Some fundamental aspects of Levy flights

Ralf Metzler ∗,1

NORDITA, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark

Aleksei V. Chechkin

Institute for Theoretical Physics NSC KIPT, Akademicheskaya st.1, 61108

Kharkov, Ukraine

Vsevolod Yu. Gonchar

Institute for Theoretical Physics NSC KIPT, Akademicheskaya st.1, 61108

Kharkov, Ukraine

Joseph Klafter

School of Chemistry, Tel Aviv University, 69978 Tel Aviv Israel

Abstract

We investigate the physical basis and properties of Levy flights (LFs), Markovianrandom walks with a long-tailed density of jump lengths, λ(ξ) ∼ |ξ|−1−α, with0 < α < 2. In particular, we show that non-trivial boundary conditions need to becarefully posed, and the method of images fails due to the non-locality of LFs. Wediscuss the behaviour of LFs in external potentials, demonstrating the existence ofmultimodal solutions whose maxima do not coincide with the potential minimum.The Kramers escape of LFs is investigated, and the physical nature of the a prioridiverging kinetic energy of an LF is addressed.

Key words: Stochastic processes, Levy flights, stable distributions, Kramersproblem, first passage, fractional Fokker-Planck equation, Langevin equationPACS: 05.40.Fb, 02.50.-Ey, 82.39.-k

∗ Corresponding author. Address: Physics Department, University of Ottawa, 150Louis Pasteur, Ottawa, Ontario K1N 6N5, Canada

Email address: metz@nordita.dk (Ralf Metzler).1 Present address: Physics Department, University of Ottawa, 150 Louis Pasteur,Ottawa, Ontario K1N 6N5, Canada

Preprint submitted to Elsevier Science 10 July 2006

1 Introduction

For sums of independent, identically distributed random variables with propernormalization to the sample size, the generalized central limit theorem guar-antees the convergence of the associated probability density to a Levy stabledensity (LSD) even though the variance of these random variables diverges[1–4]. Well-known examples for LSDs are the one-sided (defined for x ≥ 0)Levy-Smirnov distribution

f1/2,−1/2(x) =

√

1

4πx3exp

(

− 1

4x

)

, (1)

related to the first passage time density of a Gaussian random walk processof passing the origin (see below), and the Cauchy (or Lorentz) distribution

f1,0(x) =1

π(12 + x2). (2)

In general, an LSD is defined through its characteristic function of the prob-ability density f(x)

ϕ(z) ≡ F {f(x)} =∫

∞

−∞

fα,β(x)eikxdx (3)

where

logϕ(z) = −|z|α exp

{

iπβ

2sign(z)

}

, (4)

for α 6= 1. Here, the skewness (or asymmetry) parameter β is restricted to thefollowing region:

|β| ≤

α, if 0 < α < 1

2 − α, if 1 < α < 2.(5)

For β = 0, the corresponding LSD is symmetric around x = 0, while forβ = −α and 0 < α < 1, it is one-sided. In general, an LSD follows thepower-law asymptotic behaviour

fα,β(x) ∼ Aα,β

|x|1+α, α < 2, (6)

with Aα,β being a constant, such that for all LSDs with α < 2 the variancediverges

〈x2〉 = ∞. (7)

Conversely, all fractional moments 〈|x|δ〉 < ∞ for all 0 < δ < α ≤ 2. Fromabove definitions it is obvious that the LSD f2,0 corresponds to the Gaussiannormal distribution

f2,0(x) =

√

1

4πexp

(

−1

4x2)

(8)

2

possessing finite moments of any order. In this limit, the generalized centrallimit theorem coincides with the more traditional central limit theorem.

Random processes whose spatial coordinate x or clock time t are distributedaccording to an LSD exhibit anomalies, that is, no longer follows the lawsof Bronian motion. Consider a continuous time random walk process definedin terms of the jump length and waiting time distributions λ(ξ) and ψ(τ) [5].Each jump event of this random walk, that is, is characterized by a jump lengthξ drawn from the distribution λ, and the time τ between two jump events isdistributed according to ψ. (Note that an individual jump is supposed to occurspontaneously.) In absence of an external bias, continuous time random walktheory connects λ(ξ) and ψ(τ) with the probability distribution P (x, t)dxto find the random walker at a position in the interval (x, x + dx) at timet. In Fourier-Laplace space, P (k, u) ≡ F {L {P (x, t); t→ u} ; x→ k}, thisrelation reads [6]

P (k, u) =1 − ψ(u)

u

1

1 − λ(k)ψ(u), (9)

where L {f(t)} ≡ ∫

∞

0 exp(−ut)f(t)dt. We here neglect potential complicationsdue to ageing effects. The following cases can be distinguished:

(i) λ(ξ) is Gaussian with variance σ2 and ψ(τ) = δ(τ−τ0). Then, to leading or-der in k2 and u, respectively, one obtains λ(k) ≃ 1−σ2k2 and ψ(u) ≃ 1−uτ0.From relation (9) one recovers the Gaussian probability density P (x, t) =√

1/(4πKt) exp{−x2/(4Kt)} with diffusion constant K = σ2/τ0. The corre-sponding mean squared displacement grows linearly with time:

〈x2(t)〉 = 2Kt. (10)

This case corresponds to the continumm limit of regular Brownian motion.Note that here and in the following, we restrict the discussion to one dimension.

(ii) Assume λ(ξ) still to be Gaussian, while for the waiting time distributionψ(τ) we choose a one-sided LSD with stable index 0 < α < 1. Consequently,ψ(u) ≃ 1 − (uτ0)

α, and the characteristic waiting time∫

∞

0 ψ(τ)τdτ diverges.Due to this lack of a time scale separating microscopic (single jump events)and macroscopic (on the level of P (x, t)) scales, P (x, t) is no more Gaussian,but given by a more complex H-function [7–9]. In Fourier space, however, onefinds the quite simple analytical form [8]

P (k, t) = Eα

(

−Kαk2tα)

=∞∑

0

(Kαk2tα)

n

Γ(1 + αn)(11)

in terms of the Mittag-Leffler function. This generalized relaxation functionof the Fourier modes turns from an initial stretched exponential (KWW) be-haviour P (k, t) ∼ 1 −Kαk

2tα/Γ(1 + α) ∼ exp {−Kαk2tα/Γ(1 + α)} to a ter-

3

minal power-law behaviour P (k, t) ∼(

Kαk2tαΓ(1 − α)

)

−1

[8]. In the limit

α → 1, it reduces to the traditional exponential P (k, t) = exp(−Kk2t) withfinite characteristic waiting time. Also the mean squared displacement changesfrom its linear to the power-law time dependence

〈x2(t)〉 = 2Kαtα, (12)

with Kα = σ2/τα0 . This is the case of subdiffusion. We note that in x, t space

the dynamical equation is the fractional diffusion equation [7]. In the presenceof an external potential, it generalizes to the time-fractional Fokker-Planckequation [10,8,9].

(iii) Finally, take ψ(τ) = δ(τ − τ0) sharply peaked, but λ(ξ) of Levy stableform with index 0 < α < 2. The resulting process is Markovian, but withdiverging variance. It can be shown that the fractional moments scale like [11]

〈|x(t)|δ〉 ∝ (Kαt)δ/α , (13)

were Kα = σα/τ0. From Eq. (9) one can immediately obtain the Fourier imageof the associated probability density function,

P (k, t) = exp{

−Kα|k|αt}

. (14)

From Eq. (4) this is but a symmetric LSD with stable index α, and this typeof random walk process is most aptly coined a Levy flight. A Levy flight man-ifestly has regular exponential mode relaxation and is in fact Markovian (seebelow). However, the modes in position space are no more sharply localizedlike in the Gaussian or subdiffusive case. Instead, individual modes bear thehallmark of an LSD, that is, the diverging variance. We will see below how thepresence of steeper than harmonic external potentials cause a finite variance ofthe Levy flight, although a power-law form of the probability density remains.

In the remainder of this paper, we deal with the physical and mathemati-cal properties of Levy flights in the presence of external force fields. Whilemostly we will be concerned with the overdamped case, in the last section wewill address the dynamics in velocity space in the presence of Levy noise, inparticular, the question of the diverging variance of Levy flights.

2 Underlying random walk process

To derive the dynamic equation of a Levy flight in the presence of an externalforce field F (x) = −dV (x)/dx, we pursue two different routes. One starts with

4

a generalized version of the continuous time random walk, compare Ref. [12]for details.

To include the local asymmetry of the jump length distribution due to the forcefield F (x), we introduce [12,13] the generalized transfer kernel Λ(x, x′) = λ(x−x′) [A(x′)Θ(x− x′) +B(x′)Θ(x′ − x)] (and therefore Λ(x, x′) = Λ(x′; x− x′)).As in standard random walk theory (compare [14]), the coefficients A(x) andB(x) define the local asymmetry for jumping left and right, depending on thevalue of F (x). Here, Θ(x) is the Heaviside jump function. With the normaliza-tion

∫

Λ(x′,∆)d∆ = 1, the fractional Fokker-Planck equation (FFPE) ensues[12]:

∂

∂tP (x, t) =

(

− ∂

∂x

F (x)

mη+Kα

∂α

∂|x|α)

P (x, t). (15)

Remarkably, the presence of the Levy stable λ(ξ) only affects the diffusionterm, while the drift term remains unchanged [15,12]. The fractional spatialderivative represents an integrodifferential operator defined through

∂α

∂|x|αP (x, t) =−1

2 cos(πα/2)Γ(2− α)

∂2

∂x2

∫

∞

−∞

P (x′, t)

|x− x′|α−1, (16)

for 1 < α < 2, and a similar form for 0 < α < 1 [16–18]. In Fourier space, forall 0 < α ≤ 2 the simple relation

F

{

∂α

∂|x|αP (x, t)

}

= −|k|αP (k, t) (17)

holds. In the Gaussian limit α = 2, all relations above reduce to the familiarsecond-order derivatives in x and thus the corresponding P (x, t) is governedby the standard Fokker-Planck equation.

The FFPE (15) can also be derived from the Langevin equation [15,19,16]

dx(t)

dt= − 1

mγ

dV (x)

dx+ ξα(t), (18)

driven by white Levy stable noise ξα(t), defined through L(∆t) =∫ t+∆tt ξα (t′) dt′

being a symmetric LSD of index α with characteristic function p(k,∆t) =exp (−Kα|k|α∆t) for 0 < α ≤ 2. As with standard Langevin equations, Kα

denotes the noise strength, m is the mass of the diffusing (test) particle, andγ is the friction constant characterizing the dissipative interaction with thebath of surrounding particles.

A subtle point about the FFPE (15) is that it does not uniquely define the un-derlying trajectory [20]; however, starting from our definition of the process interms of the stable jump length distribution λ(ξ) ∼ |x|−1−α, or its generalizedpendant Λ(x, x′), the FFPE (15) truly represents a Levy flight in the presence

5

0.01

0.1

1

-3 -2 -1 0 1 2 3

P(x

,t)

x

Cauchy, t=0.1Cauchy, t=0.5Gauss, t=0.1Gauss, t=0.5

Fig. 1. Cauchy distribution (α = 1) for two times in comparison to the Gaussian(α = 2). We chose K1 = K2 = 1. Note that the Cauchy distribution is narrower atthe origin, and after crossing the Gaussian falls off in the much slower power-lawfashion.

of the force F (x). This poses certain difficulties when non-trivial boundaryconditions are involved, as shown below.

3 Propagator and symmetries

In absence of an external force, F (x) = 0, the exact solution of the FFPE isreadily obtained as the LSD P (k, t) = exp (−Kα|k|αt) in Fourier space. Back-transformed to position space, an analytical solution is given in terms of theFox H-function [8,21,19]

P (x, t) =1

α|x|H1,12,2

|x|(Kαt)

1/α

∣

∣

∣

∣

∣

∣

∣

(1, 1/α), (1, 1/2)

(1, 1), (1, 1/2)

, (19)

from which the series expansion

P (x, t) =1

α(Kαt)1/α

∞∑

ν=0

Γ([1 + ν]/α)

Γ([1 + ν]/2)Γ(1 − [1 + ν]/2)

×(−1)ν

ν!

(

|x|(Kαt)1/α

)ν

(20)

derives. For α = 1, the propagator reduces to the Cauchy LSD

P (x, t) =1

π (K1t+ x2/[K1t]). (21)

We plot the time evolution of P (x, t) for the Cauchy case α = 1 in Fig. 1 incomparison to the limiting Gaussian case α = 2.

6

0.01

0.1

1

-2 0 2 4 6 8 10

P(x

,t)

x

Cauchy, t=0.1Cauchy, t=0.5Cauchy, t=2.0

Fig. 2. Cauchy distribution with K1 = 1 advected along a field F0/(mγ) = 2, fordifferent times.

Due to the point symmetry of the FFPE (15) for F (x) = 0, the propagatorP (x, t) is invariant under change of sign, and it is monomodal, i.e., it has itsglobal maximum at x = 0, the point where the initial distribution P (x, 0) =δ(x) was launched at t = 0. The latter property is lost in the case of stronglyconfined Levy flights discussed below. Due to their Markovian character, Levyflights also possess a Galilei invariance [22,8]. Thus, under the influence of aconstant force field F (x) = F0, the solution of the FFPE can be expressed interms of the force-free solution by introducing the wave variable x − F0t, toobtain

PF0(x, t) = P0

(

x− F0t

mγ, t

)

. (22)

This result follows from the FFPE (15), that in Fourier domain becomes [19]

∂

∂tP (k, t) =

(

−ik F0

mγ−Kα|k|α

)

P (k, t), (23)

with solution

P (k, t) = exp

(

−[

ikF0

mγ+Kα|k|α

]

t

)

. (24)

By the translation theorem of the Fourier transform, Eq. (22) yields. We showan example of the drift superimposed to the dispersional spreading of thepropagator in Fig. 2.

7

4 Presence of external potentials

4.1 Harmonic potential

In an harmonic potential V (x) = 12λx2, an exact form for the characteristic

function can be found. Thus, from the corresponding FFPE in Fourier space,

∂

∂tP (k, t) = − λ

mγk∂

∂kP (k, t) −Kα|k|αP (k, t), (25)

by the method of characteristics one obtains

P (k, t) = exp

(

−mγKα|k|αλα

[

1 − e−αλt/(mγ)]

)

(26)

for an initially central δ-peak, P (x, 0) = δ(x) [19]. This is but the charac-teristic function of an LSD with time-varying width. For short times, 1 −exp(−αλt/[mγ]) ∼ αλt/[mγ] grows linearly in time, such that P (k, t) ∼exp (−Kα|k|αt) as for a free Levy flight. At long times, the stationary solutiondefined through

Pst(k) = exp

(

−mγKα|k|αλα

)

, (27)

is reached. Interestingly, it has the same stable index α as the driving Levynoise. By separation of variables, a summation formula for P (x, t) can beobtained similarly to the solution of the Ornstein-Uhlenbeck process in thepresence of white Gaussian noise, however, with the Hermite polynomials re-placed by H-functions [19].

We note that in the Gaussian limit α = 2, the stationary solution by necessityhas to match the Boltzmann distribution corresponding to exp (−kBTk

2/[2λ]).This requires that the Einstein-Stokes relation K2 = kBT/[mγ] is fulfilled [23].One might therefore speculate whether for a system driven by external Levynoise a generalized Einstein-Stokes relation should hold, as was established forthe subdiffusive case [10,8]. As will be shown now, in steeper than harmonicexternal potentials, the stationary form of P (x, t) even leaves the basin ofattraction of LSDs.

4.2 Steeper than harmonic potentials

To investigate the behaviour of Levy flights in potentials, that are steeper thanthe harmonic case considered above, we introduce the non-linear oscillator

8

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-3 -2 -1 0 1 2 3

Pst

(x)

x

Fig. 3. Bimodal stationary probability density Pst(x) from Eq. (32). The maximaare at ±

√

1/2.

potential

V (x) =a

2x2 +

b

4x4, (28)

that can be viewed as a next order approximation to a general confining, sym-metric potential. It turns out that the resulting process differs from abovefindings if a suitable choice of the ratio a/b is made. For simplicity, we intro-duce dimensionless variables through

x→ x/x0; t→ t/t0; a→ at0/(mγ), (29)

where

x0 =(

mγKα

b

)1/(2+α)

; t0 =xα

0

Kα, (30)

arriving at the FFPE

∂

∂tP (k, t) + |k|α =

(

k∂3

∂k3− ak

∂

∂k

)

P (k, t). (31)

Consider first the simplest case of a quartic oscillator with a = 0 in thepresence of Cauchy noise (α = 1). In this limit, the stationary solution can beobtained exactly, yielding the expression

Pst(x) =1

π

1

1 − x2 + x4(32)

plotted in Fig. 3. Two distinct new features in comparison to the free Levyflight, and the Levy flight in an harmonic potential: (1) Instead of the maxi-mum at x = 0, one observes two maxima positioned at

xm = ±√

1/2; (33)

9

at x = 0, we find a local minimum. (2) There occurs a power-law asymptote

Pst(x) ∼1

πx4(34)

for x≫ 1; consequently, this stationary solution no longer represents an LSD,and the associated mean squared displacement is finite, 〈x2〉 <∞.

A more detailed analysis of Eq. (31) reveals [24,16], that (i) the bimodalityof P (x, t) occurs only if the amplitude of the harmonic term, a, is belowa critical value ac; (ii) for general α, the asymptotic behaviour is Pst(x) ∼π−1 sin(πα/2)Γ(α)|x|−α−3; (iii) and there exists a finite bifurcation time tc atwhich the initially monomodal form of P (x, t) acquires a zero curvature atx = 0, before settling in the terminal bimodal form.

Interestingly, in the more general power-law behaviour

V (x) =|x|cc, (35)

the turnover from monomodal to bimodal form of P (x, t) occurs exactly whenc > 2. The harmonic potential is therefore a limiting case when the solution ofthe FFPE still belongs to the class of LSDs and follows the generalized centrallimit theorem. This is broken in a superharmonic (steeper than harmonic)potential. The corresponding bifurcation time tc is finite for all c > 2 [16]. Anadditional effect appears when c > 4: there exists a transient trimodal statewhen the relaxing δ(x)-peak overlaps with the forming humps at x = ±xm.At the same time, the variance is finite, if only c > 4 − α, following from theasymptotic stationary solution

Pst(x) ∼sin(πα/2)Γ(α)

π|x|α+c−1. (36)

Details of the asymptotic behaviour and the bifurcations can be found inRefs. [24,16]. From a reverse engineering point of view, Levy flights in confiningpotentials are studied in [25].

5 Non-trivial boundary conditions

One might naively expect that a jump process of Levy type, whose variancediverges (unless confined in a steep potential) may lead to ambiguities whenboundary conditions are introduced, such as an absorbing boundary at finitex. Indeed, it is conceivable that for a jump process with extremely long jumps,it becomes ambiguous how to properly define the boundary condition: should

10

the test particle be absorbed when it arrives exactly at the boundary, or whenit crosses it anyplace during a non-local jump?

This question is trivial in the case of a narrow jump length distribution: allsteps are small, and the particle cannot jump across a point (in the continuumlimit considered herein). For such processes, one enforces a Cauchy boundarycondition P (0, t) = 0 at the point x = 0 of the absorbing boundary, removingthe particle once it hits the barrier after starting at x0, where the dynamicsis governed by Eq. (15) with F (x) = 0. Its solution can easily be obtainedby standard methods, for instance, the method of images. This is completelyequivalent to considering the first arrival to the point x = 0, expressed interms of the diffusion equation with sink term:

∂

∂tP(x, t) = K

∂2

∂x2P(x, t) − pfa(t)δ(x), (37)

defined such that P (0, t) = 0. Note that the quantity P is no longer a proba-bility density, as probability decays to zero; for this reason, we use the notationP. From Eq. (37) by integration we obtain the survival probability

S (t) =∫

P(x, t)dx (38)

with S (0) = 1 and limt→∞ S (t) = 0. Then, the first arrival density becomes

pfa(t) = − d

dtS (t). (39)

Eq. (37) can be solved by standard methods (determining the homogeneousand inhomogeneous solutions). It is then possible to express P(x, t) in termsof the propagator P (x, t), the solution of Eq. (15) with F (x) = 0 with thesame initial condition, P (x, 0) = δ(x− x0) and natural boundary conditions.One obtains

P (0, t) =∫ t

0pfa(τ)P (x0, t− τ)dτ, (40)

such that the first arrival density corresponds to the waiting time distribu-tion to jump from x0 to 0 (or, vice versa, since the problem is symmetric).In Laplace space, this relation takes on the simple algebraic form pfa(u) =P (0, u)/P (x0, u). Both methods the explicit boundary value problem and thefirst arrival problem for Gaussian processes produce the well-known first pas-sage (or arrival) density of Levy-Smirnov type (1),

p(t) = pfa(t) =x0√

4πKt3exp

(

− x20

4Kt

)

∼ x0√4πKt3

, (41)

with the asymptotic power-law decay p(t) ∼ t−3/2, such that no mean firstpassage time exists [4,26].

11

0.01

0.1

100 1000 10000

t*p f

a(t)

t

α=1.2α=1.4α=1.6α=1.8

Analyt., α=1.2Analyt., α=1.8

t-3/2

Fig. 4. First arrival density for various values of α and trap width w = 0.25. Forα = 1.2 and α = 1.8 we compare to analytically predicted power-law (42). In allcases the decay is slower than for the Brownian result, ∼ t−3/2, that is also shown.

Long-tailed jump length distributions of Levy stable form, however, endow thetest particle with the possibility to jump across a certain point repeatedly. Thefirst arrival necessarily becomes less efficient. Indeed, as shown in Ref. [27],the Gaussian result (41) is generalized to

pfa(t) ∼ C(α)xα−1

0

K1−1/αα t1−1/α

, as t→ ∞ (42)

with C(α) = αΓ(2 − α)Γ(2 − 1/α) sin(π[2 − α]/2) sin2(π/α)/(π2[α− 1]), and1 < α ≥ 2 [27]. The long-time decay ∼ t−2+1/α is slower than in (41). InFig. 4, we show simulations results of the first arrival problem, corroboratingthe analytic result for various α. The insensitivity of the power-law slope tothe initial condition is demonstrated in Fig. 5. However, we cannot realize aδ-point removal in the simulations but need to define a small but finite intervalfor removing the particle; Fig. 5 also shows that if this interval w is chosentoo wide, for increasing width w the result slowly becomes steeper, eventuallyit should approach the −3/2 slope. Note that in both figures, we plot tpfa(t)for clarity.

One might naively assume that the first passage problem (the particle is re-moved once it crosses the boundary) for Levy flights should be more efficient,that is, the first passage density p(t) should decay quicker, than for a nar-row jump length distribution. However, as we have a symmetric jump lengthdistribution λ(ξ), the long outliers characteristic for these Levy flights canoccur both toward and away from the absorbing barrier. From this point ofview it is not totally surprising to see the simulations result in Figs. 6 and 7,that clearly indicate a universal asymptotic decay ∼ t−3/2, exactly as for theGaussian case.

In fact, for all Markovian processes with a symmetric jump length distri-bution, the Sparre Andersen theorem [28,2,29] proves without knowing any

12

0.01

10 100 1000 10000

t*p(

t)

t

x0=0.0, w=0.3x0=0.3, w=0.3x0=1.0, w=0.3

x0=10.0, w=0.3x0=10.0, w=1.0x0=10.0, w=3.0x0=10.0, w=5.0

Analyt.t-3/2

Fig. 5. First arrival density for α = 1.2 for various initial positions x0 and trap widthw = 0.3, that turned out to be optimal in the simulations. All data follow the samescaling law predicted by Eq. (42). If w is chosen too wide, the slope increases towardthe t−3/2 behaviour. Note that we plot tpfa(t), and that pfa(t) is not normalized.

0.001

0.01

0.1

1 10 100 1000 10000

t*p(t

)

t

x_0=0.10x_0=1.00x_0=10.0

x_0=100.0t**(-1.5)t**(-1.5)t**(-1.5)t**(-1.5)

First arrivalImages method

Fig. 6. First passage density for α = 1.2 for various initial positions x0 away from theabsorbing boundary. Each time the particle crosses the boundary, it is removed. Inall cases, the universal ∼ t−3/2 scaling is observed. The two additional lines representthe result of the corresponding first arrival problem and the images method.

details about λ(ξ) the asymptotic behaviour of the first passage time densityuniversally follows p(t) ∼ t−3/2. The details of the specific form of λ(ξ) onlyenter the prefactor, and the pre-asymptotic behaviour. A special case of theSparre Andersen theorem was proved in Ref. [30] when the particle is releasedat x0 = 0 at time t = 0, and after the first jump an absorbing boundary isinstalled at x = 0. This latter case was simulated extensively in Ref. [31].From a fractional diffusion equation point of view, it was shown in Ref. [27]that the fractional operator ∂α/∂|x|α needs to be modified, to account for thefact that P(x, t) ≡ 0 beyond the absorbing boundary, such that long-rangecorrelations are present exclusively for all x in the semi-axis containing x0.The fractional diffusion equation in the presence of the absorbing boundary

13

0.001

0.01

100 1000 10000

t*p(t

)

t

alpha=2.0alpha=1.5alpha=1.0alpha=0.6

t**(-1.5)dittodittoditto

Fig. 7. First passage density for various stable indices α and initial position x0 = 10.0away from the absorbing boundary. Again, the universal ∼ t−3/2 scaling is distinct.

therefore has to be modified to [27]

∂

∂P(x, t) =

Kα

κ

∂2

∂x2

∫

∞

0

P(x′, t)

|x− x′|α−1dx′, (43)

where κ = 2Γ(2 − α) |cos(πα/2)|, such that the first term on the right handside no longer represents a Fourier convolution. An approximate solution withCauchy boundary condition reveals p(u) ∼ 1 − cu1/2, where c is a constant,indeed leading to the Sparre Andersen behaviour p(t) ∼ t−3/2.

This also demonstrates that the method of images no longer applies whenLevy flights are considered, for the images solution

Pim(x, t) = P (x− x0, t) − P (x+ x0, t) (44)

would be governed by the full fractional diffusion equation, and not Eq. (43),and the result for the first passage density, p(t) ∼ t−1−1/α would decay fasterthan the Sparre Andersen universal behaviour. A detailed discussion of theapplicability of the method of images is given in terms of a subordinationargument in Ref. [20]. We emphasize that this subtle failure of the method ofimages has been overlooked in literature previously [32,33], and care shouldtherefore be taken when working with results based on such derivations. Wealso note that the method of images works in cases of subdiffusion, as the steplength is narrow [34].

6 Kramers problem for Levy flights

Many physical and chemical problems are related to the thermal fluctuationsdriven crossing of an energetic barrier, such as dissociation of molecules, nu-

14

cleation processes, or the escape from an external, confining potential of finiteheight [35]. A particular example of barrier crossing in a double-well potentialdriven by Levy noise was proposed for a long time series of paleoclimatic data[36]. Further cases where the crossing of a potential barrier driven by Levynoise is of interest is in the theory of plasma devices [37], among others [9].

To investigate the detailed behaviour of barrier crossing under the influenceof external Levy noise, we choose the rather generic double well shape

V (x) = −a2x2 +

b

4x4. (45)

Integrating the Langevin equation (18) with white Levy noise, we find anexponential decay of the survival density in the initial well:

p(t) =1

Tcexp

(

− t

Tc

)

, (46)

as demonstrated in Fig. 8. Levy flight processes being Markovian, this is notsurprising, since the mode relaxation is exponential [8,9]. More interesting isthe question how the mean escape time Tc behaves as function of the char-acteristic noise parameters D and α. While in the regular Kramers problemwith Gaussian driving noise the Arrhenius-type activation Tc = C exp(h/D)is followed, where h is the barrier height, and the prefactor C includes detailsof the potential, in the case of Levy noise, a power-law form

Tc(α,D) =C(α)

Kµ(α)α

(47)

was assumed [38]. Detailed investigations [39] show that the scaling exponentµ(α) = 1 for all α strictly smaller than 2. As already proposed in Ref. [40] andderived in [41] in a somewhat different model, this means that, apart from aprefactor, the Levy flight is insensitive to the external potential for the barriercrossing. This behaviour is distinctly visible in Fig. 9 in form of the parallellines in the log-log scale. Note that in comparison to Ref. [38], also values ofα in the range (0, 1) are included. For large values of D, deviations from thescaling are observed: eventually it will only take a single jump to cross thebarrier when D → ∞. Detailed studies show indeed that eventually the unittime step is reached, i.e., Tc → 1.

7 More on the ”pathology”

Despite their mathematical foundation due to the generalized central limittheorem and their broad use in the sciences and beyond as description forstatistical quantities, and despite the existence of systems (for instance, the

15

-13

-12

-11

-10

-9

-8

-7

0 1000 2000 3000 4000 5000

ln p

(t)

t

Fig. 8. Probability density function p(t) of barrier crossing times for α = 1.0and D = 10−2.5 ≈ 0.00316. The dashed line is a fit to equation (46) with MCTTc = 1057.8 ± 17.7.

1

1.5

2

2.5

3

3.5

4

4.5

5

0.5 1 1.5 2 2.5 3 3.5

lg T

-lg D

α=0.10α=0.25α=0.50α=0.75α=1.00α=1.20α=1.40α=1.60α=1.70α=1.80α=1.90α=1.95α=1.97α=1.98α=1.99α=2.00

Fig. 9. Characteristic escape time as function of the diffusivity D for the doublewell potential.

diffusion on a polymer in chemical space mediated by jumps where the poly-mer loops back on itself [42–44]), the divergence of the fluctuations of Levyprocesses is sometimes considered a pathology. This was already put forwardby West and Seshadri [45], who pointed out that a Levy flight in velocity spacewould be equivalent to a diverging kinetic energy. Here, we show that higherorder dissipation effects lead to natural cutoffs in Levy processes.

At higher velocities the friction experienced by a moving body starts to dependon the velocity itself [46]. Such non-linear friction is known from the classicalRiccati equation Mdv(t)/dt = Mg−Kv(t)2 for the fall of a particle of mass Min a gravitational field with acceleration g [47], or autonomous oscillatory sys-tems with a friction that is non-linear in the velocity [46,48]. The occurrenceof a non-constant friction coefficient γ(V ) leading to a non-linear dissipativeforce −γ(V )V was highlighted in Klimontovich’s theory of non-linear Brow-nian motion [49]. It is therefore natural that higher order, non-linear frictionterms also occur in the case of Levy processes.

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0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

200

400

600

800

1000

1200

1400

1600

<V

2 (t)>

<V

2 (t)>

(α

=1.

2, γ

2=0)

t

α=2.0; γ2=0.0α=1.2; γ2=0.1α=1.5; γ2=0.1α=1.8; γ2=0.1α=1.2; γ2=0.0

Fig. 10. Variance 〈V 2(t)〉 as function of time t, with the quartic term set to zero,γ4 = 0 and γ0 = 1.0 for all cases. The variance is finite for the cases α = 2.0, γ2 = 0.0;α = 1.2, γ2 = 0.1; α = 1.5, γ2 = 0.1; and α = 1.8, γ2 = 0.1. These correspond tothe left ordinate. For the case α = 1.2, γ2 = 0.0, the variance diverges, strongfluctuations are visible; note the large values of this curve corresponding to theright ordinate.

We consider the velocity-dependent dissipative non-linear form (necessarily aneven function) [50]

γ(V ) = γ0 + γ2V2 + . . .+ γ2NV

2N∴ γ2N > 0 (48)

for the friction coefficient of the Levy flight in velocity space as governed bythe Langevin equation

dV (t) + γ(V )V (t)dt = dL(t) (49)

with the constant friction γ0 = γ(0). L(t) is the α-stable Levy noise defined interms of a characteristic function p∗(ω, t) = F{p(L, t)} ≡ ∫

∞

−∞p(L, t) exp (iωL) dL

of the form p∗(ω, t) = exp (−D|ω|αt) [1,51], where D of dimension cmα/sec isthe generalised diffusion constant. This is equivalent to the fractional FokkerPlanck equation [15,37,16,8,9]

∂P (V, t)

∂t=

∂

∂V

(

V γ(V )P)

+D∂αP

∂|V |α . (50)

As we showed in Sec. 4.2 by the example of the Levy flights in position space,the presence of the first higher order correction, γ2V

2 in the friction coefficientγ(V ) rectifies the Levy motion such that the asymptotic power-law becomessteeper and the variance finite. When even higher order corrections are takeninto consideration, also higher order moments become finite. We show an ex-ample in Fig. 10 for the second moment.

The effect on the velocity distribution of the process defined by Eqs. (49) and(50) for higher order corrections are demonstrated in Fig. 11 for the stationary

17

-14

-12

-10

-8

-6

-4

-2

1 2 3 4 5 6 7

ln V

Pst

(V)

ln V

γ2=0.0001, γ4=0γ2=0, γ4=0.000001

Slope -1Slope -3Slope -5

Fig. 11. Stationary PDF Pst(V ) for γ0 = 1.0 and (i) γ2 = 0.0001 and γ4 = 0; and(ii) γ2 = 0 and γ4 = 0.000001; with α = 1.0. The lines indicate the slopes −1, −3,and −5.

limit, Pst(V ) = limt→∞ P (V, t): while for smaller V the character of the originalLevy stable behaviour is preserved (the original power-law behaviour, that is,persists to intermediately large V ), for even larger V the corrections due tothe dissipative non-linearity are visible in the transition(s) to steeper slope(s).

These dissipative non-linearities remove the divergence of the kinetic energyfrom the measurable subsystem of the random walker. In the ideal mathe-matical language, the surrounding bath provides an infinite amount of energythrough the Levy noise, and the coupling via the non-linear friction dissipatesan infinite amount of energy into the bath, and thereby introduces a natu-ral cutoff in the kinetic energy distribution of the random walker subsystem.Physically, such divergencies are not expected, but correspond to the limit-ing procedure of large numbers in probability theory. We showed that bothstatements can be reconciled, and that Levy processes are indeed physical.

Also Gaussian continuum diffusion exhibits non-physical features, possibly themost prominent being the infinite propagation speed inherent of the parabolicnature of the diffusion equation: even at very short times after system prepa-ration in, say, a state P (x, 0) = δ(x), there has already arrived a finite portionof probability at large x. This problem can be corrected by changing from thediffusion to the Cattaneo (telegrapher’s) equation. Still, for most purposes,the uncorrected diffusion equation is used. Similarly, one often uses naturalboundary conditions even though the system under consideration is finite,since one might not be interested in the behaviour at times when a signifi-cant portion of probability has reached the boundaries. In a similar sense, weshowed that ”somewhere out in the wings” Levy flights are naturally cut off bydissipative non-linear effects. However, instead of introducing artificial cutoffs,knowing that for all purposes Levy flights are a good quantitative descriptionand therefore meaningful, we use ”pure” Levy stable laws in physical models.

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8 Discussion

Despite their popularity, comparatively long history, and their Markovian na-ture, Levy flights are not completely understood. The proper formulation inthe presence of non-trivial boundary conditions, their behaviour in externalpotentials both infinitely high and finite, as well as their thermodynamicalmeaning are under ongoing investigation.

While the continuous time random walk model for Levy flights in the ab-sence of non-trivial boundary conditions or external potentials is a convenientdescription, in all other cases the fractional Fokker-Planck equation or, equiva-lently, the Langevin equation with white Levy stable noise are the descriptionof choice. These equations in most cases cannot be solved exactly, however, itis usually straightforward to obtain the asymptotic behaviour, estimates forinflection points etc., or to solve them numerically.

Acknowledgments

RM acknowledges partial financial support through the Natural Sciences andEngineering Research Council (NSERC) of Canada and the Canada ResearchChairs program of the Government of Canada. AVC acknowledges partialsupport from the Deutsche Forschungsgemeinschaft (DFG).

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