563 research outputs found
Localization and ballistic diffusion for the tempered fractional Brownian-Langevin motion
This paper further discusses the tempered fractional Brownian motion, its
ergodicity, and the derivation of the corresponding Fokker-Planck equation.
Then we introduce the generalized Langevin equation with the tempered
fractional Gaussian noise for a free particle, called tempered fractional
Langevin equation (tfLe). While the tempered fractional Brownian motion
displays localization diffusion for the long time limit and for the short time
its mean squared displacement has the asymptotic form , we show that
the asymptotic form of the mean squared displacement of the tfLe transits from
(ballistic diffusion for short time) to , and then to
(again ballistic diffusion for long time). On the other hand, the overdamped
tfLe has the transition of the diffusion type from to
(ballistic diffusion). The tfLe with harmonic potential is also considered.Comment: 19 pages, 9 figure
Langevin dynamics for L\'evy walk with memory
Memory effects, sometimes, can not be neglected. In the framework of
continuous time random walk, memory effect is modeled by the correlated waiting
times. In this paper, we derive the two-point probability distribution of the
stochastic process with correlated increments as well as the one of its inverse
process, and present the Langevin description of L\'evy walk with memory, i.e.,
correlated waiting times. Based on the built Langevin picture, the properties
of aging and nonstationary are discussed. The Langevin system exhibits
sub-ballistic superdiffusion if the friction force is involved, while it
displays super-ballistic diffusion or hyperdiffusion if there is no friction.
It is discovered that the correlation of waiting times suppresses the diffusion
behavior whether there is friction or not, and the stronger the correlation of
waiting times becomes, the slower the diffusion is. In particular, the
correlation function, correlation coefficient, ergodicity, and scaling property
of the corresponding stochastic process are also investigated.Comment: 11 pages, 4 figure
L\'{e}vy-walk-like Langevin dynamics
Continuous time random walks and Langevin equations are two classes of
stochastic models for describing the dynamics of particles in the natural
world. While some of the processes can be conveniently characterized by both of
them, more often one model has significant advantages (or has to be used)
compared with the other one. In this paper, we consider the weakly damped
Langevin system coupled with a new subordinator|-dependent subordinator
with . We pay attention to the diffusion behaviour of the
stochastic process described by this coupled Langevin system, and find the
super-ballistic diffusion phenomena for the system with an unconfined potential
on velocity but sub-ballistic superdiffusion phenomenon with a confined
potential, which is like L\'{e}vy walk for long times. One can further note
that the two-point distribution of inverse subordinator affects mean square
displacement of this coupled weakly damped Langevin system in essential.Comment: 24 pages, 4 figure
Aging two-state process with L\'{e}vy walk and Brownian motion
With the rich dynamics studies of single-state processes, the two-state
processes attract more and more interests of people, since they are widely
observed in complex system and have effective applications in diverse fields,
say, foraging behavior of animals. This report builds the theoretical
foundation of the process with two states: L\'{e}vy walk and Brownian motion,
having been proved to be an efficient intermittent search process. The sojourn
time distributions in two states are both assumed to be heavy-tailed with
exponents . The dynamical behaviors of this two-state
process are obtained through analyzing the ensemble-averaged and time-averaged
mean squared displacements (MSDs) in weak and strong aging cases. It is
discovered that the magnitude relationship of decides the fraction
of two states for long times, playing a crucial role in these MSDs. According
to the generic expressions of MSDs, some inherent characteristics of the
two-state process are detected. The effects of the fraction on these
observables are detailedly presented in six different cases. The key of getting
these results is to calculate the velocity correlation function of the
two-state process, the techniques of which can be generalized to other
multi-state processes.Comment: 6 pages, 1 figur
Subdiffusion in an external force field
The phenomena of subdiffusion are widely observed in physical and biological
systems. To investigate the effects of external potentials, say, harmonic
potential, linear potential, and time dependent force, we study the
subdiffusion described by subordinated Langevin equation with white Gaussian
noise, or equivalently, by the single Langevin equation with compound noise. If
the force acts on the subordinated process, it keeps working all the time;
otherwise, the force just exerts an influence on the system at the moments of
jump. Some common statistical quantities, such as, the ensemble and time
averaged mean squared displacement, position autocorrelation function,
correlation coefficient, generalized Einstein relation, are discussed to
distinguish the effects of various forces and different patterns of acting. The
corresponding Fokker-Planck equations are also presented. All the stochastic
processes discussed here are non-stationary, non-ergodicity, and aging.Comment: 15 pages, 10 figure
Tempered fractional Langevin-Brownian motion with inverse -stable subordinator
Time-changed stochastic processes have attracted great attention and wide
interests due to their extensive applications, especially in financial time
series, biology and physics. This paper pays attention to a special stochastic
process, tempered fractional Langevin motion, which is non-Markovian and
undergoes ballistic diffusion for long times. The corresponding time-changed
Langevin system with inverse -stable subordinator is discussed in
detail, including its diffusion type, moments, Klein-Kramers equation, and the
correlation structure. Interestingly, this subordination could result in both
subdiffusion and superdiffusion, depending on the value of . The
difference between the subordinated tempered fractional Langevin equation and
the subordinated Langevin equation with external biasing force is studied for a
deeper understanding of subordinator. The time-changed tempered fractional
Brownian motion by inverse -stable subordinator is also considered, as
well as the correlation structure of its increments. Some properties of the
statistical quantities of the time-changed process are discussed, displaying
striking differences compared with the original process.Comment: 19 pages, 5 figure
Feynman-Kac equation revisited
Functionals of particles' paths have diverse applications in physics,
mathematics, hydrology, economics, and other fields. Under the framework of
continuous time random walk (CTRW), the governing equations for the probability
density functions (PDFs) of the functionals, including the ones of the paths of
stochastic processes of normal diffusion, anomalous diffusion, and even the
diffusion with reaction, have been derived. Sometimes, the stochastic processes
in physics and chemistry are naturally described by Langevin equations. The
Langevin picture has the advantages in studying the dynamics with an external
force field and analyzing the effect of noise resulting from a fluctuating
environment. We derive the governing equations of the PDFs of the functionals
of paths of Langevin system with both space and time dependent force field and
arbitrary multiplicative noise; and the backward version is proposed for the
system with arbitrary additive noise or multiplicative Gaussian white noise
together with a force field. For the newly built equations, their applications
of solving the PDFs of the occupation time and area under the trajectory curve
are provided, and the results are confirmed by simulations.Comment: 13 pages, 5 figure
Strong anomalous diffusion in two-state process with L\'{e}vy walk and Brownian motion
Strong anomalous diffusion phenomena are often observed in complex physical
and biological systems, which are characterized by the nonlinear spectrum of
exponents by measuring the absolute -th moment . This paper investigates the strong anomalous diffusion behavior
of a two-state process with L\'{e}vy walk and Brownian motion, which usually
serves as an intermittent search process. The sojourn times in L\'{e}vy walk
and Brownian phases are taken as power law distributions with exponents
and , respectively. Detailed scaling analyses are
performed for the coexistence of three kinds of scalings in this system.
Different from the pure L\'{e}vy walk, the phenomenon of strong anomalous
diffusion can be observed for this two-state process even when the distribution
exponent of L\'{e}vy walk phase satisfies , provided that
. When , the probability density function (PDF)
in the central part becomes a combination of stretched L\'{e}vy distribution
and Gaussian distribution due to the long sojourn time in Brownian phase, while
the PDF in the tail part (in the ballistic scaling) is still dominated by the
infinite density of L\'{e}vy walk.Comment: 10 pages, 2 figure
Resonant behavior of the generalized Langevin system with tempered Mittag-Leffler memory kernel
The generalized Langevin equation describes anomalous dynamics. Noise is not
only the origin of uncertainty but also plays a positive role in helping to
detect signal with information, termed stochastic resonance (SR). This paper
analyzes the anomalous resonant behaviors of the generalized Langevin system
with a multiplicative dichotomous noise and an internal tempered Mittag-Leffler
noise. For the system with fluctuating harmonic potential, we obtain the exact
expressions of several SR, such as, the first moment, the amplitude and the
autocorrelation function for the output signal as well as the signal-noise
ratio. We analyze the influence of the tempering parameter and memory exponent
on the bona fide SR and the general SR. Moreover, it is detected that the
critical memory exponent changes regularly with the increase of tempering
parameter. Almost all the theoretical results are validated by numerical
simulations.Comment: 22 pages, 7 figure
Theory of relaxation dynamics for anomalous diffusion processes in harmonic potential
Optical tweezers setup is often used to probe the motion of individual tracer
particle, which promotes the study of relaxation dynamics of a generic process
confined in a harmonic potential. We uncover the dependence of ensemble- and
time-averaged mean square displacements of confined processes on the velocity
correlation function of the original process. With two different
scaling forms of for small and large , the
stationary value and the relaxation behaviors can be obtained immediately. The
gotten results are valid for a large amount of anomalous diffusion processes,
including fractional Brownian motion, scaled Brownian motion, and the
multi-scale L\'{e}vy walk with different exponents of running time
distribution.Comment: 5 pages, 3 figure
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