arXiv subjects

Gleb Oshanin

Publications and source records attributed to Gleb Oshanin.

69 records · Page 4Linked to original sources

arXiv2014

Approach to asymptotically diffusive behavior for Brownian particles in periodic potentials : extracting information from transients

A Langevin process diffusing in a periodic potential landscape has a time dependent diffusion constant which means that its average mean squared displacement (MSD) only becomes linear at late times. The long time, or effective diffusion constant, can be estimated from the slope of a linear fit of the MSD at late times. Due to the cross over between a short time microscopic diffusion constant, which is independent of the potential, to the effective late time diffusion constant, a linear fit of the MSD will not in general pass through the origin and will have a non-zero constant term. Here we address how to compute the constant term and provide explicit results for Brownian particles in one dimension in periodic potentials. We show that the constant is always positive and that at low temperatures it depends on the curvature of the minimum of the potential. For comparison we also consider the same question for the simpler problem of a symmetric continuous time random walk in discrete space. Here the constant can be positive or negative and can be used to determine the variance of the hopping time distribution.

cond-mat.stat-mech↗

arXiv2013

Geometry-induced superdiffusion in driven crowded systems

Recent Molecular Dynamics simulations of glass-forming liquids revealed superdiffusive fluctuations associated with the position of a tracer particle (TP) driven by an external force. Such anomalous response, whose mechanism remains elusive, has been observed up to now only in systems close to their glass transition, suggesting that this could be one of its hallmarks. Here, we show that the presence of superdiffusion is in actual fact much more general, provided that the system is crowded and geometrically confined. We present and solve analytically a minimal model consisting of a driven TP in a dense, crowded medium in which the motion of particles is mediated by the diffusion of packing defects, called vacancies. For such non glass-forming systems, our analysis predicts a long-lived superdiffusion which ultimately crosses over to giant diffusive behavior. We find that this trait is present in confined geometries, for example long capillaries and stripes, and emerges as a universal response of crowded environments to an external force. These findings are confirmed by numerical simulations of systems as varied as lattice-gases, dense liquids and granular fluids.

cond-mat.stat-mech↗

arXiv2013

Anomalous fluctuations of currents in Sinai-type random chains with strongly correlated disorder

We study properties of a random walk in a generalized Sinai model, in which a quenched random potential is a trajectory of a fractional Brownian motion with arbitrary Hurst parameter H, 0< H <1, so that the random force field displays strong spatial correlations. In this case, the disorder-average mean-square displacement grows in proportion to log^{2/H}(n), n being time. We prove that moments of arbitrary order k of the steady-state current J_L through a finite segment of length L of such a chain decay as L^{-(1-H)}, independently of k, which suggests that despite a logarithmic confinement the average current is much higher than its Fickian counterpart in homogeneous systems. Our results reveal a paradoxical behavior such that, for fixed n and L, the mean square displacement decreases when one varies H from 0 to 1, while the average current increases. This counter-intuitive behavior is explained via an analysis of representative realizations of disorder.

cond-mat.dis-nn↗

arXiv2013

Optimal least-squares estimators of the diffusion constant from a single Brownian trajectory

Modern developments in microscopy and image processing are revolutionising areas of physics, chemistry, and biology as nanoscale objects can be tracked with unprecedented accuracy. However, the price paid for having a direct visualisation of a single particle trajectory with high temporal and spatial resolution is a consequent lack of statistics. This naturally calls for reliable analytical tools which will allow one to extract the properties specific to a statistical ensemble from just a single trajectory. In this article we briefly survey different analytical methods currently used to determine the ensemble average diffusion coefficient from single particle data and then focus specifically on weighted least-squares estimators, seeking the weight functions for which such estimators are ergodic. Finally, we address the question of the effects of disorder on such estimators.

cond-mat.stat-mech↗

arXiv2013

Two-temperature Langevin dynamics in a parabolic potential

We study a planar two-temperature diffusion of a Brownian particle in a parabolic potential. The diffusion process is defined in terms of two Langevin equations with two different effective temperatures in the X and the Y directions. In the stationary regime the system is described by a non-trivial particle position distribution P(x,y), which we determine explicitly. We show that this distribution corresponds to a non-equilibrium stationary state, characterised by the presence of space-dependent particle currents which exhibit a non-zero rotor. Theoretical results are confirmed by the numerical simulations.

cond-mat.stat-mech↗

arXiv2013

On ergodic least-squares estimators of the generalized diffusion coefficient for fractional Brownian motion

We analyse a class of estimators of the generalized diffusion coefficient for fractional Brownian motion $B_t$ of known Hurst index $H$, based on weighted functionals of the single time square displacement. We show that for a certain choice of the weight function these functionals possess an ergodic property and thus provide the true, ensemble-averaged, generalized diffusion coefficient to any necessary precision from a single trajectory data, but at expense of a progressively higher experimental resolution. Convergence is fastest around $H\simeq0.30$, a value in the subdiffusive regime.

cond-mat.stat-mech↗

arXiv2013

Distribution of the least-squares estimators of a single Brownian trajectory diffusion coefficient

In this paper we study the distribution function $P(u_α)$ of the estimators $u_α \sim T^{-1} \int^T_0 \, ω(t) \, {\bf B}^2_{t} \, dt$, which optimise the least-squares fitting of the diffusion coefficient $D_f$ of a single $d$-dimensional Brownian trajectory ${\bf B}_{t}$. We pursue here the optimisation further by considering a family of weight functions of the form $ω(t) = (t_0 + t)^{-α}$, where $t_0$ is a time lag and $α$ is an arbitrary real number, and seeking such values of $α$ for which the estimators most efficiently filter out the fluctuations. We calculate $P(u_α)$ exactly for arbitrary $α$ and arbitrary spatial dimension $d$, and show that only for $α= 2$ the distribution $P(u_α)$ converges, as $ε= t_0/T \to 0$, to the Dirac delta-function centered at the ensemble average value of the estimator. This allows us to conclude that only the estimators with $α= 2$ possess an ergodic property, so that the ensemble averaged diffusion coefficient can be obtained with any necessary precision from a single trajectory data, but at the expense of a progressively higher experimental resolution. For any $α\neq 2$ the distribution attains, as $ε\to 0$, a certain limiting form with a finite variance, which signifies that such estimators are not ergodic.

cond-mat.stat-mech↗

arXiv2012

Optimal fits of diffusion constants from single time data points of Brownian trajectories

Experimental methods based on single particle tracking (SPT) are being increasingly employed in the physical and biological sciences, where nanoscale objects are visualized with high temporal and spatial resolution. SPT can probe interactions between a particle and its environment but the price to be paid is the absence of ensemble averaging and a consequent lack of statistics. Here we address the benchmark question of how to accurately extract the diffusion constant of one single Brownian trajectory. We analyze a class of estimators based on weighted functionals of the square displacement. For a certain choice of the weight function these functionals provide the true ensemble averaged diffusion coefficient, with a precision that increases with the trajectory resolution.

cond-mat.stat-mech↗

arXiv2012

Anomalous field-induced growth of fluctuations in dynamics of a biased intruder moving in a quiescent medium

We present exact results on the dynamics of a biased, by an external force ${\bf F}$, intruder (BI) in a two-dimensional lattice gas of unbiased, randomly moving hard-core particles. Going beyond the usual analysis of the force-velocity relation, we study the probability distribution $P({\bf R}_n)$ of the BI displacement ${\bf R}_n$ at time {\it n}. We show that despite the fact that the BI drives the gas to a non-equilibrium steady-state, $P({\bf R}_n)$ converges to a Gaussian distribution as $n \to \infty$. We find that the variance $σ_x^2$ of $P({\bf R}_n)$ along ${\bf F}$ exhibits a weakly superdiffusive growth $σ_x^2 \sim ν_1 \, n \, \ln(n)$, and a usual diffusive growth, $σ_y^2 \sim ν_2 \, n$, in the perpendicular direction. We determine $ν_1$ and $ν_2$ exactly for arbitrary bias, in the lowest order in the density of vacancies, and show that $ν_1 \sim |{\bf F}|^2$ for small bias, which signifies that superdiffusive behaviour emerges beyond the linear-response approximation. Monte Carlo simulations confirm our analytical results, and reveal a striking field-induced superdiffusive behavior $σ_x^2 \sim n^{3/2}$ for infinitely long 2D stripes and 3D capillaries.

cond-mat.stat-mech↗

arXiv2012

Optimal estimates of the diffusion coefficient of a single Brownian trajectory

Modern developments in microscopy and image processing are revolutionizing areas of physics, chemistry and biology as nanoscale objects can be tracked with unprecedented accuracy. The goal of single particle tracking is to determine the interaction between the particle and its environment. The price paid for having a direct visualization of a single particle is a consequent lack of statistics. Here we address the optimal way of extracting diffusion constants from single trajectories for pure Brownian motion. It is shown that the maximum likelihood estimator is much more efficient than the commonly used least squares estimate. Furthermore we investigate the effect of disorder on the distribution of estimated diffusion constants and show that it increases the probability of observing estimates much smaller than the true (average) value.

cond-mat.stat-mech↗

arXiv2011

Two stock options at the races: Black-Scholes forecasts

Suppose one buys two very similar stocks and is curious about how much, after some time T, one of them will contribute to the overall asset, expecting, of course, that it should be around 1/2 of the sum. Here we examine this question within the classical Black and Scholes (BS) model, focusing on the evolution of the probability density function P(w) of a random variable w = a_T^{(1)}/(a_T^{(1)} + a_T^{(2)}) where a_T^{(1)} and a_T^{(2)} are the values of two (either European- or the Asian-style) options produced by two absolutely identical BS stochastic equations. We show that within the realm of the BS model the behavior of P(w) is surprisingly different from common-sense-based expectations. For the European-style options P(w) always undergoes a transition, (when T approaches a certain threshold value), from a unimodal to a bimodal form with the most probable values being close to 0 and 1, and, strikingly, w =1/2 being the least probable value. This signifies that the symmetry between two options spontaneously breaks and just one of them completely dominates the sum. For path-dependent Asian-style options we observe the same anomalous behavior, but only for a certain range of parameters. Outside of this range, P(w) is always a bell-shaped function with a maximum at w = 1/2.

q-fin.ST↗

arXiv2010

Bias- and bath-mediated pairing of particles driven through a quiescent medium

A particle driven by an external force in a molecular crowding environment - a quiescent bath of other particles, makes their spatial distribution inhomogeneous: the bath particles accumulate in front of the biased particle (BP) and are depleted behind. In fact, a BP travels together with the inhomogeneity it creates. A natural question is what will happen with two BPs when they appear sufficiently close to each other such that the inhomogeneities around each of them start to interfere? In quest for the answer we examine here, via Monte Carlo simulations, the dynamics of two BPs in a lattice gas of bath particles. We observe that for a sufficiently dense medium, surprisingly, both BPs spend most of the time together which signifies that the interference of the microstructural inhomogeneities results in effectively attractive interactions between them. Such statistical pairing of BPs minimizes the size of the inhomogeneity and hence reduces the frictional drag force exerted on the BPs by the medium. As a result, in some configurations the center-of-mass of a pair of BPs propagates faster than a single isolated BP. These jamming-induced forces are very different from fundamental physical interactions, exist only in presence of an external force, and require the presence of a quiescent bath to mediate the interactions between the driven particles.

cond-mat.soft↗

arXiv2010

Ballistic deposition patterns beneath a growing KPZ interface

We consider a (1+1)-dimensional ballistic deposition process with next-nearest neighbor interaction, which belongs to the KPZ universality class, and introduce for this discrete model a variational formulation similar to that for the randomly forced continuous Burgers equation. This allows to identify the characteristic structures in the bulk of a growing aggregate ("clusters" and "crevices") with minimizers and shocks in the Burgers turbulence, and to introduce a new kind of equipped Airy process for ballistic growth. We dub it the "hairy Airy process" and investigate its statistics numerically. We also identify scaling laws that characterize the ballistic deposition patterns in the bulk: the law of "thinning" of the forest of clusters with increasing height, the law of transversal fluctuations of cluster boundaries, and the size distribution of clusters. The corresponding critical exponents are determined exactly based on the analogy with the Burgers turbulence and simple scaling considerations.

cond-mat.stat-mech↗

arXiv2009

Post-Tanner stages of droplet spreading: the energy balance approach revisited

The spreading of a circular liquid drop on a solid substrate can be described by the time evolution of its base radius R(t). In complete wetting the quasistationary regime (far away from initial and final transients) typically obeys the so-called Tanner law, with R t^alpha_T, alpha_T=1/10. Late-time spreading may differ significantly from the Tanner law: in some cases the drop does not thin down to a molecular film and instead reaches an equilibrium pancake-like shape; in other situations, as revealed by recent experiments with spontaneously spreading nematic crystals, the growth of the base radius accelerates after the Tanner stage. Here we demonstrate that these two seemingly conflicting trends can be reconciled within a suitably revisited energy balance approach, by taking into account the line tension contribution to the driving force of spreading: a positive line tension is responsible for the formation of pancake-like structures, whereas a negative line tension tends to lengthen the contact line and induces an accelerated spreading (a transition to a faster power law for R(t) than in the Tanner stage).

cond-mat.soft↗

arXiv2009

Efficient search by optimized intermittent random walks

We study the kinetics for the search of an immobile target by randomly moving searchers that detect it only upon encounter. The searchers perform intermittent random walks on a one-dimensional lattice. Each searcher can step on a nearest neighbor site with probability "alpha", or go off lattice with probability "1 - α" to move in a random direction until it lands back on the lattice at a fixed distance L away from the departure point. Considering "alpha" and L as optimization parameters, we seek to enhance the chances of successful detection by minimizing the probability P_N that the target remains undetected up to the maximal search time N. We show that even in this simple model a number of very efficient search strategies can lead to a decrease of P_N by orders of magnitude upon appropriate choices of "alpha" and L. We demonstrate that, in general, such optimal intermittent strategies are much more efficient than Brownian searches and are as efficient as search algorithms based on random walks with heavy-tailed Cauchy jump-length distributions. In addition, such intermittent strategies appear to be more advantageous than Levy-based ones in that they lead to more thorough exploration of visited regions in space and thus lend themselves to parallelization of the search processes.

cond-mat.stat-mech↗