Limit theorems for L´evy walks in d dimensions: rare and bulk fluctuations ∗ Itzhak Fouxon1,2,† Sergey Denisov3,4,‡ Vasily Zaburdaev3,5,§ and Eli Barkai1¶ 1 Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 52900, Israel 2 Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, South Korea 3 Institute of Supercomputing Technologies, Lobachevsky State University of Nizhny Novgorod, Gagarina Av. 23, Nizhny Novgorod, 603140, Russia 4 Sumy State University, Rimsky-Korsakov Street 2, 40007 Sumy, Ukraine and 5 Max Planck Institute for the Physics of Complex Systems, No¨thnitzer Strasse 38, D-01187 Dresden, Germany We consider super-diffusive L´evy walks in d (cid:62) 2 dimensions when the duration of a single step, 7 i.e., a ballistic motion performed by a walker, is governed by a power-law tailed distribution of 1 0 infinite variance and finite mean. We demonstrate that the probability density function (PDF) 2 of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the d−dimensional generalization of the one-dimensional L´evy n distribution and is the counterpart of central limit theorem (CLT) for random walks with finite a dispersion. In contrast with the one-dimensional L´evy distribution and the CLT this distribution J does not have universal shape. The PDF reflects anisotropy of the single-step statistics however 8 large the time is. The other scaling limit, the so-called ’infinite density’, describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats ] h the angular structure of PDF of velocity in one step. Typical realization of the walk consists of c anomalousdiffusivemotion(describedbyanisotropicd−dimensionalL´evydistribution)intermitted e by long ballistic flights (described by infinite density). The long flights are rare but due to them m the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of L´evy walks, with isotropic and anisotropic distributions of - t velocities. Furthermore,weshowthatforisotropicbutotherwisearbitraryvelocitydistributionthe a d−dimensional process can be reduced to one-dimensional L´evy walk. t s . t a I. INTRODUCTION famous example is L´evy walks (LWs) [3–27] that belong m toamoregeneralclassofstochasticprocessescalledcon- - tinuous time random walks (CTRWs) [3, 4]. In CTRWs d Itisauniversalconsequenceofmicroscopicchaosthat itisnotonlythedirectionofthedisplacementduringone n o the velocity v(t) of a moving particle has a finite corre- stepthatisrandom(asinthestandardrandomwalk)but c lation time [1, 2]. The particle’s displacement (cid:82)tv(t(cid:48))dt(cid:48) alsothelengthofthestepandthetimethatittakes. This 0 [ onlargetimescalescanbeconsideredasanoutcomeofa flexibility allows to cover a large number of real-life sit- sumofindependentrandom’steps’. Asinglestephereis uations including dynamics of an ordinary gas molecule 1 v a motion during the shortest time over which the veloc- when both the time between consecutive collisions and 7 ity of the particle is correlated. Velocity correlations at velocityofthemoleculevary. Inidealgastheprobability 5 different steps can be neglected and the integral over the of large (much larger than the mean free time) time be- 9 time interval [0,t] can be replaced with the sum of inte- tween the collisions is negligibly small so the probability 1 grals over disjoint steps. If variations of step durations density function (PDF) of the step duration decays fast 0 andvelocityfluctuationscanbeneglectedaswell, wear- forlargearguments. Thisisnotthecasefortheso-called . 1 riveatthestandardrandomwalkconsistingofstepsthat Lorentz billiard [2], in which the particle moves freely 0 take the same fixed time. The distance covered during between collisions with scatterers arranged in a spatially 7 singlestepisfixedbutthedirectionofthestepisrandom. periodic array. In the case without horizon, when in- 1 It can be, for example, a step along one of the basis vec- finitelylongcorridorsbetweenthescatterersarepresent, : v torswhenrandomwalkisperformedonad−dimensional the particle can fly freely for a very long time if its ve- Xi lattice or it can be a step in a completely random direc- locity vector aligns close to the direction of the corridor. tion in d−dimensional space as in the case of isotropic The distribution of times between consecutive collisions r a walk. In many situations though the variation of the has a power-law tail and infinite variance [28]. The dy- step’s duration and velocity cannot be disregarded. A namics of a particle can be reproduced with a L´evy walk processtogreatdetail[29,30]. L´evywalkswerefoundin diverse real-life processes including the spreading of cold atoms in optical lattices, animal foraging, and diffusion ∗SubmittedtoSmoluchowski’sspecialissueGudowska-Novak,Lin- of light in disordered glasses and hot atomic vapors (for denberg,Metzler,Editors. more examples see a recent review [20] and references †Electronicaddress: [email protected] therein). Despite of these advances and new experimen- ‡Electronicaddress: [email protected] tal findings, the theory of LWs remains mainly confined §Electronicaddress: [email protected] ¶Electronicaddress: [email protected] to the case of one-dimensional geometry. 2 Inthisworkwestudyd−dimensionalL´evywalkswhen startingfromorderαbetween1and2thereisfinitelimit, the duration τ of single steps is characterized by a PDF ψ(τ) with power-law asymptotic form, ψ(τ) ∝ τ−1−α. lim td/αP(t1/αx,t) t→∞ The most interesting and practically relevant is the so- (cid:90) (cid:20) A (cid:12) (cid:16)πα(cid:17)(cid:12) (cid:21) dk called ’sub-ballistic super-diffusive regime’ [20], 1<α< = exp ik·x− (cid:12)cos (cid:12)(cid:104)|k·v|α(cid:105) . (3) (cid:104)τ(cid:105)(cid:12) 2 (cid:12) (2π)d 2,whenthesteptimehasfinitemeanbutinfinitevariance [31]. We study the PDF P(x,t) of the walker’s coordi- This holds for arbitrary statistics of particle’s veloc- nate x(t) at time t in this regime. We show that there ity that has finite moments and obeys F(v) = F(−v). aretwowaysofrescalingP(x,t)withpowersoftimethat We demonstrate that this result smoothly connects with produce finite infinite time limits. The two limiting dis- the usual central limit theorem given by Eq. (1). In- tributionsdescribethebulkandthetailsofP(x,t). Both deed Eq. (3) reproduces Gaussian distribution at α=2. distributionsaresensitivetothemicroscopicstatisticsof Thus our result includes the usual central limit theorem the velocity of walkers and are model specific. as particular case and can be called generalized central We consider a random walk in Rd with a PDF F(v) limit theorem. We clarify in Sec. VI that Eq. (3) can of the single step velocity that obeys F(−v)=F(v), cf. be obtained from distribution of sum of many indepen- [32, 33]. Thus the process is unbiased and the average dentidenticallydistributedscalarrandomvariableswith displacement is zero. In the case of the sum of a large power-law tailed distributions, cf. [8, 12, 39–42]. number of independent and identically-distributed (i. i. ThelimitingdistributiongivenbyEq.(3)hasfeatures d.) random variables with finite dispersion there is a that distinguish it from both the one-dimensional LW well-known scaling, and d−dimensional random walks. In those cases there (cid:16) √ (cid:17) is universality: the form of the scale-invariant PDF does lim td/2P x t,t =g(x), (1) not depend on details of the single step statistics (for t→∞ instance, ordinary two-dimensional random walks on tri- where g(x) is a Gaussian distribution [34]. A stronger, angular and square lattices are described by the same large deviation limit tells that isotropic Gaussian PDF in the limit of large times [43]). In the case of d−dimensional LWs the anisotropy of the 1 singlestepstatisticsisimprintedintothestatisticsofthe lim lnP(tx,t)=s(x), (2) t→∞ t displacement–nomatterhowlargetheobservationtime t is; see the discussion of the particular case of d = 2 in where the convex function s(x) is known as large devia- Ref. [22]. tions, or entropy, or rate, or Kramer’s function [35, 36]; Inthelanguageoffieldtheory, ordinaryrandomwalks see [37] for simple derivation. This limit describes ex- are renormalizable: in the long-time limit the informa- ponential decay of the probability of large deviations of tion on macroscopic structure of the walks is reduced to finite-time value x(t)/t from its infinite time limit fixed finite number of constants [43]. Thus the Gaussian PDF by the law of large numbers, lim (1/t)x(t) = 0. It ofthesumoflargenumberofi. i. d. randomvariablesis t→∞ corresponds to the Boltzmann formula and tells that fullydeterminedbythemeanandthedispersionofthose the PDF of macroscopic thermodynamic variable (rep- variables and their total number. The rest of the infor- resentable as the sum of large number of independent mation on the statistics of these variables is irrelevant – random variables) is exponential of the entropy whose whentheGaussianbulkofthePDFisaddressed. Incon- maximum’s location gives the average. The coefficients trast, the entropy function of the large deviations theory ofthequadraticexpansionnearthemaximumcharacter- istheLegendretransformofthelogarithmofthecharac- ize thermodynamic fluctuations [36]. Thus in the case of teristicfunctionoftherandomvariableinthesum. Thus finite dispersion there is one universal scaling with the large deviations description is sensitive to the details of limiting distribution (the special case of finite dispersion statistics of one step of the walk. We conclude that the but power-law tail, α > 2, produces different limiting Gaussian bulk of the PDF of the sum of large number of distribution [38]; we do not consider this case here). i. i. d. random variables with finite dispersion is deter- Therearetwodifferentscalingsthatwerefoundinthe mined by a finite number of constants characterizing the caseofone-dimensionalsuper-diffusiveLWs[17,18]. One statisticsofthesinglestepbutthetailofthePDFisnot. of them is a continuation of the central limit theorem In this work we show that the PDF of a d - dimen- (CLT) to the case of i. i. d. random variables with in- sionalLWisnotuniversalandcannotbespecifiedwitha finite dispersion, the so-called generalized central limit finite number of constants – already in the bulk – if the theorem (gCLT) [39]. The corresponding distribution is statisticsofthevelocityofsinglesteps,givenbythePDF known as the celebrated L´evy distribution. We first gen- F(v), isanisotropic. Tostressthisfactwecallthecorre- eralize this scaling to the d−dimensional case and find sponding distributions described by Eq. (3) ’anisotropic significant difference from the one-dimensional case and L´evy distributions’. In contrast, if the velocity statistics ordinaryd−dimensionalrandomwalks. Ourresultshows is isotropic, for example, F(v) is given by the uniform thatford−dimensionalL´evywalkswithpowerlawtailed distribution on the surface of the d-dimensional sphere distributionofstepdurationτ thathasinfinitemoments of the radius υ =const (see Fig. 1a), the description of 0 3 FIG.1: Examples of L´evy walks in three dimensions. Awalkerhasaconstant(byabsolutevalue)velocityυ =|v|=1. 0 Whenperforminguniform L´evywalk(a),thewalker,aftercompletingaballisticflight,instantaneouslyselectsarandomtime τ for a new flight and randomly chooses a flight direction (it can be specified by a point on the surface of the unit sphere). ThevelocityPDFF(v)isdescribedbytheuniformdistributionovertheunitsphere’ssurface. InthecaseofanisotropicXYZ L´evywalk(b), thewalkerisallowedtomoveonlyalongoneaxisatatime. Afterthecompletionofaflight, thewalkerselects arandomtimeτ foranewflight,oneoutofsixdirections(withequalprobability)andthenmovesalongchosendirection. The velocity PDF F(v) in this case is six delta-like distributions located at the points where the unit sphere is penetrated by the frame axes. The parameter α is 3/2. In our simulations we used ψ(τ)=(3/2)τ−5/2 for τ >1 otherwise it is zero. the process can be reduced to one-dimensional case and τ. This distribution describes the tail of the PDF of the the bulk of the corresponding PDF is fully determined particle’s coordinate. In this sense it is the counterpart by a finite number of constants which can be calculated of the large deviations result for ordinary random walk from F(v). Thus in the anisotropic case there is a dra- given by Eq. (2). There is however significant difference: matic difference between the bulk of the PDFs of a LW the large deviations function describes averages of high- and the random walk with finite dispersion of the single order moments but in the case of L´evy walks the infinite step duration. density provides already dispersion of the process. This We demonstrate that besides the limiting distribution can be seen observing that as we demonstrate the distri- given by Eq. (3) there is another limiting distribution butionprovidedbyEq.(3)haspower-lawtailwithdiver- which is determined by gentsecondmoment. ThisisbecauseforL´evywalksrare eventswhenthewalkerperformsextremelylongballistic P(tv,t) A flights have a substantial impact on the total displace- lim = t→∞t1−d−α vd−1|Γ(1−α)|(cid:104)τ(cid:105) ment of the walker even in the limit of long times. The (cid:90) (cid:20) v(cid:48)α v(cid:48)α−1(cid:21) probability of long ballistic steps is not negligibly small × F(v(cid:48)vˆ)v(cid:48)d−1dv(cid:48) α −(α−1) , (4) and single ballistic steps could be discerned in a single v1+α vα v(cid:48)>v trajectory of the walker for any time t. Such steps form where v = x/t is the effective velocity of the particle, the outer regions of the PDF. v = vvˆ and the limit exists because the tail of the PDF Thus the two limit distributions describe the bulk and is determined by ballistic-type events. This distribution the tail of the LW’s PDF, respectively. The bulk is is called infinite density where the word ’infinite’ refers formed by the accumulation of typical (most probable) to the non-normalizable character of this function found steps. Theyareresponsibleforadiffusivemotion(albeit previously in one-dimensional case [17, 18, 44, 45]. This already anomalous one). In contrast, the PDF’s tails are pointwise limit holds for v (cid:54)= 0 non-contradicting nor- formedbylongballisticflightsandtheyaredescribedby malization of the PDF: in this limiting procedure the theinfinitedensity. Theseflightsarerarestepswherethe normalization is carried by v = 0 point. The existence walker moves for a long (i. e., comparable to the total oftheotherscalinglimitisuniquepropertythathasori- observation time t) time without changing its velocity. gins in the scale-invariance of the tail of distribution of In the case of the Lorentz billiard this is the situation 4 when the velocity vector of the particle aligns close to Weconsidertwointuitivemodels,theuniformLWand thedirectionofoneoftheballisticcorridors[29]. Though anisotropic XYZ... LW. In the uniform model F(v) is thishappensrelativelyrarely,thedistancecoveredbythe specified by the uniform distribution on the surface of walker during a such flight is so large that these flights d−dimensional unit sphere so velocity has fixed mag- givefinite contributiontotheprobabilitythatthewalker nitude 1; see Fig. 1a. As we demonstrate in the next displacement after the time t is of the order v t. When section, in many respects this model can be reduced to 0 the probability of long flights is, for example, exponen- the one-dimensional case. In the anisotropic XYZ... tiallysmall(asinthecaseofthestandardrandomwalks) model particle moves along one of the d basis vectors the contribution of the flights can be neglected. This is at a time; see Fig. 1b. The analysis we present be- not,however,thecaseoftheLWswithpower-lawasymp- low is valid for any PDF F(v) obeying the symmetry totic of ψ(τ), as we demonstrate in this paper. F(−v) = F(v). As an illustration, we consider a par- The paper is organized as follows. In Section II we in- ticular type of LWs in Rd with factorized velocity distri- troduce the basic definitions and the tool of the study - bution F(v) = F (|v|)·F (v/|v|). In this product PDF v d theFourier-LaplacetransformofthePDFofthewalker’s firstmultipliercontrolstheabsolutevalueofthevelocity coordinate. In the next Section we provide complete so- [thesimplestchoicesisF (|v|)=δ(|v|−v )]whilesecond v 0 lutionforthecaseofisotropicstatisticsofvelocityofthe multiplier is governs the direction statistics of steps. A walker. Central result of our work - the anisotropic CLT PDF F (v/|v|) is a subject of directional statistics [33] d for the bulk of the PDF is derived in Section IV. The and can be specified with a probability distribution on nextSectiondescribesuniversalityofthetailofthisnon- the surface of the (d−1) dimensional unit sphere in Rd. universalbulkthathelpsfindinglowmomentsofthedis- For example, in R3 the continuous transition from the tancefromtheorigin. SectionVIIprovidestheotherlim- isotropic model to the XYZ LW can be realized with iting theorem on infinite density that provides the tail of six von Mises-Fisher distributions [33] (centered at the thedistribution. ThenextSectionprovidesdetailedform points where the axes pierce the unit sphere) by tun- of the moments of arbitrary order including anomaly in ingtheconcentrationparameterofthedistributionsfrom growth due to anisotropy. In Section IX we provide the zerotoinfinity. Wedemonstrateinthefollowingsections tail of the PDF and Conclusions resume our work. that different statistics enter the PDF P(x,t) through the moments (cid:104)(k ·v)2n(cid:105). In particular, for the XYZ.. model we have II. FOURIER-LAPLACE TRANSFORM OF THE PDF P(x,t) (cid:104)(k·v)2n(cid:105)= (cid:80)di=1ki2n (v )2n. (6) d 0 In this Section we specify the considered random pro- In [22] we discuss physical models belonging to different cessandintroducethemaintooloftheanalysisonwhich classes of symmetry, e. g. the Lorentz gas with infinite all further results rest. This is the Fourier (in space) - horizon belongs to the XYZ.. class. Laplace transform (in time) of P(x,t). The remaining PDF that defines the walk process is We consider a LW as an infinite sequence of flights ψ(τ). Below we consider ψ(τ) that has the tail (steps) of random duration τ where i is the flight’s in- i dexinthechronologicallyorderedsequence. Theprocess A ψ(τ)∼ τ−1−α, 1<α<2, (7) starts at time t = 0 at the point x = 0. The velocity of Γ(−α) thewalkerduringaflightisarandomvectorv whichre- i where A > 0 and Γ(x) is the gamma function (observe mainsconstantduringtheflight. Uponthecompletionof that Γ(−α) > 0 when 1 < α < 2). The factor Γ(−α) is theflightboththevelocityv andthedurationτ of i+1 i+1 introduced in order to make the Laplace transform ψ(u) thenextflightarerandomlychosen,byusingPDFsF(v) of ψ(τ) and ψ(τ), respectively. The number of flights performed duringtheobservationtimet, N(t), isarandomnumber (cid:90) ∞ constrained by t = (cid:80)N(t)τ +τ , where τ = t−t is ψ(u)= exp[−uτ]ψ(τ)dτ, (8) i=1 i b b N so-called the backward recurrence time [14]. The time t 0 coordinate of the particle is, to have small u behavior [that is determined by the tail of ψ(τ)],   N(t) N(t) (cid:88) (cid:88) x(t)= viτi+vN(t)+1t− τi. (5) ψ(u)=1−(cid:104)τ(cid:105)u+Auα+..., (9) i=1 i=1 (cid:82)∞ where (cid:104)τ(cid:105) = tψ(t)dt is the average waiting time and 0 The simplest model is d−dimensional ’L´evy plotter’, the dots stand for higher-order terms. product of d independent one-dimensional walks along We use k and u to denote coordinates in Fourier and the basis vectors which span Rd. The PDF of this pro- Laplace space, respectively. By explicitly providing the cess is the product of the corresponding one-dimensional argument of a function, we will distinguish between the PDFs [17, 18, 20]. This case demands no further calcu- normal or transformed space, for example ψ(τ) → ψ(u) lations so we next consider non-trivial set-ups. and g(x)→g(k). 5 Thelowerlimitofτ forwhichEq.(7)holdsdependson whereF(v )isthePDFofx−componentofthevelocity. x the considered model. For instance the inverse gamma It can be written in terms of the PDF F(v) = F(v) = PDF, F (v/|v|) which obeys the normalization, d (cid:90) (cid:90) ∞ 2τ−5/2 (cid:20) 1(cid:21) F(v)dv =S vd−1F(v)dv =1. (17) ψ(τ)= √ exp − , (10) d−1 π τ 0 For d>2 we have, thetaildescribedbyEq.(7)holdsatτ (cid:29)1withα=3/2 (cid:82) δ(v −v(cid:48))F(v(cid:48))dv(cid:48) and A=8/3. The corresponding Laplace pair obeys, F(vx)= x(cid:82) F(vx(cid:48))dv(cid:48) (cid:2) √ (cid:3) (cid:2) √ (cid:3) 8u3/2 (cid:82)∞ (v(cid:48))d−1F(v(cid:48))dv(cid:48)(cid:82)πδ(v −v(cid:48)cosθ)sind−2θdθ ψ(u)= 1+2 u exp −2 u ∼1−2u+ 3 ,(11) = |vx| (cid:82)∞(v(cid:48))d−1F(v0(cid:48))dv(cid:48)x(cid:82)πsind−2θdθ 0 0 that reproduces Eq. (9) where we use (cid:82)∞ (v(cid:48))d−1F(v(cid:48))dv(cid:48)(cid:82)1 δ(v −v(cid:48)x)(1−x2)(d−3)/2dx = |vx| −1 x (cid:90) ∞ 2τ−3/2dτ (cid:20) 1(cid:21) 2Γ(1/2) S−1 (cid:82)1 (1−x2)(d−3)/2dx (cid:104)τ(cid:105)= √ exp − = √ =2. (12) d−1 −1 0 π τ π 2π(d−1)/2 (cid:90) ∞ (cid:18) v2(cid:19)(d−3)/2 = vd−2 1− x F(v)dv. (18) We introduce the key instrument of our analysis, the Γ[(d−1)/2] |vx| v2 Montroll-Weiss equation. It provides with the Laplace √ where we used (cid:82)1 (1 − x2)(d−3)/2dx = πΓ[(d − transform −1 1)/2]/Γ(d/2) and Eq. (17). In the case of two dimen- (cid:90) ∞ sionsθ variesbetween0and2π notπ butthecalculation P(k,u)= exp[−ut]P(k,t)dt, (13) still holds. Thus Eq. (18) provides the distribution of 0 x−component of velocity in arbitrary space dimension of the characteristic function of the position x(t) of the d>1. random walker at time t, Equation (18) can be simplified further in the case of uniform model with velocity v where F(v) = (cid:90) 0 P(k,t)=(cid:104)exp[ik·x(t)](cid:105)= exp[ik·x]P(x,t)dx,(14) v01−dSd−−11δ(v−v0). Integration in Eq. (18) gives F(v )=PS (v ), x d/2−2,v0 x in terms of averages over statistics of v and τ. We have wherePS (v)isthe(normalized)powersemicircle d/2−2,v0 (cid:28)1−ψ(u−ik·v)(cid:29) 1 PDF with range v0 and shape parameter d/2−2 that P(k,u)= ,(15) vanishes when |v|>v and for |v|<v is given by [46] u−ik·v 1−(cid:104)ψ(u−ik·v)(cid:105) 0 0 Γ(d/2) (cid:18) v2(cid:19)(d−3)/2 see Appendix A. Here the angular brackets denote the PS (v)= √ 1− (1.9) d/2−2,v0 πv Γ[(d−1)/2] v2 averaging over the PDF F(v). The technical problem is 0 0 to invert this formula in the limit of large time. The moments of this distribution read vγΓ[(γ+1)/2]Γ(d/2) (cid:104)|v |γ(cid:105)= 0 √ . (20) x πΓ[(d+γ)/2] III. ISOTROPIC MODEL Note that the ratio of gamma functions can be rewrit- In this Section we consider isotropic statistics of ve- ten as a product if d is an odd number. Finally, from locity where F(v) depends on |v| only. The magnitude this PDF we can derive the PDF and the moments for of velocity is a random variable drawn from the PDF arbitrary F(v). For the PDF we find from Eqs. (18) and S vd−1F(v) where S = 2πd/2/Γ(d/2) is the area (19), d−1 d−1 of unit sphere in d dimensions (2π in d = 2 and 4π in 2πd/2 (cid:90) ∞ d = 3). We show that the PDF P(x,t) of a LW in Rd F(v )= vd−1PS (v)F(v)dv,(21) x Γ(d/2) d/2−2,vx can be derived from one-dimensional distribution. Thus |vx| thewell-developedtheoryofone-dimensionalLWscanbe that can also be seen directly from the definition. For used for describing d-dimensional isotropic LWs. themoments, interchangingtheorderofintegrations, we Westartwithobservationthatforisotropicstatisticsof obtain from Eq. (18) the identity v, theaverageofanarbitraryfunctionhofk·v depends only on |k|; thus (cid:104)h(k ·v)(cid:105) can be obtained by taking (cid:90) ∞ (cid:20) 2π(d−1)/2 (cid:104)|v |γ(cid:105)= vd−1S F(v )dv k=kxˆ (where xˆ is unit vector in x−direction), x 0 d−1 0 0 Γ[(d−1)/2] 0 (cid:90) ∞ (cid:90) (cid:90) ∞ (cid:18) v2(cid:19)(d−3)/2 δ(v−v ) (cid:35) (cid:104)h(k·v)(cid:105)=(cid:104)h(kvx)(cid:105)= h(kvx)F(vx)dvx,(16) |vx|γdvx vd−2 1− vx2 vd−1S 0 dv .(22) −∞ |vx| 0 d−1 6 By noting that the term in brackets is the corresponding We can write K in terms of (cid:104)|v|α(cid:105) using Eq. (25), α moment of PS , we find d/2−2,v0 AΓ[(α+1)/2]Γ(d/2) (cid:12) (cid:16)πα(cid:17)(cid:12) K = √ (cid:104)|v|α(cid:105)(cid:12)cos (cid:12). (30) 2π(d−1)/2Γ[(γ+1)/2](cid:90) ∞ α (cid:104)τ(cid:105)Γ[(d+α)/2] π (cid:12) 2 (cid:12) (cid:104)|v |γ(cid:105)= vd+γ−1F(v)dv. (23) x Γ[(d+γ)/2] 0 Thiscoefficientreducestothediffusioncoefficientofone- dimensional walk found in [17] setting d=1. In the case For the Gaussian distribution F(v) = (2πv˜2)−d/2exp(cid:2)−v2/2v˜2(cid:3) it gives where v is a conserved constant v0 (modelling conserva- 0 0 tion of energy), we find 2γ/2Γ[(γ+1)/2]v˜γ (cid:104)|v |γ(cid:105)= √ 0, (24) AΓ[(α+1)/2]Γ(d/2) (cid:12) (cid:16)πα(cid:17)(cid:12) x π Kα = (cid:104)τ(cid:105)Γ[(d+α)/2]√π v0α(cid:12)(cid:12)cos 2 (cid:12)(cid:12). (31) that reproduces (cid:104)v2(cid:105)=v˜2 when γ =2. We observe that (cid:104)|v|γ(cid:105)=S (cid:82)∞vxd+γ−1F0(v)dv so that Eq. (23) implies UsingtheinverseFouriertransformwefindforthePDF’s d−1 0 bulk, the identity (cid:18) (cid:19) 1 x Γ[(γ+1)/2]Γ(d/2) P (x,t)∼ L , (32) (cid:104)|v |γ(cid:105)= √ (cid:104)|v|γ(cid:105). (25) cen (K t)d/α d (K t)1/α x Γ[(d+γ)/2] π α α (cid:90) dk L (x)= exp[ik·x−kα] . (33) This formula is a consequence of the isotropy of the pro- d (2π)d cess and for any random vector x whose PDF depends These formulas provide generalized CLT for on |x| only we have, d−dimensional isotropic LWs. Below these will be Γ[(γ+1)/2]Γ(d/2) generalized to the case of arbitrary (not necessarily (cid:104)|x |γ(cid:105)= √ (cid:104)|x|γ(cid:105), (26) s Γ[(d+γ)/2] π isotropic) statistics of velocity. We provide the form of L (x) in the cases of physical interest d = 2 and d = 3. d where x , s ∈ {1,2,...,d}, is one of the Cartesian coor- In the two-dimensional case we have, s dinates of x. (cid:90) ∞ dk L (x)= kJ (kx)exp[−kα] , (34) 2 0 2π 0 A. Bulk statistics: d−dimensional L´evy that can be called two-dimensional isotropic L´evy den- distributions sity. Here J (z) is the Bessel function of the first 0 kind. In three dimensions we find (L (x) is normalized We use Eq. (16) to rewrite Eq. (15) in the one- (cid:82) 3 L (x)dx=1), dimensional form, 3 (cid:28) (cid:29) 1 (cid:90) ∞dk L(cid:48)(x) P(k,u)= 1−ψ(u−ikvx) 1 ,(27) L3(x)=−x(∂x) 2π2 cos(kx)exp[−kα]= 2πx , (35) u−ikv 1−(cid:104)ψ(u−ikv )(cid:105) 0 x vx x vx where L(x)=L (x) is the standard L´evy distribution, where the averaging is taken over the distribution of v 1 x given by Eqs. (18), (21). Thus we can directly use the (cid:90) dk results for P(k,t) in the one-dimensional case. The dif- L(x)= exp(ikx−|k|α) . (36) 2π ference from the one-dimensional case is in how the real space PDF is reproduced from P(k,t): here the formula Thus in three dimensions the isotropic L´evy density can for the inverse Fourier transform of radially symmetric be obtained from the one-dimensional one by differenti- function in d dimensions has to be used. We find from ation (this is improved version of the old result of [12] [17] that the bulk of the PDF is described with, valid for arbitrary velocity distribution). This is true for any odd-dimensional case. It can be demonstrated that Pcen(k,t)∼exp[−Kαt|k|α], (28) in the case of even dimension L2n(x) can be obtained A (cid:12) (cid:16)πα(cid:17)(cid:12) fromL(x)usingderivativeoperatorofhalfintegerorder, K = (cid:104)|v |α(cid:105)(cid:12)cos (cid:12), (29) α (cid:104)τ(cid:105) x (cid:12) 2 (cid:12) see Appendix E and cf. [21]. We find from Eq. (35) that L (x) ∝ |x|−α−3 at large 3 where (cid:104)|v |α(cid:105) is given by Eq. (23) with γ = α. Here the argument where we use the well-known behavior L(x)∼ x subscriptinP wasintroducedin[17]. Itstandsforthe |x|−α−1, see e. g. [14]. (Similarly it will be demon- cen centre (or bulk) part of the PDF. Briefly, to obtain this strated below that L (x)∼|x|−α−d.) This tail must fail d result we expand P(k,u) using the scaling assumption at larger arguments because it would give divergent dis- that kα is of the order u when both are small. Later, persion(cid:104)x2(t)(cid:105)∝(cid:82) x2L (x)dx=∞,whichiswrongpro- d we will derive these results as a special case of the more videdthatthemomentsofF(v)arefinite,anassumption general non-isotropic model (see eq. (55) below). we use all along this paper. The PDF must necessarily 7 FIG. 2: Scaling limits for a three-dimensional uniform L´evy walk. a) L´evy scaling of the bulk of numerically sampled PDFs P(x,t)=(cid:104)δ(|x(t)|−x)(cid:105). Thin black line corresponds to the PDFs P(x,t)=(cid:104)δ(|x(t)|−x)(cid:105) obtain by averaging over L3 the three-dimensional L´evy distribution, Eq. (35); b) Ballistic scaling of the tails of the PDFs. Thin black line is Eq. (42). Both PDFs were sampled over 1012 realizations. The parameters are α=3/2, v =1. 0 (cid:68) (cid:69) decay fast at x > vtt where vt is the typical value of (k·x)2n = k2n(cid:104)x21n(cid:105), cf. with similar consideration velocity. Thus the L´evy density does not provide valid for velocity. description of the tail of the PDF that determines dis- Though(cid:10)x2n(cid:11)cannotbefoundcompletelyatalltimes, persion. This necessitates the study of the tail of the 1 it was discovered in[17] that this can be done asymptot- distribution performed below. ically in the limit of large times. The proper adaptation of the result tells that using small k and u expansion of the quasi-one-dimensional Montroll-Weiss equation (15) B. Infinite density when keeping the ratio k/u fixed we find, The description of the tail of the PDF is performed P(k,t)∼1+ A (cid:88)∞ Γ(2n−α)(−1)nt2n+1−α(cid:10)vx2n(cid:11)k2n.(38) using the reduction to one-dimensional case where the (cid:104)τ(cid:105) (2n−1)!|Γ(1−α)|Γ(2n+2−α) n=1 problem was solved in [17]. This solution is based on asymptotic resummation of the series for the character- Thisresultwasobtainedinthelimitoflongtimesasymp- istic function. totically that is the n−th term in the series is valid pro- We observe that for isotropic statistics of velocity vided time is large. How large this time is depends on P(k,t) obeys, n: the higher n is, the larger times are needed for the validity of the asymptotic form. Thus at however large P(k,t)=(cid:104)exp[−ik·x](cid:105)=1+(cid:88)∞ (−1)nk2n(cid:10)x21n(cid:11),(37) blaurtgefinbiutet tfinthiteetner.mTshoufst,hinesceornietsrafsati,lsttoarEtqin.g(8fr6o)mthseomree- (2n)! n=1 summation of the series given by Eq. (38) does not have to lead to P(x,t) because there is no time for which all where we used that odd moments of k · x vanish and (cid:68) (cid:69) terms in the series of Eq. (38) are valid. that isotropy implies that (k·x)2n is independent of It is the finding of [17] that resummation of the se- directionofksoitcanbeobtainedsettingk=kxˆgiving ries in Eq. (38) still produces function that has physical 8 meaning. That function called the infinite density gives IV. CLT FOR ANISOTROPIC LE´VY WALKS validdescriptionofthetailofP(x,t)butfailsinthebulk. Thisisbecausethebulkcorrespondstosmallxandlarge We start the analysis of the anisotropic LWs with k ∝ 1/x. For very small x very large k are relevant im- derivation of the generalized CLT that describes PDF plying that terms with very large n become relevant for P(x,t) of the walker. We consider in this Section ran- the sum. However these terms are not valid at finite t dom walks whose single step duration’s PDF ψ(u) obeys leaving the small x inaccessible for the sum in Eq. (38). Eq. (9) at small u but we let the range of considered α Since statistics of both x and v are isotropic then, include α=2. That is we consider ψ(u) in Eq. (9) with (cid:104)x2n(cid:105) (cid:104)v2n(cid:105) 1 < α ≤ 2. In the case of α = 2 though ψ(τ) does not 1 = x , (39) obey Eq. (7) with α = 2. This is because Eq. (9) with (cid:104)x2n(cid:105) (cid:104)v2n(cid:105) α=2describesthecaseoffinitedispersionofτ givenby see Eq. (26). We find comparing the series in Eqs. (86)- (cid:104)τ2(cid:105) = ψ(cid:48)(cid:48)(u = 0) = 2A. In contrast, for α = 2 Eq. (7) (38), gives τ−3 tail for which the dispersion is infinite. Thus 2nA(cid:10)v2n(cid:11) ψ(τ) obeying Eq. (9) with α=2 decays faster than τ−3. (cid:10)x2n(t)(cid:11)= t2n+1−α.(40) We demonstrate that for ψ(u) obeying Eq. (9) with (cid:104)τ(cid:105)|Γ(1−α)|(2n+1−α)(2n−α) 1<α≤2 there is finite limit, Remarkably this is independent of dimension and thus coincides with the one-dimensional case (isotropy im- lim td/αP(t1/αx,t) t→∞ plies that the geometry disappears from (cid:104)x2n(cid:105) because (cid:90) (cid:20) A (cid:12) (cid:16)πα(cid:17)(cid:12) (cid:21) dk of Eq. (39)). The use of these moments for formal re- = exp ik·x− (cid:12)cos (cid:12)(cid:104)|k·v|α(cid:105) ,(44) (cid:104)τ(cid:105)(cid:12) 2 (cid:12) (2π)d construction of the long-time limit of the PDF P(x,t)= (cid:104)δ(|x(t)|−x)(cid:105) of |x(t)| through P (x,t) defined by, A that holds for arbitrary statistics of v obeying F(v) = (cid:90) dk (cid:34) (cid:88)∞ (ik)2n(cid:10)x2n(t)(cid:11)(cid:35) F(−v). It is proper to call this result generalized CLT PA(x,t)= 2π exp[−ikx] 1+ (2n)! ,(41) becauseforα=2itreproducesthecentrallimittheorem, n=1 with (cid:10)x2n(t)(cid:11) given by Eq. (40) gives on resummation limtd/2P(x√t,t)=(cid:90)exp(cid:20)ik·x−Γplkpkl(cid:21) dk , (45) [17], t→∞ 2 (2π)d AS (cid:90) ∞ where the RHS defines g(x) in Eq. (1), the covariance P (x,t)= d−1 vd−1F(v)dv A (cid:104)τ(cid:105)|Γ(1−α)|tα matrix Γ is defined by, |x|/t (cid:20) |v|α |v|α−1(cid:21) (cid:104)τ2(cid:105)(cid:104)v v (cid:105) × α|x/t|1+α −(α−1)|x/t|α , (42) Γpl = (cid:104)τ(cid:105)p l , (46) that holds for x (cid:54)= 0. The function P (x,t) clearly de- A and we used A = (cid:104)τ2(cid:105)/2. In Eq. (45) we use Einstein scribes the long-time behavior of the moments of |x(t)| summation rule over the repeated indices. We observe via that the units of k in Eq. (45) are t1/2/l, units of x are (cid:90) ∞ (cid:104)x2n(t)(cid:105)∼ P (x,t)x2ndx, n≥1. (43) l/t1/2 andunitsofΓarel2/tsothattheargumentofthe A 0 exponent is dimensionless. This function however does not describe the normaliza- The form of the covariance matrix could be seen tion (obtained as n = 0) since P (x,t) ∼ x−(1+α) for considering the second moment of displacement x(t)= A x → 0. Hence PA(x,t) is not normalizable for which (cid:80)Nk=(t1)vkτk, reason it is called infinite density. However it does de- scribetheintegerordermoments(cid:82)∞P(x,t)x2ndxwhere (cid:104)x (t)x (t)(cid:105) (cid:104)N(t)v v τ2(cid:105) (cid:104)τ2(cid:105)(cid:104)v v (cid:105) 0 lim p l = lim p l = p l , P(x,t) is the PDF of the distance to the origin |x(t)|. t→∞ t t→∞ t (cid:104)τ(cid:105) Thus P(x,t)∼P (x,t) is true for integrals with integer A powers. It can be seen that this function describes the where we used that the law of large numbers implies tail of the PDF P(x,t) at x ∼ t (that is at large times lim (cid:104)N(t)(cid:105)/t=1/(cid:104)τ(cid:105). t→∞ P(x,t) ∼ P (x,t) holds for large x ∝ t) while the bulk In the Gaussian α = 2 case the details of statistics A corresponds to x ∼ t1/α. This fits that the moments of individual steps of the walk become irrelevant in the of integer order are determined by the tail of P(x,t) as long-time limit: they get summarized in d(d+1)/2 inde- clarifiedinthecomingSections. Belowwederivetheinfi- pendentcoefficientsofthecovariancematrixΓ. Thesec- nitedensityinddimensionsindifferentformprovingthe ond moments of velocity (cid:104)v v (cid:105) determine the long-time p l existence of finite long-time limit lim td−1+αP(tx,t) statistics of the displacement uniquely. In contrast in t→∞ for arbitrary (anisotropic) statistics of velocity. The de- α<2 case the details of the walk influence the displace- scriptions of the bulk and the tail of the PDF together ment’s PDF however large time is via (cid:104)|k·v|α(cid:105). This with the moments provide a complete description of the is non-trivial function of direction of k that depends on d−dimensional walk with isotropic statistics. whichdirectionsofmotionaremoreprobableinonestep. 9 This is a function of continuous variable rather than Γ This is one of our main results as it provides the gen- that depends on finite number of discrete indices. Here eralized CLT for non-isotropic LWs. For isotropic case we assume that isotropy is broken - in the isotropic case Eq. (55) reduces to Eqs. (29). the degree of universality of α < 2 and α = 2 is the We characterize different statistics of velocity with same: (cid:104)|v|α(cid:105) determines uniquely the long-time behavior structure function s(kˆ) that depends on the unit vector given by Eq. (30). Correspondingly infinite variability of kˆ =k/k, shapes of P(x,t) is possible in contrast with fixed Gaus- (cid:68) (cid:69) sian shape in α=2 case. √ |kˆ·v|α Γ[(d+α)/2] π We start the derivation of Eq. (44). The calculation s(kˆ)= . (56) Γ[(α+1)/2]Γ(d/2) (cid:104)|v|α(cid:105) below hold for 1<α≤2. We use (cid:90) dk (cid:104) (cid:105) This is defined so that for isotropic statistics s(kˆ) = 1 td/αP(t1/αx,t)=td/α exp it1/αk·x P(k,t) (cid:68) (cid:69) (2π)d (in that case we have |kˆ·v|α = (cid:104)|vx|α(cid:105) where (cid:104)|vx|α(cid:105) (cid:90) dk isdeterminedbyEq.(25)). Wehavewiththisdefinition, = exp[ik·x]P(t−1/αk,t). (47) (2π)d (cid:90) (cid:104) (cid:105) dk lim td/αP(t1/αx,t)= exp ik·x−K kαs(kˆ) , Thus we have to prove the existence of the limit, t→∞ α (2π)d (cid:104) (cid:105) (cid:90) du 1 (cid:18) k u(cid:19) P(k,t)∼exp −Kαtkαs(kˆ) , (57) limP(t−1/αk,t)= exp[u] lim P , .(48) t→∞ 2πi t→∞t t1/α t where we use K defined in Eq. (30) (in anisotropic case α K does not have direct interpretation of diffusion coef- We use that ((cid:104)k·v(cid:105)=0), α ficient so this is to be taken as mathematical definition). (cid:28)1−ψ(ut−1−ik·vt−1/α)(cid:29) The PDF of the displacement obeys, =(cid:104)τ(cid:105)+o(t), (49) ut−1−ik·vt−1/α 1 (cid:18) x (cid:19) P(x,t)∼ Lˆ , (58) (cid:28) (cid:18)u ik·v(cid:19)(cid:29) (cid:104)τ(cid:105)u (cid:28)(cid:18)u ik·v(cid:19)α(cid:29) (K t)d/α d (K t)1/α α α 1− ψ − = −A − (50) t t1/α t t t1/α (cid:90) (cid:104) (cid:105) dk Lˆ (x)= exp ik·x−kαs(kˆ) . (59) (cid:104)τ(cid:105)u+A|cos(πα/2)|(cid:104)|k·v|α(cid:105) d (2π)d +o(t)= , (51) t For isotropic model there is no modulation and Lˆ (x) is d where we used that F(v)=F(−v) implies [17], the universal function Ld(x) introduced previously, see Eq. (33). Thus different isotropic statistics produces (cid:28) (cid:20) (cid:21)(cid:29) iπα the same long-time PDF in the bulk that differ only (cid:104)(−ik·v)α(cid:105)= |k·v|αexp − sign(k·v) 2 by the value of Kα. For XYZ... model (cid:104)|k·v|α(cid:105) = (cid:16)πα(cid:17) vα(cid:80)d |k |α/d we find that the distribution factorizes =(cid:104)|k·v|α(cid:105)cos . (52) 0 i=1 i 2 in the product of one-dimensional distributions. Thus in this case the bulk of the PDF coincides with that of in- This formula holds in α = 2 case as well. We find using dependent walks along different axes. In these cases the Eqs. (49)-(51) in the Montroll-Weiss equation that, functional shape is universal. (cid:18) (cid:19) The chief feature introduced by the passage from 1 k u 1 lim P , = , one dimension to the higher-dimensional case is that t→∞ t t1/α t u+A˜|cos(πα/2)|(cid:104)|k·v|α(cid:105) quite arbitrary angular structure of the distribution be- comes possible. The structure function s(kˆ) describes where A˜ = A/(cid:104)τ(cid:105), cf. one-dimensional case in [17]. We positive angular modulation in k−space that changes conclude that, correspondingly the functional form in real space, see (cid:90) du exp[u] Eq. (59). This function does not seem to obey strong limP(t−1/αk,t)= ,(53) constraints that would strongly limit the possible forms t→∞ 2πiu+A˜|cos(πα/2)|(cid:104)|k·v|α(cid:105) of Lˆ (x). We stress this fact calling distribution Lˆ (x) d d ’anisotropicd−dimensionalL´evydistribution’incontrast see Eq. (48). We find performing the integration, with d−dimensional isotropic L´evy distributions intro- (cid:20) A (cid:12) (cid:16)πα(cid:17)(cid:12) (cid:21) duced previously [12] in the context of L´evy flights that tl→im∞P(t−1/αk,t)=exp −(cid:104)τ(cid:105)(cid:12)(cid:12)cos 2 (cid:12)(cid:12)(cid:104)|k·v|α(cid:105) ,(54) have universal shape. Considering marginal distributions of components of x one reduces the problem to one dimension restoring completingtheproofofEq.(44). Wefindasymptotically universality of the distribution. Integrating Eq. (58), at large times that, (cid:20) tA (cid:12) (cid:16)πα(cid:17)(cid:12) (cid:21) P(x ,t)=(cid:90)P(x,t)(cid:89)dx ∼ 1 L(cid:18) xi (cid:19)(,60) P(k,t)∼exp −(cid:104)τ(cid:105)(cid:12)(cid:12)cos 2 (cid:12)(cid:12)(cid:104)|k·v|α(cid:105) . (55) i k(cid:54)=i k (Kit)1/α (Kit)1/α 10 FIG. 3: Probability density functions of Levy walks in three-dimensions. The distributions for the time t/τ = 104 0 were obtained by sampling over 1011 realizations. They are represented by the set of two-dimensional ’slices’ along z-axis, P(x,y,z=const), plotted on a log scale. The other parameters are as in Fig. 1. where L(x) is defined in Eq. (36) and K is ”diffusion where xd−1 factor describes the contribution of the sur- i coefficient in i−th direction”, face of the sphere of radius x. In the isotropic case we have, A (cid:12) (cid:16)πα(cid:17)(cid:12) K = (cid:12)cos (cid:12)(cid:104)|v |α(cid:105). (61) i (cid:104)τ(cid:105)(cid:12) 2 (cid:12) i (cid:90) dk S Γ[(2n+d)/α] c = k2nexp[−K kα]= d−1 . Thus marginal PDFs are given by the standard one- n (2π)d/2 α (2π)d/2αKα(2n+d)/α dimensional symmetric L´evy stable law. In contrast the (cid:82) We find using that in isotropic case P (x) = PDF P(x,t) = δ(|x(t)|−x)P(x,t)dx of the distance 0 P (x)/[xd−1S ] the Taylor series for P (x,t) defined x(t) = |x(t)| from the origin is not universal. We find 0 d−1 cen in Eq. (33), integrating Eq. (58) that, 1 (cid:16) x (cid:17) (cid:88)∞ (−1)nΓ[(2n+d)/α] P(x,t)∼ t1/αP0 t1/α , (62) Ld(x)= n!Γ(n+d/2)22n+d−1πd/2αx2n, (65) (cid:90) xd/2J (kx)dk (cid:104) (cid:105) n=0 P (x)= d/2−1 exp −K kαs(kˆ) , (63) 0 (2π)d/2kd/2−1 α The factor of Γ(n + d/2) can be simplified in cases of odd and even dimensions using Γ(n+1)=n! and Γ(n+ √ (see further details for notation choices in the next Sec- 1/2) = 2−n π(2n−1)!!. In the case of d = 1 the series tion) where we used reproduces the one-dimensional formula [14]. As mentioned, for arbitrary statistics of velocity there (cid:90) (2πx)d/2J (kx) d/2−1 seems to be no constraint on the Taylor coefficients c exp[ik·x]δ(|x(t)|−x)dx= , n kd/2−1 that would determine uniquely the functional form of P (x). However despite that the small x expansion of whereJ (z)istheBesselfunctionofthefirstkindoforder 0 ν P (x) is not universal, the large x behavior of P (x) is ν. We observe that integration over angles implied by 0 0 universal. ThiswillbedemonstratedinthenextSection. the definition of the PDF of the distance from the origin does not bring universal form of the PDF (that would then coincide with the PDF for isotropic statistics). The V. UNIVERSAL TAIL OF ANISOTROPIC LE´VY structurefunctionispresentinEq.(63)andcanproduce DISTRIBUTION AND LOW-ORDER MOMENTS quite different forms of P(x,t). We have using Taylor series for the Bessel function in Eq. (63), In this Section we demonstrate that the PDF of the P0(x)=xd−1(cid:88)∞ n!Γ(d/(−21+)nnc)n2d/2−1 (cid:16)x2(cid:17)2n, (64) dlaiwstatnaiclewoifththueniwvaelrksearl(firnodmeptehnedwenatlko’sfsotraitgiisntihcsasofpvoewloecr-- n=0 ity) exponent. This has the implication that moments of (cid:90) dk (cid:104) (cid:105) distancefromtheoriginwithordersmallerthanαarede- c = k2nexp −K kαs(kˆ) , n (2π)d/2 α termined by the bulk of the PDF but moments of higher