Langevin formulation of a subdiffusive continuous-time random walk in physical time

Anomalous diffusion in external-force-affected deterministic systems

This study investigates the impact of external forces on the movement of particles, specifically focusing on a type of box piecewise linear map that generates normal diffusion akin to Brownian motion. Through numerical methods, the research delves into the effects of two distinct external forces: linear forces linked to the particle's current position and periodic sinusoidal forces related to time. The results uncover anomalous dynamical behavior characterized by nonlinear growth in the ensemble-averaged mean-squared displacement (EAMSD), aging, and ergodicity breaking. Notably, the diffusion pattern of particles under linear external forces resembles an Ehrenfest double urn model, with its asymptotic EAMSD coinciding with the Langevin equation under linear potential. Meanwhile, particle movement influenced by periodic sinusoidal forces corresponds to an inhomogeneous Markov chain, with its external force amplitude and diffusion coefficient function exhibiting a "multipeak" fractal structure. The study also provides insights into the formation of this structure through the turnstiles dynamics.

Aging and confinement in subordinated fractional Brownian motion

We study the effects of aging properties of subordinated fractional Brownian motion (FBM) with drift and in harmonic confinement, when the measurement of the stochastic process starts a time t_{a}>0 after its original initiation at t=0. Specifically, we consider the aged versions of the ensemble mean-squared displacement (MSD) and the time-averaged MSD (TAMSD), along with the aging factor. Our results are favorably compared with simulations results. The aging subordinated FBM exhibits a disparity between MSD and TAMSD and is thus weakly nonergodic, while strong aging is shown to effect a convergence of the MSD and TAMSD. The information on the aging factor with respect to the lag time exhibits an identical form to the aging behavior of subdiffusive continuous-time random walks (CTRW). The statistical properties of the MSD and TAMSD for the confined subordinated FBM are also derived. At long times, the MSD in the harmonic potential has a stationary value, that depends on the Hurst index of the parental (nonequilibrium) FBM. The TAMSD of confined subordinated FBM does not relax to a stationary value but increases sublinearly with lag time, analogously to confined CTRW. Specifically, short aging times t_{a} in confined subordinated FBM do not affect the aged MSD, while for long aging times the aged MSD has a power-law increase and is identical to the aged TAMSD.

Langevin picture of subdiffusion in nonuniformly expanding medium

Anomalous diffusion phenomena have been observed in many complex physical and biological systems. One significant advance recently is the physical extension of particle's motion in a static medium to a uniformly and even nonuniformly expanding medium. The dynamic mechanism of the anomalous diffusion in the nonuniformly expanding medium has only been investigated by the approach of continuous-time random walk. To study more physical observables and to supplement the physical models of the anomalous diffusion in the expanding mediums, we characterize the nonuniformly expanding medium with a spatiotemporal dependent scale factor a(x,t) and build the Langevin picture describing the particle's motion in the nonuniformly expanding medium. Besides the existing comoving and physical coordinates, by introducing a new coordinate and assuming that a(x,t) is separable at a long-time limit, we build the relation between the nonuniformly expanding medium and the uniformly expanding one and further obtain the moments of the comoving and physical coordinates. Different forms of the scale factor a(x,t) are considered to uncover the combined effects of the particle's intrinsic diffusion and the nonuniform expansion of medium. The theoretical analyses and simulations provide the foundation for studying more anomalous diffusion phenomena in the expanding mediums.

Langevin picture of anomalous diffusion processes in expanding medium

The expanding medium is very common in many different fields, such as biology and cosmology. It brings a nonnegligible influence on particle's diffusion, which is quite different from the effect of an external force field. The dynamic mechanism of a particle's motion in an expanding medium has only been investigated in the framework of a continuous-time random walk. To focus on more diffusion processes and physical observables, we build the Langevin picture of anomalous diffusion in an expanding medium, and conduct detailed analyses in the framework of the Langevin equation. With the help of a subordinator, both subdiffusion process and superdiffusion process in the expanding medium are discussed. We find that the expanding medium with different changing rate (exponential form and power-law form) leads to quite different diffusion phenomena. The particle's intrinsic diffusion behavior also plays an important role. Our detailed theoretical analyses and simulations present a panoramic view of investigating anomalous diffusion in an expanding medium under the framework of the Langevin equation.

Different effects of external force fields on aging Lévy walk

Aging phenomena have been observed in numerous physical systems. Many statistical quantities depend on the aging time t_a for aging anomalous diffusion processes. This paper pays more attention to how an external force field affects the aging Lévy walk. Based on the Langevin picture of the Lévy walk and the generalized Green-Kubo formula, we investigate the quantities that include the ensemble- and time-averaged mean-squared displacements in both weak aging t_a≪t and strong aging t_a≫t cases and compare them to the ones in the absence of any force field. Two typical force fields, constant force F and time-dependent periodic force F(t)=f₀sin⁡(ωt), are considered for comparison. The generalized Einstein relation is also discussed in the case with the constant force. We find that the constant force is the key to causing the aging phenomena and enhancing the diffusion behavior of the aging Lévy walk, while the time-dependent periodic force is not. The different effects of the two kinds of forces on the aging Lévy walk are verified by both theoretical analyses and numerical simulations.

Langevin picture of Lévy walk in a constant force field

Lévy walk is a practical model and has wide applications in various fields. Here we focus on the effect of an external constant force on the Lévy walk with the exponent of the power-law-distributed flight time α∈(0,2). We add the term Fη(s) [η(s) is the Lévy noise] on a subordinated Langevin system to characterize such a constant force, as it is effective on the velocity process for all physical time after the subordination. We clearly show the effect of the constant force F on this Langevin system and find this system is like the continuous limit of the collision model. The first moments of velocity processes for these two models are consistent. In particular, based on the velocity correlation function derived from our subordinated Langevin equation, we investigate more interesting statistical quantities, such as the ensemble- and time-averaged mean-squared displacements. Under the influence of constant force, the diffusion of particles becomes faster. Finally, the superballistic diffusion and the nonergodic behavior are verified by the simulations with different α.

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 the 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 displacements, position autocorrelation function, correlation coefficient, and 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 nonstationary, nonergodic, and aging.

Weak Galilean invariance as a selection principle for coarse-grained diffusive models

How does the mathematical description of a system change in different reference frames? Galilei first addressed this fundamental question by formulating the famous principle of Galilean invariance. It prescribes that the equations of motion of closed systems remain the same in different inertial frames related by Galilean transformations, thus imposing strong constraints on the dynamical rules. However, real world systems are often described by coarse-grained models integrating complex internal and external interactions indistinguishably as friction and stochastic forces. Since Galilean invariance is then violated, there is seemingly no alternative principle to assess a priori the physical consistency of a given stochastic model in different inertial frames. Here, starting from the Kac-Zwanzig Hamiltonian model generating Brownian motion, we show how Galilean invariance is broken during the coarse-graining procedure when deriving stochastic equations. Our analysis leads to a set of rules characterizing systems in different inertial frames that have to be satisfied by general stochastic models, which we call "weak Galilean invariance." Several well-known stochastic processes are invariant in these terms, except the continuous-time random walk for which we derive the correct invariant description. Our results are particularly relevant for the modeling of biological systems, as they provide a theoretical principle to select physically consistent stochastic models before a validation against experimental data.