First passage times for a tracer particle in single file diffusion and fractional Brownian motion

Langevin Equation in Heterogeneous Landscapes: How to Choose the Interpretation

The Langevin equation is a common tool to model diffusion at a single-particle level. In nonhomogeneous environments, such as aqueous two-phase systems or biological condensates with different diffusion coefficients in different phases, the solution to a Langevin equation is not unique unless the interpretation of stochastic integrals involved is selected. We analyze the diffusion of particles in such systems and evaluate the mean, the mean square displacement, and the distribution of particles, as well as the variance of the time-averaged mean-square displacements. Our analytical results provide a method to choose the interpretation parameter from single-particle tracking experiments.

Understanding Short-Range Memory Effects in Deep Neural Networks

Stochastic gradient descent (SGD) is of fundamental importance in deep learning. Despite its simplicity, elucidating its efficacy remains challenging. Conventionally, the success of SGD is ascribed to the stochastic gradient noise (SGN) incurred in the training process. Based on this consensus, SGD is frequently treated and analyzed as the Euler-Maruyama discretization of stochastic differential equations (SDEs) driven by either Brownian or Lévy stable motion. In this study, we argue that SGN is neither Gaussian nor Lévy stable. Instead, inspired by the short-range correlation emerging in the SGN series, we propose that SGD can be viewed as a discretization of an SDE driven by fractional Brownian motion (FBM). Accordingly, the different convergence behavior of SGD dynamics is well-grounded. Moreover, the first passage time of an SDE driven by FBM is approximately derived. The result suggests a lower escaping rate for a larger Hurst parameter, and thus, SGD stays longer in flat minima. This happens to coincide with the well-known phenomenon that SGD favors flat minima that generalize well. Extensive experiments are conducted to validate our conjecture, and it is demonstrated that short-range memory effects persist across various model architectures, datasets, and training strategies. Our study opens up a new perspective and may contribute to a better understanding of SGD.

Exit times of totally asymmetric simple exclusion processes

We address the question of the time needed by N particles, initially located on the first sites of a finite one-dimensional lattice of size L, to exit that lattice when they move according to a TASEP transport model. Using analytical calculations and numerical simulations, we show that when N≪L, the mean exit time of the particles is asymptotically given by T_{N}(L)∼L+β_{N}sqrt[L] for large lattices. Building upon exact results obtained for two particles, we devise an approximate continuous space and time description of the random motion of the particles that provides an analytical recursive relation for the coefficients β_{N}. The results are shown to be in very good agreement with numerical results. This approach sheds some light on the exit dynamics of N particles in the regime where N is finite while the lattice size L→∞. This complements previous asymptotic results obtained by Johansson [Commun. Math. Phys. 209, 437 (2000)0010-361610.1007/s002200050027] in the limit where both N and L tend to infinity while keeping the particle density N/L finite.

Occupation time statistics of the fractional Brownian motion in a finite domain

We study statistics of occupation times for a fractional Brownian motion (fBm), which is a typical model of a non-Markov process. Due to the non-Markovian nature, recurrence times to the origin depend on the history. Numerical simulations indicate that dependence on the sum of successive recurrence times becomes weak. As a result, the distribution of the occupation time in a finite domain follows the Mittag-Leffler distribution when the Hurst exponent of the fBm is close to 1/2. We show this distributional behavior of a time-averaged observable by renewal theory. This result is an extension of the distributional limit theorem known as the Darling-Kac theorem in general Markov processes to non-Markov processes.

Variational inference of fractional Brownian motion with linear computational complexity

We introduce a simulation-based, amortized Bayesian inference scheme to infer the parameters of random walks. Our approach learns the posterior distribution of the walks' parameters with a likelihood-free method. In the first step a graph neural network is trained on simulated data to learn optimized low-dimensional summary statistics of the random walk. In the second step an invertible neural network generates the posterior distribution of the parameters from the learned summary statistics using variational inference. We apply our method to infer the parameters of the fractional Brownian motion model from single trajectories. The computational complexity of the amortized inference procedure scales linearly with trajectory length, and its precision scales similarly to the Cramér-Rao bound over a wide range of lengths. The approach is robust to positional noise, and generalizes to trajectories longer than those seen during training. Finally, we adapt this scheme to show that a finite decorrelation time in the environment can furthermore be inferred from individual trajectories.

Driven transport of soft Brownian particles through pore-like structures: Effective size method

Single-file transport in pore-like structures constitutes 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 penetration and particle crossing in a controlled manner. We find that even weak deviations from the hardcore case can have a strong impact on the particle current. Despite this fact, knowledge about the transport in a corresponding hardcore system is shown to be useful to describe and interpret our findings for the softcore case. This is achieved by assigning a thermodynamic effective size to the particles based on the equilibrium density functional of hard spheres.

Sampling first-passage times of fractional Brownian motion using adaptive bisections

We present an algorithm to efficiently sample first-passage times for fractional Brownian motion. To increase the resolution, an initial coarse lattice is successively refined close to the target, by adding exactly sampled midpoints, where the probability that they reach the target is non-negligible. Compared to a path of N equally spaced points, the algorithm achieves the same numerical accuracy N_{eff}, while sampling only a small fraction of all points. Though this induces a statistical error, the latter is bounded for each bridge, allowing us to bound the total error rate by a number of our choice, say P_{error}^{tot}=10^{-6}. This leads to significant improvements in both memory and speed. For H=0.33 and N_{eff}=2^{32}, we need 5000 times less CPU time and 10000 times less memory than the classical Davies-Harte algorithm. The gain grows for H=0.25 and N_{eff}=2^{42} to 3×10^{5} for CPU and 10^{6} for memory. We estimate our algorithmic complexity as C^{ABSec}(N_{eff})=O[(lnN_{eff})^{3}], to be compared to Davies-Harte, which has complexity C^{DH}(N)=O(NlnN). Decreasing P_{error}^{tot} results in a small increase in complexity, proportional to ln(1/P_{error}^{tot}). Our current implementation is limited to the values of N_{eff} given above, due to a loss of floating-point precision. Our algorithm can be adapted to other extreme events and arbitrary Gaussian processes. It enables one to numerically validate theoretical predictions that were hitherto inaccessible.

Single-file transport in periodic potentials: The Brownian asymmetric simple exclusion process

Single-file Brownian motion in periodic structures is an important process in nature and technology, which becomes increasingly amenable for experimental investigation under controlled conditions. To explore and understand generic features of this motion, the Brownian asymmetric simple exclusion process (BASEP) was recently introduced. The BASEP refers to diffusion models where hard spheres are driven by a constant drag force through a periodic potential. Here we derive general properties of the rich collective dynamics in the BASEP. Average currents in the steady state change dramatically with the particle size and density. For an open system coupled to particle reservoirs, extremal current principles predict various nonequilibrium phases, which we verify by Brownian dynamics simulations. For general pair interactions we discuss connections to single-file transport by traveling-wave potentials and prove the impossibility of current reversals in systems driven by a constant drag and by traveling waves.

Probability density of the fractional Langevin equation with reflecting walls

We investigate anomalous diffusion processes governed by the fractional Langevin equation and confined to a finite or semi-infinite interval by reflecting potential barriers. As the random and damping forces in the fractional Langevin equation fulfill the appropriate fluctuation-dissipation relation, the probability density on a finite interval converges for long times towards the expected uniform distribution prescribed by thermal equilibrium. In contrast, on a semi-infinite interval with a reflecting wall at the origin, the probability density shows pronounced deviations from the Gaussian behavior observed for normal diffusion. If the correlations of the random force are persistent (positive), particles accumulate at the reflecting wall while antipersistent (negative) correlations lead to a depletion of particles near the wall. We compare and contrast these results with the strong accumulation and depletion effects recently observed for nonthermal fractional Brownian motion with reflecting walls, and we discuss broader implications.

Survival probability of stochastic processes beyond persistence exponents

For many stochastic processes, the probability [Formula: see text] of not-having reached a target in unbounded space up to time [Formula: see text] follows a slow algebraic decay at long times, [Formula: see text]. This is typically the case of symmetric compact (i.e. recurrent) random walks. While the persistence exponent [Formula: see text] has been studied at length, the prefactor [Formula: see text], which is quantitatively essential, remains poorly characterized, especially for non-Markovian processes. Here we derive explicit expressions for [Formula: see text] for a compact random walk in unbounded space by establishing an analytic relation with the mean first-passage time of the same random walk in a large confining volume. Our analytical results for [Formula: see text] are in good agreement with numerical simulations, even for strongly correlated processes such as Fractional Brownian Motion, and thus provide a refined understanding of the statistics of longest first-passage events in unbounded space.

Hydration water dynamics around a protein surface: a first passage time approach

A stochastic noise-driven dynamic model is proposed to study the diffusion of water molecules around a protein surface, under the effect of thermal fluctuations that arise due to the collision of water molecules with the surrounding environment. The underlying dynamics of such a system may be described in the framework of the generalized Langevin equation, where the thermal fluctuations are assumed to be algebraically correlated in time, which governs the non-Markovian behavior of the system. Results of the calculations of mean-square displacement and the velocity autocorrelation function reveal that the hydration water around the protein surface follows subdiffusive dynamics at long times. Analytical expressions for the first passage time distribution, survival probability, mean residence time and mean first passage time of water molecules are derived for different boundary conditions, to analyze hydration water dynamics under the effect of thermally correlated noise. The results depict a unimodal distribution of the first passage time unlike Brownian motion. The survival probability of hydration water follows a stretched exponential decay for both boundary conditions. The mean residence time of the hydration water molecule for different initial positions increases with increase in the complexity/heterogeneity of the surrounding environment for both boundary conditions. The mean first passage time of the water molecule to reach the absorbing/reflecting boundary follows an asymptotic power law with respect to the thickness of the hydration layer, and increases with increase in the complexity/heterogeneity of the environment.

First-passage times in multiscale random walks: The impact of movement scales on search efficiency

An efficient searcher needs to balance properly the trade-off between the exploration of new spatial areas and the exploitation of nearby resources, an idea which is at the core of scale-free Lévy search strategies. Here we study multiscale random walks as an approximation to the scale-free case and derive the exact expressions for their mean-first-passage times in a one-dimensional finite domain. This allows us to provide a complete analytical description of the dynamics driving the situation in which both nearby and faraway targets are available to the searcher, so the exploration-exploitation trade-off does not have a trivial solution. For this situation, we prove that the combination of only two movement scales is able to outperform both ballistic and Lévy strategies. This two-scale strategy involves an optimal discrimination between the nearby and faraway targets which is only possible by adjusting the range of values of the two movement scales to the typical distances between encounters. So, this optimization necessarily requires some prior information (albeit crude) about target distances or distributions. Furthermore, we found that the incorporation of additional (three, four, …) movement scales and its adjustment to target distances does not improve further the search efficiency. This allows us to claim that optimal random search strategies arise through the informed combination of only two walk scales (related to the exploitative and the explorative scales, respectively), expanding on the well-known result that optimal strategies in strictly uninformed scenarios are achieved through Lévy paths (or, equivalently, through a hierarchical combination of multiple scales).

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).

Tracer dynamics in a single-file system with absorbing boundary

The paper addresses the single-file diffusion in the presence of an absorbing boundary. The emphasis is on an interplay between the hard-core interparticle interaction and the absorption process. The resulting dynamics exhibits several qualitatively new features. First, starting with the exact probability density function for a given particle (a tracer), we study the long-time asymptotics of its moments. Both the mean position and the mean-square displacement are controlled by dynamical exponents which depend on the initial order of the particle in the file. Second, conditioning on nonabsorption, we study the distribution of long-living particles. In the conditioned framework, the dynamical exponents are the same for all particles, however, a given particle possesses an effective diffusion coefficient which depends on its initial order. After performing the thermodynamic limit, the conditioned dynamics of the tracer is subdiffusive, the generalized diffusion coefficient D(1/2) being different from that reported for the system without absorbing boundary.

Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking

This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.

Transient aging in fractional Brownian and Langevin-equation motion

Stochastic processes driven by stationary fractional Gaussian noise, that is, fractional Brownian motion and fractional Langevin-equation motion, are usually considered to be ergodic in the sense that, after an algebraic relaxation, time and ensemble averages of physical observables coincide. Recently it was demonstrated that fractional Brownian motion and fractional Langevin-equation motion under external confinement are transiently nonergodic-time and ensemble averages behave differently-from the moment when the particle starts to sense the confinement. Here we show that these processes also exhibit transient aging, that is, physical observables such as the time-averaged mean-squared displacement depend on the time lag between the initiation of the system at time t=0 and the start of the measurement at the aging time t(a). In particular, it turns out that for fractional Langevin-equation motion the aging dependence on t(a) is different between the cases of free and confined motion. We obtain explicit analytical expressions for the aged moments of the particle position as well as the time-averaged mean-squared displacement and present a numerical analysis of this transient aging phenomenon.

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).