Diffusion on two-dimensional percolation clusters: Influence of cluster anisotropy

Generalization of the Kubo relation for confined motion and ergodicity breakdown

A time-dependent generalized Kubo relation is derived by introducing the notion of a diffusion function for a particle confined in a harmonic potential. The relation reduces to the standard Kubo relation as a special case but holds for anomalous diffusion, nonergodic processes, and bounded motion. We analyze in detail the behaviors of the diffusion and memory functions and report a generalized Stokes-Einstein relation concerning anomalous diffusion. Furthermore, we demonstrate that when a high finite-frequency cutoff is imposed on the noise spectral density, a breakdown in ergodicity accompanied by the appearance of nonstationarity in the velocity autocorrelation function occurs in forced systems. This breakdown is taken as explicit evidence for either decay-spring-memory or recovering-force effects leading to nonexponential relaxation kinematics.

Generalized Einstein relations and conditions for anomalous relaxation

The generalized Einstein relation (GER) for nonergodic processes is investigated within the framework of the generalized Langevin equation. The conditions for anomalous relaxation such as long-tail decay and non-vanishing velocity autocorrelation function (VAF) are proposed and distinguished. For the stationary nonergodic process, if the initial preparation of the particle velocity is non-thermal, an asymptotic GER occurs in a departure from the usual result. It is shown that the GER holding is a necessary condition rather than a full condition for the system being close to equilibrium. For the nonergodic process of the second type due to cutoff of high frequencies, the VAF oscillates with time, the GER holds but the equilibrium fails in the long-time limit. Applications to some practical examples confirm the present theoretical findings.

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.

State transition of a non-Ohmic damping system in a corrugated plane

Anomalous transport of a particle subjected to non-Ohmic damping of the power delta in a tilted periodic potential is investigated via Monte Carlo simulation of the generalized Langevin equation. It is found that the system exhibits two relative motion modes: the locked state and the running state. In an environment of sub-Ohmic damping (0<delta<1) , the particle should transfer into a running state from a locked state only when local minima of the potential vanish; hence a synchronization oscillation occurs in the particle's mean displacement and mean square displacement (MSD). In particular, the two motion modes are allowed to coexist in the case of super-Ohmic damping (1<delta<2) for moderate driving forces, namely, where double centers exist in the velocity distribution. This causes the particle to have faster diffusion, i.e., its MSD reads <Deltax(2)(t)>=2D_(eff)(delta){t(delta_eff} . Our result shows that the effective power index delta_(eff) can be enhanced and is a nonmonotonic function of the temperature and the driving force. The mixture of the two motion modes also leads to a breakdown of the hysteresis loop of the mobility.

Time-dependent barrier passage of a non-Ohmic damping system

We consider a particle passing over the saddle point of an inverse harmonic potential, which is described by a generalized Langevin equation with a non-Ohmic damping of power exponent delta. The time-dependent passing probability and transmission coefficient are obtained analytically by using the reaction flux method. It is shown that the overshooting phenomenon for the passing probability appears in the regime 0<delta<1 and the backflow recrossing over the saddle point is observed, where a nonmonotonous time dependence of the passage probability is detected. The long memory aspect of friction is at the origin of this behavior. Thus the steady transmission coefficient is also a nonmonotonous function of delta.

Harmonic velocity noise: non-Markovian features of noise-driven systems at long times

We propose a harmonic velocity noise with a broadband feature, which is the time derivative of the harmonic noise. If this noise is regarded as a thermal one, the system has a vanishing effective friction and it should induce ballistic diffusion of a free particle at long times. The effective temperature of the system coupled to such a structured heat bath represented by the harmonic velocity noise is introduced. This means that any initial preparation will approach asymptotically a preparation-dependent variance and mean value for velocity variable. Thus the fluctuation-dissipation theorem does not hold as there is no unique stationary state being connected with a breakdown of ergodicity. This noise can show greenness when it is taken as an external noise source to drive a correlation ratchet.

Anomalous dissipation: strong non-Markovian effect and its dynamical origin

We report the effects of anomalous dissipation with a vanishing effective friction, which can induce ballistic diffusion and dissipative acceleration, where the fluctuation-dissipation theorem is fulfilled. An influence factor is introduced in order to describe the role of non-Markovian friction and the force-folded effect on the long-time results. The velocity-dependent coupling and force might be a dynamical origin of this dissipation. The steady acceleration of a particle moving in periodic and magnetic-force potentials are calculated.

Anomalous transport in unbound and ratchet potentials

A non-Markovian Langevin equation with a broadband noise is proposed to describe anomalous transport of a particle passing over a potential saddle or moving in a ratchet potential. In the presence of thermal broadband noise, the asymptotic mean square displacement of a free particle is proportional to the square of time; this is called ballistic diffusion. The passing probability of a particle driven by this broadband noise over the saddle of an inverted harmonic potential is obtained analytically. It is shown that the passing probability increases with the kinetic energy, which is slower than that of normal case. The mechanisms of ballistic diffusion and mobility are also applied to the rocking (a square-wave driving force acting on the potential) and flashing (the potential fluctuating between on and off) ratchets. Phenomena such as acceleration and double-peak mean velocity are observed.