Generalized Langevin equation with fractional derivative and long-time correlation function

Time-fractional Caputo derivative versus other integrodifferential operators in generalized Fokker-Planck and generalized Langevin equations

Fractional diffusion and Fokker-Planck equations are widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be derived as continuum limits of continuous-time random walks and are associated with the Mittag-Leffler relaxation of Fourier modes, interpolating between a short-time stretched exponential and a long-time inverse power-law scaling. More recently, a number of other integrodifferential operators have been proposed, including the Caputo-Fabrizio and Atangana-Baleanu forms. Moreover, the conformable derivative has been introduced. We study here the dynamics of the associated generalized Fokker-Planck equations from the perspective of the moments, the time-averaged mean-squared displacements, and the autocovariance functions. We also study generalized Langevin equations based on these generalized operators. The differences between the Fokker-Planck and Langevin equations with different integrodifferential operators are discussed and compared with the dynamic behavior of established models of scaled Brownian motion and fractional Brownian motion. We demonstrate that the integrodifferential operators with exponential and Mittag-Leffler kernels are not suitable to be introduced to Fokker-Planck and Langevin equations for the physically relevant diffusion scenarios discussed in our paper. The conformable and Caputo Langevin equations are unveiled to share similar properties with scaled and fractional Brownian motion, respectively.

Velocity autocorrelation of a free particle driven by a Mittag-Leffler noise: fractional dynamics and temporal behaviors

We investigate the dynamical phase diagram of the generalized Langevin equation of the free particle driven by a Mittag-Leffler noise and show critical curves and a critical value of the exponent parameter of the Mittag-Leffler function that mark different dynamical regimes. By considering that the modeling of a Mittag-Leffer memory kernel corresponds to a power-law second-order memory kernel, we show that the generalized Langevin equation of the velocity autocorrelation function (VACF) is transformed in a fractional Langevin equation. In the superdiffusive case our results exhibit oscillations and negative correlations of the VACF that are not provided by the usual power-law noise model.

Fractional fluctuation effects on the light scattered by a viscoelastic suspension

We generalize fluctuating hydrodynamics to study the effect of fractional time derivatives on the light-scattering spectrum of a suspension in a viscoelastic solvent under an external density gradient. Viscoelasticity introduces additional memory effects into the fluctuating hydrodynamic equations, causing the time scales associated with the mesoscopic variables and those of the microscopic events to be no longer well separated. This situation is taken into account by introducing Caputo's fractional time derivative into the description. The structure factor of the suspension is calculated, and we find that its nonequilibrium correction is an odd function of the frequency. It exhibits a shift towards negative frequencies proportional to the magnitude of the imposed gradient. We consider solvents that are described by Maxwell's or power-law rheological equations of state. The fractional structure factor is compared with the nonfractional one, and it is found that the ratio of the former to the latter may be positive and up to two orders of magnitude for both types of viscoelasticity. This prediction of our model calculation suggests that this relative change might be measurable.

Superdiffusion induced by a long-correlated external random force

We consider a particle immersed in a thermal reservoir and simultaneously subjected to an external random force that drives the system to a nonequilibrium situation. Starting from a Langevin equation description, we derive exact expressions for the mean-square displacement and the velocity autocorrelation function of the diffusing particle. An effective temperature is introduced to characterize the deviation from the internal equilibrium situation. Using a power-law force autocorrelation function, the mean-square displacement and the velocity autocorrelation function are analytically obtained in terms of Mittag-Leffler functions. In this case, we show that the present model exhibits a superdiffusive regime as a consequence of the competition between passive and active processes.

Continuous-time random walk: crossover from anomalous regime to normal regime

We consider decoupled continuous time random walk with finite characteristic waiting time and jump length variance. We take approximate jump length probability distribution and waiting time probability distribution given by a product of power-law and exponential function. Using this waiting time probability distribution we study diffusion behaviors for all the time. Due to the finite characteristic waiting time and jump length variance the model presents normal diffusive behavior in the long-time limit. However, the model can describe anomalous behavior at the short and intermediate times. In particular, the model can describe subdiffusive, normal, and superdiffusive behaviors at the short times. Moreover, exact solution for probability distribution of the system is also investigated.

A lower bound to the survival probability and an approximate first passage time distribution for Markovian and non-Markovian dynamics in phase space

We derive a very general expression of the survival probability and the first passage time distribution for a particle executing Brownian motion in full phase space with an absorbing boundary condition at a point in the position space, which is valid irrespective of the statistical nature of the dynamics. The expression, together with the Jensen's inequality, naturally leads to a lower bound to the actual survival probability and an approximate first passage time distribution. These are expressed in terms of the position-position, velocity-velocity, and position-velocity variances. Knowledge of these variances enables one to compute a lower bound to the survival probability and consequently the first passage distribution function. As examples, we compute these for a Gaussian Markovian process and, in the case of non-Markovian process, with an exponentially decaying friction kernel and also with a power law friction kernel. Our analysis shows that the survival probability decays exponentially at the long time irrespective of the nature of the dynamics with an exponent equal to the transition state rate constant.

Anomalous diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise

The diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise is studied. Using the Laplace analysis we derive exact expressions for the relaxation functions of the particle in terms of generalized Mittag-Leffler functions and its derivatives from a generalized Langevin equation. Our results show that the oscillator displays an anomalous diffusive behavior. In the strictly asymptotic limit, the dynamics of the harmonic oscillator corresponds to an oscillator driven by a noise with a pure power-law autocorrelation function. However, at short and intermediate times the dynamics has qualitative difference due to the presence of the characteristic time of the noise.

Fractional Langevin equation and Riemann-Liouville fractional derivative

In this present work we consider a fractional Langevin equation with Riemann-Liouville fractional time derivative which modifies the classical Newtonian force, nonlocal dissipative force, and long-time correlation. We investigate the first two moments, variances and position and velocity correlation functions of this system. We also compare them with the results obtained from the same fractional Langevin equation which uses the Caputo fractional derivative.

Anomalous diffusion induced by a Mittag-Leffler correlated noise

We introduce a Mittag-Leffler correlated random force leading to anomalous diffusion. Starting from a generalized Langevin equation, and using Laplace analysis we derive exact expressions for the mean values, variances and diffusion coefficient for a free particle in terms of generalized Mittag-Leffler functions and its derivatives. The asymptotic behavior of these quantities are obtained, from which the anomalous diffusion behavior of the particle is displayed.