Kramers' escape problem for fractional Klein-Kramers equation with

PHYSICAL REVIEW E 84, 021137 (2011)

Kramers’ escape problem for fractional Klein-Kramers equation with 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 the α-stable laws, to the case of waiting times belonging to the class of tempered α-stable distributions. We introduce a generalized version of the Klein-Kramers equation, in which the fractional Riemman-Liouville derivative is replaced with a more general integro-differential operator. This allows a transition from the initial subdiffusive character of motion to the standard diffusion for long times to be modeled. Taking advantage of the corresponding Langevin equation, we study some properties of the tempered dynamics, in particular, we approximate solutions of the tempered Klein-Kramers equation via Monte Carlo methods. Also, we study the distribution of the escape time from the potential well and compare it to the classical results in the Kramers escape theory. Finally, we derive the analytical formula for the first-passage-time distribution for the case of free particles. We show that the 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) F (x) ∂ ∂ ηv − = −v + ∂t ∂x ∂v m 2 kB T ∂ W (x,v,t), +η m ∂v 2

(1)

describes position x and velocity v of the particle diffusing in the external force field F (x) = −V (x). Here, V (x) is the external potential, η denotes the friction coefficient, kB T is the Boltzmann temperature, and m is the mass of the particle. The initial condition is assumed here, W (x,v,0) = δ(x,v). This model was used by Kramers to study the dependence of the escape rate on the temperature and viscosity. Metzler and Klafter [2] introduced a fractional extension of the classical Klein-Kramers equation (FKKE) in the form ∂W (x,v,t) ∂ ∂ F (x) = 0 Dt1−α − γ v +γ ηv − ∂t ∂x ∂v m 2 kB T ∂ W (x,v,t). (2) +γη m ∂v 2 The operator 0 Dt1−α , 0 < α < 1 stands for the fractional Riemman-Liouville derivative, and is responsible for memory effects in the model. Moreover, the constant γ is the ratio of the intertrapping time scale and the internal waiting time scale [2]. For other fractional generalizations of the Klein-Kramers equation, see [3–6]. The Klein-Kramers equation plays a fundamental role in modeling the particle escape over a barrier and many other physical and chemical processes. The fractional extension suits well in investigating properties of systems characterized by subdiffusive dynamics. Equation (2) describes the position

* †

[email protected] [email protected]

1539-3755/2011/84(2)/021137(8)

and velocity of the Brownian particle, which is successively immobilized in traps. The waiting times when the particle stays motionless are drawn from the power-law probability density function (PDF). It is assumed that after each trapping event the particle is released with the same position and velocity that it had prior to the immobilization. The heavy-tailed waiting times considerably slow the overall motion and lead to the sublinear in time mean-square displacement of the particle, which is typical for subdiffusion. The subdiffusive regime was confirmed in a variety of physical systems (see [7] and the references therein). However, in a number of cases we observe a characteristic transition from the initial subdiffusive character of motion to the standard diffusion for long times. Such a transition was experimentally confirmed in [8] for the case of random motion of photospheric bright points. The transition from anomalous to normal diffusion was also observed in the motion of molecules inside living cells [9,10]. Similar effects were discovered very recently in the dynamics of lipid granules in living fission yeast cells [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 of the heavy-tailed α-stable waiting times in the underlying continuous-time random walk (CTRW) scenario. This truncation 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 the problem of truncating heavy-tailed distributions was presented in [12] (see also [13]). Recently, the family of tempered α-stable distributions was introduced independently in [14] and [15]. The truncation of stable laws proposed in [14,15] was on the level of the Lévy measure, which resulted in many desired properties of the introduced tempered laws. Specifically, the tempered α-stable distributions belong to the mathematically relevant family of infinitely divisible distributions. Moreover, they have finite moments of all orders; on the other hand, they resemble stable laws in many ways.

021137-1

©2011 American Physical Society

JANUSZ GAJDA AND MARCIN MAGDZIARZ


Applications of the tempered stable distributions in physics in the context of astrophysics and relaxation can be found in [16]. For the Fokker-Planck equation describing tempered dynamics, see [16,17]. Other important applications related to finance and geophysics can be found in [18,19] and [20], respectively. In this paper we use the tempered α-stable laws to describe the transition from anomalous (subdiffusive) to normal motion on the level of the Klein-Kramers equation. Using the subordination method, we introduce a model which describes the position and velocity of the Brownian particle, whose motion is interrupted by the trapping events distributed according to the tempered stable laws. We show that the PDF of the introduced model corresponds to the generalized FKKE, where the Riemman-Liouville derivative is replaced by a more general memory operator. Moreover, by the use of Monte Carlo methods we approximate the trajectories of the tempered subdiffusion and detect some relevant statistical properties of the tempered dynamics. In Sec. III we investigate the survival probability of a Brownian particle immersed in a potential hole and slowed down by tempered α-stable waiting times. We compare our results to the ones known from the classical and fractional Kramers escape theory. Finally, we derive the analytical formula for the first-passage-time distribution for the case of free particle. II. LANGEVIN PICTURE OF THE FKKE

The underlying stochastic structure of Eq. (2) was presented in [21], where it was shown that the PDF P (x,v,t) of the two-dimensional stochastic process ˜ Y(t) = [X(St ),V (St )]

(3)

solves the FKKE (2). The process [X(τ ),V (τ )] is given by the following Itô stochastic differential equation: kB T F [X(τ )] dτ + 2γ η dB(τ ), dV (τ ) = γ −ηV (τ ) + m m (4) dX(τ ) = γ V (τ )dτ, where B(τ ) is the standard Brownian motion. The process St is the so-called inverse α-stable subordinator defined as St = inf{τ : U (τ ) > t}. Here, U (τ ) is the strictly increasing α-stable Lévy motion with the Laplace transform of the form α E(e−sU (τ ) ) = e−τ s , 0 < α < 1, [22]. The inverse subordinator is a new operational time of the system. It introduces the random periods of time when the particle stays motionless. These waiting times slow down the overall motion, which results in subdiffusion. In what follows, we propose a modification of (3) in which the α-stable waiting times are replaced with the tempered α-stable laws. This modification allows the transition from the initial subdiffusive dynamics to the standard diffusion for long times to be modelled. The tempered α-stable random variable Tα,λ > 0 is defined via its Laplace transform [15] E(e−uTα,λ ) = e−[(u+λ)

α

−λα )]

,

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

tempered stable random variable has the form gα,λ (x) = ce−λx fα (x), where fα (x) is the PDF of the one-sided α-stable random variable and c is the appropriate normalizing constant. Exponential truncating makes the moments of all orders corresponding to gα,λ (x) finite, which is particularly attractive for physical applications of tempered laws. Moreover, taking λ = 0 we obtain the ordinary α-stable distribution. For each tempered α-stable random variable Tα,λ , we introduce the corresponding stochastic process Tα,λ (τ ) via the Laplace 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 describing the position and velocity of a Brownian particle, which is successively immobilized in traps distributed according to the tempered stable laws. As shown in [23], the mean of the inverse subordinator satisfies Sα,λ (t) ∝ t α for small t. Moreover, Sα,λ (t) ∝ t as t → ∞. Thus, for small times X[Sα,λ (t)] displays subdiffusive dynamics, whereas for large time scales the process behaves as the standard diffusion. Thus, by applying the tempered stable waiting times, we are able to recover the desired transition from anomalous to normal diffusion. Having defined a new Langevin process Y(t), we introduce the corresponding generalized version of fractional KleinKramers equation ∂W (x,v,t) ∂ ∂ F (x) = t − γ v +γ ηv − ∂t ∂x ∂v m 2 kB T ∂ +γη W (x,v,t). (7) m ∂v 2 Here, W (x,v,0) = δ(x,v) and t is the integro-differential operator defined as t d t f (t) = M(t − y)f (y)dy, dt 0 with the memory kernel given via its Laplace transform ∞ 1 ˆ e−ut M(t)dt = . M(u) = (u + λ)α − λα 0 The constant γ is the ratio γ = τ ∗ /τ of the intertrapping time scale τ ∗ and the internal waiting time scale τ [2]. It is worth mentioning that for λ = 0 the integro-differential operator t (·) is equal to the fractional Riemman-Liouville derivative Dt1−α . For λ = 0, α = 1 Eq. (7) reduces to the ∂ ordinary FKKE equation. Moreover, the operator −1 t ( ∂t ) is the tempered fractional derivative, which was introduced in [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

021137-2

KRAMERS’ ESCAPE PROBLEM FOR FRACTIONAL KLEIN- . . .

By some standard calculations we get that the Laplace transform of g(τ,t) is equal to (u + λ)α − λα −τ [(u+λ)α −λα ] . (10) e u Consequently, in the Laplace space we have ∞ ˆ p(x,v,u) = e−ut p(x,v,t)dt 0 ∞ ˆ f (x,v,τ )g(τ,u)dτ = 0 ∞ (u + λ)α − λα −τ [(u+λ)α −λα ] e dτ = f (x,v,τ ) u 0 (u + λ)α − λα ˆ f [x,v,(u + λ)α − λα ]. = (11) u Since the process [X(τ ),V (τ )] is given by (4), its PDF f (x,v,τ ) obeys the ordinary Klein-Kramers equation ∂ ∂ F (x) ∂f (x,v,t) = − γv +γ ηv − ∂t ∂x ∂v m 2 kB T ∂ f (x,v,t), (12) +γη m ∂v 2 ˆ g(τ,u) =

Equation (12) in the Laplace space u yields ∂ ∂ F (x) ˆ uf (x,v,u) − f (x,v,0) = − γ v +γ ηv − ∂x ∂v m 2 kB T ∂ fˆ(x,v,u). (13) +γη m ∂v 2 The above formula after the change of variables u → (u + λ)α − λα together with (11) gives ∂ u ˆ − γv up(x,v,u) − p(x,v,0) = (u + λ)α − λα ∂x F (x) kB T ∂ 2 ∂ ˆ p(x,v,u). (14) ηv − + γη +γ ∂v m m ∂v 2 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 KleinKramers equation (7). Analogously to the case of FKKE (2), the assumption of the tempered model is that following every trapping event the particle is released with the same velocity which it had prior to the immobilization. Consequences of this assumption can be seen in Fig. 1. During the trapping periods the process V [Sα,λ (t)] is constant and nonzero, although the position of the particle does not change. Thus V [Sα,λ (t)] should not be

(t))

α,λ

V(S

0 −2 0

2

4

6

8

10

6

8

10

t 4

(t))

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 of Tα,λ (τ ) by h(t,τ ), then t ∂ g(τ,t) = − h(t ,τ )dt . (9) ∂τ −∞

2

2 0

α,λ

0

4 (a)

X(S

advantage of total probability formula, the PDF p(x,v,t) of the process (6) has the form ∞ f (x,v,τ )g(τ,t)dτ. (8) p(x,v,t) =


−2 0

(b)

2

4 t

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 constant intervals represent the tempered α-stable waiting times in which the particle stays motionless. Due to the assumption that following every trapping event the particle is released with the same velocity which it had prior to the immobilization, the process V [Sα,λ (t)] is nonzero while position of the particle does not change. Thus, V [Sα,λ (t)] should not be considered a usual particle velocity. The parameters are α = 0.95,λ = 0.01,kB T = m = η = 1,F (x) = 0.

considered a usual particle velocity. To overcome this problem one could possibly modify the model by putting V [Sα,λ (t)] = 0 during every constant period. However, such a modification would make the trajectories of V [Sα,λ (t)] discontinuous. Another consequence of the aforementioned assumption is the violation of the classical Newtonian relation between position and velocity. It is straightforward to verify that the following relationship between the position and velocity processes holds (d/dt)X[Sα,λ (t)] = t γ V [Sα,λ (t)]. This clearly violates the Newtonian relation. As argued in [25], this violation of Newton’s law is due to the additional waiting time average, which camouflages the Newtonian, Langevindominated events. The Langevin picture (6) allows us to introduce an efficient algorithm of simulating sample paths of the tempered subdiffusion. The algorithm is analogous to the one presented recently in [17]. Every two-dimensional trajectory of Y(t) is obtained 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 the trajectory of the inverse subordinator Sα,λ (t). In the second step 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 the details of the algorithm, see [17,24]. The above algorithm allows us to simulate efficiently the trajectories of Y(t). In Fig. 1 a sample path of the tempered subdiffusion is depicted. The constant intervals represent tempered α-stable trapping events when the particle stays motionless. 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 dynamics of Y(t). The quantile lines were estimated using Monte Carlo

021137-3



3 2 90% 80% 70% 60% 50% 40% 30% 20% 10%

0

V(S

α,λ

(t))

1

−1 −2 −3 0

5

10

15

20

25

30

t FIG. 2. (Color online) Sample path [blue (gray) line] and the estimated quantile lines (10%,20%, . . . ,90%) of the process V [Sα,λ (t)] in the presence of the external force F (x) = −V (x) = −(x 3 − 16x)/20. The results were obtained via Monte Carlo simulation on the basis of 104 simulated trajectories. The parameters are α = 0.95,λ = 0.01,kB T = m = η = 1.

methods for the case of external force F (x) = −V (x) = −(x 3 − 16x)/20. In Fig. 4 we show the estimated stationary solution of (7) compared with the theoretical one. III. ESCAPE FROM A POTENTIAL WELL

In this section we consider the survival probability of a particle trapped in a potential hole. Initially the particle is sitting deep in the potential well and through the irregular shuttling action of noise can escape over a barrier (see Fig. 5). In the simplest situation the noise is modeled by Brownian motion [1]. More recently, the fractional extension of the classical Klein-Kramers equation was proposed [2], 6

FIG. 4. (Color online) Theoretical and estimated stationary solution of FKKE (7) in the case of double-well potential. Parameters as in Fig. 2. The numerical results presented in Figs. 1–4 are very similar to the ones obtained for the case of stable waiting times in [21]. However, the difference between the stable and tempered stable case is 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 long waiting times coexisting with long jumps was considered in [26]. The survival probability corresponding to the fractional Gaussian noise was considered in detail in [27].

90% 80% 70% 60%

4

V(x)

X(Sα,λ(t))

2 V(xmax)

0

50%

−2

ΔV

40% 30% 20% 10%

−4 −6 0

5

10

15

20

25

V(x

)

min

30

x

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

min

x

max

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

021137-4


where Kα is the generalized diffusion constant. Equivalently, the position of the particle is described by the Langevin(t) = X[Sα,λ (t)], where Sα,λ (t) is the inverse like process Y subordinator, and X(τ ) is the standard diffusion process satisfying F [X(τ )] dX(τ ) = dτ + 2Kα dB(τ ). mη Denote by τX the first passage time of the process X(τ ) over the potential barrier. Since X(τ ) is a standard diffusion process, τX is exponentially distributed. The corresponding probability 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 . ˆ In what follows, by Z(u) we denote the Laplace transform of a random variable Z, i.e., ∞ ˆ Z(u) = E(e−uZ ) = e−uz g(z)dz, (17) 0

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 process (t) = X[Sα,λ (t)]. Denote by τY the first passage time of the Y (t) over the potential barrier. Then, we get process Y d

τY = Tα,λ (τX ).

(18)

Taking advantage of the above fact, the Laplace transform of τY is then given by rK τˆY(u) = . [(u + λ)α − λα ] + rK Thus, for the survival probability, we have ∞ K 1 − [(u+λ)αr−λ α ]+r K −ut e P (τY > t)dt = u 0 [(u + λ)α − λα ] = . u(rK + [(u + λ)α − λα ]) Using the Taylor expansion, we obtain for small u ∞ 1 e−ut P (τY > t)dt ∝ . rK /(αλα−1 ) + u 0

(19)

(t) Inverting the above, we see that the survival probability of Y satisfies pY(t) ∝ e−t[rK /(αλ

α−1

)]

(20)

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

0

10

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

−1

10

−2

10 ln (p(t))

In this section we study the Kramers escape problem corresponding to the tempered subdiffusion process. The motion of the particle sitting in the potential well is described by the position process in the introduced tempered KleinKramers model (7). Thus, in the high-friction limit, the particle position follows the generalized Fokker-Planck equation: ∂W (x,t) ∂ F (x) ∂2 = t − + Kα 2 W (x,t), (15) ∂t ∂x mη ∂x


−3

10

−4

10

−5

10

0

200

400

600

800

1000

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

We have performed Monte Carlo simulations for the classical and for both α-stable and tempered α-stable waitingtimes models. We have investigated the survival probability for the potential of the form V (x) = −x 3 /3 + x (see Fig. 5). Initially a test particle was placed in a potential hole of coordinate xmin = −1 and started its motion. When the particle crossed 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 survival probability was calculated via the Monte Carlo method using the collected first passage times. Figure 6 depicts the survival probability for different parameters λ. One can see that for very small tempering parameter λ the exponential decay is observed only for large times. In Fig. 7 we observe the behavior of the survival probability for the case of tempered stable waiting times with different parameters α. In each case we observe the exponential decay. In Fig. 8 we have compared the survival probability of the tempered case with the classical and stable ones. For smaller values of λ the tempered model is closer to the model with stable waiting times. However, the greater values of λ yield the classical exponential decay of the survival probability.

IV. FIRST PASSAGE TIMES FOR THE FORCE-FREE CASE

In this section we calculate first passage times for the tempered subdiffusion. For recent advances in the area of first passage times for anomalous diffusion, see [29] and references therein. We consider here the case of free particles (no external force or confinement is present). Thus, the analyzed process = B[Sα,λ (t)], where B is the standard has the form B(t) Brownian motion and Sα,λ (t) is the inverse subordinator.

021137-5



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

0

10

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

−1

10

> d}. τB(d) = inf{t 0 : B(t) Let us recall that the Laplace transform of τB (d) is given by [30] E(e−uτB (d) ) = e−d

−2

ln (p(t))

10

√ √ 2 u

.

Thus, using the following relation from the previous section, d

τB(d) = Tα,λ [τB (d)],

−3

10

(21)

we get that the Laplace transform of τB(d) is given by E(e−uτB(d) ) = e−[(u+λ)

α

−4

10

−5

10

0

200

400

600

800

1000

t FIG. 7. Behavior of the survival probability for the tempered waiting times with different values of α. For each case the exponential decay is observed.

√ −λα ](1/2) d 2

(22)

.

Therefore, we have ∞ 1 − e−uτB(d) e−ut P [τB(d) > t]dt = u 0 1 − e−[(u+λ) −λ ] d = u Using the Taylor expansion, we obtain ∞ √ e−ut P [τB(d)] > tdt ∝ d 2αλα−1 u−1/2 α

α (1/2)

√

2

.

(23)

0

Denote the first passage time over the distance d for Brownian motion B by τB (d) = inf{t 0 : B(t) > d}.

for small u. Finally, from the Tauberian theorem [30] we obtain the following asymptotics, √ d 2αλα−1 −1/2 t , (24) P [τB(d) > t] ∝ √ π for large times t. This proves that the first passage time distribution of the tempered subdiffusion follows the Sparre Andersen universality [31]. Numerical results presented in Figs. 9 and 10 confirm the correctness of the derived formulas.

0

10

−1

10

0

10 −2

−3

10

∫

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

−4

10

−5

10

−6

10

B

ln (P(τ (d)>t))

ln (p(t))

10

−1

10

α=0.9, λ=0.1, d=3 Analytical result α=0.9, λ=0.01, d=3 Analytical result −2

1

10

2

10

10

3

10

1

10

ln (t)

2

10

3

10

ln (t)

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

FIG. 9. (Color online) Behavior of P [τB(d) > t] with different values of the tempering parameter λ. The red solid lines represent the theoretical results for the asymptotics of the first-passage-time distribution. One can observe a very good agreement between theory and simulations.

021137-6



0

2

10

1

10 −1

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

2

10

ln ()

∫

B

ln (P( τ (d)>t))

10

0

10

Tempered case asymptotics ~ tα asymptotics ~ t

−2 1

2

10

10 ln (t)

−1

3

10

10

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

However, deviations from the Sparre Andersen scaling can be observed for very small values of the parameter λ. Note that for λ 0 the Laplace transform in √(22) takes the α/2 well-known stretched exponential form e−u d 2 . The tail of the corresponding first passage time distribution decays as t −α/2 , which is clearly different from the Sparre Andersen behavior. 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—for greater values of the tempering parameter λ the classical Sparre Andersen scaling is observed. This fact was confirmed numerically; the obtained results were similar in nature to the ones presented in Fig. 8 for the survival probability. One of the most characteristic properties of the tempered subdiffusion is the crossover from the initial anomalous behavior to the standard diffusive one for large times. The mean square displacement of the tempered subdiffusion process for the force-free case is given by [16] t E(B 2 [Sα,λ (t)]) = e−λy y α−1 Eα,α (λα y α )dy, 0

where Eα,β (z) is the Mittag-Leffler function [32]. Consequently, E(B 2 [Sα,λ (t)]) ∝ t α

−1

10

0

1

10

2

10

10

ln (t)

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

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

In this paper we have introduced the generalized KleinKramers model, in which the standard diffusion process is slowed down by the waiting times, during which the particle gets immobilized. The introduced model is a modification of the subdiffusive Klein-Kramers model discussed in [2]. 0

10

−α/2

t ln (1−CDF(FPT(t))

10

−1

10

t−1/2

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

for t 0 and E(B 2 [Sα,λ (t)]) ∝ t for t ∞. This crossover from anomalous to normal dynamics is depicted in Fig. 11. Consequently, in the initial phase the process B[Sα,λ (t)] behaves as a subdiffusion, whereas for large times we observe the standard diffusive dynamics. This property strongly affects the behavior of the first passage time τB(d) of the tempered subdiffusion. As for the case of mean square displacement, we observe a crossover from subdiffusive

−2

10

0

10

1

2

10

10

3

10

ln (t)

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

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The modification consists of replacing the stable waiting times with the tempered stable ones in the underlying CTRW scenario. We have derived the corresponding Langevin and Klein-Kramers equations. These fundamental equations were used to investigate the dynamics of the process. We have shown that the survival probability of the tempered model decays exponentially; however, for small values of the tempering parameter, the subdiffusive power-law behavior is recovered. Moreover, we have derived the analytical formula

for the first-passage-time distribution for the case of free particle. Our results confirmed that the Sparre Andersen scaling holds also for the tempered subdiffusion. The proposed model of tempered subdiffusion is capable of describing the transition from the initial subdiffusive character of motion to the standard diffusion for long times. Since such a transition was observed in a number of physical and biological systems, we believe that our investigations will provide additional tools for the analysis of such systems.

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