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 "pushing" (all background particles behind the tagged-particle) scenarios for which we find non-trivial and qualitatively different relaxation behaviors. Notably and somewhat unexpectedly, at a fixed particle number, we find that the higher the barrier, the stronger the memory effects are. The fact that the external potential alters the memory is important more generally and should be taken into account in applications of generalized Langevin equations. Our results can readily be tested experimentally and may be relevant for understanding transport in biological ion-channels.
We consider a gas of point particles moving in a one-dimensional channel with a hard-core inter-particle interaction that prevents particle crossings --- this is called single-file motion. Starting from equilibrium initial conditions we observe the motion of a tagged particle. It is well known that if the individual particle dynamics is diffusive, then the tagged particle motion is sub-diffusive, while for ballistic particle dynamics, the tagged particle motion is diffusive. Here we compute exactly the large deviation function for the tagged particle displacement and show that this is universa…
Single-file diffusion is a one-dimensional interacting infinite-particle system in which the order of particles never changes. An intriguing feature of single-file diffusion is that the mean-square displacement of a tagged particle exhibits an anomalously slow sub-diffusive growth. We study the full statistics of the displacement using a macroscopic fluctuation theory. For the simplest single-file system of impenetrable Brownian particles we compute the large deviation function and provide an independent verification using an exact solution based on the microscopic dynamics. For an arbitrary s…
We derive and study a theoretical description for single file diffusion, i.e., diffusion in a one dimensional lattice of particles with hard core interaction. It is well known that for this system a tagged particle has anomalous diffusion for long times. The novelty of the present approach is that it allows for the derivation of correlations between a tagged particle and other particles of the system at a given distance with empty sites in between. The behavior of the correlation gives deeper insights into the processes involved. Numerical integration of differential equations are in good agre…
We study diffusion of (fluorescently) tagged hard-core interacting particles of finite size in a finite one-dimensional system. We find an exact analytical expression for the tagged particle probability density using a coordinate Bethe-ansatz, from which the mean square displacement is calculated. The analysis show the existence of three regimes of drastically different behavior for short, intermediate and large times. The results show excellent agreement with stochastic simulations (Gillespie algorithm). The findings of the Letter holds promise for the development of novel bio-nano sensors.
We investigate the single-file dynamics of a tagged particle in a system consisting of N hardcore interacting particles (the particles cannot pass each other) which are diffusing in a one-dimensional system where the particles have different diffusion constants. For the two particle case an exact result for the conditional probability density function (PDF) is obtained for arbitrary initial particle positions and all times. The two-particle PDF is used to obtain the tagged particle PDF. For the general N-particle case (N large) we perform stochastic simulations using our new computationally ef…
We study the statistics of a tagged particle in single-file diffusion, a one-dimensional interacting infinite-particle system in which the order of particles never changes. We compute the two-time correlation function for the displacement of the tagged particle for an arbitrary single-file system. We also discuss single-file analogs of the arcsine law and the law of the iterated logarithm characterizing the behavior of Brownian motion. Using a macroscopic fluctuation theory we devise a formalism giving the cumulant generating functional. In principle, this functional contains the full statisti…
Strong positional correlations between particles render the diffusion of a tracer particle in a single file anomalous and non-Markovian. While ensemble average observables of tracer particles are nowadays well understood, little is known about the statistics of the corresponding functionals, i.e. the time-average observables. It even remains unclear how the non-Markovian nature emerges from correlations between particle trajectories at different times. Here, we first present rigorous results for fluctuations and two-tag correlations of general bounded functionals of ergodic Markov processes wi…
We compute the full probability distribution of the positions of a tagged particle exactly for given arbitrary initial positions of the particles and for general single-particle propagators. We consider the thermodynamic limit of our exact expressions in quenched and annealed settings. For a particular class of single-particle propagators, the exact formula is expressed in a simple integral form in the quenched case whereas in the annealed case, it is expressed as a simple combination of Bessel functions. In particular, we focus on the step and the power-law initial configurations. In the form…
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…
Single-file diffusion is a paradigm for strongly correlated classical stochastic many-body dynamics and has widespread applications in soft condensed matter and biophysics. However, exact results for {single-file} systems are sparse and limited to the simplest scenarios. We present an algorithm for computing the non-Markovian time-dependent conditional probability density function of a {tagged-particle} in a {single-file} of $N$ particles diffusing in a confining external potential. The algorithm implements an eigenexpansion of the full interacting many-body problem obtained by means of the co…
We consider a gas of point particles moving on the one-dimensional line with a hard-core inter-particle interaction that prevents particle crossings --- this is usually referred to as single-file motion. The individual particle dynamics can be arbitrary and they only interact when they meet. Starting from initial conditions such that particles are uniformly distributed, we observe the displacement of a tagged particle at time $t$, with respect to the initial position of another tagged particle, such that their tags differ by $r$. For $r=0$, this is the usual well studied problem of the tagged …
We study the single file diffusion problem on a one-dimensional lattice with a self-similar distribution of hopping rates. We find that the time dependence of the mean-square displacement of both a tagged particle and the center of mass of the system present anomalous power laws modulated by logarithmic periodic oscillations. The anomalous exponent of a tagged particle is one half of the exponent of the center of mass, and always smaller than 1/4. Using heuristic arguments, the exponents and the periods of oscillation are analytically obtained and confirmed by Monte Carlo simulations.
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.
We investigate the long-time behavior of the survival probability of a tagged particle in a single-file diffusion in a finite interval. The boundary conditions are of two types: 1) one boundary is absorbing the second is reflecting, 2) both boundaries are absorbing. For each type of the boundary conditions we consider two types of initial conditions: a) initial number of particles N is given, b) initial concentration of particles is given (N is random). In all four cases the tagged-particle survival probability exhibits different asymptotic behavior. When the both boundaries are absorbing we a…
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.
We study the dynamics of a tagged particle in an environment of point Brownian particles with hard-core interactions in an infinite one dimensional channel (a single-file model). In particular we examine the influence of initial conditions on the dynamic of the tagged particle. We compare two initial conditions: equal distances between particles and uniform density distribution. The effect is shown by the differences of mean-square-displacement and correlation function for the two ensembles of initial conditions. We discuss the violation of Einstein relation, and its dependence on the initial …
The term single file (SF) dynamics refers to the motion of an assembly of particles through a channel with cross-section comparable to the particles' diameter. Single file diffusion (SFD) is then the diffusion of a tagged particle in a \emph{single file}, i.e., under the condition that particle passing is not allowed. SFD accounts for a large variety of processes in nature, including diffusion of colloids in synthetic and natural-shaped channels, biological motors along molecular chains, electrons in proteins and liquid helium, ions through membrane, just to mention a few examples. Albeit intr…
The dynamics of hard-core interacting Brownian particles in an external potential field is studied in one dimension. Using the Jepsen line we find a very general and simple formula relating the motion of the tagged center particle, with the classical, time dependent single particle reflection ${\cal R}$ and transmission ${\cal T}$ coefficients. Our formula describes rich physical behaviors both in equilibrium and the approach to equilibrium of this many body problem.
We study the stochastic thermodynamics of an overdamped harmonic chain, which can be viewed equivalently as a 1D Rouse chain or as an approximate model of single file diffusion. We discuss mainly two levels of description of this system: the Markovian level for which the trajectories of all the particles of the chain are known and the non-Markovian level in which only the motion of a tagged particle is available. For each case, we analyze the energy dissipation and its dependence on initial conditions. Surprisingly, we find that the average coarse-grained entropy production rate can become tra…
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 solve a non-equilibrium statistical mechanics problem exactly, namely, the single-file dynamics of N hard-core interacting particles (the particles cannot pass each other) of size Δdiffusing in a one dimensional system of finite length L with reflecting boundaries at the ends. We obtain an exact expression for the conditional probability density function P_T(y_T,t|y_{T,0}) that a tagged particle T (T=1,...,N) is at position y_T at time t given that it at time t=0 was at position y_{T,0}. Going beyond previous studies, we consider the asymptotic limit of large N, maintaining L finite, using …
We study the single-file dynamics of three classes of active particles: run-and-tumble particles, active Brownian particles and active Ornstein-Uhlenbeck particles. At high activity values, the particles, interacting via purely repulsive and short-ranged forces, aggregate into several motile and dynamical clusters of comparable size, and do not display bulk phase-segregation. In this dynamical steady-state, we find that the cluster size distribution of these aggregates is a scaled function of the density and activity parameters across the three models of active particles with the same scaling …
We study the diffusion of tagged hard core interacting particles under the influence of an external force field. Using the Jepsen line we map this many particle problem onto a single particle one. We obtain general equations for the distribution and the mean square displacement $<(x_T)^2>$ of the tagged center particle valid for rather general external force fields and initial conditions. A wide range of physical behaviors emerge which are very different than the classical single file sub-diffusion $<(x_T)^2 > \sim t^{1/2}$ found for uniformly distributed particles in an infinite space and in …
We study single-file diffusion on a one-dimensional lattice with a random fractal distribution of hopping rates. For finite lattices, this problem shows three clearly different regimes, namely, nearly independent particles, highly interacting particles, and saturation. The mean-square displacement of a tagged particle as a function of time follows a power law in each regime. The first crossover time $t_s$, between the first and the second regime, depends on the particle density. The other crossover time $t_l$, between the second and the third regime, depends on the lattice length. We find anal…
The standard setup for single-file diffusion is diffusing particles in one dimension which cannot overtake each other, where the dynamics of a tracer (tagged) particle is of main interest. In this article we generalise this system and investigate first-passage properties of a tracer particle when flanked by crowder particles which may, besides diffuse, unbind (rebind) from (to) the one-dimensional lattice with rates $k_{\rm off}$ ($k_{\rm on}$). The tracer particle is restricted to diffuse with rate $k_D$ on the lattice. Such a model is relevant for the understanding of gene regulation where r…
Single-file transport in pore-like structures constitute an important topic for both theory and experiment. For hardcore interacting particles, a good understanding of the collective dynamics has been achieved recently. Here we study how softness in the particle interaction affects the emergent transport behavior. To this end, we investigate the driven Brownian motion of particles in a periodic potential. The particles interact via a repulsive softcore potential with a shape corresponding to a smoothed rectangular barrier. This shape allows us to elucidate effects of mutual particle penetratio…
We present a study on the dynamics of a system consisting of a pair of hardcore particles diffusing with different rates. We solved the drift-diffusion equation for this model in the case when one particle, labeled F, drifts and diffuses slowly towards the second particle, labeled M. The displacements of particle M exhibits a crossover from diffusion to drift at a characteristic time which depends on the rate constants. We show that the positional fluctuation of M exhibits an intermediate crossover regime of subdiffusion separating initial and asymptotic diffusive behavior; this is in agreemen…
