164 research outputs found
Anomalous fluctuation relations
We study Fluctuation Relations (FRs) for dynamics that are anomalous, in the
sense that the diffusive properties strongly deviate from the ones of standard
Brownian motion. We first briefly review the concept of transient work FRs for
stochastic dynamics modeled by the ordinary Langevin equation. We then
introduce three generic types of dynamics generating anomalous diffusion:
L\'evy flights, long-time correlated Gaussian stochastic processes and
time-fractional kinetics. By combining Langevin and kinetic approaches we
calculate the work probability distributions in the simple nonequilibrium
situation of a particle subject to a constant force. This allows us to check
the transient FR for anomalous dynamics. We find a new form of FRs, which is
intimately related to the validity of fluctuation-dissipation relations.
Analogous results are obtained for a particle in a harmonic potential dragged
by a constant force. We argue that these findings are important for
understanding fluctuations in experimentally accessible systems. As an example,
we discuss the anomalous dynamics of biological cell migration both in
equilibrium and in nonequilibrium under chemical gradients.Comment: book chapter; 25 pages, 10 figures. see
http://www.maths.qmul.ac.uk/~klages/smallsys/smallsys_rk.htm
Stationary states for underdamped anharmonic oscillators driven by Cauchy noise
Using methods of stochastic dynamics, we have studied stationary states in
the underdamped anharmonic stochastic oscillators driven by Cauchy noise. Shape
of stationary states depend both on the potential type and the damping. If the
damping is strong enough, for potential wells which in the overdamped regime
produce multimodal stationary states, stationary states in the underdamped
regime can be multimodal with the same number of modes like in the overdamped
regime. For the parabolic potential, the stationary density is always unimodal
and it is given by the two dimensional -stable density. For the mixture
of quartic and parabolic single-well potentials the stationary density can be
bimodal. Nevertheless, the parabolic addition, which is strong enough, can
destroy bimodlity of the stationary state.Comment: 9 page
Einstein-Smoluchowsky equation handled by complex fractional moments
In this paper the response of a non linear half oscillator driven by a-stable white noise in terms of probability density function (PDF) is investigated. The evolution of the PDF
of such a system is ruled by the so called Einstein-Smoluchowsky equation involving, in the diffusive term, the Riesz fractional derivative. The solution is obtained by the use of complex fractional moments of the PDF, calculated with the aid of Mellin
transform operator. It is shown that solution can be found for various values of stability index a and for any nonlinear function f (X; t)
Probabilistic characterization of nonlinear systems under α-stable white noise via complex fractional moments
The probability density function of the response of a nonlinear system under external α-stable Lévy white noise is ruled by the so called Fractional Fokker-Planck equation. In such equation the diffusive term is the Riesz fractional derivative of the probability density function of the response. The paper deals with the solution of such equation by using the complex fractional moments. The analysis is performed in terms of probability density for a linear and a non-linear half oscillator forced by Lévy white noise with different stability indexes α. Numerical results are reported for a wide range of non-linearity of the mechanical system and stability index of the Lévy white noise
First passage time moments of asymmetric L\'evy flights
We investigate the first-passage dynamics of symmetric and asymmetric L\'evy
flights in a semi-infinite and bounded intervals. By solving the
space-fractional diffusion equation, we analyse the fractional-order moments of
the first-passage time probability density function for different values of the
index of stability and the skewness parameter. A comparison with results using
the Langevin approach to L\'evy flights is presented. For the semi-infinite
domain, in certain special cases analytic results are derived explicitly, and
in bounded intervals a general analytical expression for the mean first-passage
time of L\'evy flights with arbitrary skewness is presented. These results are
complemented with extensive numerical analyses.Comment: 47 pages, 13 figures, IOP LaTe
Slow manifolds for stochastic koper models with stable LĂ©vy noises
The Koper model is a vector field in which the differential equations describe the electrochemical oscillations appearing in diffusion processes. This work focuses on the understanding of the slow dynamics of a stochastic Koper model perturbed by stable LĂ©vy noise. We establish the slow manifold for a stochastic Koper model with stable LĂ©vy noise and verify exponential tracking properties. We also present two practical examples to demonstrate the analytical results with numerical simulations
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