Drift-Controlled Anomalous Diffusion: A Solvable Gaussian Model Fabrizio Lillo and Rosario N. Mantegna Istituto Nazionale per la Fisica della Materia, Unità di Palermo and Dipartimento di Energetica ed Applicazioni di Fisica, Università di Palermo, Viale delle Scienze, I-90128, Palermo, Italia 0 0 0 WeintroduceaLangevinequationcharacterizedbyatimedependentdrift. Byassumingatemporal 2 power-law dependence of the drift we show that a great variety of behavior is observed in the n dynamics of the variance of the process. In particular diffusive, subdiffusive, superdiffusive and a stretched exponentially diffusive processes are described by this model for specific values of the J two control parameters. The model is also investigated in the presence of an external harmonic 4 potential. We prove that the relaxation to the stationary solution is power-law in time with an 1 exponent controlled by one of model parameters. ] h 02.50.Ey, 05.40.Jc, 05.70.Ln c e m Diffusive stochastic processes,i.e. stochastic processes anomalous regimes discussed above for Gaussian pro- - t x(t) characterized by a linear grows in time of the vari- cesses. Specifically, we study the properties of the class a t ance < x2(t) > t are quite common in physical sys- of Langevin equations s ∝ tems. However deviations from a diffusive process are . at observedinseveralstochasticsystems. Superdiffusive(< x˙ +γ(t)x=Γ(t), (1) m x2(t)> tν withν >1)andsubdiffusive ( tν withν <∝1)randomprocesseshavebeendetectedand∝in- where γ(t) is a function of time t and Γ(t) is a Langevin - d vestigated in physical and complex systems. A classical force with zero mean and with a correlation function n exampleofsuperdiffusiverandomprocessisRichardson’s given by < Γ(t2)Γ(t1) >= Dδ(t2 t1). Equation (1) − o describes an Ornstein-Uhlenbeck process [15] in the par- observationthattwoparticlesmovinginaturbulentfluid c ticular case of γ(t) = γ. This equation is linear and which at time t = 0 are originally placed very close the [ one with the other have a relative separation ℓ at time t solvable. For the sake of simplicity we set the boundary 1 that follows the relation < ℓ2(t) > t3 [1]. Most recent conditionof Eq. (1) att=0. The formalsolutionof Eq. v ∝ (1) is examplesincludeanomalouskineticsinchaoticdynamics 1 due to flights and trapping [2,3], anomalous diffusion in 1 t Γ(s) 2 aggregateofamphiphilicmolecules[4]andanomalousdif- x(t)=x(0)G(t)+G(t) ds, (2) 1 fusioninatwo-dimensionalrotatingflow[9]. Subdiffusive Z0 G(s) 0 stochastic processes have also been detected and investi- t 0 gated. Examplesincludeschargetransportinamorphous where G(t) exp[ γ(s)ds]. By using this formal so- 0 semiconductors[5,6]andthedynamicsofabeadinpoly- lution and a≡ll orde−r co0rrelation functions of Γ(t) we ob- t/ mers [7]. Another class of stochastic processeswhich are tain all central momRents of x(t). The first two central a notdiffusiveinasimplewayistheonecharacterizedbya moments are given by m variance with a stretched exponential time dependence. - When a such process is Gaussian distributed the prob- =x(0)G(t) µ(t), d ≡ on aKboihliltryauosfchretluawrnPt0o(tt)he oreixgpin[ Ptν0](tw)iitshdνesc=DG2(t) t G21(s)ds≡σ2(t). (3) c behaviors are observed∝in glas−sy systems and in random Z0 : v walks in ultrametric spaces [8]. The general relation between higher-order even central i The modeling of some of the above discussed anoma- moments and the second central moment of the investi- X lousdiffusingstochasticprocesseshasbeendonebyusing gated processes is the one observed in a Gaussian pro- r avarietyofapproaches. Tocitesomeexamples,werecall cess. Moreover odd central moments are zero hence we a that superdiffusive and subdiffusive processes have been conclude that the stochastic processes described by Eq. modeledbywritingdownageneralizeddiffusionequation (1) is Gaussian. [1,10,11],byintroducingLévywalksmodels[12],byusing We now consider the two-time correlationfunctions of a fractional Fokker-Planck equation approach [13] or by the process x(t) and of its time derivative x˙(t). In the using “ad hoc” stochastic models such as, for example, following we label the two times t1 and t2 of the corre- the fractional Brownianmotion [14]. lation functions in such a way that t2 t1. By using ≥ In this rapid communication we introduce a class of theformalsolutionofEq. (2)wedeterminethetwo-time Langevin equations able to describe all the different correlationfunction for the random variable x(t) 1 =µ(t1)µ(t2)+ G(t2)σ2(t1). (4) Specifically we focus on the asymptotic temporal evo- G(t1) lution of the variance and of the two-time correlation function of x˙(t). We recall that for a = 0 Eq. (1) de- Ingeneral,the correlationfunction isnot scribes a Wiener process with a variance increasing in a function of t2−t1 and therefore the process is usually a diffusive way, σ2(t) t, and a delta correlated x˙(t). non-stationary. ∼ When β = 0 Eq. (1) describes an Ornstein-Uhlenbeck By starting from the correlation function of x(t) and process and two regimes are observed depending on the from the formal solution of the Langevin equation we sign of a. When a > 0 the stochastic process has a sta- obtain the two-time correlation function of x˙(t) as tionary Gaussian solution, whereas when a < 0 there is =µv(t1)µv(t2)+Dδ(t2 t1) no stationarystate andthe varianceincreasesasymptot- − ically in an exponential way, σ(t) exp(2at) [15,16]. +γ(t2)G(t2)F(t1), (5) The two-time correlation of the velo∼city dec|re|ases in an where µ (t) = γ(t)µ(t) indicates the mean of the time exponential way as exp( a(t2 t1)). derivativve x˙(t)−and F(t1) γ(t1)σ2(t1) D /G(t1). The cases considered a−b|ov|e a−re known, in addition to The two-time correlationfun≡ction of x˙(t) is−the sum of a these cases we observe a large variety of new behaviors delta function and of a smooth(cid:2)function. (cid:3) controlled by the specific values of parameters a and β. The Fokker-Planck equation associated to the By investigating the (β,a) set of parameters, we detect Langevin equation given in Eq. (1) is different anomalous behavior that we discuss below sys- tematically by considering different regions of the β pa- ∂ρ ∂ D∂2ρ rameter. = (γ(t)xρ)+ . (6) ∂t ∂x 2 ∂x2 (i)Regionwithβ >1. Theprocessx(t)isdiffusiveand itsvarianceincreaseslinearlywithtimeforanyvalueofa. This Fokker-Planck equation is the same as the Smolu- Thetwo-timecorrelationfunctionofx˙(t)canbeobtained chowskiequationofaBrownianparticlemovinginahar- starting from Eq. (5). A direct calculation gives monicoscillatorwithatime dependentpotentialU(x) x2γ(t). Inourstudy weconsiderbothpositiveandnega∝- a at1−β tivevaluesofγ(t). Forpositivevaluesofγ(t)theposition exp − 2 F(t1). (9) x=0 is a stable equilibrium position whereas in the op- ∼ tβ2 1−β ! posite case x=0 is an unstable equilibrium position. Wecalculatethetwo-timeconditionalprobabilityden- The process x˙(t) can be positively or negatively corre- sity P(x2,t2 x1,t1) as the Green function of the Fokker- lated depending on the sign of a. When a > 0 (a < 0) | Planckequation. Inourformalismt1 t2. Ourdetermi- the correlation is negative (positive). This property is ≤ nation is done by working with the Fourier transform of valid for any value of β parameter. By investigating the P(x2,t2 x1,t1)with respectto the xvariable. The equa- explicitformofEq. (9)oneobservesthatthe correlation | tion for the Fourier transformof P(x2,t2 x1,t1) is a first function decreases as a function of t2 with a power-law order partial differential equation, which| can be solved dependence, 1/tβ2. ∼ by the methods of characteristics. We obtain (ii) Region with β = 1. In this case we observe two regimes. When a > 1/2 the variance increases in a P(x2,t2 x1,t1)= 1 diffusive way σ2(t) −t. We find a different behavior | 2πs2(t2,t1) when a < 1/2. In f∼act, by using Eq. (3) one can show (x2 m(pt2,t1)x1)2 that − exp − , (7) (cid:18)− 2s2(t2,t1) (cid:19) σ2(t) t2|a|. (10) ∼ where m(t2,t1) = exp t2γ(y)dy , and s2(t2,t1) = − t1 ThereforetheparticleperformsaGaussiansuperdiffusive D t2exp 2 t2γ(y)dyh dRz. Hence tihe transitionprob- random process. At the boundary value a = 1/2 the t1 − z varianceincreasesinalogdivergentwayasσ2(t)− tlogt. abRility ofhEq.R (7) isia Gaussian transition proba- The two-time correlation function of x˙(t) is det∼ermined bility. Moreover, Eq. (7) satisfies the Chapman- starting from Eq. (5). An explicit calculation gives Kolmogorov equation. In fact from a direct in- tegration one can verify that P(x3,t3 x1,t1) = a P(x3,t3|x2,t2)P(x2,t2|x1,t1)dx2. | ∼ t12+aF(t1). (11) Inthe restofthisrapidcommunicationwerestrictour R attention to the class of Langevin equations with a drift Thetwo-timecorrelationfunctionofx˙(t)showsapower- term which has a temporal behavior of the form law time dependence and the x˙(t) process is a strongly dependent random process [17]. We wish to point out γ(t) a/tβ (8) ∼ that when a = 1 the diffusion of the x(t) process is − for large time values. We study the stochastic process ballistic. Thisspecific casehasbeenalreadyinvestigated of Eq. (1) for different values of parameters a and β. by E. Nelson in the framework of stochastic mechanics. 2 TheItoequationdescribingthestochasticprocessassoci- The results obtained above refer to x(t) random pro- atedwiththefreeevolutionofaGaussianquantumwave cesses which are not stationary. We now consider the packet is [18] problem of a process x(t) whose dynamics is controlled by a modified version of Eq. (1) in which the effects of t c the presence of an “external” time-independent poten- dx(t)= − x dt+dw(t), (12) t2+c2 tial are taken into account. To this end we consider the specific case of a overdamped particle of mass M mov- wherew(t)isaWienerprocessandcisaconstant. This ing in a viscous medium in the presence of a potential stochastic equation describes the same random process having a time-dependence of the kind of Eq. (8) and a of Eq. (1) for large values of t. time-independent part. The equation of motion of such (iii) In the region 0 < β < 1 we observe two regimes, a system is which depend onthe sign ofa. When a>0 the variance increases as Mηx˙ +g(t)x F(x)=MΓ˜(t), (16) − σ2(t)∼tβ. (13) where η is the friction constantand Γ˜(t) is the Langevin force with diffusion constant 2ηk T/M. This equation Thisbehavioristhecustomarybehaviorobservedinsub- B is formally equivalent to diffusiverandomprocess. Thetwo-timecorrelationfunc- tion of x˙(t) behaves asymptotically as x˙ +γ(t)x+V′(x)=Γ(t), (17) aD a ∼− tβ exp −1 β(t12−β −t11−β) . w2kheTn/VM′(ηx.)T=he−pFri(mx)e/MinηV,(γx()t)in=dicga(tte)s/MspaηtiaanlddeDriva=- 2 (cid:18) − (cid:19) B tive. Itisworthpointingoutthatwhenγ(t)goestozero (14) as t increases (as, for example, in the case γ(t) a/tβ ∼ with β > 0) Eq. (17) might have a stationary solution. If the time interval τ t2 t1 is shorter than t1 ≡ − The presenceof astationarysolutiondepends onthe ex- [19] the power-law term dominates in this equation and act shape of V(x). the stochastic process is power-law anti-correlated. For Toinvestigateinaconcreteexampletherelaxationdy- τ >> t1 the two-time correlation function decreases ex- namicsoftheprobabilitydensityfunctionofx(t)towards ponentially. the stationary solution we study Eq. (17) in the pres- When a < 0 the variance increases as a stretched ex- enceofanexternalharmonicpotential,V(x)= 1kx2. In ponential 2 this case the process has a stationary state. A general 2a solution of Eq. (17) is found by using the substitution σ2(t) exp | | t1−β . (15) ∼ 1 β γ(t) γ(t)+k/Mη inEq. (2). In this casethe variance (cid:20) − (cid:21) of th→e process is equal to Since the process is Gaussian, the probability of return to the origin follows the Kohlrausch law, ρ(x(0),t) = σ2(t)=De−2kt/MηG2(t) t e2ks/Mηds. (18) 1/√2πσ(t) exp t1−β . This kind of anomalous dif- 0 G2(s) ∼ − Z fusion has been observedin glassesand in random walks on an ultrametric(cid:2)space(cid:3)[8]. The two-time correlation The asymptotic stationary value of σ2(t) is σs2t ≡ function of x˙(t) increases with time as Eq. (14). DMη/2k = kBT/k which is independent from the pa- (iv) Region β < 0. This region is essentially different rameters a and β. However, we observe a relaxation dy- from the previous ones because the absolute value of the namics whose functional form is controlledby the values drift term increases in time and eventually diverges. In of a and β parameters. To detect the different relax- this case we also observe two regimes depending on the ation dynamics we evaluate numerically the integral in sign of a. When a < 0 the time evolution of the vari- Eq. (18)bysettingγ(t)=a/(τβ+tβ). InFig. 1weshow ance is formally the same as Eq. (15) of case (iii). In inalog-logplotthequantity∆(t)≡|σ2(t)−σs2t|/σs2t asa this region of β parameter the variance increases more function of time. The quantity ∆(t) provides a measure than exponentially in time. When a > 0 we find that of the distance of the system from the stationary behav- the variance decreases with time with a power-law de- ior. In Fig. 1 we show that the quantity ∆(t) decreases pendence σ2(t) 1/t|β|. By using the Smoluchowski following the power-law behavior ∆(t) 1/tβ for large picture, we can i∼nterpret this behavior as the motion of values of t and for all the investigated v∝alues of the pa- Brownian particle moving in a time dependent potential rameters a and β. In particular, when a>0, σ2(t)−σs2t which leads to a localization of the particle in the point goes to zero as a negative value whereas when a < 0 x = 0. For both regimes the two-time correlation func- the same quantity goesto zerotakingpositive values. In tion of x˙(t) is given by Eq. (9). order to illustrate this result we show σ2(t) as a func- We summarize the above discussed variety of diffusive tion of time when β = 0.6 and a = 1 in the inset of ± behavior of σ2(t) in Table I. Fig. 1. For 0 < β 1 (therefore including subdiffusive, ≤ 3 superdiffusive and stretched exponentially diffusing pro- 422 (1968). cesses) it is not possible to define a characteristic time [15] G.E. Uhlenbeck and L.S. Ornstein, Phys. Rev. 36, 823 scale for the convergence of σ2(t) during the process of (1930). relaxation. This is due to the fact that the integral of [16] H.Risken,TheFokker-PlanckEquation,Springer-Verlag, ist∞1st∆ill(to)/b∆se(rtv1e)ddtw=he∞n.βA>lth1o,ugithias pwoowrtehr-lpaowinbtienhgavoiuotr [17] M(19.8C9a).ssandro and G. Jona-Lasinio, Adv. Phys. 27, 913 R (1978). that in this interval of β parameter a typical time scale [18] E. Nelson, Dynamical Theories of Brownian Motion, might be determined by considering the above discussed Princeton Univ.Press, Princeton (1967). integral which is finite in this region of β parameter. [19] M. S.Keshner, Proc. of the IEEE 70, 212 (1982). InconclusiontheLangevinequations(1)and(17)with thechoiceofEq. (8)describenon-stationaryandstation- ary random processes showing a wide class of (normal 100 and anomalous) diffusion. When a stationary state ex- ists, the relaxation dynamics to the stationary state has a power-law time dependence. The processes modeled 10−2 by Eqs (1) and (17) are characterized by a time dependent drift term in the associated Fokker-Planck equation. Our model is complementary to the Batchelor’s 10−4 descriptionofanomalousdiffusion obtainedby assuming) t ( a time-dependent diffusion term [10]. Equations (1) and∆ 1 (17) can be used to model metastable systems in which 10−6 oneofthephysicalobservables,suchas,forexample,the viscosity, is time dependent. They can also be used to 0.5 10−8 develop simple and efficient algorithms generating real- izations of random processes with controlled anomalous 0 diffusion. 10−10 0 2.5 5 7.5 10 The authors thank INFM and MURST for financial 100 102 104 106 t support. F.LilloacknowledgesFSE-INFMforhisfellow- ship. FIG.1. Theoreticalestimationofthenormalizeddifference ofthevarianceattimetfromthestationaryvalueσ2,∆(t)as st afunctionoftime. Differentcurvesrefertodifferentvaluesof thecontrolparametersaandβ. Theavaluesarea=1(solid lines) and a = −1 (dashed lines). The parameter β assumes thevalues0.2,0.6,1.0,1.4and1.8fromtoptobottom. Inthe 2 inset we show a typical time evolution of σ (t) obtained by [1] L.F. Richardson, Proc. R. Soc. London Ser. A 110, 709 setting β = 0.6 and two different values of a, a= 1 (bottom (1926). curve) and a = −1 (top curve). The other parameters are [2] T. Geisel, J. Nierwetberg, and A. Zacherl, Phys. Rev. τ =1, k/Mη=1 and D=1 Lett.54, 616 (1985). [3] M.F.Shlesinger, G.M. Zaslavsky, and J. Klafter, Nature 363, 31 (1993). TABLEI. Summaryofthedifferentdiffusionregimes. The [4] A. Ott, J.-P. Bouchaud, D. Langevin, and W. Urbach, constant C ≡2|a|/(1−β) Phys.Rev.Lett. 65, 2201 (1990). [5] H.Scherand E. Montroll, Phys. Rev.B12, 2455 (1975). β a σ2(t) Description [6] Q.Gu, et al., Phys.Rev.Lett. 76, 3196 (1996). β a=0 t Wiener (diffusive) [7] F. Amblard,et al., Phys.Rev.Lett. 77, 4470 (1996). β >1 a>0 t diffusive [8] A.T.OgielskiandD.L.Stein,Phys.Rev.Lett.55,1634 β >1 a<0 t diffusive (1985). β =1 a>−1/2 t diffusive [9] T.H.Solomon,E.R.Weeks,andH.L.Swinney,Phys.Rev. β =1 a=−1/2 tlogt log divergent Lett.71, 3975 (1993). β =1 a<−1/2 t2|a| superdiffusive [10] G.K. Batchelor, Proc. Cambridge Philos. Soc. 48, 345 0<β<1 a>0 tβ subdiffusive (1952). 0<β<1 a<0 exp[Ct1−β] less than [11] H.G.E.HentschelandI.Procaccia,Phys.Rev.A29,1461 exponentially diffusive (1984). β =0 a>0 1−exp(−2at) Ornstein-Uhlenbeck [12] M.F. Shlesinger, J. Klafter, and Y.M. Wong, J. Stat. β =0 a<0 exp(2|a|t) exponentially diffusive Phys.27, 499 (1982). β <0 a>0 1/t|β| localized [13] R. Metzler, E. Barkai, and J. Klafter, Phys. Rev. Lett. β <0 a<0 exp[Ct1−β] more than 82, 3563 (1999). exponentially diffusive [14] B.B.MandelbrotandJ.W.vanNess,SIAMReview10, 4

Drift-Controlled Anomalous Diffusion: A Solvable Gaussian Model

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