Correlations in a generalized elastic model: fractional Langevin equation approach

Anomalous phase diagram of the elastic interface with nonlocal hydrodynamic interactions in the presence of quenched disorder

We investigate the influence of quenched disorder on the steady states of driven systems of the elastic interface with nonlocal hydrodynamic interactions. The generalized elastic model (GEM), which has been used to characterize numerous physical systems such as polymers, membranes, single-file systems, rough interfaces, and fluctuating surfaces, is a standard approach to studying the dynamics of elastic interfaces with nonlocal hydrodynamic interactions. The criticality and phase transition of the quenched generalized elastic model are investigated numerically and the results are presented in a phase diagram spanned by two tuning parameters. We demonstrate that in the one-dimensional disordered driven GEM, three qualitatively different behavior regimes are possible with a proper specification of the order parameter (mean velocity) for this system. In the vanishing order parameter regime, the steady-state order parameter approaches zero in the thermodynamic limit. A system with a nonzero mean velocity can be in either the continuous regime, which is characterized by a second-order phase transition, or the discontinuous regime, which is characterized by a first-order phase transition. The focus of this research is to investigate the critical scaling features near the pinning-depinning threshold. The behavior of the quenched generalized elastic model at the critical depinning force is explored. Near the depinning threshold, the critical exponent is obtained numerically.

Scaling characteristics of one-dimensional fractional diffusion processes in the presence of power-law distributed random noise

Here, we present results of numerical simulations and the scaling characteristics of one-dimensional random fluctuations with heavy-tailed probability distribution functions. Assuming that the distribution function of the random fluctuations obeys Lévy statistics with a power-law scaling exponent, we investigate the fractional diffusion equation in the presence of μ-stable Lévy noise. We study the scaling properties of the global width and two-point correlation functions and then compare the analytical and numerical results for the growth exponent β and the roughness exponent α. We also investigate the fractional Fokker-Planck equation for heavy-tailed random fluctuations. We show that the fractional diffusion processes in the presence of μ-stable Lévy noise display special scaling properties in the probability distribution function (PDF). Finally, we numerically study the scaling properties of the heavy-tailed random fluctuations by using the diffusion entropy analysis. This method is based on the evaluation of the Shannon entropy of the PDF generated by the random fluctuations, rather than on the measurement of the global width of the process. We apply the diffusion entropy analysis to extract the growth exponent β and to confirm the validity of our numerical analysis.

Numerical approach to unbiased and driven generalized elastic model

From scaling arguments and numerical simulations, we investigate the properties of the generalized elastic model (GEM) that is used to describe various physical systems such as polymers, membranes, single-file systems, or rough interfaces. We compare analytical and numerical results for the subdiffusion exponent β characterizing the growth of the mean squared displacement 〈(δh)(2)〉 of the field h described by the GEM dynamic equation. We study the scaling properties of the qth order moments 〈∣δh∣(q)〉 with time, finding that the interface fluctuations show no intermittent behavior. We also investigate the ergodic properties of the process h in terms of the ergodicity breaking parameter and the distribution of the time averaged mean squared displacement. Finally, we study numerically the driven GEM with a constant, localized perturbation and extract the characteristics of the average drift for a tagged probe.

Analytical calculation of four-point correlations for a simple model of cages involving numerous particles

Dynamics of a one-dimensional system of Brownian particles with short-range repulsive interaction (diameter σ) is studied with a liquid-theoretical approach. The mean square displacement, the two-particle displacement correlation, and the overlap-density-based generalized susceptibility are calculated analytically by way of the Lagrangian correlation of the interparticulate space, instead of the Eulerian correlation of density that is commonly used in the standard mode-coupling theory. In regard to the mean square displacement, the linear analysis reproduces the established result on the asymptotic subdiffusive behavior of the system. A finite-time correction is given by incorporating the effect of entropic nonlinearity with a Lagrangian version of mode-coupling theory. The notorious difficulty in derivation of the mode-coupling theory concerning violation of the fluctuation-dissipation theorem is found to disappear by virtue of the Lagrangian description. The Lagrangian description also facilitates analytical calculation of four-point correlations in the space-time, such as the two-particle displacement correlation. The two-particle displacement correlation, which is asymptotically self-similar in the space-time, illustrates how the cage effect confines each particle within a short radius on one hand and creates collective motion of numerous particles on the other hand. As the time elapses, the correlation length grows unlimitedly, and the generalized susceptibility based on the overlap density converges to a finite value which is an increasing function of the density. The distribution function behind these dynamical four-point correlations and its extension to three-dimensional cases, respecting the tensorial character of the two-particle displacement correlation, are also discussed.

Anomalous transport in the crowded world of biological cells

A ubiquitous observation in cell biology is that the diffusive motion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarizing their densely packed and heterogeneous structures. The most familiar phenomenon is a sublinear, power-law increase of the mean-square displacement (MSD) as a function of the lag time, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations in time, non-Gaussian distributions of spatial displacements, heterogeneous diffusion and a fraction of immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarize some widely used theoretical models: Gaussian models like fractional Brownian motion and Langevin equations for visco-elastic media, the continuous-time random walk model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Particular emphasis is put on the spatio-temporal properties of the transport in terms of two-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even if the MSDs are identical. Then, we review the theory underlying commonly applied experimental techniques in the presence of anomalous transport like single-particle tracking, fluorescence correlation spectroscopy (FCS) and fluorescence recovery after photobleaching (FRAP). We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where a variety of model systems mimic physiological crowding conditions. Finally, computer simulations are discussed which play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.

Two-point approximation to the Kramers problem with coloured noise

We present a method, founded on previous renewal approaches as the classical Wilemski-Fixman approximation, to describe the escape dynamics from a potential well of a particle subject to non-Markovian fluctuations. In particular, we show how to provide an approximated expression for the distribution of escape times if the system is governed by a generalized Langevin equation (GLE). While we show that the method could apply to any friction kernel in the GLE, we focus here on the case of power-law kernels, for which extensive literature has appeared in the last years. The method presented (termed as two-point approximation) is able to fit the distribution of escape times adequately for low potential barriers, even if conditions are far from Markovian. In addition, it confirms that non-exponential decays arise when a power-law friction kernel is considered (in agreement with related works published recently), which questions the existence of a characteristic reaction rate in such situations.