# Directed transport driven by Lévy flights coexisting with subdiffusion

Bao-quan Ai1<EMAIL_ADDRESS>Ya-feng He2 1Laboratory of Quantum Information
Technology, ICMP and SPTE, South China Normal University, 510006 Guangzhou,
China.
2College of Physics Science and Technology, Hebei University, 071002 Baoding,
China

###### Abstract

Transport of the Brownian particles driven by Lévy flights coexisting with
subdiffusion in asymmetric periodic potentials is investigated in the absence
of any external driving forces. Using the Langevin-type dynamics with
subordination techniques, we obtain the group velocity which can measure the
transport. It is found that the group velocity increases monotonically with
the subdiffusive index and there exists an optimal value of the Lévy index at
which the group velocity takes its maximal value. There is a threshold value
of the subdiffusive index below which the ratchet effects will disappear. The
nonthermal character of the Lévy flights and the asymmetry of the potential
are necessary to obtain the directed transport. Some peculiar phenomena
induced by the competition between Lévy flights and subdiffusion are also
observed. The pseudo-normal diffusion will appear on the level of the median.

Ratchet, Lévy flights, subdiffusion

###### pacs:

05\. 40. Fb, 05. 10. Gg, 05. 40. -a

##  I Introduction

Directed Brownian motion induced by zero-mean non-equilibrium fluctuations in
the absence of macroscopic forces and potential gradients is presently under
intense investigation a1 . This comes from the desire to understand
unidirectional transport in biological systemsa2 , as well as their potential
technological applications ranging from classical non-equilibrium modelsa3  to
quantum systemsa4 . A ratchet system is generally defined as a system that is
able to transport particles in a periodic structure with nonzero macroscopic
velocity in the absence of macroscopic force on average. Broadly speaking,
ratchet devices fall into three categories depending on how the applied
perturbation couples to the substrate asymmetry: rocking ratchetsa5 , flashing
ratchetsa6 , and correlation ratchets a7 . Additionally, entropic ratchets, in
which Brownian particles move in a confined structure, instead of a potential,
were also extensively investigated a8 . These studies on directed transport of
the Brownian particles focused on the normal diffusion.
However, anomalous diffusion has attracted growing attention, being observed
in various fields of physics and related sciencesa9 . Description of physical
models in terms of Lévy flights and subdiffusion becomes more and more popular
a10 ; a11 ; a12 ; a13 ; a14 ; a15 ; a16 ; a17 ; a18 ; a19 ; a20 ; a21 ; a22 ;
a23 . In the complex systems the distinct class of subdiffusion processes was
reported in condensed phasesa10 , ecologya11 , and biologya12 . Superdiffusion
driven by Levy flights is actually observed in various real systems and is
used to model a variety of processes such as bulk mediated surface diffusion
a13 , exciton and charge transport in polymers under conformational motion a14
, transport in micelle systems or heterogeneous rocks a15 ,two-dimensional
rotating flow a16 , and many othersa9 . Goychuk and coworkers a17  studied the
subdiffusive transport in tilted periodic potentials and established a
universal scaling relation for diffusive transport. Dybiec and coworkers a18
studied the minimal setup for a Lévy ratchet and found that due to the
nonthermal character of the Lévy noise, the net current can be obtained even
in the absence of whatever additional time-dependent forces. Del-Castillo-
Negrete and coworkers a19  also found the similar results in constant force-
driven Lévy ratchet. Rosa and Beims a20  studied the optimal transport and its
relation to superdiffusive transport and Lévy walks for Brownian particles in
ratchet potential in the presence of modulated environment and external
oscillating forces. We also studied the transport of Brownian particles in the
presence of ac-driving forces and Lévy flights and multiple current reversals
were observeda21

Recently, much attention has been devoted to the competition between
subdiffusion and Lévy flights. The competition is conveniently described by
the fractional Fokker-Planck equation with temporal and spatial fractional
derivatives a9 . It is very difficult to see this competition in the framework
of the fractional Fokker-Planck dynamics. Magdziarz and coworkers a22
proposed a equivalent approach based on the subordinated Langevin method to
visualize the competition on the level of sample paths as well as on the level
of probability density functions. Based on this approach Dybiec and coworkers
a23  found that due to the competition between Lévy flights and subdiffusion,
the standard measure used to discriminate between anomalous and normal
behavior cannot be applied straightforwardly. Koren and coworkersa24  have
investigated the first passage times in one-dimensional system displaying a
competition between subdiffusion and Lévy flights and found some peculiar
phenomena.

What will happen when the particles move in a ratchet potential subjected to
subdiffusion and Lévy flights? In order to answer this question we use the
subordinated Langevin method proposed by the Magdziarz and coworkers a22  to
investigate this competition in a minimal Lévy ratchet without any external
driving forces. We emphasize on visualizing the competition on the level of
the group velocity and diffusion and finding how this competition affects the
directed transport.

##  II Model and Methods

We consider the transport of the Brownian particles driven by Lévy flights and
subdiffusion in the absence of whatever additional time-dependent forces. The
competition between Lévy flights and subdiffusion in a ratchet potential
$V(x)$ can be described by the fractional Fokker-Planck equation with temporal
and spatial fractional derivatives a9 ; a22

$\frac{\partial p(x,t)}{\partial
t}={{}_{0}}D_{t}^{1-\alpha}\left[\frac{\partial}{\partial
x}\frac{V^{{}^{\prime}}(x)}{\eta}+D\frac{\partial^{\mu}}{\partial|x|^{\mu}}\right]p(x,t),$
(1)

where $p(x,t)$ is the probability density for particles at position $x$ and
time $t$. The prime stands for the derivative with respect to the space
variable $x$. $D$ is the anomalous diffusion coefficient which describes the
noise intensity in the subordinated process. $\eta$ denotes the generalized
friction constant. Here ${{}_{0}}D_{t}^{1-\alpha}$ is the fractional of the
Riemann-Liouville operator ($0<\alpha\leq 1$) defined through a9 ; a22

${{}_{0}}D_{t}^{1-\alpha}g(t)=\frac{1}{\Gamma(\alpha)}\frac{d}{dt}\int^{t}_{0}(t-s)^{\alpha-1}g(s)ds,$
(2)

where $\Gamma(.)$ is the Gamma function. From the definition, it becomes
apparent that subdiffusion corresponds to a slowly decaying memory integral in
the dynamical equation for $p(x,t)$. The operator
$\frac{\partial^{\mu}}{\partial|x|^{\mu}}$, $0<\mu\leq 2$, stands for the
Riesz fractional derivative a9 ; a22  with the Fourier transform
$\mathcal{F}\\{\frac{\partial^{\mu}}{\partial|x|^{\mu}}f(x)\\}=-|k|^{\mu}\tilde{f}(k)$.

The occurrences of the operator ${{}_{0}}D_{t}^{1-\alpha}$ and
$\frac{\partial^{\mu}}{\partial|x|^{\mu}}$ are induced by the heavy-tailed
waiting times between successive jumps and the heavy-tailed distributions of
the jumps, respectively, in the underlying continuous-time random walk scheme.
The case of $\alpha=1$, $\mu=2$ corresponds to the standard Fokker-Planck
equation. $V(x)$ is an asymmetric periodic potential

$V(x)=\frac{V_{0}}{2\pi}\left[\sin(2\pi x)+\frac{\Delta}{4}\sin(4\pi
x)\right],$ (3)

where $V_{0}$ and $\Delta$ are the amplitude and the asymmetric parameter of
the potential, respectively.

Because it is very difficult to solve Eq. (1) analytically and numerically, we
used the subordinated Langevin method proposed by Magdziarz and coworkers a22
to investigate the transport. In their method, the solution $p(x,t)$ of Eq.
(1) is equal to the probability density function of the subordinated process

$Y(t)=X(S_{t}),$ (4)

where the parent process $X(\tau)$ is defined as the solution the stochastic
differential equation

$dX(\tau)=-\frac{V^{{}^{\prime}}(X(\tau))}{\eta}d\tau+D^{1/\mu}dL_{\mu}(\tau),$
(5)

where $L_{\mu}(\tau)$ is the symmetric $\mu$-stable Lévy motion with the
Fourier transform $\mathcal{F}\\{L_{\mu}(\tau)\\}=e^{-\tau|k|^{\mu}}$.
Employing the Euler scheme to Eq. (5), one can obtain

$X(\tau_{0})=0,$ (6)
$X(\tau_{i})=X(\tau_{i-1})-\frac{V^{{}^{\prime}}(X(\tau_{i-1}))}{\eta}\Delta\tau+(D\Delta\tau)^{1/\mu}\xi_{i},$
(7)

where $i=1,2,3...$ and $\xi_{i}$ are the random variables with standard
symmetric $\mu$-stable distribution. The procedure of generating realizations
$\xi_{i}$ is the following a22 ; b1

$\xi_{i}=\frac{\sin(\mu V)}{(\cos
V)^{1/\mu}}\left[\frac{\cos([1-\mu]V)}{W}\right]^{\frac{1-\mu}{\mu}},$ (8)

where the random variable $V$ is uniformly distributed on$(-\pi/2,\pi/2)$, $W$
has exponential distribution with mean one.

The inverse-time $\alpha$-stable subordinator $S_{t}$, which is assumed to be
independent of $X(\tau)$, is defined as

$S_{t}=\inf\\{\tau:U(\tau)>t\\},$ (9)

where $U(\tau)$ is the strictly increasing $\alpha$-stable Lévy motion with
Laplace transform $\mathcal{L}\\{U(\tau)\\}=e^{-\tau k^{\alpha}}$.

Using the standard method of summing increments of the process $U(\tau)$ one
can get

$U(\tau_{0})=0,$ (10)
$U(\tau_{j})=U(\tau_{j-1})+\Delta\tau^{1/\alpha}\zeta_{j},$ (11)

where $j=1,2,3...$ and $\zeta_{j}$ are the skewed positive $\alpha$-stable
random variables a22 ; b1 . The method to generate the random variables is

$\zeta_{j}=\frac{\sin(\alpha(V+\frac{\pi}{2}))}{[\cos(V)]^{\frac{1}{\alpha}}}\left[\frac{\cos(V-\alpha(V+\frac{\pi}{2}))}{W}\right]^{\frac{1-\alpha}{\alpha}},$
(12)

where $V$ and $W$ have the same definitions as that in Eq. (8). From the above
procedures, one can obtain the subordinated process $Y(t)$ and its probability
distribution function is equal to the solution of Eq. (1). For more detailed
information on the algorithm, please see the Ref. (22).

In the classical ratchets, one can use the average velocity and effective
diffusion coefficient to describe the transport. However, for the noise with
distribution of a Lévy-stable law, the mean of the noise and the second moment
may do not exist. As a consequence, the classical stochastic theory (average
velocity and effective diffusion coefficient), which is based on the ordinary
central limit theorem, is no longer valid. To overcome this problem, Dybiec
and coworkers a18  recently proposed a different approach to the Lévy ratchet
problem based on the quantile line analysis for $0<\mu<2$.

Quantile line is a very useful tool for investigation of the overall motion of
the probability density of finding a particle in the vicinity of $Y(t)$ a18 ;
a23 . A median line for a stochastic process $Y(t)$ is a function of
$q_{0.5}(t)$ given by the relationship $Pr(Y(t)\leq q_{0.5}(t))=0.5$.
Therefore, one can use the derivative of the median to define the group
velocity of the particle packeta18 ,

$V_{g}=\frac{dq_{0.5}(t)}{dt},$ (13)

and this definition is valid even for the case of lacking average velocity.

##  III Numerical results and discussion

In order to investigate the competition between Lévy flights and subdiffusion
in a ratchet potential, we carried out extensively numerical simulations based
on the subordinated Langevin method a22 . For simplicity we set $\eta=1.0$ and
$1<\mu\leq 2$ throughout the work. In our simulations, we have considered more
than $10^{5}$ realizations to obtain the accurate median. In order to provide
the requested accuracy of the system the dynamics time step was chosen to be
smaller than $10^{-3}$. We have checked that these are sufficient for the
system to obtain consistent results.

Firstly, we will investigate the diffusive properties of the Brownian
particles. Usually, the types of the diffusion processes are determined by the
spread of the distance traveled by a random walker. The diffusion is
characterized through the power law form of the mean-square displacement
$\langle x^{2}(t)\rangle\propto t^{\delta}$. According to the value of the
index $\delta$, one can distinguish subdiffusion ($0<\delta<1$), normal
diffusion ($\delta=1$) and superdiffusion ($\delta>1$). Here, we use the
median of square displacement $M(x^{2})$, instead of mean-square displacement,
to characterize the diffusion. Fig. 1 (a) shows the time dependence of
$M(x^{2})/t$ for different combinations of $\mu$ and $\alpha$ without any
external potential. It is found that the linear time dependence of the median
of square displacement, $M(x^{2})\propto t$, will occur for the case of
$\frac{2\alpha}{\mu}$=1, which indicates the normal diffusion. However, this
is not true, for example $\mu=1.8$ and $\alpha=0.9$, the process is still non-
Markov and non-Gaussian. This pseudo-normal diffusion is due to the
competition between Lévy flights and subdiffusion. Dybiec and coworkers a23
have presented discussions in detail on this paradoxical diffusion. Fig. 1 (b)
presents the time dependence of $M(x^{2})/t$ in the presence of a ratchet
potential. Interestingly, the pseudo-normal diffusion for $\mu=1.8$ and
$\alpha=0.9$ with external potentials is not normal.

Next, we will study the rectified mechanism of the Lévy ratchets. Usually, the
ratchet mechanism demands three key ingredients b2  which are (a) nonlinear
periodic potential: it is necessary since the system will produce a zero mean
output from zero-mean input in a linear system; (b)asymmetry of the potential,
it can violate the symmetry of the response; (c)fluctuating: Lévy flights can
break thermodynamical equilibrium. In Fig. 2 (a), we studied the time
dependence of the median for different values of the asymmetry parameter
$\Delta$ at $\mu=1.5$ and $\alpha=1.0$. The median is positive for $\Delta>0$,
zero at $\Delta=0$, and negative for $\Delta<0$. Therefore, the asymmetry of
the potential will determine the direction of the transport and no directed
transport occurs in a symmetric potential. Now we will give the physical
interpretation of the directed transport for the case of $\Delta=1$. Firstly,
the particles stay in the minima of the potential awaiting large noise pulse
to be catapulted out. The particles will be thrown out to the left and the
right with the equal probabilities. In this case, the distance from minima to
maxima is shorter from the right side than that from the left side.
Consequently, most of the particles are thrown out from the right side,
resulting in positive transport. This gives rise to the overall preferred
motion to the right.

Figure 2 (b) gives the time dependence of the median for different
combinations of $\mu$ and $\alpha$. We find that Lévy flights are necessary to
obtain the directed transport. For Gaussian case ($\mu=2.0$), directed
transport disappears. This is due to the nonthermal character of the Lévy
flights that can break thermodynamical equilibrium. From Fig. 2(a) and (b) we
can see that the asymmetry of the potential and the non-equilibrium character
of the Lévy flights are the two necessary conditions for directed transport.
The direction of the transport is determined by the direction of the steeper
slope of of the potential and the Lévy flights can break thermaldynamical
equilibrium. These two key ingredients can realize the ratchet effects.

Figure 3 illustrates the dependence of the group velocity $V_{g}$ on the
subdiffusive index $\alpha$ for different values of the Lévy index $\mu$. One
can see that group velocity $V_{g}$ increases monotonically with the
subdiffusive index $\alpha$. For small values of $\alpha$, the waiting time
between successive jumps is very long and it is not easy for particles to pass
across the barrier. Thus, most particles will stay in their original minima of
the potential and the group velocity becomes very small. Especially, we also
find that there exists a threshold value of $\alpha$ below which no directed
transport can be obtained. The subdiffusion dominates the transport for small
values of $\alpha$ ($\alpha<0.7$), while the effects of the Lévy flights
become preponderant for large values of $\alpha$.

Figure 4 shows the dependence of the group velocity $V_{g}$ on the Lévy index
$\mu$ for different values of $\alpha$. When $\mu\rightarrow 2.0$, the system
is under thermodynamical equilibrium and no directed transport appears. For
small values of $\mu$, Lévy flights are longer and the outliers in the Lévy
noise are larger. In this case, the effects of the asymmetry of the potential
become very small, resulting in small group velocity. Therefore, there exists
a optimal value of $\mu$ at which the group velocity takes its maxima. This
can also be confirmed by Fig. 3. For very small values of $\alpha$, for
example $\alpha=0.5$, the group velocity is zero for all values of $\mu$ and
the transport is absolutely dominated by subdiffusion.

The group velocity $V_{g}$ as a function of noise intensity $D$ is shown in
Fig. 5 for different combinations of $\mu$ and $\alpha$. The curve is observed
to be bell shaped which shows the feature of resonance. When $D\rightarrow 0$,
the particles cannot pass across the barrier and there is no directed current.
When $D\rightarrow\infty$ so that the noise is very large, the effect of the
potential disappears and the group velocity tends to zero, also. There is an
optimal value of $D$ at which the group velocity is maximal. There are two
intersections ($D_{c1}$ and $D_{c2}$) between the line of $\mu=1.9$ and
$\alpha=1.0$ and the line of $\mu=1.5$ and $\alpha=0.9$. For simplicity, we
define $V_{g}(\mu,\alpha)$ as the group velocity for different values of $\mu$
and $\alpha$. When $D<D_{c1}$, $V_{g}(1.9,1.0)>V_{g}(1.5,0.9)$, Lévy flights
dominates the transport. When $D_{c1}<D<D_{c2}$,
$V_{g}(1.9,1.0)<V_{g}(1.5,0.9)$, the transport is governed by subdiffusion.
For the case of $D>D_{c2}$, all particles can easily pass across the barrier
and Lévy flights will mainly contribute to the transport and
$V_{g}(1.9,1.0)>V_{g}(1.5,0.9)$

In Fig. 6, we plot the dependence of the group velocity $V_{g}$ on the
amplitude $V_{0}$ of the potential for different combinations of $\mu$ and
$\alpha$. When $V_{0}\rightarrow 0$, the effects of the potential disappear
and the group velocity tends to zero. When $V_{0}\rightarrow\infty$, the
particles cannot pass across the barrier and the group velocity goes to zero,
also. Thus, the curve shows a peak. Remarkably, there is an intersection
between the line of $\mu=1.5$ and $\alpha=0.9$ and the line of $\mu=1.9$ and
$\alpha=1.0$. This is due to the competition between Lévy flights and
subdiffusion. When $V_{0}<V_{c}$, Lévy flights are predominant and
$V_{g}(1.5,0.9)>V_{g}(1.9,1.0)$. When $V_{0}>V_{c}$, subdiffusion dominates
the transport and $V_{g}(1.5,0.9)<V_{g}(1.9,1.0)$. In this case, the height of
the barrier is very high and few particles driven by Lévy flights can pass
across the barrier, the effects of the Lévy flights will disappear and
subdiffusion will play a major role.

##  IV Concluding Remarks

In this paper, we have investigated the directed transport of the Brownian
particles in a ratchet potential driven by Lévy flights coexisting with
subdiffusion. We used recently developed framework of Monte Carlo simulation
a22  which is equal to the solution of the fractional Fokker-Planck equation.
The group velocity proposed by Dybiec and coworkers a18  is used to measure
the transport. It is found that the group velocity increases monotonically
with the subdiffusive index, while the group velocity as a function of the
Lévy index is nonmonotonic. The former is caused by the increase of the
waiting time between successive jumps and the latter is owing to the interplay
between Lévy flights and the height of the barriers. There is a threshold
value of $\alpha$ below which the transport is absolutely dominated by
subdiffusion and the directed transport disappears. The dependences of the
group velocity on the noise intensity and the amplitude of the potential are
also investigated. There is an optimal value of the noise intensity (the
amplitude of the potential) at which the group velocity is maximal. The
competition between Levy fights and subdiffusion in the ratchet potential is
observed on the level of the group velocity as well as the median of square
displacement. The nonthermal character of the Lévy flights and the asymmetry
of the potential are the necessary conditions for directed transport when the
system is in the absence of any external driving forces. Because of this
competition, we also found the pseudo-normal diffusion reported by Dybiec and
coworkers a23 , in which time dependence of the median of square displacement
is linear, $M(x^{2})\propto t$, while the process is still non-Markov and non-
Gaussian.

Anomalous transport is becoming widely recognized in a variety of the fields.
Beyond its intrinsic theoretical interest, the results we have presented may
have wide applications in some complex systems, such as diffusive transport in
plasmas, particles separation with non-Gaussian diffusion, and ratchet
transport in biology systems that are intrinsically out of equilibrium.

We would like to thank Dr. Magdziarz for enthusiastic help on numerical
algorithm. This work was supported in part by National Natural Science
Foundation of China with Grant Nos. 30600122 and 10947166 and GuangDong
Provincial Natural Science Foundation with Grant No. 06025073. Y. F. He also
acknowledges the Research Foundation of Education Bureau of Hebei Province,
China (Grant No. 2009108)

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##  V Caption list

Fig. 1. Time dependence of $M(x^{2})/t$ for different combinations of $\mu$
and $\alpha$: (a)without external potential at $D=0.4$, solid lines present
$t^{\frac{2\alpha}{\mu}-1}$ scaling; (b)with external potential at $D=0.4$,
$V_{0}=5.0$, and $\Delta=1.0$.

Fig. 2. Time dependence of the median: (a) for different values of the
asymmetry parameter $\Delta$ at $D=0.4$, $V_{0}=5.0$, $\mu=1.5$, and
$\alpha=1.0$, the inset shows the potential profile; (b)for different
combinations of $\mu$ and $\alpha$ at $D=0.4$, $V_{0}=5.0$, and $\Delta=1.0$.

Fig. 3. Group velocity $V_{g}$ versus subdiffusive index $\alpha$ for
different values of $\mu$ at $D=0.4$, $V_{0}=5.0$, and $\Delta=1.0$.

Fig. 4. Group velocity $V_{g}$ versus Lévy index $\mu$ for different values of
$\alpha$ at $D=0.4$, $V_{0}=5.0$, and $\Delta=1.0$.

Fig. 5. Group velocity $V_{g}$ as a function of noise intensity $D$ for
different combinations of $\mu$ and $\alpha$ at $V_{0}=5.0$ and $\Delta=1.0$.

Fig. 6. Group velocity $V_{g}$ versus the amplitude $V_{0}$ of the potential
for different combinations of $\mu$ and $\alpha$ at $D=0.4$ and $\Delta=1.0$.

Figure 1:

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Figure 6: