In biased and soft-walled channels: Insights into transport phenomena and damped modulation.

The motion of a particle along a channel of finite width is known to be affected by either the presence of energy barriers or changes in the bias forces along the channel direction. By using the lateral equilibrium hypothesis, we have successfully derived the effective diffusion coefficient for soft-walled channels, and the diffusion is found to be influenced by the curvature profile of the potential. A typical phenomenon of diffusion enhancement is observed under the appropriate parameter conditions. We first discovered an anomalous phenomenon of quasi-periodic enhancement of oscillations, which cannot be captured by the one-dimensional effective potential, under the combination of sub-Ohmic damping with two-dimensional restricted channels. We innovatively develop the effective potential and the formation mechanism of velocity variance under super-Ohmic and ballistic damping, and meanwhile, ergodicity is of concern. The theoretical framework of a ballistic system can be reinterpreted through the folding acceleration theory. This comprehensive analysis significantly enhances our understanding of diffusion processes in constrained geometries.

Multiple diffusive behaviors of the random walk in inhomogeneous environments.

Anomalous diffusive behaviors are observed in highly inhomogeneous but relatively stable environments such as intracellular media and are increasingly attracting attention. In this paper we develop a coupled continuous-time random walk model in which the waiting time is power-law coupled with the local environmental diffusion coefficient. We provide two forms of the waiting time density, namely, a heavy-tailed density and an exponential density. For different waiting time densities, anomalous diffusions with the diffusion exponent between 0 and 2 and Brownian yet non-Gaussian diffusion can be realized within the present model. The diffusive behaviors are analyzed and discussed by deriving the mean-squared displacement and probability density function. In addition we derive the effective jump length density corresponding to the decoupled form to help distinguish the diffusion types. Our model unifies two kinds of anomalous diffusive behavior with different characteristics in the same inhomogeneous environment into a theoretical framework. The model interprets the random motion of particles in a complex inhomogeneous environment and reproduces the experimental results of different biological and physical systems.

Quantifying the energy landscape in weakly and strongly disordered frictional media.

We investigate the "roughness" of the energy landscape of a system that diffuses in a heterogeneous medium with a random position-dependent friction coefficient α(x). This random friction acting on the system stems from spatial inhomogeneity in the surrounding medium and is modeled using the generalized Caldira-Leggett model. For a weakly disordered medium exhibiting a Gaussian random diffusivity D(x) = kBT/α(x) characterized by its average value ⟨D(x)⟩ and a pair-correlation function ⟨D(x1)D(x2)⟩, we find that the renormalized intrinsic diffusion coefficient is lower than the average one due to the fluctuations in diffusivity. The induced weak internal friction leads to increased roughness in the energy landscape. When applying this idea to diffusive motion in liquid water, the dissociation energy for a hydrogen bond gradually approaches experimental findings as fluctuation parameters increase. Conversely, for a strongly disordered medium (i.e., ultrafast-folding proteins), the energy landscape ranges from a few to a few kcal/mol, depending on the strength of the disorder. By fitting protein folding dynamics to the escape process from a metastable potential, the decreased escape rate conceptualizes the role of strong internal friction. Studying the energy landscape in complex systems is helpful because it has implications for the dynamics of biological, soft, and active matter systems.

Ergodic Measure and Potential Control of Anomalous Diffusion.

In statistical mechanics, the ergodic hypothesis (i.e., the long-time average is the same as the ensemble average) accompanying anomalous diffusion has become a continuous topic of research, being closely related to irreversibility and increasing entropy. While measurement time is finite for a given process, the time average of an observable quantity might be a random variable, whose distribution width narrows with time, and one wonders how long it takes for the convergence rate to become a constant. This is also the premise of ergodic establishment, because the ensemble average is always equal to the constant. We focus on the time-dependent fluctuation width for the time average of both the velocity and kinetic energy of a force-free particle described by the generalized Langevin equation, where the stationary velocity autocorrelation function is considered. Subsequently, the shortest time scale can be estimated for a system transferring from a stationary state to an effective ergodic state. Moreover, a logarithmic spatial potential is used to modulate the processes associated with free ballistic diffusion and the control of diffusion, as well as the minimal realization of the whole power-law regime. The results presented suggest that non-ergodicity mimics the sparseness of the medium and reveals the unique role of logarithmic potential in modulating diffusion behavior.

Coexistence of ergodicity and nonergodicity in the aging two-state random walks.

The two-state stochastic phenomenon is observed in various systems and is attracting more interest, and it can be described by the two-state random walk (TSRW) model. The TSRW model is a typical two-state renewal process alternating between the continuous-time random walk state and the Lévy walk state, in both of which the sojourn time distributions follow a power law. In this paper, by discussing the statistical properties and calculating the ensemble averaged and time averaged mean squared displacement, the ergodic property and the ultimate diffusive behavior of the aging TSRW is studied. Results reveal that because of the two-state intermittent feature, ergodicity and nonergodicity can coexist in the aging TSRW, which behave as the time scalings of the time averages and ensemble averages not being identically equal. In addition, we find that the unique state occupation mechanism caused by the diverging mean of the sojourn times of one state, determines the aging TSRW's ultimate diffusive behavior at extremely large timescales, i.e., instead of the term with the larger diffusion exponent, the diffusion is surprisingly characterized by the term with the smaller one, which is distinctly different from previous conclusions and known results. At last, we note that the Lévy walk with rests model which also displays aging and ergodicity breaking, can be generalized by the TSRW model.

Consistent Hamiltonian models for space-momentum diffusion.

We develop a unified Hamiltonian approach to the diffusion of a particle coupled to a dissipative environment, an archetypal model widely invoked to interpret condensed phase phenomena, such as polymerization and cold-atom diffusion in optical lattices. By appropriate choices of the coupling functions, we reformulate phenomenological diffusion models by adding otherwise ignored space-momentum terms. We thus numerically predict a variety of diffusion regimes, from diffusion saturation to superballistic diffusion. With reference to ultracold atoms in optical lattices, we also show that time correlated external noises prevent superdiffusion from exceeding Richardson's law. Some of these results are unexpected and call for experimental validation.

Strong anomalous diffusive behaviors of the two-state random walk process.

The phenomenon of the two-state process is observed in various systems and is increasingly attracting attention, such that there is a need for a theoretical model of the process. In this paper, we present a prototypal two-state random walk (TSRW) model of a renewal process alternating between the continuous-time random walk (CTRW) state and Lévy walk (LW) state. The jump length distribution of the CTRW state is assumed to be Gaussian whereas the time distributions of the two states are both considered to follow a power law. The diffusive behavior is analyzed and discussed by calculating the mean squared displacement (MSD) analytically and numerically. The results reveal that it displays strong anomalous diffusive behaviors caused by random motions of both states, i.e., two anomalous diffusion terms coexist in the expression of the MSD, and the time distribution which has the heavier tail determines their forms. Moreover, because the two diffusion terms originate from different mechanisms, we find that the diffusion can be characterized by either the term with the largest diffusion exponent or the term with the largest diffusion coefficient at long timescales, which shows very different properties from the single-state process. In addition, the two-state nature of the process of the particle moving in a velocity field makes the TSRW model applicable to describe it. Results obtained from the two-state model reveal that the diffusion can even exhibit subdiffusive behavior, which is significantly different from known results obtained using the single-state model.

Correlated continuous-time random walk in the velocity field: the role of velocity and weak asymptotics.

Within the framework of a space-time correlated continuous-time random walk model, anomalous diffusion of particles moving in the velocity field is studied in this paper. The weak asymptotic form ω(t) ∼ t-(1+α), 1 < α < 2 for large t, is considered to be the waiting time distribution. The analytical results reveal that the diffusion in the velocity field, i.e., the mean squared displacement, can display a multi-fractional form caused by dispersive bias and space-time correlation. The numerical results indicate that the multi-fractional diffusion leads to a crossover phenomenon in-between the process at an intermediate timescale, followed by a steady state which is always determined by the largest diffusion exponent term. In addition, the role of velocity and weak asymptotics is discussed. The extremely small fluid velocity can characterize the diffusion by a diffusion coefficient instead of diffusion exponent, which is distinctly different from the former definition. In particular, for the waiting time displaying a weak asymptotic property, if the anomalous part is suppressed by the normal part, a second crossover phenomenon appears at an intermediate timescale, followed by a steady normal diffusion, which implies that the anomalies underlying the process are smoothed out at large timescales. Moreover, we discuss that the consideration of bias and correlation could help to avoid a possible not readily noticeable mistake in studying the topic concerned in this paper, which may be helpful in the relevant experimental research.

Debye Brownian oscillator and Debye-type noise: A series solution versus Monte Carlo simulation.

For the Debye Brownian oscillator, we present a series solution to the generalized Langevin equation describing the motion of a particle. The external potential is considered to be a harmonic potential and the spectral density of driven noise is a hard cutoff at high finite frequencies. The results are in agreement with both numerical calculations and Monte Carlo simulations. We demonstrate abnormal weak ergodic breaking; specifically, the long-time average of the observable vanishes but the corresponding ensemble average continues to oscillate with time. This Debye Brownian oscillator does not arrive at an equilibrium state and undergoes underdamped-like motion for any model parameter. Nevertheless, ergodic behavior and equilibrium can be recovered concurrently using a strong bound potential. We give an understanding of the behavior as being the consequence of discrete breather modes in the lattices similar to the formation of an additional periodic signal. Furthermore, we compare the results calculated by cutting off separately the spectral density and the correlation function of colored noise.

Ergodic time scale and transitive dynamics in single-particle tracking.

We investigate ergodic time scales in single-particle tracking by introducing a covariance measure Ω(Δ;t) for the time-averaged relative square displacement recorded in lag-time Δ at elapsed time t. The present model is established in the generalized Langevin equation with a power-law memory function. The ratio Ω(Δ;Δ)/Ω(Δ;t) is shown to obey a universal scaling law for long but finite times and is used to extract the effective ergodic time. We derive a finite-time-averaged Green-Kubo relation and find that, to control the deviations in measurement results from ensemble averages, the ratio Δ/t must be neither too small nor close to unity. Our paper connects the experimental self-averaging property of a tracer with the theoretic velocity autocorrelation function and sheds light on the transition to ergodicity.

Effect of Self-Oscillation on Escape Dynamics of Classical and Quantum Open Systems.

We study the effect of self-oscillation on the escape dynamics of classical and quantum open systems by employing the system-plus-environment-plus-interaction model. For a damped free particle (system) with memory kernel function expressed by Zwanzig (J. Stat. Phys. 9, 215 (1973)), which is originated from a harmonic oscillator bath (environment) of Debye type with cut-off frequency wd, ergodicity breakdown is found because the velocity autocorrelation function oscillates in cosine function for asymptotic time. The steady escape rate of such a self-oscillated system from a metastable potential exhibits nonmonotonic dependence on wd, which denotes that there is an optimal cut-off frequency makes it maximal. Comparing results in classical and quantum regimes, the steady escape rate of a quantum open system reduces to a classical one with wd decreasing gradually, and quantum fluctuation indeed enhances the steady escape rate. The effect of a finite number of uncoupled harmonic oscillators N on the escape dynamics of a classical open system is also discussed.

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.

Correlated continuous-time random walk in a velocity field: Anomalous bifractional crossover.

The diffusion of space-time correlated continuous-time random walk moving in the velocity field, which includes the fluid flowing freely and the fluid flowing through porous media, is investigated in this paper. Results reveal that it presents anomalous diffusion merely caused by space-time correlation in the freely flowing fluid, and the bias from the velocity field only supplies a standard advection, which is verified by the corresponding generalized diffusion equation which includes a standard advection term. However, the diffusion in the fluid flowing through porous media, i.e., the mean squared displacement, can display a bifractional form of which one originates from space-time correlation and the other one originates from dispersive bias caused by sticking of the porous media. The fractional advection term emerging in the corresponding generalized diffusion equation confirms the results. Moreover, the coexistence of correlation and dispersive bias result in crossover phenomenon in-between the diffusive process at an intermediate timescale, but just as the definition of diffusion, the one owning the largest diffusion exponent always prevails at large timescales. However, since the two fractional diffusions originate from a different mechanism, even if it owns the smaller diffusion exponent, that one can dominate the diffusion if it fluctuates much stronger than the other one, which no longer obeys the previous conclusion.

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.

Diffusion crossing over a barrier in a random rough metastable potential.

We carry out a detailed study of escape dynamics of a particle driven by a white noise over a metastable potential corrugated by spatial disorder in the form of zero-mean random correlated potential. The approach of double-averaging over test particles and statistic ensemble is proposed to calculate the escape rate in a finite-size random rough metastable potential, moreover, the interference mechanism of test particles multi-passing over the saddle point is considered. Through analyzing the dependence of the steady escape rate on various modelled potentials and parameters, we demonstrate that the obstruction induced by roughness leads to a decrease in the steady escape rate with the increase of rough intensity. We also add the random correlated potential into the vicinity of the saddle-point of metastable potentials of three kinds and show an enhancement phenomenon of escape rate similar to the previous study of a surmounting fluctuating sharp barrier.

Anomalous barrier escape: The roles of noise distribution and correlation.

We study numerically and analytically the barrier escape dynamics of a particle driven by an underlying correlated Lévy noise for a smooth metastable potential. A "quasi-monochrome-color" Lévy noise, i.e., the first-order derivative variable of a linear second-order differential equation subjected to a symmetric α-stable white Lévy noise, also called the harmonic velocity Lévy noise, is proposed. Note that the time-integral of the noise Green function of this kind is equal to zero. This leads to the existence of underlying negative time correlation and implies that a step in one direction is likely followed by a step in the other direction. By using the noise of this kind as a driving source, we discuss the competition between long flights and underlying negative correlations in the metastable dynamics. The quite rich behaviors in the parameter space including an optimum α for the stationary escape rate have been found. Remarkably, slow diffusion does not decrease the stationary rate while a negative correlation increases net escape. An approximate expression for the Lévy-Kramers rate is obtained to support the numerically observed dependencies.

Transition of multidiffusive states in a biased periodic potential.

We study a frequency-dependent damping model of hyperdiffusion within the generalized Langevin equation. The model allows for the colored noise defined by its spectral density, assumed to be proportional to ω^{δ-1} at low frequencies with 0<δ<1 (sub-Ohmic damping) or 1<δ<2 (super-Ohmic damping), where the frequency-dependent damping is deduced from the noise by means of the fluctuation-dissipation theorem. It is shown that for super-Ohmic damping and certain parameters, the diffusive process of the particle in a titled periodic potential undergos sequentially four time regimes: thermalization, hyperdiffusion, collapse, and asymptotical restoration. For analyzing transition phenomenon of multidiffusive states, we demonstrate that the first exist time of the particle escaping from the locked state into the running state abides by an exponential distribution. The concept of an equivalent velocity trap is introduced in the present model; moreover, reformation of ballistic diffusive system is also considered as a marginal situation but does not exhibit the collapsed state of diffusion.

Group superballistic diffusion: bimodal velocity inducing coexistence of two states in a corrugated plane.

We consider anomalous diffusion of a particle moving in a tilted periodic potential in the presence of Lévy noise and nonlinear friction. Using Monte Carlo simulations, we have found some interesting characteristics of diffusion in such a nonlinear system: when the noise intensity is weak and the external force is close to the critical value at which local minima of the potential just vanish, the nonmonotonic behavior of the effective diffusion index and the superballistic diffusion are observed. This is due to the bimodal nature of the velocity distribution, and thus the test particles exist in either a running state or a long-tailed behind state in the spatial coordinate; the latter is disintegrated into small pieces of the probability peaks. We provide a relation between the group diffusion coefficient and the phase diffusion coefficient. It is shown that the distance between the above two-state centers increasing with time plays the definitive role in the superballistic group diffusion.

Multidimensional master equation and its Monte-Carlo simulation.

We derive an integral form of multidimensional master equation for a markovian process, in which the transition function is obtained in terms of a set of discrete Langevin equations. The solution of master equation, namely, the probability density function is calculated by using the Monte-Carlo composite sampling method. In comparison with the usual Langevin-trajectory simulation, the present approach decreases effectively coarse-grained error. We apply the master equation to investigate time-dependent barrier escape rate of a particle from a two-dimensional metastable potential and show the advantage of this approach in the calculations of quantities that depend on the probability density function.

Inertial Lévy flight.

Lévy flights with inertial term in various potentials are studied analytically and numerically in terms of the fractional Fokker-Planck equation. The probability density functions of the Lévy flight particle in the linear and harmonic potentials are exactly obtained and a transient contribution of inertial term is found in the case of linear potential, which can be neglected when evolution time is much longer than γ(-1) (γ is the damping coefficient); for the harmonic potential, where the stationary state has still infinite variance of coordinate and its width is larger than that of the inertialess case for arbitrary Lévy index µ in the region of 0<µ<2, we still find that the distribution of velocity of the particle is also the Lévy-type one. Moreover, a crossover from bimodal to unimodal shape for the spatial distribution is shown in the anharmonic potential. Finally, we represent an analytical expression of the rate constant for the inertial Lévy flight particle escaping from a metastable potential by using the reactive flux method and show that the rate is a nonmonotonic function of the Lévy index.