Brownian motion with alternately fluctuating diffusivity: Stretched-exponential and power-law relaxation

Generalized two-state random walk model: Nontrivial anomalous diffusion, aging, and ergodicity breaking

The intermittent stochastic motion is a dichotomous process that alternates between two distinct states. This phenomenon, observed across various physical and biological systems, is attracting increasing interest and highlighting the need for comprehensive theories to describe it. In this paper, we introduce a generalized intermittent random walk model based on a renewal process that alternates between the continuous time random walk (CTRW) state and the generalized Lévy walk (gLW) state. Notably, the nonlinear space-time coupling inherent in the gLW state allows this generalized model to encompass a variety of random walk models and makes it applicable to diverse systems. By deriving the velocity correlation function and utilizing the scaling Green-Kubo relation, the ensemble-averaged and time-averaged mean-squared displacement (MSD) is calculated, and the anomalous diffusive behavior, aging effect, and ergodic property of the model are further analyzed and discussed. The results reveal that, due to the intermittent nature, there are two diffusive terms in the expression of the MSD, and the diffusion can be intermediately characterized by the diffusive term with the largest diffusion coefficient instead of the diffusive term with the largest diffusion exponent, which is significantly different from single-state stochastic process. We demonstrate that, due to the power-law distribution of sojourn times, nonlinear space-time coupling, and intermittent characteristics, both ergodicity and nonergodicity can coexist in intermittent stochastic processes.

Generalized Langevin dynamics for single beads in linear elastic networks

We derive generalized Langevin equations (GLEs) for single beads in linear elastic networks. In particular, the derivations of the GLEs are conducted without employing normal modes, resulting in two distinct representations in terms of resistance and mobility kernels. The fluctuation-dissipation relations are also confirmed for both GLEs. Subsequently, we demonstrate that these two representations are interconnected via Laplace transforms. Furthermore, another GLE is derived by utilizing a projection operator method, and it is shown that the equation obtained through the projection scheme is consistent with the GLE with the resistance kernel. As simple examples, the general theory is applied to the Rouse model and the ring polymer, where the GLEs with the resistance and mobility kernels are explicitly derived for arbitrary positions of the tagged bead in these models. Finally, the GLE with the mobility kernel is also derived for the elastic network with hydrodynamic interactions under the pre-averaging approximation.

Statistics of the number of renewals, occupation times, and correlation in ordinary, equilibrium, and aging alternating renewal processes

The renewal process is a point process where an interevent time between successive renewals is an independent and identically distributed random variable. Alternating renewal process is a dichotomous process and a slight generalization of the renewal process, where the interevent time distribution alternates between two distributions. We investigate statistical properties of the number of renewals and occupation times for one of the two states in alternating renewal processes. When both means of the interevent times are finite, the alternating renewal process can reach an equilibrium. On the other hand, an alternating renewal process shows aging when one of the means diverges. We provide analytical calculations for the moments of the number of renewals, occupation time statistics, and the correlation function for several case studies in the interevent-time distributions. We show anomalous fluctuations for the number of renewals and occupation times when the second moment of interevent time diverges. When the mean interevent time diverges, distributional limit theorems for the number of events and occupation times are shown analytically. These are known as the Mittag-Leffler distribution and the generalized arcsine law in probability theory.

Anomalous diffusion, nonergodicity, non-Gaussianity, and aging of fractional Brownian motion with nonlinear clocks

How do nonlinear clocks in time and/or space affect the fundamental properties of a stochastic process? Specifically, how precisely may ergodic processes such as fractional Brownian motion (FBM) acquire predictable nonergodic and aging features being subjected to such conditions? We address these questions in the current study. To describe different types of non-Brownian motion of particles-including power-law anomalous, ultraslow or logarithmic, as well as superfast or exponential diffusion-we here develop and analyze a generalized stochastic process of scaled-fractional Brownian motion (SFBM). The time- and space-SFBM processes are, respectively, constructed based on FBM running with nonlinear time and space clocks. The fundamental statistical characteristics such as non-Gaussianity of particle displacements, nonergodicity, as well as aging are quantified for time- and space-SFBM by selecting different clocks. The latter parametrize power-law anomalous, ultraslow, and superfast diffusion. The results of our computer simulations are fully consistent with the analytical predictions for several functional forms of clocks. We thoroughly examine the behaviors of the probability-density function, the mean-squared displacement, the time-averaged mean-squared displacement, as well as the aging factor. Our results are applicable for rationalizing the impact of nonlinear time and space properties superimposed onto the FBM-type dynamics. SFBM offers a general framework for a universal and more precise model-based description of anomalous, nonergodic, non-Gaussian, and aging diffusion in single-molecule-tracking observations.

Temperature and friction fluctuations inside a harmonic potential

In this article we study the trapped motion of a molecule undergoing diffusivity fluctuations inside a harmonic potential. For the same diffusing-diffusivity process, we investigate two possible interpretations. Depending on whether diffusivity fluctuations are interpreted as temperature or friction fluctuations, we show that they display drastically different statistical properties inside the harmonic potential. We compute the characteristic function of the process under both types of interpretations and analyze their limit behavior. Based on the integral representations of the processes we compute the mean-squared displacement and the normalized excess kurtosis. In the long-time limit, we show for friction fluctuations that the probability density function (PDF) always converges to a Gaussian whereas in the case of temperature fluctuations the stationary PDF can display either Gaussian distribution or generalized Laplace (Bessel) distribution depending on the ratio between diffusivity and positional correlation times.

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.

Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and models

How does a systematic time-dependence of the diffusion coefficient $D (t)$ affect the ergodic and statistical characteristics of fractional Brownian motion (FBM)? Here, we examine how the behavior of the...

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.

Orientational memory of active particles in multistate non-Markovian processes

The orientational memory of particles can serve as an effective measure of diffusivity, spreading, and search efficiency in complex stochastic processes. We develop a theoretical framework to describe the decay of directional correlations in a generic class of stochastic active processes consisting of distinct states of motion characterized by their persistence and switching probabilities between the states. For exponentially distributed sojourn times, the orientation autocorrelation is analytically derived and the characteristic times of its crossovers are obtained in terms of the persistence of each state and the switching probabilities. We show how nonexponential sojourn-time distributions of interest, such as Gaussian and power-law distributions, can result from history-dependent transitions between the states. The relaxation behavior of the correlation function in such non-Markovian processes is governed by the history dependence of the switching probabilities and cannot be solely determined by the mean sojourn times of the states.

Time averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does stochastic resetting impact nonergodicity? We examine, both analytically and by simulations, the implications of resetting on the MSD- and TAMSD-based spreading dynamics of particles executing fractional Brownian motion (FBM) with a long-time memory, heterogeneous diffusion processes (HDPs) with a power-law space-dependent diffusivity D(x)=D_{0}|x|^{γ} and their "combined" process of HDP-FBM. We find, inter alia, that the resetting dynamics of originally ergodic FBM for superdiffusive Hurst exponents develops disparities in scaling and magnitudes of the MSDs and mean TAMSDs indicating weak ergodicity breaking. For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD and observe a new trimodal form of the probability density function. For reset FBM, HDPs and HDP-FBM we compute analytically and verify by simulations the short-time MSD and TAMSD asymptotes and long-time plateaus reminiscent of those for processes under confinement. We show that certain characteristics of these reset processes are functionally similar despite a different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicity-breaking parameter EB as a function of the resetting rate r. For all reset processes studied we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB∼(1/r)-decay at large r. Alongside the emerging MSD-versus-TAMSD disparity, this r-dependence of EB can be an experimentally testable prediction. We conclude by discussing some implications to experimental systems featuring resetting dynamics.

Fast-field-cycling ultralow-field nuclear magnetic relaxation dispersion

Optically pumped magnetometers (OPMs) based on alkali-atom vapors are ultra-sensitive devices for dc and low-frequency ac magnetic measurements. Here, in combination with fast-field-cycling hardware and high-resolution spectroscopic detection, we demonstrate applicability of OPMs in quantifying nuclear magnetic relaxation phenomena. Relaxation rate dispersion across the nT to mT field range enables quantitative investigation of extremely slow molecular motion correlations in the liquid state, with time constants > 1 ms, and insight into the corresponding relaxation mechanisms. The 10-20 fT/\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{{\rm{H}}}{\rm{z}}$$\end{document}Hz sensitivity of an OPM between 10 Hz and 5.5 kHz ¹H Larmor frequency suffices to detect magnetic resonance signals from ~ 0.1 mL liquid volumes imbibed in simple mesoporous materials, or inside metal tubing, following nuclear spin prepolarization adjacent to the OPM. High-resolution spectroscopic detection can resolve inter-nucleus spin-spin couplings, further widening the scope of application to chemical systems. Expected limits of the technique regarding measurement of relaxation rates above 100 s⁻¹ are discussed.

Nuclear spin polarization and relaxation can be studied using nuclear magnetic resonance (NMR). Here the authors demonstrate a combination of fast-field cycling and optical magnetometry techniques, to realize a NMR sensor that operates in the region of very low frequency and high relaxation rate.

Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

We study a two state "jumping diffusivity" model for a Brownian process alternating between two different diffusion constants, D+>D-, with random waiting times in both states whose distribution is rather general. In the limit of long measurement times, Gaussian behavior with an effective diffusion coefficient is recovered. We show that, for equilibrium initial conditions and when the limit of the diffusion coefficient D-⟶0 is taken, the short time behavior leads to a cusp, namely a non-analytical behavior, in the distribution of the displacements P(x,t) for x⟶0. Visually this cusp, or tent-like shape, resembles similar behavior found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state D+ to a δ-function. The short time behavior of the same quantity converges to a uniform distribution, which leads to the non-analyticity in P(x,t). We demonstrate how super-statistical framework is a zeroth order short time expansion of P(x,t), in the number of transitions, that does not yield the cusp like shape. The latter, considered as the key feature of experiments in the field, is found with the first correction in perturbation theory.

Piecewise linear model of self-organized hierarchy formation

The Bonabeau model of self-organized hierarchy formation is studied by using a piecewise linear approximation to the sigmoid function. Simulations of the piecewise-linear agent model show that there exist two-level and three-level hierarchical solutions and that each agent exhibits a transition from nonergodic to ergodic behaviors. Furthermore, by using a mean-field approximation to the agent model, it is analytically shown that there are asymmetric two-level solutions, even though the model equation is symmetric (asymmetry is introduced only through the initial conditions) and that linearly stable and unstable three-level solutions coexist. It is also shown that some of these solutions emerge through supercritical-pitchfork-like bifurcations in invariant subspaces. Existence and stability of the linear hierarchy solution in the mean-field model are also elucidated.

Coarse-graining of microscopic dynamics into a mesoscopic transient potential model

We show that a mesoscopic coarse-grained dynamics model which incorporates the transient potential can be formally derived from an underlying microscopic dynamics model. As a microscopic dynamics model, we employ the overdamped Langevin equation. By utilizing the path probability and the Onsager-Machlup type action, we calculate the path probability for the coarse-grained mesoscopic degrees of freedom. The action for the mesoscopic degrees of freedom can be simplified by incorporating the transient potential. Then the dynamic equation for the mesoscopic degrees of freedom can be simply described by the Langevin equation with the transient potential (LETP). As a simple and analytically tractable approximation, we introduce additional degrees of freedom which express the state of the transient potential. Then we approximately express the dynamics of the system as the the combination of the LETP and the dynamics model for the transient potential. The resulting dynamics model has the same dynamical structure as the responsive particle dynamics type models [W. J. Briels, Soft Matter 5, 4401 (2009)1744-683X10.1039/b911310j] and the multichain slip-spring type models [T. Uneyama and Y. Masubuchi, J. Chem. Phys. 137, 154902 (2012)JCPSA60021-960610.1063/1.4758320]. As a demonstration, we apply our coarse-graining method with the LETP to a single particle dynamics in a supercooled liquid, and compare the results of the LETP with the molecular dynamics simulations and other coarse-graining models.