We review the latest advances in the analytical modelling of single file diffusion. We focus first on the derivation of the fractional Langevin equation that describes the motion of a tagged file particle. We then propose an alternative derivation of the very same stochastic equation by starting from the diffusion-noise formalism for the time evolution of the file density.
Single-file diffusion behaves as normal diffusion at small time and as anomalous subdiffusion at large time. These properties can be described by fractional Brownian motion with variable Hurst exponent or multifractional Brownian motion. We introduce a new stochastic process called Riemann-Liouville step fractional Brownian motion which can be regarded as a special case of multifractional Brownian motion with step function type of Hurst exponent tailored for single-file diffusion. Such a step fractional Brownian motion can be obtained as solution of fractional Langevin equation with zero dampi…
We introduce a stochastic equation for the microscopic motion of a tagged particle in the single file model. This equation provides a compact representation of several of the system's properties such as Fluctuation-Dissipation and Linear Response relations, achieved by means of a diffusion noise approach. Most important, the proposed Langevin Equation reproduces quantitatively the \emph{three} temporal regimes and the corresponding time scales: ballistic, diffusive and subdiffusive.
In the simplest model of single-file diffusion, $N$ point particles wander on a segment of the $x$ axis of length $L$, with hard core interactions, which prevent passing, and with overdamped Brownian dynamics, $λ\dot{x}=η(t)$, where $η(t)$ has the form of Gaussian white noise with zero mean. In 1965 Harris showed that in the limit $N\to\infty$, $L\to\infty$ with constant $ρ=N/L$, the mean square displacement of a tagged particle grows subdiffusively, as $t^{1/2}$, for long times. Recently, it has been shown that the proportionality constants of the $t^{1/2}$ law for randomly-distributed initia…
We discuss two independent methods of solution of a master equation whose biased jump transition rates account for long jumps of Lévy-stable type and nonetheless admit a Boltzmannian (thermal) equilibrium to arise in the large time asymptotics of a probability density function $ρ(x,t)$. Our main goal is to demonstrate a compatibility of a {\it direct} solution method (an explicit, albeit numerically assisted, integration of the master equation) with an {\it indirect} path-wise procedure, recently proposed in [Physica {\bf A 392}, 3485, (2013)] as a valid tool for a dynamical analysis of non-La…
We present a comparison between the random motion of an adiabatic and a diathermal piston sliding in a perfect gas. In particular, their dynamical behaviour, if investigated by means of Langevin's approach, shows the amplitude of the adiabatic-piston random displacements around the equilibrium position to be much larger (by a factor (M/m)^(1/2), where M and m are the piston mass and the mass of the single gas molecule) than that of the diathermal piston. The origin of this intriguing difference, which is accounted for in the frame of Langevin's approach, is also explored in terms of a space-ph…
The Generalized Elastic Model is a linear stochastic model which accounts for the behaviour of many physical systems in nature, ranging from polymeric chains to single-file systems. If an external perturbation is exerted \emph{only} on a single point $\vec{x}^\star$ (\emph{tagged probe}), it propagates throughout the entire system. Within the fractional Langevin equation framework, we study the effect of such a perturbation, in cases of a constant force applied. We report most of the results arising from our previous analysis and, in the present work, we show that the Fox $H$-functions formali…
We investigate the non-Langevin relative of the Lévy-driven Langevin random system, under an assumption that both systems share a common (asymptotic, stationary, steady-state) target pdf. The relaxation to equilibrium in the fractional Langevin-Fokker-Planck scenario results from an impact of confining conservative force fields on the random motion. A non-Langevin alternative has a built-in direct response of jump intensities to energy (potential) landscapes in which the process takes place. We revisit the problem of Lévy flights in superharmonic potential wells, with a focus on the extremally…
Langevin equation with a multiplicative stochastic force is considered. That force is uncorrelated, it has the Lévy distribution and the power-law intensity. The Fokker-Planck equations, which correspond both to the Itô and Stratonovich interpretation of the stochastic integral, are presented. They are solved for the case without drift and for the harmonic oscillator potential. The variance is evaluated; it is always infinite for the Itô case whereas for the Stratonovich one it can be finite and rise with time slower that linearly, which indicates subdiffusion. Analytical results are compared …
We study the nonequilibrum steady states in a unidirectional {or driven} single file motion (DSFM) of a collection of particles with hard-core repulsion in a closed system. For driven propulsion that is {spatially} smoothly varying with a few discontinuities, we show that the steady states are broadly classified into two classes, independent of any system detail: (i) when the steady state current depends explicitly on the {conserved} number density $n$, and (ii) when it is independent of $n$. This manifests itself in the universal topology of the phase diagrams {and fundamental diagrams (i.e.,…
We have proposed a new stochastic interpretation of the sudiffusion described by the Sharma-Mittal entropy formalism which generates a nonlinear subdiffusion equation with natural order derivatives. We have shown that the solution to the diffusion equation generated by Gauss entropy (which is the particular case of Sharma-Mittal entropy) is the same as the solution of the Fokker-Planck (FP) equation generated by the Langevin generalised equation where the `long memory effect' is taken into account. The external noise which pertubates the subdiffusion coefficient (occuring in the solution of FP…
In the context of the phenomenon of Stochastic Resonance (SR) we study the correlation function, the signal-to-noise ratio (SNR) and the ratio of output over input SNR, i.e. the gain, which is associated to the nonlinear response of a bistable system driven by time-periodic forces and white Gaussian noise. These quantifiers for SR are evaluated using the techniques of Linear Response Theory (LRT) beyond the usually employed two-mode approximation scheme. We analytically demonstrate within such an extended LRT description that the gain can indeed not exceed unity. We implement an efficient algo…
The path-integral representation of Smoluchowski equation is exploited to explore the stochastic dynamics of a tagged Brownian particle within an interacting system where hydrodynamic effects are neglected. In particular, this formalism is applied to a particle system confined to a one-dimensional infinite line aiming to investigate the single-file diffusion phenomenon in this scenario. In particular, the path-integral method is contrasted against the standard many-particle Langevin equation for a system of interacting Brownian particles in a harmonic chain model, exhibiting excellent agreemen…
We investigate memory effects in barrier-crossing in the overdamped setting. We focus on the scenario where the hidden degrees of freedom relax on exactly the same time scale as the observable. As a prototypical model, we analyze tagged-particle diffusion in a single file confined to a bi-stable potential. We identify the signatures of memory and explain their origin. The emerging memory is a result of the projection of collective many-body eigenmodes onto the motion of a tagged-particle. We are interested in the "confining" (all background particles in front of the tagged-particle) and "pushi…
Based on the generalized Langevin equation for the momentum of a Brownian particle a generalized asymptotic Einstein relation is derived. It agrees with the well-known Einstein relation in the case of normal diffusion but continues to hold for sub- and super-diffusive spreading of the Brownian particle's mean square displacement. The generalized asymptotic Einstein relation is used to analyze data obtained from molecular dynamics simulations of a two-dimensional soft disk fluid. We mainly concentrated on medium densities for which we found super-diffusive behavior of a tagged fluid particle. A…
Single particle tracking has become a standard tool to investigate diffusive properties, especially in small systems such as biological cells. Usually the resulting time series are analyzed in terms of time averages over individual trajectories. Here we study confined normal as well as anomalous diffusion modeled by fractional Brownian motion and the fractional Langevin equation, and show that even for such ergodic systems time-averaged quantities behave differently from their ensemble averaged counterparts, irrespective of how long the measurement time becomes. Knowledge of the exact behavior…
Analytical expressions are put forward to investigate the forced spiking activity of abstract neuron models such as the driven leaky integrate-and-fire (LIF) model. The method is valid in a wide parameter regime beyond the restraining limits of weak driving (linear response) and/or weak noise. The novel approximation is based on a discrete state Markovian modeling of the full dynamics with time-dependent rates. The scheme yields very good agreement with numerical Langevin and Fokker-Planck simulations of the full non-stationary dynamics for both, the first-passage time statistics and the inter…
Recently, a novel framework to handle stochastic processes has emerged from a series of studies in biology, showing situations beyond 'Itô versus Stratonovich'. Its internal consistency can be demonstrated via the zero mass limit of a generalized Klein-Kramers equation. Moreover, the connection to other integrations becomes evident: the obtained Fokker-Planck equation defines a new type of stochastic calculus that in general differs from the α-type interpretation. A unique advantage of this new approach is a natural correspondence between stochastic and deterministic dynamics, which is useful …
We use a Generalised Langevin Equation (GLE) scheme to study the thermal transport of low dimensional systems. In this approach, the central classical region is connected to two realistic thermal baths kept at two different temperatures [H. Ness et al., Phys. Rev. B {\bf 93}, 174303 (2016)]. We consider model Al systems, i.e. one-dimensional atomic chains connected to three-dimensional baths. The thermal transport properties are studied as a function of the chain length $N$ and the temperature difference $ΔT$ between the baths. We calculate the transport properties both in the linear response …
We present a simple point process model of $1/f^β$ noise, covering different values of the exponent $β$. The signal of the model consists of pulses or events. The interpulse, interevent, interarrival, recurrence or waiting times of the signal are described by the general Langevin equation with the multiplicative noise and stochastically diffuse in some interval resulting in the power-law distribution. Our model is free from the requirement of a wide distribution of relaxation times and from the power-law forms of the pulses. It contains only one relaxation rate and yields $1/f^ β$ spectra in a…
