Langevin formulation for single-file diffusion

Single-file diffusion of active Brownian particles

Single-file diffusion (SFD) is a key mechanism underlying transport phenomena in confined physical and biological systems. In a typical SFD process, microscopic particles are restricted to moving in a narrow channel where they cannot pass one another, resulting in constrained motion and anomalous long-time diffusion. In this study, we use Brownian dynamics simulations and analytical theory to investigate the SFD of athermal active Brownian particles (ABPs)-a minimal model of active colloids. Building on prior work [Schiltz-Rouse et al., Phys. Rev. E 108, 064601 (2023)], where the kinetic temperature, pressure, and compressibility of the single-file ABP system were derived, we develop an accurate analytical expression for the mean square displacement (MSD) of a tagged particle. We find that the MSD exhibits ballistic behavior at short times, governed by the reduced kinetic temperature of the system. At long times, the characteristic subdiffusive scaling of SFD, [⟨(Δx)2⟩∼ t1/2], is preserved. However, self-propulsion introduces significant changes to the 1D-mobility, which we directly relate to the system's compressibility. Furthermore, we demonstrate that the generalized 1D-mobility, originally proposed by Kollmann for equilibrium systems [M. Kollmann, Phys. Rev. Lett. 90, 180602 (2003)], can be extended to active systems with minimal modification. These findings provide a framework for understanding particle transport in active systems and for tuning transport properties at the microscale, particularly in geometries where motion is highly restricted.

Diffusion enhancement and autoparametric resonance

The possibility of an autoparametric resonance in an isolated many-particle system induces a specific behavior of the particles in the presence of thermal noise. In particular, the variance associated with a resonant mode, and consequently that of the associated particles, is strongly increased compared to what it would have in the absence of parametric resonance. In this paper we consider a dimer submitted to a periodic potential for which there are only two modes, the center of mass motion and the internal vibration mode. This is the simplest system which is dynamically rich enough to exhibit an autoparametric excitation of the internal vibrations by the center of mass motion. The consequences of this autoparametric excitation on the particles diffusion will be discussed according to the stiffness of the interaction and to the initial energy of the dimer, the relevant parameters which characterize this dynamics.

In biased and soft-walled channels: Insights into transport phenomena and damped modulation

The motion of a particle along a channel of finite width is known to be affected by either the presence of energy barriers or changes in the bias forces along the channel direction. By using the lateral equilibrium hypothesis, we have successfully derived the effective diffusion coefficient for soft-walled channels, and the diffusion is found to be influenced by the curvature profile of the potential. A typical phenomenon of diffusion enhancement is observed under the appropriate parameter conditions. We first discovered an anomalous phenomenon of quasi-periodic enhancement of oscillations, which cannot be captured by the one-dimensional effective potential, under the combination of sub-Ohmic damping with two-dimensional restricted channels. We innovatively develop the effective potential and the formation mechanism of velocity variance under super-Ohmic and ballistic damping, and meanwhile, ergodicity is of concern. The theoretical framework of a ballistic system can be reinterpreted through the folding acceleration theory. This comprehensive analysis significantly enhances our understanding of diffusion processes in constrained geometries.

Nuclear magnetic resonance signal decay in the presence of a background gradient: Normal and anomalous diffusion

A novel way for calculating the diffusion-weighted nuclear magnetic resonance (NMR) attenuation signal expression in the presence of a background gradient is developed. This method is easily applicable to NMR-attenuated signals arising from any pulse field gradient sequence experiments. Here, we provide detailed calculations for the classical pulsed gradient stimulated echo and the pulsed gradient spin echo, as the particular cases. Within this general theoretical framework, devised for Gaussian processes with stationary increments, we recover and extend the previous Stejskal-Tanner results in the case of normal diffusion and we furnish a new expression in the case of anomalous diffusion.

A new perspective of molecular diffusion by nuclear magnetic resonance

The diffusion-weighted NMR signal acquired using Pulse Field Gradient (PFG) techniques, allows for extrapolating microstructural information from porous materials and biological tissues. In recent years there has been a multiplication of diffusion models expressed by parametric functions to fit the experimental data. However, clear-cut criteria for the model selection are lacking. In this paper, we develop a theoretical framework for the interpretation of NMR attenuation signals in the case of Gaussian systems with stationary increments. The full expression of the Stejskal-Tanner formula for normal diffusing systems is devised, together with its extension to the domain of anomalous diffusion. The range of applicability of the relevant parametric functions to fit the PFG data can be fully determined by means of appropriate checks to ascertain the correctness of the fit. Furthermore, the exact expression for diffusion weighted NMR signals pertaining to Brownian yet non-Gaussian processes is also derived, accompanied by the proper check to establish its contextual relevance. The analysis provided is particularly useful in the context of medical MRI and clinical practise where the hardware limitations do not allow the use of narrow pulse gradients.

Generalized Correlation Profiles in Single-File Systems

Single-file diffusion refers to the motion in narrow channels of particles which cannot bypass each other, and leads to tracer subdiffusion. Most approaches to this celebrated many-body problem were restricted to the description of the tracer only. Here, we go beyond this standard description by introducing and providing analytical results for generalized correlation profiles (GCPs) in the frame of the tracer. In addition to controlling the statistical properties of the tracer, these quantities fully characterize the correlations between the tracer position and the bath particles density. Considering the hydrodynamic limit of the problem, we determine the scaling form of the GCPs with space and time, and unveil a nonmonotonic dependence with the distance to the tracer despite the absence of any asymmetry. Our analytical approach provides several exact results for the GCPs for paradigmatic models of single-file diffusion, such as Brownian particles with hardcore repulsion, the symmetric exclusion process and the random average process. The range of applicability of our approach is further illustrated by considering (i) extensions to general interactions between particles, (ii) the out-of-equilibrium situation of an initial step of density, and (iii) beyond the hydrodynamic limit, the GCPs at arbitrary time in the dense limit.

Cumulant generating functions of a tracer in quenched dense symmetric exclusion processes

The symmetric exclusion process (SEP), where particles hop on a one-dimensional lattice with the restriction that there can only be one particle per site, is a paradigmatic model of interacting particle systems. Recently, it has been shown that the nature of the initial conditions-annealed or quenched-has a quantitative impact on the long-time properties of tracer diffusion. However, so far, the cumulant generating function in the quenched case was only determined in the low-density limit and for the specific case of a half-filled system. Here, we derive it in the opposite dense limit with quenched initial conditions. Importantly, our approach also allows us to consider the nonequilibrium situations of (i) a biased tracer in the SEP and (ii) a symmetric tracer in a step of density. In the former situation, we show that the initial conditions have a striking impact, and change the very dependence of the cumulants on the bias.

Two modes of cluster dynamics govern the viscoelasticity of colloidal gels

Colloidal gels formed by strongly attractive particles at low particle volume fractions are composed of space-spanning networks of uniformly sized clusters. We study the thermal fluctuations of the clusters using differential dynamic microscopy by decomposing them into two modes of dynamics, and link them to the macroscopic viscoelasticity via rheometry. The first mode, dominant at early times, represents the localized, elastic fluctuations of individual clusters. The second mode, pronounced at late times, reflects the collective, viscoelastic dynamics facilitated by the connectivity of the clusters. By mixing two types of particles of distinct attraction strengths in different proportions, we control the transition time at which the collective mode starts to dominate, and hence tune the frequency dependence of the linear viscoelastic moduli of the binary gels.

Fock-space methods for diffusion: Capturing volume exclusion via fermionic statistics

Volume exclusion and single-file diffusion play an important role at very small scales, such as those associated with molecular machines, ion channels, and transport in zeolites, while introducing fundamental differences compared to Brownian motion, such as changes to the power-law relation between the mean square displacement and time. In this work we map the chemical master equation for excluded diffusion onto a Schrödinger equation via annihilation and creation ladder operators with fermionic statistics, together with linear and symbolic algebra with the resulting Fock-space representation to describe the effect of volume-exclusion processes in finite one-dimensional chains. We contrast the dynamics with the nonexclusive (bosonic) diffusion for crowded, intermediate, and dilute particle populations. Motivated by shuttling in molecular machines, we proceed to investigate differences in exit time distributions introduced by volume exclusion, incorporating the presence of transport bias. More generally, this study demonstrates how one can analyze volume-excluded transport for small stochastic systems, without the need for stochastic simulation and ensemble averaging, simply by considering anticommutation relations and fermionic statistics in a Fock-space representation of the stochastic dynamics.

Tracer diffusion in crowded narrow channels

We summarise different results on the diffusion of a tracer particle in lattice gases of hard-core particles with stochastic dynamics, which are confined to narrow channels-single-files, comb-like structures and quasi-one-dimensional channels with the width equal to several particle diameters. We show that in such geometries a surprisingly rich, sometimes even counter-intuitive, behaviour emerges, which is absent in unbounded systems. This is well-documented for the anomalous diffusion in single-files. Less known is the anomalous dynamics of a tracer particle in crowded branching single-files-comb-like structures, where several kinds of anomalous regimes take place. In narrow channels, which are broader than single-files, one encounters a wealth of anomalous behaviours in the case where the tracer particle is subject to a regular external bias: here, one observes an anomaly in the temporal evolution of the tracer particle velocity, super-diffusive at transient stages, and ultimately a giant diffusive broadening of fluctuations in the position of the tracer particle, as well as spectacular multi-tracer effects of self-clogging of narrow channels. Interactions between a biased tracer particle and a confined crowded environment also produce peculiar patterns in the out-of-equilibrium distribution of the environment particles, very different from the ones appearing in unbounded systems. For moderately dense systems, a surprising effect of a negative differential mobility takes place, such that the velocity of a biased tracer particle can be a non-monotonic function of the force. In some parameter ranges, both the velocity and the diffusion coefficient of a biased tracer particle can be non-monotonic functions of the density. We also survey different results obtained for a tracer particle diffusion in unbounded systems, which will permit a reader to have an exhaustively broad picture of the tracer diffusion in crowded environments.

Two-tag correlations and nonequilibrium fluctuation-response relation in ageing single-file diffusion

Spatiotemporally correlated motions of interacting Brownian particles, confined in a narrow channel of infinite length, are studied in terms of statistical quantities involving two particles. A theoretical framework that allows analytical calculation of two-tag correlations is presented on the basis of the Dean-Kawasaki equation describing density fluctuations in colloidal systems. In the equilibrium case, the time-dependent Einstein relation holds between the two-tag displacement correlation and the response function corresponding to it, which is a manifestation of the fluctuation-dissipation theorem for the correlation of density fluctuations. While the standard procedure of closure approximation for nonlinear density fluctuations is known to be obstructed by inconsistency with the fluctuation-dissipation theorem, this difficulty is naturally avoided by switching from the standard Fourier representation of the density field to the label-based Fourier representation of the vacancy field. In the case of ageing dynamics started from equidistant lattice configuration, the time-dependent Einstein relation is violated, as the two-tag correlation depends on the waiting time for equilibration while the response function is not sensitive to it. Within linear approximation, however, there is a simple relation between the density (or vacancy) fluctuations and the corresponding response function, which is valid even if the system is out of equilibrium. This non-equilibrium fluctuation-response relation can be extended to the case of nonlinear fluctuations by means of closure approximation for the vacancy field.

Collective Motion of Repulsive Brownian Particles in Single-File Diffusion with and without Overtaking

Subdiffusion is commonly observed in liquids with high density or in restricted geometries, as the particles are constantly pushed back by their neighbors. Since this “cage effect” emerges from many-body dynamics involving spatiotemporally correlated motions, the slow diffusion should be understood not simply as a one-body problem but as a part of collective dynamics, described in terms of space–time correlations. Such collective dynamics are illustrated here by calculations of the two-particle displacement correlation in a system of repulsive Brownian particles confined in a (quasi-)one-dimensional channel, whose subdiffusive behavior is known as the single-file diffusion (SFD). The analytical calculation is formulated in terms of the Lagrangian correlation of density fluctuations. In addition, numerical solutions to the Langevin equation with large but finite interaction potential are studied to clarify the effect of overtaking. In the limiting case of the ideal SFD without overtaking, correlated motion with a diffusively growing length scale is observed. By allowing the particles to overtake each other, the short-range correlation is destroyed, but the long-range weak correlation remains almost intact. These results describe nested space–time structure of cages, whereby smaller cages are enclosed in larger cages with longer lifetimes.

N-tag probability law of the symmetric exclusion process

The symmetric exclusion process (SEP), in which particles hop symmetrically on a discrete line with hard-core constraints, is a paradigmatic model of subdiffusion in confined systems. This anomalous behavior is a direct consequence of strong spatial correlations induced by the requirement that the particles cannot overtake each other. Even if this fact has been recognized qualitatively for a long time, up to now there has been no full quantitative determination of these correlations. Here we study the joint probability distribution of an arbitrary number of tagged particles in the SEP. We determine analytically its large-time limit for an arbitrary density of particles, and its full dynamics in the high-density limit. In this limit, we obtain the time-dependent large deviation function of the problem and unveil a universal scaling form shared by the cumulants.

Procedures for Model-Guided Data Analysis of Chromosomal Loci Dynamics at Short Time Scales

This chapter provides theoretical background and practical procedures for model-guided analysis of mobility of chromosomal loci from movies of many single trajectories. We guide the reader through existing physical models and measurable quantities, illustrating how this knowledge is useful for the interpretation of the measurements.

Single file dynamics in soft materials

The term single file (SF) dynamics refers to the motion of an assembly of particles through a channel with cross-sections comparable to the particles' diameter. Single file diffusion (SFD) is then the diffusion of a tagged particle in a 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 channels, biological motors along molecular chains, electrons in proteins and liquid helium, ions through membranes, just to mention a few examples. Albeit introduced in 1965s, over the last decade the classical notion of SF dynamics has been generalised to account for a more realistic modelling of the particle properties, file geometry, particle-particle and channel-particle interactions, which paves the way to remarkable applications of the SF model, for instance, in the technology of bio-integrated nanodevices. We provide here a comprehensive review of the recent advances in the theory of SF dynamics with the purpose of spurring further experimental work.

Calculation of displacement correlation tensor indicating vortical cooperative motion in two-dimensional colloidal liquids

As an indicator of cooperative motion in a system of Brownian particles that models two-dimensional colloidal liquids, a displacement correlation tensor is calculated analytically and compared with numerical results. The key idea for the analytical calculation is to relate the displacement correlation tensor, which is a kind of four-point space-time correlation, to the Lagrangian two-time correlation of the deformation gradient tensor. Tensorial treatment of the statistical quantities, including the displacement correlation itself, allows capturing the vortical structure of the cooperative motion. The calculated displacement correlation also implies a negative long-time tail in the velocity autocorrelation, which is a manifestation of the cage effect. Both the longitudinal and transverse components of the displacement correlation are found to be expressible in terms of a similarity variable, suggesting that the cages are nested to form a self-similar structure in the space-time.

Single-File Escape of Colloidal Particles from Microfluidic Channels

Single-file diffusion is a ubiquitous physical process exploited by living and synthetic systems to exchange molecules with their environment. It is paramount to quantify the escape time needed for single files of particles to exit from constraining synthetic channels and biological pores. This quantity depends on complex cooperative effects, whose predominance can only be established through a strict comparison between theory and experiments. By using colloidal particles, optical manipulation, microfluidics, digital microscopy, and theoretical analysis we uncover the self-similar character of the escape process and provide closed-formula evaluations of the escape time. We find that the escape time scales inversely with the diffusion coefficient of the last particle to leave the channel. Importantly, we find that at the investigated microscale, bias forces as tiny as 10^{-15}  N determine the magnitude of the escape time by drastically reducing interparticle collisions. Our findings provide crucial guidelines to optimize the design of micro- and nanodevices for a variety of applications including drug delivery, particle filtering, and transport in geometrical constrictions.

Insights from Single-File Diffusion into Cooperativity in Higher Dimensions

Diffusion in colloidal suspensions can be very slow due to the cage effect, which confines each particle within a short radius on one hand, and involves large-scale cooperative motions on the other. In search of insight into this cooperativity, here the authors develop a formalism to calculate the displacement correlation in colloidal systems, mainly in the two-dimensional (2D) case. To clarify the idea for it, studies are reviewed on cooperativity among the particles in the one-dimensional (1D) case, i.e. the single-file diffusion (SFD). As an improvement over the celebrated formula by Alexander and Pincus on the mean-square displacement (MSD) in SFD, it is shown that the displacement correlation in SFD can be calculated from Lagrangian correlation of the particle interval in the one-dimensional case, and also that the formula can be extended to higher dimensions. The improved formula becomes exact for large systems. By combining the formula with a nonlinear theory for correlation, a correction to the asymptotic law for the MSD in SFD is obtained. In the 2D case, the linear theory gives description of vortical cooperative motion.

Single file diffusion into a semi-infinite tube

We investigate single file diffusion (SFD) of large particles entering a semi-infinite tube, such as luminal diffusion of proteins into microtubules or flagella. While single-file effects have no impact on the evolution of particle density, we report significant single-file effects for individually tracked tracer particle motion. Both exact and approximate ordering statistics of particles entering semi-infinite tubes agree well with our stochastic simulations. Considering initially empty semi-infinite tubes, with particles entering at one end starting from an initial time t = 0, tracked particles are initially super-diffusive after entering the system, but asymptotically diffusive at later times. For finite time intervals, the ratio of the net displacement of individual single-file particles to the average displacement of untracked particles is reduced at early times and enhanced at later times. When each particle is numbered, from the first to enter (n = 1) to the most recent (n = N), we find good scaling collapse of this distance ratio for all n. Experimental techniques that track individual particles, or local groups of particles, such as photo-activation or photobleaching of fluorescently tagged proteins, should be able to observe these single-file effects. However, biological phenomena that depend on local concentration, such as flagellar extension or luminal enzymatic activity, should not exhibit single-file effects.

Correlations in Single File Diffusion: Open and Closed Systems

We present a discussion of positional and velocity correlations of particles in single-file diffusion, based on some earlier work. We consider two physical situations: (a) An open system of N hard-core particles on an infinite line. (b) A large system with a fixed density of hard-core particles at an arbitrary temperature. In the first case (a), moments and correlations show unusual behavior. The average displacement of a particle is nonzero and grows as t1/2. Furthermore it depends on the position of the particle. Particles on the right of center are pushed right and those on the left are pushed left. The mean-square displacement also depends on the position. The diffusion constant is small for particles around the center but grows rapidly toward edges. Certain correlations in particle displacement grow with separation. For the second case (b) we give exact results for velocity-velocity auto-correlator of a tagged particle and establish that with time this correlator becomes negative and approaches zero as a power-law t-3/2 at long times. The mobility of the tagged particle is shown to decrease rapidly with density as has been observed in experiments.

[Formula: see text] Special Issue Comments: This article presents mathematical results on the dynamics in expanding files, and constant density files. This article is connected to the Special Issue articles about advanced statistical properties in single file dynamics29 and files with force and advanced formulations.30

Non-Markovian effects in the first-passage dynamics of obstructed tracer particle diffusion in one-dimensional systems

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 generalize this system and investigate first-passage properties of a tracer particle when flanked by identical crowder particles which may, besides diffuse, unbind (rebind) from (to) the one-dimensional lattice with rates k(off) (k(on)). The tracer particle is restricted to diffuse with rate k(D) on the lattice and the density of crowders is constant (on average). The unbinding rate k(off) is our key parameter and it allows us to systematically study the non-trivial transition between the completely Markovian case (k(off) ≫ k(D)) to the non-Markovian case (k(off) ≪ k(D)) governed by strong memory effects. This has relevance for several quasi one-dimensional systems. One example is gene regulation where regulatory proteins are searching for specific binding sites on a crowded DNA. We quantify the first-passage time distribution, f(t) (t is time), numerically using the Gillespie algorithm, and estimate f(t) analytically. In terms of k(off) (keeping k(D) fixed), we study the transition between the two known regimes: (i) when k(off) ≫ k(D) the particles may effectively pass each other and we recover the single particle result f(t) ∼ t(-3/2), with a reduced diffusion constant; (ii) when k(off) ≪ k(D) unbinding is rare and we obtain the single-file result f(t) ∼ t(-7/4). The intermediate region displays rich dynamics where both the characteristic f(t) - peak and the long-time power-law slope are sensitive to k(off).

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.

Everlasting effect of initial conditions on single-file diffusion

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 dynamics 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 condition, and the difference between time and ensemble averaging. More specifically, using the Jepsen line, we will discuss how transport coefficients, like diffusivity, depend on the initial state. Our work shows that initial conditions determine the long time limit of the dynamics, and in this sense the system never forgets its initial state in complete contrast with thermal systems (i.e., a closed system that attains equilibrium independent of the initial state).

Evolution equation for tagged-particle density and correlations in single-file diffusion

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. The numerical integration of differential equations are in good agreement with Monte Carlo simulations.

Active transport in dense diffusive single-file systems

We study a minimal model of active transport in crowded single-file environments which generalizes the emblematic model of single-file diffusion to the case when the tracer particle (TP) performs either an autonomous directed motion or is biased by an external force, while all other particles of the environment (bath) perform unbiased diffusions. We derive explicit expressions, valid in the limit of high density of bath particles, of the full distribution P((n))(X) of the TP position and of all its cumulants, for arbitrary values of the bias f and for any time n. Our analysis reveals striking features, such as the anomalous scaling [proportionality] √[n] of all cumulants, the equality of cumulants of the same parity characteristic of a Skellam distribution and a convergence to a Gaussian distribution in spite of asymmetric density profiles of bath particles. Altogether, our results provide the full statistics of the TP position and set the basis for a refined analysis of real trajectories of active particles in crowded single-file environments.

Tunable diffusion of magnetic particles in a quasi-one-dimensional channel

The diffusion of a system of ferromagnetic dipoles confined in a quasi-one-dimensional parabolic trap is studied using Brownian dynamics simulations. We show that the dynamics of the system is tunable by an in-plane external homogeneous magnetic field. For a strong applied magnetic field, we find that the mobility of the system, the exponent of diffusion, and the crossover time among different diffusion regimes can be tuned by the orientation of the magnetic field. For weak magnetic fields, the exponent of diffusion in the subdiffusive regime is independent of the orientation of the external field.

Diffusion in a quasi-one-dimensional system on a periodic substrate

The diffusion of charged particles interacting through a repulsive Yukawa potential, exp(-r/λ)/r, confined by a parabolic potential in the y direction and subjected to a periodic substrate potential in the x direction is investigated. Langevin dynamic simulations are used to investigate the effect of the particle density, the amplitude of the periodic substrate, and the range of the interparticle interaction potential on the diffusive behavior of the particles. We found that in general the diffusion is suppressed with increasing the amplitude of the periodic potential, but for specific values of the strength of the substrate potential a remarkable increase of the diffusion is found with increasing the periodic potential amplitude. In addition, we found a strong dependence of the diffusion on the specific arrangement of the particles, e.g., single-chain versus multichain configuration. For certain particle configurations, a reentrant behavior of the diffusion is found as a function of the substrate strength due to structural transitions in the ordering of the particles.

Different diffusive regimes, generalized Langevin and diffusion equations

We investigate a generalized Langevin equation (GLE) in the presence of an additive noise characterized by the mixture of the usual white noise and an arbitrary one. This scenario lead us to a wide class of diffusive processes, in particular the ones whose noise correlation functions are governed by power laws, exponentials, and Mittag-Leffler functions. The results show the presence of different diffusive regimes related to the spreading of the system. In addition, we obtain a fractional diffusionlike equation from the GLE, confirming the results for long time.

Single-file diffusion of particles with long-range interactions: damping and finite-size effects

We study the single file diffusion of a cyclic chain of particles that cannot cross each other, in a thermal bath, with long-ranged interactions and arbitrary damping. We present simulations that exhibit new behaviors specifically associated with systems of small numbers of particles and with small damping. In order to understand those results, we present an original analysis based on the decomposition of the particles' motion in the normal modes of the chain. Our model explains all dynamic regimes observed in our simulations and provides convincing estimates of the crossover times between those regimes.

Correlations in a generalized elastic model: fractional Langevin equation approach

The generalized elastic model (GEM) provides the evolution equation which governs the stochastic motion of several many-body systems in nature, such as polymers, membranes, and growing interfaces. On the other hand a probe (tracer) particle in these systems performs a fractional Brownian motion due to the spatial interactions with the other system's components. The tracer's anomalous dynamics can be described by a fractional Langevin equation (FLE) with a space-time correlated noise. We demonstrate that the description given in terms of GEM coincides with that furnished by the relative FLE, by showing that the correlation functions of the stochastic field obtained within the FLE framework agree with the corresponding quantities calculated from the GEM. Furthermore we show that the Fox H -function formalism appears to be very convenient to describe the correlation properties within the FLE approach.

Effect of correlated noise on quasi-one-dimensional diffusion

Single-file diffusion (SFD) of an infinite one-dimensional chain of interacting particles has a long-time mean-square displacement ∝t(1/2), independent of the type of interparticle repulsive interaction. This behavior is also observed in finite-size chains, although only for certain intervals of time t depending on the chain length L, followed by the ∝t for t→∞, as we demonstrate for a closed circular chain of diffusing interacting particles. Here, we show that spatial correlation of noise slows down SFD and can result, depending on the amount of correlated noise, in either subdiffusive behavior ∝tα, where 0<α<1/2, or even in a total suppression of diffusion (in the limit N→∞). Spatial correlation can explain the subdiffusive behavior in recent SFD experiments in circular channels.

Effective Edwards-Wilkinson equation for single-file diffusion

In this work, we present an effective discrete Edwards-Wilkinson equation aimed to describe the single-file diffusion process. The key physical properties of the system are captured defining an effective elasticity, which is proportional to the single particle diffusion coefficient and to the inverse squared mean separation between particles. The effective equation gives a description of single-file diffusion using the global roughness of the system of particles, which presents three characteristic regimes, namely, normal diffusion, subdiffusion, and saturation, separated by two crossover times. We show how these regimes scale with the parameters of the original system. Additional repulsive interaction terms are also considered and we analyze how the crossover times depend on the intensity of the additional terms. Finally, we show that the roughness distribution can be well characterized by the Edwards-Wilkinson universal form for the different single-file diffusion processes studied here.

Foundation of fractional Langevin equation: harmonization of a many-body problem

In this study we derive a single-particle equation of motion, from first principles, starting out with a microscopic description of a tracer particle in a one-dimensional many-particle system with a general two-body interaction potential. Using a harmonization technique, we show that the resulting dynamical equation belongs to the class of fractional Langevin equations, a stochastic framework which has been proposed in a large body of works as a means of describing anomalous dynamics. Our work sheds light on the fundamental assumptions of these phenomenological models and a relation derived by Kollmann.

Generalized elastic model yields a fractional Langevin equation description

Starting from a generalized elastic model which accounts for the stochastic motion of several physical systems such as membranes, (semi)flexible polymers, and fluctuating interfaces among others, we derive the fractional Langevin equation (FLE) for a probe particle in such systems, in the case of thermal initial conditions. We show that this FLE is the only one fulfilling the fluctuation-dissipation relation within a new family of fractional Brownian motion equations. The FLE for the time-dependent fluctuations of the donor-acceptor distance in a protein is shown to be recovered. When the system starts from nonthermal conditions, the corresponding FLE, which does not fulfill the fluctuation-dissipation relation, is derived.

Diffusion of tagged particle in an exclusion process

We study the diffusion of tagged hard-core interacting Brownian point particles under the influence of an external force field in one dimension. 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 <(xT)2> of the tagged center particle valid for rather general external force fields and initial conditions. The case of symmetric distribution of initial conditions around the initial position of the tagged particle on x=0 and symmetric potential fields V(x)=V(-x) yields zero drift <xT>=0 and is investigated in detail. We find <(xT)2>=R(1-R)/2Nr2 where 2N is the (large) number of particles in the system. R is a single particle reflection coefficient, i.e., the probability that a particle free of collisions starts on x0>0 and remains in x>0 while r is the probability density of noninteracting particles on the origin. We show that this equation is related to the mathematical theory of order statistics and it can be used to find <(xT)2> even when the motion between collision events is not Brownian (e.g., it might be ballistic or anomalous diffusion). As an example we derive the Percus relation for non-Gaussian diffusion. A wide range of physical behaviors emerge which are very different than the classical single file subdiffusion <(xT)2> approximately t1/2 found for uniformly distributed particles in an infinite space and in the absence of force fields.

Velocity correlations and mobility in single-file diffusion

We study the velocity correlations of a tagged particle in an infinite assembly of interacting particles with a given density in one dimension. The assembly is in contact with a heat bath, and the particles interact via a hard-core repulsion with each other. We evaluate the two-time velocity correlation function exactly as function of time when an ensemble average is taken over initial conditions. This correlation function decays rapidly with time and becomes negative, with the rate of decay increasing with the density. This is followed by a slow decay toward zero through a power-law behavior of the form -t(-3/2) at large times for all densities. We also consider mobility of the assembly in the presence of a constant force acting on the particles, as well as the mobility of a tagged particle when only the tagged particle is driven by the force. The power spectrum of velocity fluctuations is also presented.

Diffusion of finite-sized hard-core interacting particles in a one-dimensional box: Tagged particle dynamics

We solve a nonequilibrium statistical-mechanics problem exactly, namely, the single-file dynamics of N hard-core interacting particles (the particles cannot pass each other) of size Delta diffusing 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 rhoT(yT,t|yT,0) that a tagged particle T (T=1,...,N) is at position yT at time t given that it at time t=0 was at position yT,0. Using a Bethe ansatz we obtain the N -particle probability density function and, by integrating out the coordinates (and averaging over initial positions) of all particles but particle T , we arrive at an exact expression for rhoT(yT,t|yT,0) in terms of Jacobi polynomials or hypergeometric functions. Going beyond previous studies, we consider the asymptotic limit of large N , maintaining L finite, using a nonstandard asymptotic technique. We derive an exact expression for rhoT(yT,t|yT,0) for a tagged particle located roughly in the middle of the system, from which we find that there are three time regimes of interest for finite-sized systems: (A) for times much smaller than the collision time t<<taucoll=1/(rho2D) , where rho=N/L is the particle concentration and D is the diffusion constant for each particle, the tagged particle undergoes a normal diffusion; (B) for times much larger than the collision time t >>taucoll but times smaller than the equilibrium time t<<taueq=L2/D, we find a single-file regime where rhoT(yT,t|yT,0) is a Gaussian with a mean-square displacement scaling as t1/2; and (C) for times longer than the equilibrium time t>>taue , rhoT(yT,t|yT,0) approaches a polynomial-type equilibrium probability density function. Notably, only regimes (A) and (B) are found in the previously considered infinite systems.