PHYSICAL REVIEW E 84, 021137 (2011)

Kramers’ escape problem for fractional Klein-Kramers equationwith tempered α-stable waiting times

Janusz Gajda* and Marcin Magdziarz†

Hugo Steinhaus Center, Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27,PL-50-370 Wroclaw, Poland

(Received 10 January 2011; revised manuscript received 16 June 2011; published 25 August 2011)

In this paper we extend the subdiffusive Klein-Kramers model, in which the waiting times are modeled by theα-stable laws, to the case of waiting times belonging to the class of tempered α-stable distributions. We introducea generalized version of the Klein-Kramers equation, in which the fractional Riemman-Liouville derivative isreplaced with a more general integro-differential operator. This allows a transition from the initial subdiffusivecharacter of motion to the standard diffusion for long times to be modeled. Taking advantage of the correspondingLangevin equation, we study some properties of the tempered dynamics, in particular, we approximate solutionsof the tempered Klein-Kramers equation via Monte Carlo methods. Also, we study the distribution of the escapetime from the potential well and compare it to the classical results in the Kramers escape theory. Finally, wederive the analytical formula for the first-passage-time distribution for the case of free particles. We show thatthe well-known Sparre Andersen scaling holds also for the tempered subdiffusion.

DOI: 10.1103/PhysRevE.84.021137 PACS number(s): 05.40.Fb, 02.70.−c, 05.10.−a, 02.50.Ey

I. INTRODUCTION

The classical Klein-Kramers equation [1],

∂W (x,v,t)

∂t=

[− v

∂

∂x+ ∂

∂v

(ηv − F (x)

m

)

+ ηkBT

m

∂2

∂v2

]W (x,v,t), (1)

describes position x and velocity v of the particle diffusingin the external force field F (x) = −V ′(x). Here, V (x) isthe external potential, η denotes the friction coefficient, kBT

is the Boltzmann temperature, and m is the mass of theparticle. The initial condition is assumed here, W (x,v,0) =δ(x,v). This model was used by Kramers to study thedependence of the escape rate on the temperature and viscosity.Metzler and Klafter [2] introduced a fractional extension of theclassical Klein-Kramers equation (FKKE) in the form

∂W (x,v,t)

∂t= 0D

1−αt

[− γ v

∂

∂x+ γ

∂

∂v

(ηv − F (x)

m

)

+ γ ηkBT

m

∂2

∂v2

]W (x,v,t). (2)

The operator 0D1−αt , 0 < α < 1 stands for the fractional

Riemman-Liouville derivative, and is responsible for memoryeffects in the model. Moreover, the constant γ is the ratio ofthe intertrapping time scale and the internal waiting time scale[2]. For other fractional generalizations of the Klein-Kramersequation, see [3–6].

The Klein-Kramers equation plays a fundamental role inmodeling the particle escape over a barrier and many otherphysical and chemical processes. The fractional extension suitswell in investigating properties of systems characterized bysubdiffusive dynamics. Equation (2) describes the position

*janusz.gajda@pwr.wroc.pl†marcin.magdziarz@pwr.wroc.pl

and velocity of the Brownian particle, which is successivelyimmobilized in traps. The waiting times when the particle staysmotionless are drawn from the power-law probability densityfunction (PDF). It is assumed that after each trapping eventthe particle is released with the same position and velocity thatit had prior to the immobilization. The heavy-tailed waitingtimes considerably slow the overall motion and lead to thesublinear in time mean-square displacement of the particle,which is typical for subdiffusion.

The subdiffusive regime was confirmed in a variety ofphysical systems (see [7] and the references therein). However,in a number of cases we observe a characteristic transition fromthe initial subdiffusive character of motion to the standarddiffusion for long times. Such a transition was experimentallyconfirmed in [8] for the case of random motion of photosphericbright points. The transition from anomalous to normaldiffusion was also observed in the motion of moleculesinside living cells [9,10]. Similar effects were discovered veryrecently in the dynamics of lipid granules in living fission yeastcells [11].

To capture such a transition observed in physical systems,we propose a modification of the waiting-time distribution.The modification consists of the appropriate truncation ofthe heavy-tailed α-stable waiting times in the underlyingcontinuous-time random walk (CTRW) scenario. This trunca-tion eventually abolishes the anomalous character of motion,and thus for long times the standard diffusion is observed.

In physics, probably the first successful approach to theproblem of truncating heavy-tailed distributions was presentedin [12] (see also [13]). Recently, the family of temperedα-stable distributions was introduced independently in [14]and [15]. The truncation of stable laws proposed in [14,15]was on the level of the Levy measure, which resulted inmany desired properties of the introduced tempered laws.Specifically, the tempered α-stable distributions belong tothe mathematically relevant family of infinitely divisibledistributions. Moreover, they have finite moments of all orders;on the other hand, they resemble stable laws in many ways.

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JANUSZ GAJDA AND MARCIN MAGDZIARZ PHYSICAL REVIEW E 84, 021137 (2011)

Applications of the tempered stable distributions in physicsin the context of astrophysics and relaxation can be foundin [16]. For the Fokker-Planck equation describing tempereddynamics, see [16,17]. Other important applications relatedto finance and geophysics can be found in [18,19] and [20],respectively.

In this paper we use the tempered α-stable laws todescribe the transition from anomalous (subdiffusive) tonormal motion on the level of the Klein-Kramers equation.Using the subordination method, we introduce a model whichdescribes the position and velocity of the Brownian particle,whose motion is interrupted by the trapping events distributedaccording to the tempered stable laws. We show that the PDFof the introduced model corresponds to the generalized FKKE,where the Riemman-Liouville derivative is replaced by a moregeneral memory operator. Moreover, by the use of MonteCarlo methods we approximate the trajectories of the temperedsubdiffusion and detect some relevant statistical properties ofthe tempered dynamics. In Sec. III we investigate the survivalprobability of a Brownian particle immersed in a potentialhole and slowed down by tempered α-stable waiting times.We compare our results to the ones known from the classicaland fractional Kramers escape theory. Finally, we derive theanalytical formula for the first-passage-time distribution forthe case of free particle.

II. LANGEVIN PICTURE OF THE FKKE

The underlying stochastic structure of Eq. (2) was presentedin [21], where it was shown that the PDF P (x,v,t) of thetwo-dimensional stochastic process

Y(t) = [X(St ),V (St )] (3)

solves the FKKE (2). The process [X(τ ),V (τ )] is given by thefollowing Ito stochastic differential equation:

dV (τ ) = γ

(−ηV (τ ) + F [X(τ )]

m

)dτ +

√2γ η

kBT

mdB(τ ),

(4)dX(τ ) = γV (τ )dτ,

where B(τ ) is the standard Brownian motion. The processSt is the so-called inverse α-stable subordinator defined asSt = inf{τ : U (τ ) > t}. Here, U (τ ) is the strictly increasingα-stable Levy motion with the Laplace transform of the formE(e−sU (τ )) = e−τsα

, 0 < α < 1, [22]. The inverse subordinatoris a new operational time of the system. It introduces the ran-dom periods of time when the particle stays motionless. Thesewaiting times slow down the overall motion, which results insubdiffusion. In what follows, we propose a modification of(3) in which the α-stable waiting times are replaced with thetempered α-stable laws. This modification allows the transitionfrom the initial subdiffusive dynamics to the standard diffusionfor long times to be modelled.

The tempered α-stable random variable Tα,λ > 0 is definedvia its Laplace transform [15]

E(e−uTα,λ ) = e−[(u+λ)α−λα )],

where 0 < α < 1, λ > 0. The constant λ is responsible fortruncating the α-stable distribution. This can be best seenon the level of the corresponding PDFs. The PDF of the

tempered stable random variable has the form gα,λ(x) =ce−λxfα(x), where fα(x) is the PDF of the one-sided α-stablerandom variable and c is the appropriate normalizing constant.Exponential truncating makes the moments of all orderscorresponding to gα,λ(x) finite, which is particularly attractivefor physical applications of tempered laws. Moreover, takingλ = 0 we obtain the ordinary α-stable distribution.

For each tempered α-stable random variable Tα,λ, weintroduce the corresponding stochastic process Tα,λ(τ ) via theLaplace transform

E(e−uTα,λ(τ )) = e−τ [(u+λ)α−λα ]. (5)

Consequently, we define the first-passage-time process Sα,λ(t),called the the inverse subordinator, as

Sα,λ(t) = inf{τ > 0 : Tα,λ > t}, t � 0.

Finally, we replace St with Sα,λ(t) in (3) to get

Y(t) = [X(Sα,λ(t)],V [Sα,λ(t)]. (6)

In this way we obtain the Langevin-like process describingthe position and velocity of a Brownian particle, which issuccessively immobilized in traps distributed according tothe tempered stable laws. As shown in [23], the mean ofthe inverse subordinator satisfies 〈Sα,λ(t)〉 ∝ tα for smallt . Moreover, Sα,λ(t) ∝ t as t → ∞. Thus, for small timesX[Sα,λ(t)] displays subdiffusive dynamics, whereas for largetime scales the process behaves as the standard diffusion. Thus,by applying the tempered stable waiting times, we are ableto recover the desired transition from anomalous to normaldiffusion.

Having defined a new Langevin process Y(t), we introducethe corresponding generalized version of fractional Klein-Kramers equation

∂W (x,v,t)

∂t= t

[− γ v

∂

∂x+ γ

∂

∂v

(ηv − F (x)

m

)

+ γ ηkBT

m

∂2

∂v2

]W (x,v,t). (7)

Here, W (x,v,0) = δ(x,v) and t is the integro-differentialoperator defined as

tf (t) = d

dt

∫ t

0M(t − y)f (y)dy,

with the memory kernel given via its Laplace transform

M(u) =∫ ∞

0e−utM(t)dt = 1

(u + λ)α − λα.

The constant γ is the ratio γ = τ ∗/τ of the intertrappingtime scale τ ∗ and the internal waiting time scale τ [2]. Itis worth mentioning that for λ = 0 the integro-differentialoperator t (·) is equal to the fractional Riemman-Liouvillederivative D1−α

t . For λ = 0, α = 1 Eq. (7) reduces to theordinary FKKE equation. Moreover, the operator −1

t ( ∂∂t

)is the tempered fractional derivative, which was introducedin [14] (see also [20,24]).

In what follows, we prove that the PDF of the process Y(t)defined in (6) is the solution of generalized FKKE (7). Taking

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advantage of total probability formula, the PDF p(x,v,t) ofthe process (6) has the form

p(x,v,t) =∫ ∞

0f (x,v,τ )g(τ,t)dτ. (8)

Here, by f (x,v,τ ) and g(τ,t) we denote the PDFs of[X(τ ),V (τ )] and Sα,λ(t), respectively. Denote the PDF ofTα,λ(τ ) by h(t,τ ), then

g(τ,t) = − ∂

∂τ

∫ t

−∞h(t ′,τ )dt ′. (9)

By some standard calculations we get that the Laplacetransform of g(τ,t) is equal to

g(τ,u) = (u + λ)α − λα

ue−τ [(u+λ)α−λα]. (10)

Consequently, in the Laplace space we have

p(x,v,u) =∫ ∞

0e−utp(x,v,t)dt

=∫ ∞

0f (x,v,τ )g(τ,u)dτ

=∫ ∞

0f (x,v,τ )

(u + λ)α − λα

ue−τ [(u+λ)α−λα]dτ

= (u + λ)α − λα

uf [x,v,(u + λ)α − λα]. (11)

Since the process [X(τ ),V (τ )] is given by (4), its PDFf (x,v,τ ) obeys the ordinary Klein-Kramers equation

∂f (x,v,t)

∂t=

[− γ v

∂

∂x+ γ

∂

∂v

(ηv − F (x)

m

)

+ γ ηkBT

m

∂2

∂v2

]f (x,v,t), (12)

Equation (12) in the Laplace space u yields

uf (x,v,u) − f (x,v,0) =[

− γ v∂

∂x+ γ

∂

∂v

(ηv − F (x)

m

)

+ γ ηkBT

m

∂2

∂v2

]f (x,v,u). (13)

The above formula after the change of variables u → (u +λ)α − λα together with (11) gives

up(x,v,u) − p(x,v,0) = u

(u + λ)α − λα

[− γ v

∂

∂x

+ γ∂

∂v

(ηv − F (x)

m

)+ γ η

kBT

m

∂2

∂v2

]p(x,v,u). (14)

By inverting the Laplace transform we obtain Eq. (7). Thus,we have proved that the PDF p(x,v,t) of the process[X(Sα,λ),V (Sα,λ)] is the solution of the fractional Klein-Kramers equation (7).

Analogously to the case of FKKE (2), the assumption ofthe tempered model is that following every trapping event theparticle is released with the same velocity which it had priorto the immobilization. Consequences of this assumption canbe seen in Fig. 1. During the trapping periods the processV [Sα,λ(t)] is constant and nonzero, although the position ofthe particle does not change. Thus V [Sα,λ(t)] should not be

0 2 4 6 8 10−2

0

2

4

t

V(S

α,λ(t

))

0 2 4 6 8 10−2

0

2

4

t

X(S

α,λ(t

))

(a)

(b)

FIG. 1. Exemplary realizations of the velocity process V [Sα,λ(t)](panel a) and the position process X[Sα,λ(t)] (panel b). The constantintervals represent the tempered α-stable waiting times in which theparticle stays motionless. Due to the assumption that following everytrapping event the particle is released with the same velocity whichit had prior to the immobilization, the process V [Sα,λ(t)] is nonzerowhile position of the particle does not change. Thus, V [Sα,λ(t)] shouldnot be considered a usual particle velocity. The parameters are α =0.95,λ = 0.01,kBT = m = η = 1,F (x) = 0.

considered a usual particle velocity. To overcome this problemone could possibly modify the model by putting V [Sα,λ(t)] = 0during every constant period. However, such a modificationwould make the trajectories of V [Sα,λ(t)] discontinuous.Another consequence of the aforementioned assumption isthe violation of the classical Newtonian relation betweenposition and velocity. It is straightforward to verify thatthe following relationship between the position and velocityprocesses holds (d/dt)〈X[Sα,λ(t)]〉 = tγ 〈V [Sα,λ(t)]〉. Thisclearly violates the Newtonian relation. As argued in [25],this violation of Newton’s law is due to the additional waitingtime average, which camouflages the Newtonian, Langevin-dominated events.

The Langevin picture (6) allows us to introduce an effi-cient algorithm of simulating sample paths of the temperedsubdiffusion. The algorithm is analogous to the one presentedrecently in [17]. Every two-dimensional trajectory of Y(t) isobtained by subordinating the two-dimensional trajectory of[X(τ ),V (τ )] with the subordinator Sα,λ(t).

In the first step of the algorithm one approximates thetrajectory of the inverse subordinator Sα,λ(t). In the secondstep the trajectories of diffusion [X(τ ),V (τ )] are simulated.Finally, by putting together the trajectories of [X(τ ),V (τ )]and Sα,λ(t), one obtains the tempered subdiffusion. For thedetails of the algorithm, see [17,24].

The above algorithm allows us to simulate efficiently thetrajectories of Y(t). In Fig. 1 a sample path of the temperedsubdiffusion is depicted. The constant intervals representtempered α-stable trapping events when the particle staysmotionless. In Figs. 2 and 3 we present the quantile lines [22]of X[Sα,λ(t)],V [Sα,λ(t)], which give insight into the dynamicsof Y(t). The quantile lines were estimated using Monte Carlo

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JANUSZ GAJDA AND MARCIN MAGDZIARZ PHYSICAL REVIEW E 84, 021137 (2011)

0 5 10 15 20 25 30−3

−2

−1

0

1

2

3

t

V(S

α,λ(t

))

90%

10%20%30%40%50%60%70%80%

FIG. 2. (Color online) Sample path [blue (gray) line] and the es-timated quantile lines (10%,20%, . . . ,90%) of the process V [Sα,λ(t)]in the presence of the external force F (x) = −V ′(x) = −(x3 −16x)/20. The results were obtained via Monte Carlo simulationon the basis of 104 simulated trajectories. The parameters areα = 0.95,λ = 0.01,kBT = m = η = 1.

methods for the case of external force F (x) = −V ′(x) =−(x3 − 16x)/20. In Fig. 4 we show the estimated stationarysolution of (7) compared with the theoretical one.

III. ESCAPE FROM A POTENTIAL WELL

In this section we consider the survival probability of aparticle trapped in a potential hole. Initially the particle issitting deep in the potential well and through the irregularshuttling action of noise can escape over a barrier (seeFig. 5). In the simplest situation the noise is modeled byBrownian motion [1]. More recently, the fractional extensionof the classical Klein-Kramers equation was proposed [2],

0 5 10 15 20 25 30−6

−4

−2

0

2

4

6

t

X(S

α,λ(t

))

10%20%30%40%

50%

60%

80%70%

90%

FIG. 3. (Color online) Two sample paths [blue (gray) lines] andthe estimated quantile lines (10%,20%, . . . ,90%) of the processX[Sα,λ(t)]. One observes two equilibrium states at x = ±4. Parame-ters as in Fig. 2.

FIG. 4. (Color online) Theoretical and estimated stationary solu-tion of FKKE (7) in the case of double-well potential. Parametersas in Fig. 2. The numerical results presented in Figs. 1–4 are verysimilar to the ones obtained for the case of stable waiting times in [21].However, the difference between the stable and tempered stable caseis clearly visible in the behavior of survival probability (see Figs.6–8).

inducing new considerations on the Kramers escape theory.The heavy-tailed waiting times in the CTRW underlying (2)led to the Mittag-Leffler relaxation of the survival probability,replacing the classical exponential one. The case of longwaiting times coexisting with long jumps was consideredin [26]. The survival probability corresponding to the fractionalGaussian noise was considered in detail in [27].

V(xmin

)

Δ V

V(xmax

)

xmax

xmin

V(x)

FIG. 5. (Color online) Potential well in the Kramers escapemodel. Initially the particle is caught in the potential hole of thecoordinate xmin and can only escape by jumping over a barrier ofheight V = V (xmax) − V (xmin) placed in xmax.

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KRAMERS’ ESCAPE PROBLEM FOR FRACTIONAL KLEIN- . . . PHYSICAL REVIEW E 84, 021137 (2011)

In this section we study the Kramers escape problemcorresponding to the tempered subdiffusion process. Themotion of the particle sitting in the potential well is describedby the position process in the introduced tempered Klein-Kramers model (7). Thus, in the high-friction limit, the particleposition follows the generalized Fokker-Planck equation:

∂W (x,t)

∂t= t

[− ∂

∂x

F (x)

mη+ Kα

∂2

∂x2

]W (x,t), (15)

where Kα is the generalized diffusion constant. Equivalently,the position of the particle is described by the Langevin-like process Y (t) = X[Sα,λ(t)], where Sα,λ(t) is the inversesubordinator, and X(τ ) is the standard diffusion processsatisfying

dX(τ ) = F [X(τ )]

mηdτ +

√2KαdB(τ ).

Denote by τX the first passage time of the process X(τ )over the potential barrier. Since X(τ ) is a standard diffusionprocess, τX is exponentially distributed. The correspondingprobability of finding the particle within the potential well(survival probability) is given by [25,28]

pX(t) = P (τX > t) = e−rK t , (16)

for the appropriate Kramers rate rK .In what follows, by Z(u) we denote the Laplace transform

of a random variable Z, i.e.,

Z(u) = E(e−uZ) =∫ ∞

0e−uzg(z)dz, (17)

where g(z) is the PDF of Z.By knowing the survival probability of X(τ ), we will derive

the survival probability for the tempered subdiffusion processY (t) = X[Sα,λ(t)]. Denote by τY the first passage time of theprocess Y (t) over the potential barrier. Then, we get

τY

d= Tα,λ(τX). (18)

Taking advantage of the above fact, the Laplace transform ofτY is then given by

τY (u) = rK

[(u + λ)α − λα] + rK

.

Thus, for the survival probability, we have∫ ∞

0e−utP (τY > t)dt =

1 − rK

[(u+λ)α−λα ]+rK

u

= [(u + λ)α − λα]

u(rK + [(u + λ)α − λα]).

Using the Taylor expansion, we obtain for small u∫ ∞

0e−utP (τY > t)dt ∝ 1

rK/(αλα−1) + u. (19)

Inverting the above, we see that the survival probability of Y (t)satisfies

pY (t) ∝ e−t[rK/(αλα−1)] (20)

for large t . The exponential decay of pY (t) can also be derivedfrom the fact that Sα,λ(t) ∝ t for large t .

0 200 400 600 800 100010

−5

10−4

10−3

10−2

10−1

100

t

ln (

p(t)

)

Temp. stab. α=0.80, λ=0.1Temp. stab. α=0.80, λ=0.01Temp. stab. α=0.80, λ=0.001

FIG. 6. Behavior of the survival probability for the temperedwaiting times with different values of the tempering parameter λ.For very small λ, the exponential decay is observed only for largetimes.

We have performed Monte Carlo simulations for theclassical and for both α-stable and tempered α-stable waiting-times models. We have investigated the survival probabilityfor the potential of the form V (x) = −x3/3 + x (see Fig. 5).Initially a test particle was placed in a potential hole ofcoordinate xmin = −1 and started its motion. When the particlecrossed the barrier of coordinate xmax = 1 for the first time,the iteration stopped and the time instant was remembered.Such simulations were performed 105 times. The survivalprobability was calculated via the Monte Carlo method usingthe collected first passage times. Figure 6 depicts the survivalprobability for different parameters λ. One can see that for verysmall tempering parameter λ the exponential decay is observedonly for large times. In Fig. 7 we observe the behavior of thesurvival probability for the case of tempered stable waitingtimes with different parameters α. In each case we observe theexponential decay. In Fig. 8 we have compared the survivalprobability of the tempered case with the classical and stableones. For smaller values of λ the tempered model is closerto the model with stable waiting times. However, the greatervalues of λ yield the classical exponential decay of the survivalprobability.

IV. FIRST PASSAGE TIMES FOR THE FORCE-FREE CASE

In this section we calculate first passage times for thetempered subdiffusion. For recent advances in the area of firstpassage times for anomalous diffusion, see [29] and referencestherein. We consider here the case of free particles (no externalforce or confinement is present). Thus, the analyzed processhas the form B(t) = B[Sα,λ(t)], where B is the standardBrownian motion and Sα,λ(t) is the inverse subordinator.

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JANUSZ GAJDA AND MARCIN MAGDZIARZ PHYSICAL REVIEW E 84, 021137 (2011)

0 200 400 600 800 100010

−5

10−4

10−3

10−2

10−1

100

t

ln (

p(t)

)

Temp. stab. α=0.75, λ=0.01Temp. stab. α=0.80, λ=0.01Temp. stab. α=0.85, λ=0.01Temp. stab. α=0.90, λ=0.01Temp. stab. α=0.95, λ=0.01

FIG. 7. Behavior of the survival probability for the temperedwaiting times with different values of α. For each case the exponentialdecay is observed.

Denote the first passage time over the distance d for Brownianmotion B by

τB(d) = inf{t � 0 : B(t) > d}.

101

102

103

10−6

10−5

10−4

10−3

10−2

10−1

100

ln (t)

ln (

p(t)

)

Temp. stab. α=0.80, λ=0.1Temp. stab. α=0.80, λ=0.01Temp. stab. α=0.80, λ=0.001Class. KK. modelStab. α=0.80

FIG. 8. Comparison of the survival probability corresponding tothe classical, stable, and tempered stable models. One can observethat for smaller values of λ the results are closer to the α-stable case.However, for greater values of the tempering parameter λ, the survivalprobability displays the classical exponential behavior.

Similarly, the first passage time over the distance d for thetempered subdiffusion B(t) = B[Sα,λ(t)] will be denoted by

τB(d) = inf{t � 0 : B(t) > d}.Let us recall that the Laplace transform of τB(d) is givenby [30]

E(e−uτB (d)) = e−d√

2√

u.

Thus, using the following relation from the previous section,

τB(d)d= Tα,λ[τB(d)], (21)

we get that the Laplace transform of τB(d) is given by

E(e−uτB (d)) = e−[(u+λ)α−λα ](1/2)d√

2. (22)

Therefore, we have∫ ∞

0e−utP [τB(d) > t]dt = 1 − e−uτB (d)

u

= 1 − e−[(u+λ)α−λα ](1/2)d√

2

u.

Using the Taylor expansion, we obtain∫ ∞

0e−utP [τB(d)] > tdt ∝ d

√2αλα−1u−1/2 (23)

for small u. Finally, from the Tauberian theorem [30] we obtainthe following asymptotics,

P [τB(d) > t] ∝ d√

2αλα−1

√π

t−1/2, (24)

for large times t .This proves that the first passage time distribution of the

tempered subdiffusion follows the Sparre Andersen universal-ity [31]. Numerical results presented in Figs. 9 and 10 confirmthe correctness of the derived formulas.

101

102

103

10−2

10−1

100

ln (t)

ln (

P(τ

B(d

)>t)

)

α=0.9, λ=0.1, d=3Analytical resultα=0.9, λ=0.01, d=3Analytical result

∫

FIG. 9. (Color online) Behavior of P [τB (d) > t] with differentvalues of the tempering parameter λ. The red solid lines representthe theoretical results for the asymptotics of the first-passage-timedistribution. One can observe a very good agreement between theoryand simulations.

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101

102

103

10−2

10−1

100

ln (t)

ln (

P(

τ B(d

)>t)

)

α=0.70, λ=0.01, d=3Analytical resultα=0.80, λ=0.01, d=3Analytical resultα=0.90, λ=0.01, d=3Analytical result

∫

FIG. 10. (Color online) Depicted is the behavior of P [τB (d) > t]with different values of the parameter α. The red solid lines arethe theoretical results for the asymptotic behavior of the first-passage-time distribution. A very good agreement between theoryand simulations is observed. This confirms that the standard SparreAndersen scaling holds also for the tempered subdiffusion.

However, deviations from the Sparre Andersen scalingcan be observed for very small values of the parameter λ.Note that for λ ↘ 0 the Laplace transform in (22) takes thewell-known stretched exponential form e−uα/2d

√2. The tail of

the corresponding first passage time distribution decays ast−α/2, which is clearly different from the Sparre Andersenbehavior. Thus, depending on the choice of the parameter λ

one can control the scaling properties of the first passage time.For very small λ the results are closer to the α-stable case—forgreater values of the tempering parameter λ the classicalSparre Andersen scaling is observed. This fact was confirmednumerically; the obtained results were similar in nature to theones presented in Fig. 8 for the survival probability.

One of the most characteristic properties of the temperedsubdiffusion is the crossover from the initial anomalousbehavior to the standard diffusive one for large times. The meansquare displacement of the tempered subdiffusion process forthe force-free case is given by [16]

E(B2[Sα,λ(t)]) =∫ t

0e−λyyα−1Eα,α(λαyα)dy,

where Eα,β (z) is the Mittag-Leffler function [32]. Conse-quently,

E(B2[Sα,λ(t)]) ∝ tα

for t ↘ 0 and

E(B2[Sα,λ(t)]) ∝ t

for t ↗ ∞. This crossover from anomalous to normal dynam-ics is depicted in Fig. 11. Consequently, in the initial phasethe process B[Sα,λ(t)] behaves as a subdiffusion, whereas forlarge times we observe the standard diffusive dynamics. Thisproperty strongly affects the behavior of the first passage timeτB(d) of the tempered subdiffusion. As for the case of meansquare displacement, we observe a crossover from subdiffusive

10−1

100

101

102

10−1

100

101

102

ln (t)

ln (

<x

2 (t)>

)

Tempered case

asymptotics ~ tα

asymptotics ~ t

FIG. 11. (Color online) (Color online). The behavior of the meansquare displacement of the tempered subdiffusion. One observes acrossover from the initial anomalous to the eventual linear in timebehavior of the mean square displacement. Here, α = 0.75, λ = 0.1.

to normal regime. The tail of the first passage time distributionP [τB(d) > t] agrees with the subdiffusive regime for small t ,whereas for large t the classical Sparre Andersen scaling isobserved. This crossover is depicted in Fig. 12.

V. CONCLUSIONS

In this paper we have introduced the generalized Klein-Kramers model, in which the standard diffusion process isslowed down by the waiting times, during which the particlegets immobilized. The introduced model is a modificationof the subdiffusive Klein-Kramers model discussed in [2].

100

101

102

103

10−2

10−1

100

ln (t)

ln (

1−C

DF

(FP

T(t

))

Brownian Motion, d=1Tempered case λ=0.01, α=0.8, d=1Stable case α=0.8, d=1Theoretical asymptotics

t−1/2

t−α/2

FIG. 12. (Color online) Shown is the behavior of the first passagetime of the tempered subdiffusion. For small t the tail of the firstpassage time distribution agrees with the subdiffusive one, whereasfor large t the classical Sparre Andersen scaling is recovered.

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JANUSZ GAJDA AND MARCIN MAGDZIARZ PHYSICAL REVIEW E 84, 021137 (2011)

The modification consists of replacing the stable waitingtimes with the tempered stable ones in the underlying CTRWscenario. We have derived the corresponding Langevin andKlein-Kramers equations. These fundamental equations wereused to investigate the dynamics of the process.

We have shown that the survival probability of the temperedmodel decays exponentially; however, for small values of thetempering parameter, the subdiffusive power-law behavior isrecovered. Moreover, we have derived the analytical formula

for the first-passage-time distribution for the case of freeparticle. Our results confirmed that the Sparre Andersenscaling holds also for the tempered subdiffusion.

The proposed model of tempered subdiffusion is capable ofdescribing the transition from the initial subdiffusive characterof motion to the standard diffusion for long times. Sincesuch a transition was observed in a number of physical andbiological systems, we believe that our investigations willprovide additional tools for the analysis of such systems.

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