe h he scling form r } follows direcly from he srucure of he diffusion equion. Th equion is second order in he spil coordines nd firs order in ime: Chnging he spil scle by fcor of 3 corresponds o chnging he ime scle by fcor of 9. In vriey of physicl sysems, however, he simple scling perinen o Fickin diffusion is violed. 3,4 The men squred displcemen grows s r } wih he eponen Þ. A consisen generlizion of he diffusion equion could sill be second order in he spil coordine nd hve frcionl-order emporl derivive for emple, ] () P( r, ) k ] P( r, ), where he dimension of he frcionl diffusion coefficien k is [L /T ]. Equion looks rher unusul. Does i mke ny sense? I does, s we now proceed o show. Coninuous-ime rndom wlks Rndom wlks nd diffusion serve s n inerfce beween kineics on one hnd nd derivives nd inegrls of frcionl order on he oher. The simples model leding o norml diffusion is he rndom wlk. Reled models were inroduced in Lord Ryleigh s sudies of isoperiodic vibrions (88) nd in Louis Bchelier s nlysis of sockmrke flucuions (9). If rndom wlker srolling in one dimension moves sep of lengh in eiher direcion precisely when ech uni ime elpses, hen he displcemen fer lrge number of seps (h is, fer long ime) will be disribued ccording o he Gussin, equion. In coninuous-ime rndom wlks (CTRWs) inroduced in physics by Ellio Monroll nd George Weiss he condiion h he seps occur fied imes is reled. 5 Rher, he ime inervls beween consecuive seps re governed by wiing-ime disribuion c(). In describing rnspor, he c() disribuions my sem from possible obscles nd rps h dely he pricle s jumps nd hus inroduce memory effecs ino he moion. If he men wiing ime beween consecuive seps is finie, c()d < F, he CTRW is described by Fick s diffusion equion, wih he diffusion coefficien k equl o /. The siuion chnges drsiclly if he men wiing ime diverges, s is he cse for power-lw wiing-ime disribuions of he form c( ) } /( / ), (3) wih < <. 6 (See he ricle by Hrvey Scher, Michel F. Shlesinger, nd John T. Bendler, PHYSICS TODAY, Jnury 99, pge 6.) Figure compres he displcemens of wlkers undergoing simple rndom wlks wih hose following CTRWs. The behvior for CTRW is subdiffusive. The men squre displcemen grows s () }. (4) Trnspor phenomen in sysems ehibiing subdiffusion hve <, wheres sysems h ehibi superdiffusion hve >. Frcionl modificions of he commonly used diffusion nd Fokker Plnck equions genere he scling behvior seen in subdiffusive sysems. Consider, for emple, he frcionl equion firs inroduced 5 NOVEMBER PHYSICS TODAY hp://

4 P(, =).8 P(, =.5).8 P(, =) P(, =4) FIGURE. PROBABILITY DISTRIBUTIONS EVOLVE s pricles governed by Fokker Plnck equions wih hrmonic poenils rel owrd Bolzmnn equilibrium. The grphs lef (wih posiion nd ime in rbirry unis) show he evoluion of he probbiliy disribuion for pricle collecion iniilly prepred. For boh he regulr (blue) nd / frcionl Fokker Plnck (red) equions, he disribuions sympoiclly pproch Gussins wih men vlue ending o zero. For he regulr equion, he disribuion is lwys Gussin. The behvior of he soluion in he frcionl cse is srikingly differen. I s no jus h he speed of relion is considerbly slower. The form of he disribuion is chrcerisic of subdiffusive sysems, showing cusp singulriy, he iniil vlue of, h remins visible even very long imes.. 3 by Venkrmn Blkrishnn nd by W. R. Schneider nd W. Wyss: 6 ] P (,) (5) ] D k P (,). Equion 5 cn be derived from he CTRW scheme long he lines used in Einsein s work on Brownin moion, 7 nd hus pplies o ll siuions discussed dequely by CTRW. I res sysems h behve nomlously in frmework very much like he frmework used for sysems wih norml diffusion, so h known soluions of he simple cse cn be esily generlized o he nomlous cse. 8 Frcionl Fokker Plnck equion The sndrd diffusion equion ccouns for pricle s moion due o uncorreled moleculr impcs. In mny cses, n eernl deerminisic force is imposed on sysem in ddiion o such rndom impcs. The Fokker Plnck equion considers boh conribuions. 9 I cn be derived by combining Fick s firs lw epressed in erms of probbiliy curren nd king ino ccoun he eernl force wih he coninuiy equion. The probbiliy curren is jr (, ) k P( r, ) mfr (, ) P( r, ), (6) where f is he eernl force cing on he pricle nd m is he pricle s mobiliy. When he coninuiy equion is pplied o he probbiliy curren, he Fokker Plnck equion follows: ]P(,) r ( mfp( r, ) k P( r, )). (7) ] In he bsence of n eernl force, f, he Fokker Plnck equion reduces o he sndrd diffusion equion. In prllel o he diffusion cse, one cn generlize he Fokker Plnck equion o. ]P( r, ) D ( mfp( r, ) k P( r, )), ] (8) where m is he frcionl mobiliy. In equilibrium, he curren j mus vnish. Afer epressing he force in erms of poenil 3 funcion U(r), one my wrie he equilibrium probbiliy disribuion h sisfies equion 8 in he form P(r) } ep( m U(r)/k ). Now, for independen pricles, he equilibrium probbiliy is Bolzmnn disribuion, so k /m k B T, generlizion of Einsein s relion k/m k B T. For emple, consider consn eernl force cing in he direcion. The force leds o men drif, () f m f / G( ), which is reled o he force-free men squre displcemen () f k / G( ) hrough () f () f f. (9) kt B The epression bove is he flucuion-dissipion heorem, which holds for subdiffusion in he frcionl Fokker Plnck frmework. Ornsein Uhlenbeck processes in one dimension provide second emple. Such processes involve diffusion in he hrmonic poenil U() b / so h he force is f b. The corresponding frcionl Fokker Plnck equion is ]P(, ) ] ] () ] D m b [ P(, )] k ] ] P (, ). Figure shows snpshos of he ime-dependen probbiliy disribuion for pricles originlly well-defined locion, ccording o boh he frcionl nd convenionl Fokker Plnck equions wih hrmonic poenils. In boh cses, he sympoic forms of he disribuion re Gussin, becuse he Fokker Plnck equions describe he relion of hermodynmic sysems o equilibrium. In he frcionl cse, hough, he disribuion finie imes hs cusp chrcerisic of brod CTRW disribuions. In hermodynmic pplicions, one is mosly ineresed in men vlues for emple, he men pricle posiion <()> P(, ) d. In he regulr cse of simple diffusion in he hrmonic poenil U(), here is chrcerisic ime given by bm. The evoluion of he men posiion sisfies d ( ) d (). () hp:// NOVEMBER PHYSICS TODAY 5

5 4 log E ( ( / / ) / ) 4 U log / 4 FIGURE 3. THE MITTAG LEFFLER FUNCTION describes he relion owrd equilibrium of pricles governed by he frcionl Fokker Plnck equion. For close o zero, he funcion behves like sreched eponenil, E ( (/) ) ep ( (/) ) /G( ). For lrge, he funcion pproches power-lw, E ( (/) ) (/) /G( ). The red curve shows he Mig Leffler funcion for /. The dshed lines show he shor-ime (blue) nd he long-ime (blck) forms. Noe he double logrihmic scles. sein Uhlenbeck equion o describe he dynmics of proein molecules probed by elecron rnsfer. One cn epec pplicions of he frcionl Fokker Plnck equion in chemicl nd biologicl sysems. As erly s 96, Mrin Smoluchowski showed how he res of chemicl recions cn be deermined by imposing boundry condiions on diffusion equions. Becuse he frcionl Fokker Plnck equion hndles boundry vlue problems in he sme wy s is regulr counerpr does, i is vluble ool for describing recions in comple sysems, s recenly repored by groups led by Kj Lindenberg he Universiy of Cliforni, Sn Diego, nd Bob Silbey he Msschuses Insiue of Technology. Mny environmenl sudies re compliced by poor undersnding of he diffusion of conminns in comple geologicl formions. Eperimens poin o nomlous diffusion. They sugges he need for frcionl diffusion dvecion equions, which my help scieniss o undersnd nd predic he long-erm impc of polluion on ecosysems. Deiled sudies of he frcionl Krmers problem represen noher possible ren for he pplicion of frcionl kineics. The problem concerns he escpe of pricle over poenil brrier. Firs seps owrd is so- I follows h () decys eponenilly owrd equilibrium: () () ep( /). In he cse of diffusion governed by he frcionl Fokker Plnck equion, he men displcemen obeys d ( ) D (), () d wih bm, s is redily verified by muliplicion of equion by nd inegrion, followed by n inegrion by prs of he righ-hnd side. The soluion of equion cn be epressed in erms of he Mig Leffler funcion E vi <()> <()> E ( (/) ). The Mig Leffler funcion, illusred in figure 3 for /, is nurl generlizion of he eponenil funcion; in priculr, E ( /) ep( /). Figure 4 compres he eponenil relion of he men posiion obined in he cse of norml diffusion wih he slower relion described by he Mig Leffler funcion. FIGURE 4. A PARTICLE DIFFUSING in hrmonic poenil eemplifies hermodynmic relion. For regulr diffusion, he men vlue of he pricle s posiion decys eponenilly wih ime (blue). In he frcionl, subdiffusive cse, he relion is described by Mig Leffler funcion. The priculr Mig Leffler funcion leding o he relion illusred in red hs s is frcionl prmeer /. Applicions The regulr Ornsein Uhlenbeck process is good model for he behvior of diffusing pricle rpped in opicl weezers: The weezers cree n pproimely hrmonic well. The corresponding relion perns disply eponenil decys, s shown by Roy Br-Ziv nd collegues from he Weizmnn Insiue of Science in Rehovo, Isrel. Pricles moving ccording o he frcionl Ornsein Uhlenbeck equion (equion ) should ehibi Mig Leffler relion. Rony Grnek, from Ben-Gurion Universiy, suggesed h such relion would be observed for beds ched o vesicle nd held by opicl weezers. Wler Glöckle nd Theo Nonnenmcher of he Universiy of Ulm, Germny, used Mig Leffler relion in heir nlyses of rheology in polymeric sysems nd of rebinding eperimens in proeins. This ps yer, group led by Hrvrd Universiy s Sunny Xie used he frcionl Orn- 5 NOVEMBER PHYSICS TODAY hp://

6 Bo. From Oliver Heviside o Sress Moduli of Polymers Frcionl derivives ofen emerge in he descripion of self-similr, hierrchiclly orgnized sysems. Consider discree model for rnslnic elegrph cble, s proposed by Lord Kelvin. The model consiss of idenicl resisnces R nd idenicl cpciors C, illusred in he figure righ. The response of he cble o volge V() pplied, sy, is righ end my be reled o he impednce Z of he sysem. The relion beween impednce, curren I, nd he volge is mos simply epressed in erms of he Fourier-rnsformed funcions: Z( w) I( w) V( w). Clculing Z(w) is sndrd problem in he heory of elecricl circuis. For he rnslnic cble model, he soluion is Z(w) R [iwc /Z(w)], or Z(w) R/ (R /4 R/iwC ) /. In he limi in which boh R nd C end o zero (boh depend on he subdivision lengh of he cble), bu he quoien R/C z sys consn, one obins Z(w) O z(iw) /. Hence I( w) V( w)/ Z( w) ( iw) V( w), where, for convenience, we hve se z. In erms of he corresponding funcions of ime, I() F D / V(), involving semi-differeniion, s poined ou by Oliver Heviside. The response of he cble o volge V() v()q() swiched on is given by I () D V(). As cn be confirmed by referring o equion 5, which defines he frcionl inegrl, he bove epression for he curren describes rerded response, I() d/d M( )V( )d, wih he memory funcion M() decying slowly, s /. Frcionl derivives, s he rnslnic cble model shows, re nurl ool for describing he liner response of sysems wih long, power-lw memory. The mechnicl equivlen of Kelvin s model, chin consising of springs nd beds immersed in viscous fluid, is he sndrd Rouse model in polymer dynmics. Th model, illusred below he rnslnic cble, ccouns for he flucuing forces due o solven molecules, nd for viscous fricion. If force f () f q() cs on one of he monomers of he chin, hen he men displcemen of he monomer sisfies <()> }, wih /. Inroducing hydrodynmic inercions forces medied by he solven leds o he Zimm model, for which he eponen in he men displcemen s ime evoluion is /3. Helmu Schiessel, Chrisin Friedrich, nd Ale Blumen obined similr scling lws when considering he dynmics of oher hierrchicl srucures, such s frcl neworks (see ref., p. 33). More compliced behvior rises when severl forces c / / simulneously for emple, when he monomers of nework re rndomly chrged nd eposed o n eernl volge. In such cses, one sill observes scling, bu wih vlues of h depend on he disribuion of he chrges on he polymer. Mny compliced sysems cn be modeled using frcionl differenil equions whose prmeers depend on. For emple, he figure below presens mesuremens for he mechnicl sorge (circles) nd mechnicl loss (squres) moduli for wo ehne-co--buenecopolymer compounds responding o hrmonic eernl srin field. The horizonl is gives he field s frequency nd he vericl is he resuling sress moduli, boh in ppropriely normlized unis. (For furher deils of he mesuremens by Chrisin Friedrich of he Universiy of Freiburg in Germny, see ref., p. 33.) The blck curves, derived from he so-clled frcionl Mwell model, fi he d well. STRESS MODULUS T = 3 C ref EB64 EB8 3 4 FREQUENCY G G G FMM G FMM C R luion hve been chieved by Rlf Mezler nd Yossi Klfer, boh working Tel Aviv Universiy. Frcionl clculus cn be pplied o mny res of physics, oher hn frcionl kineics. In fc, frcionl clculus ws inroduced in heurisic mnner long go in rheology (for emple, by Andrew Gemn in 936) in n effor o describe liner viscoelsiciy hrough he eension of he meril-dependen consiuive equions used in he field. 3 Bo discusses emples of hierrchicl srucures whose dynmics re convenienly described wih frcionl derivives. Mny oher emples of pplicions of frcionl clculus o modern physics re presened in reference. Wh bou superdiffusion? Frcionl generlizions of he diffusion nd Fokker Plnck equions hve been inroduced for superdiffusion s well. Those equions, which pply spil frcionl derivives rher hn emporl ones, re inimely reled o Lévy processes in spce. George Zslvsky hs dvoced using such equions o describe choic diffusion in Hmilonhp:// NOVEMBER PHYSICS TODAY 53

7 in sysems, nd oher pplicions hve been discussed by Hns Fogedby nd by Bruce Wes nd Polo Grigolini.,4 We hsen o noe, however, h superdiffusion is fr from being compleely undersood. The clssicl diffusion lws derived by Fick domined physiciss views on diffusion nd rnspor for more hn cenury. Bu recen observions hve clerly demonsred h Fick s lws hve ecepions. Those ecepions, which hve been ermed srnge kineics, 3 require compleely fresh view of kineic processes, bsed on rndomwlk pproches nd on unconvenionl disribuion funcions (see he ricle by Joseph Klfer, Michel F. Shlesinger, nd Ger Zumofen, PHYSICS TODAY, Februry 996, pge 33). Frcionl clculus helps formule he problems of srnge kineics in simple nd elegn wy. We hnk our collegues Eli Brki, Chrisin Friedrich, Rlf Mezler nd Helmu Schiessel for fruiful discussions. References. R. Hilfer, ed., Applicions of Frcionl Clculus in Physics, World Scienific, River Edge, N. J. ().. K. B. Oldhm, J. Spnier, The Frcionl Clculus, Acdemic Press, Sn Diego, Clif. (974); K. S. Miller, B. Ross, An Inroducion o he Frcionl Clculus nd Frcionl Differenil Equions, Wiley, New York (993). 3. M. F. Shlesinger, G. M. Zslvsky, J. Klfer, Nure 363, 3 (993). 4. J.-P. Bouchud, A. Georges, Phys. Rep. 95, 7 (99). 5. E. W. Monroll, G. H. Weiss, J. Mh. Phys., 753 (969); E. W. Monroll, M. F. Shlesinger, in Nonequilibrium Phenomen II: From Sochsics o Hydrodynmics, J. L. Lebowiz, E. W. Monroll, eds., Norh Hollnd, New York (984); E. W. Monroll, B. J. Wes, in Flucuion Phenomen, E. W. Monroll, J. L. Lebowiz, eds., Norh Hollnd, New York (987). 6. V. Blkrishnn, Physic A 3, 569 (985); W. R. Schneider, W. Wyss, J. Mh. Phys. 3, 34 (989). 7. R. Mezler, J. Klfer, Phys. Rep. 339, (). 8. E. Brki, Phys. Rev. E 63, 468 (); I. M. Sokolov, Phys. Rev. E 63, 56 (). 9. H. Risken, The Fokker Plnck Equion: Mehods of Soluion nd Applicions, Springer-Verlg, New York (996).. S. B. Yuse, K. Lindenberg, Phys. Rev. Le. 87, 83 (); J. Sung, E. Brki, R. J. Silbey, S. Lee, J. Chem. Phys. 6, 338 ().. J. W. Kirchner, X. Feng, C. Nel, Nure 43, 54 ().. R. Mezler, J. Klfer, I. M. Sokolov, Phys. Rev. E 58, 6 (998). 3. C. Friedrich, H. Schiessel, A. Blumen, in Advnces in he Flow nd Rheology of Non-Newonin Fluids, D. A. Siginer, D. De Kee, R. P. Chhbr, eds., Elsevier, New York (999), p G. M. Zslvsky, Chos 4, 5 (994); H. C. Fogedby, Phys. Rev. Le. 73, 57 (994). Righs & Permissions You my mke single copies of ricles or deprmens for prive use or for reserch. Auhorizion does no eend o sysemic or muliple reproducion, o copying for promoionl purposes, o elecronic sorge or disribuion (including on he Web), or o republicion in ny form. In ll such cses, you mus obin specific, wrien permission from he Americn Insiue of Physics. Conc he AIP Righs nd Permissions Office, Suie NO, Huningon Qudrngle, Melville, NY F: Telephone: E-mil: righs@ip.org Circle number 9 on Reder Service Crd 54 NOVEMBER PHYSICS TODAY