Relation between generalized diffusion equations and subordination schemes

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.

Parameter estimation of the fractional Ornstein-Uhlenbeck process based on quadratic variation

Modern experiments routinely produce extensive data of the diffusive dynamics of tracer particles in a large range of systems. Often, the measured diffusion turns out to deviate from the laws of Brownian motion, i.e., it is anomalous. Considerable effort has been put in conceiving methods to extract the exact parameters underlying the diffusive dynamics. Mostly, this has been done for unconfined motion of the tracer particle. Here, we consider the case when the particle is confined by an external harmonic potential, e.g., in an optical trap. The anomalous particle dynamics is described by the fractional Ornstein-Uhlenbeck process, for which we establish new estimators for the parameters. Specifically, by calculating the empirical quadratic variation of a single trajectory, we are able to recover the subordination process governing the particle motion and use it as a basis for the parameter estimation. The statistical properties of the estimators are evaluated from simulations.

Anomalous diffusion, non-Gaussianity, and nonergodicity for subordinated fractional Brownian motion with a drift

The stochastic motion of a particle with long-range correlated increments (the moving phase) which is intermittently interrupted by immobilizations (the trapping phase) in a disordered medium is considered in the presence of an external drift. In particular, we consider trapping events whose times follow a scale-free distribution with diverging mean trapping time. We construct this process in terms of fractional Brownian motion with constant forcing in which the trapping effect is introduced by the subordination technique, connecting "operational time" with observable "real time." We derive the statistical properties of this process such as non-Gaussianity and nonergodicity, for both ensemble and single-trajectory (time) averages. We demonstrate nice agreement with extensive simulations for the probability density function, skewness, kurtosis, as well as ensemble and time-averaged mean-squared displacements. We place a specific emphasis on the comparisons between the cases with and without drift.

Non-normalizable quasiequilibrium states under fractional dynamics

Particles anomalously diffusing in contact with a thermal bath are initially released from an asymptotically flat potential well. For temperatures that are sufficiently low compared to the potential depth, the dynamical and thermodynamical observables of the system remain almost constant for long times. We show how these stagnated states are characterized as non-normalizable quasiequilibrium (NNQE) states. We use the fractional-time Fokker-Planck equation (FTFPE) and continuous-time random walk approaches to calculate ensemble averages. We obtain analytical estimates of the durations of NNQE states, depending on the fractional order, from approximate theoretical solutions of the FTFPE. We study and compare two types of observables, the mean square displacement typically used to characterize diffusion, and the thermodynamic energy. We show that the typical timescales for transient stagnation depend exponentially on the value of the depth of the potential well, in units of temperature, multiplied by a function of the fractional exponent.

Subdiffusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiffusion to superdiffusion

We use a subdiffusion equation with fractional Caputo time derivative with respect to another function g (g-subdiffusion equation) to describe a smooth transition from ordinary subdiffusion to superdiffusion. Ordinary subdiffusion is described by the equation with the "ordinary" fractional Caputo time derivative, superdiffusion is described by the equation with a fractional Riesz-type spatial derivative. We find the function g for which the solution (Green's function, GF) to the g-subdiffusion equation takes the form of GF for ordinary subdiffusion in the limit of small time and GF for superdiffusion in the limit of long time. To solve the g-subdiffusion equation we use the g-Laplace transform method. It is shown that the scaling properties of the GF for g-subdiffusion and the GF for superdiffusion are the same in the long time limit. We conclude that for a sufficiently long time the g-subdiffusion equation describes superdiffusion well, despite a different stochastic interpretation of the processes. Then, paradoxically, a subdiffusion equation with a fractional time derivative describes superdiffusion. The superdiffusive effect is achieved here not by making anomalously long jumps by a diffusing particle, but by greatly increasing the particle jump frequency which is derived by means of the g-continuous-time random walk model. The g-subdiffusion equation is shown to be quite general, it can be used in modeling of processes in which a kind of diffusion change continuously over time. In addition, some methods used in modeling of ordinary subdiffusion processes, such as the derivation of local boundary conditions at a thin partially permeable membrane, can be used to model g-subdiffusion processes, even if this process is interpreted as superdiffusion.

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.

The Fokker-Planck equation of the superstatistical fractional Brownian motion with application to passive tracers inside cytoplasm

By collecting from literature data experimental evidence of anomalous diffusion of passive tracers inside cytoplasm, and in particular of subdiffusion of mRNA molecules inside live Escherichia coli cells, we obtain the probability density function of molecules' displacement and we derive the corresponding Fokker-Planck equation. Molecules' distribution emerges to be related to the Krätzel function and its Fokker-Planck equation to be a fractional diffusion equation in the Erdélyi-Kober sense. The irreducibility of the derived Fokker-Planck equation to those of other literature models is also discussed.

General approach to stochastic resetting

We address the effect of stochastic resetting on diffusion and subdiffusion process. For diffusion we find that mean square displacement relaxes to a constant only when the distribution of reset times possess finite mean and variance. In this case, the leading order contribution to the probability density function (PDF) of a Gaussian propagator under resetting exhibits a cusp independent of the specific details of the reset time distribution. For subdiffusion we derive the PDF in Laplace space for arbitrary resetting protocol. Resetting at constant rate allows evaluation of the PDF in terms of H function. We analyze the steady state and derive the rate function governing the relaxation behavior. For a subdiffusive process the steady state could exist even if the distribution of reset times possesses only finite mean.

Real-space imaging of nanoparticle transport and interaction dynamics by graphene liquid cell TEM

Thermal motion of colloidal nanoparticles and their cohesive interactions are of fundamental importance in nanoscience but are difficult to access quantitatively, primarily due to the lack of the appropriate analytical tools to investigate the dynamics of individual particles at nanoscales. Here, we directly monitor the stochastic thermal motion and coalescence dynamics of gold nanoparticles smaller than 5 nm, using graphene liquid cell (GLC) transmission electron microscopy (TEM). We also present a novel model of nanoparticle dynamics, providing a unified, quantitative explanation of our experimental observations. The nanoparticles in a GLC exhibit non-Gaussian, diffusive motion, signifying dynamic fluctuation of the diffusion coefficient due to the dynamically heterogeneous environment surrounding nanoparticles, including organic ligands on the nanoparticle surface. Our study shows that the dynamics of nanoparticle coalescence is controlled by two elementary processes: diffusion-limited encounter complex formation and the subsequent coalescence of the encounter complex through rotational motion, where surface-passivating ligands play a critical role.

Integral decomposition for the solutions of the generalized Cattaneo equation

We present the integral decomposition for the fundamental solution of the generalized Cattaneo equation with both time derivatives smeared through convoluting them with some memory kernels. For power-law kernels t^{-α}, α∈(0,1] this equation becomes the time fractional one governed by the Caputo derivatives in which the highest order is 2. To invert the solutions from the Fourier-Laplace domain to the space-time domain we use analytic methods based on the Efross theorem and find out that solutions looked for are represented by integral decompositions which tangle the fundamental solution of the standard Cattaneo equation with nonnegative and normalizable functions being uniquely dependent on the memory kernels. Furthermore, the use of methodology arising from the theory of complete Bernstein functions allows us to assign such constructed integral decompositions the interpretation of subordination. This fact is preserved in two limit cases built into the generalized Cattaneo equations, i.e., either the diffusion or the wave equations. We point out that applying the Efross theorem enables us to go beyond the standard approach which usually leads to the integral decompositions involving the Gaussian distribution describing the Brownian motion. Our approach clarifies puzzling situation which takes place for the power-law kernels t^{-α} for which the subordination based on the Brownian motion does not work if α∈(1/2,1].

Subdiffusion equation with Caputo fractional derivative with respect to another function

We show an application of a subdiffusion equation with Caputo fractional time derivative with respect to another function g to describe subdiffusion in a medium having a structure evolving over time. In this case a continuous transition from subdiffusion to other type of diffusion may occur. The process can be interpreted as "ordinary" subdiffusion with fixed subdiffusion parameter (subdiffusion exponent) α in which timescale is changed by the function g. As an example, we consider the transition from "ordinary" subdiffusion to ultraslow diffusion. The g-subdiffusion process generates the additional aging process superimposed on the "standard" aging generated by "ordinary" subdiffusion. The aging process is analyzed using coefficient of relative aging of g-subdiffusion with respect to "ordinary" subdiffusion. The method of solving the g-subdiffusion equation is also presented.