G. Oshanin

First passages in bounded domains: When is the mean first passage time meaningful?

We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability P(ω) distribution of the random variable ω=τ(1)/(τ(1)+τ(2)), which is a measure for how similar the first passage times τ(1) and τ(2) are of two independent realizations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether P(ω) represents a unimodal, bell-shaped form, or a bimodal, M-shaped behavior. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behavior, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of P(ω), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in two-dimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behavior even in bounded domains, in which all moments of the first passage distribution exist.

First passages for a search by a swarm of independent random searchers

In this paper we study some aspects of search for an immobile target by a swarm of N non-communicating, randomly moving searchers (numbered by the index k, k = 1, 2,..., N), which all start their random motion simultaneously at the same point in space. For each realization of the search process, we record the unordered set of time moments {?k}, where ?k is the time of the first passage of the kth searcher to the location of the target. Clearly, ?ks are independent, identically distributed random variables with the same distribution function ?(?). We evaluate then the distribution P(?) of the random variable , where is the ensemble-averaged realization-dependent first passage time. We show that P(?) exhibits quite a non-trivial and sometimes a counterintuitive behavior. We demonstrate that in some well-studied cases (e.g. Brownian motion in finite d-dimensional domains) the mean first passage time is not a robust measure of the search efficiency, despite the fact that ?(?) has moments of arbitrary order. This implies, in particular, that even in this simplest case (not to mention complex systems and/or anomalous diffusion) first passage data extracted from a single-particle tracking should be regarded with appropriate caution because of the significant sample-to-sample fluctuations.

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 an 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 nonglass-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.

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 α 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 α and L as optimization parameters, we seek to enhance the chances of successful detection by minimizing the probability PN 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 PN by orders of magnitude upon appropriate choices of α 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.

Molecular Weight Dependence of Spreading Rates of Ultrathin Polymeric Films.

We study experimentally the molecular weight

Dynamics and conformational properties of polyampholytes in external electrical fields

M

Anomalous Fluctuations of Currents in Sinai-Type Random Chains with Strongly Correlated Disorder

dependence of spreading rates of molecularly thin precursor films, growing at the bottom of droplets of polymer liquids. In accord with previous observations, we find that the radial extension R(t) of the film grows with time as R(t) = (D_{exp} t)^{1/2}. Our data substantiate the M-dependence of D_{exp}; we show that it follows D_{exp} \sim M^{-\gamma}, where the exponent \gamma is dependent on the chemical composition of the solid surface, determining its frictional properties with respect to the molecular transport. In the specific case of hydrophilic substrates, the frictional properties can be modified by the change of the relative humidity (RH). We find that \gamma \approx 1 at low RH and tends to zero when RH gets progressively increased. We propose simple theoretical arguments which explain the observed behavior in the limits of low and high RH.

Survival probability of a particle in a sea of mobile traps: a tale of tails.

In this work we report first analytical results on the dynamics and conformational properties of polyampholytes (PAs, polymers containing positive and negative charges) in the presence of external electrical fields. In terms of the Rouse model of polymer dynamics and in the so‐called weak coupling limit we obtain for PAs explicitly the mean‐square displacement both of the center of mass and of individual beads, and also determine the PAs’ equilibrium end‐to‐end distance. For a singly charged PA we also relate the findings to a fractional differential equation.

Ultraslow vacancy-mediated tracer diffusion in two dimensions: the Einstein relation verified.

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 counterintuitive behavior is explained via an analysis of representative realizations of disorder.

Motion of a driven tracer particle in a one-dimensional symmetric lattice gas.

We study the long-time tails of the survival probability P(t) of an A particle diffusing in d-dimensional media in the presence of a concentration rho of traps B that move subdiffusively, such that the mean square displacement of each trap grows as tgamma with 0 < or = gamma < or =1. Starting from a continuous time random walk description of the motion of the particle and of the traps, we derive lower and upper bounds for P(t) and show that for gamma < or =2/(d+2) these bounds coincide asymptotically, thus determining asymptotically exact results. The asymptotic decay law in this regime is exactly that obtained for immobile traps. This means that for sufficiently subdiffusive traps, the moving A particle sees the traps as essentially immobile, and Lifshitz or trapping tails remain unchanged. For gamma >2/(d+2) and d< or =2 the upper and lower bounds again coincide, leading to a decay law equal to that of a stationary particle. Thus, in this regime the moving traps see the particle as essentially immobile. For d>2 , however, the upper and lower bounds in this gamma regime no longer coincide, and the decay law for the survival probability of the A particle remains ambiguous.