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Fractional Brownian motion (fBm) is an experimentally-relevant, non-Markovian Gaussian stochastic process with long-ranged correlations between the increments, parametrised by the so-called Hurst exponent $H$; depending on its value the process can be sub-diffusive $(0 < H < 1/2)$, diffusive $(H = 1/2)$ or super-diffusive $(1/2 < H < 1)$. There exist three alternative equally often used definitions of fBm -- due to L\'evy and due to Mandelbrot and van Ness (MvN), which differ by the interval on which the time variable is formally defined. Respectively, the covariance functions of these fBms have different functional forms. Moreover, the MvN fBms have stationary increments, while for the L\'evy fBm this is not the case. One may therefore be tempted to conclude that these are, in fact, different processes which only accidentally bear the same name. Recently determined explicit path integral representations also appear to have very different functional forms, which only reinforces the latter conclusion. Here we develope a unifying equivalent path integral representation of all three fBms in terms of Riemann-Liouville fractional integrals, which links the fBms and proves that they indeed belong to the same family. We show that the action in such a representation involves the fractional integral of the same form and order (dependent on whether $H < 1/2$ or $H > 1/2$) for all three cases, and differs only by the integration limits., Comment: 24 pages

In many practically important problems which rely on particles' transport in realistic corrugated channels, one is interested to know the probability that either of the extremities, (e.g., the one containing a chemically active site, or connected to a broader channel), is reached before the other one. In mathematical literature, the latter are called the "splitting" probabilities (SPs). Here, within the Fick-Jacobs approach, we study analytically the SPs as functions of system's parameters for dynamics in three-dimensional corrugated channels, confronting standard diffusion and active Brownian motion. Our analysis reveals some similarities in the behavior and also some markedly different features, which can be seen as fingerprints of the activity of particles.

It has been shown recently that the optimal fluctuation method -- essentially geometrical optics -- provides a valuable insight into large deviations of Brownian motion. Here we extend the geometrical optics formalism to two-sided, $-\infty

The same system can exhibit a completely different dynamical behavior when it evolves in equilibrium conditions or when it is driven out-of-equilibrium by, e.g., connecting some of its components to heat baths kept at different temperatures. Here we concentrate on an analytically solvable and experimentally-relevant model of such a system -- the so-called Brownian gyrator -- a two-dimensional nanomachine that performs a systematic, on average, rotation around the origin under non-equilibrium conditions, while no net rotation takes place in equilibrium. On this example, we discuss a question whether it is possible to distinguish between two types of a behavior judging not upon the statistical properties of the trajectories of components, but rather upon their respective spectral densities. The latter are widely used to characterize diverse dynamical systems and are routinely calculated from the data using standard built-in packages. From such a perspective, we inquire whether the power spectral densities possess some "fingerprint" properties specific to the behavior in non-equilibrium. We show that indeed one can conclusively distinguish between equilibrium and non-equilibrium dynamics by analyzing the cross-correlations between the spectral densities of both components in the short frequency limit, or from the spectral densities of both components evaluated at zero frequency. Our analytical predictions, corroborated by experimental and numerical results, open a new direction for the analysis of a non-equilibrium dynamics., Comment: 23 pages, 6 figures

We study the extremal properties of a stochastic process $x_t$ defined by a Langevin equation $\dot{x}_t=\sqrt{2 D_0 V(B_t)}\,\xi_t$, where $\xi_t$ is a Gaussian white noise with zero mean, $D_0$ is a constant scale factor, and $V(B_t)$ is a stochastic "diffusivity" (noise strength), which itself is a functional of independent Brownian motion $B_t$. We derive exact, compact expressions for the probability density functions (PDFs) of the first passage time (FPT) $t$ from a fixed location $x_0$ to the origin for three different realisations of the stochastic diffusivity: a cut-off case $V(B_t) =\Theta(B_t)$ (Model I), where $\Theta(x)$ is the Heaviside theta function; a Geometric Brownian Motion $V(B_t)=\exp(B_t)$ (Model II); and a case with $V(B_t)=B_t^2$ (Model III). We realise that, rather surprisingly, the FPT PDF has exactly the L\'evy-Smirnov form (specific for standard Brownian motion) for Model II, which concurrently exhibits a strongly anomalous diffusion. For Models I and III either the left or right tails (or both) have a different functional dependence on time as compared to the L\'evy-Smirnov density. In all cases, the PDFs are broad such that already the first moment does not exist. Similar results are obtained in three dimensions for the FPT PDF to an absorbing spherical target., Comment: 8 pages, 3 figures, RevTeX

Using scaling arguments and extensive numerical simulations, we study dynamics of a tracer particle in a corrugated channel represented by a periodic sequence of broad chambers and narrow funnel-like bottlenecks enclosed by a hard-wall boundary. A tracer particle is affected by an external force pointing along the channel, and performs an unbiased diffusion in the perpendicular direction. We present a detailed analysis a) of the distribution function of the height above the funnel's boundary at which the first crossing of a given bottleneck takes place, and b) of the distribution function of the first passage time to such an event. Our analysis reveals several new features of the dynamical behaviour which are overlooked in the studies based on the Fick-Jacobs approach. In particular, trajectories passing through a funnel concentrate predominantly on its boundary, which makes first-crossing events very sensitive to the presence of binding sites and a microscopic roughness., Comment: 9 pages, 8 figures

Stochastic models based on random diffusivities, such as the diffusing-diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media. Studies of these models typically focus on the moments and the displacement probability density function. Here we develop the complementary power spectral description for a broad class of random diffusivity processes. In our approach we cater for typical single particle tracking data in which a small number of trajectories with finite duration are garnered. Apart from the diffusing-diffusivity model we study a range of previously unconsidered random diffusivity processes, for which we obtain exact forms of the probability density function. These new processes are different versions of jump processes as well as functionals of Brownian motion. The resulting behaviour subtly depends on the specific model details. Thus, the central part of the probability density function may be Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential, log-normal or even power-law forms. For all these models we derive analytically the moment-generating function for the single-trajectory power spectral density. We establish the generic $1/f^2$-scaling of the power spectral density as function of frequency in all cases. Moreover, we establish the probability density for the amplitudes of the random power spectral density of individual trajectories. The latter functions reflect the very specific properties of the different random diffusivity models considered here. Our exact results are in excellent agreement with extensive numerical simulations., Comment: 39 pages, 11 figures, IOP LaTeX

In the scenario of the narrow escape problem (NEP) a particle diffuses in a finite container and eventually leaves it through a small "escape window" in the otherwise impermeable boundary, once it arrives to this window and over-passes an entropic barrier at the entrance to it. This generic problem is mathematically identical to that of a diffusion-mediated reaction with a partially-reactive site on the container's boundary. Considerable knowledge is available on the dependence of the mean first-reaction time (FRT) on the pertinent parameters. We here go a distinct step further and derive the full FRT distribution for the NEP. We demonstrate that typical FRTs may be orders of magnitude shorter than the mean one, thus resulting in a strong defocusing of characteristic temporal scales. We unveil the geometry-control of the typical times, emphasising the role of the initial distance to the target as a decisive parameter. A crucial finding is the further FRT defocusing due to the barrier, necessitating repeated escape or reaction attempts interspersed with bulk excursions. These results add new perspectives and offer a broad comprehension of various features of the by-now classical NEP that are relevant for numerous biological and technological systems., Comment: 30 pages, 9 figures, IOPLaTeX

The basic phenomenology of experimentally observed synchronization (i.e., a stochastic phase locking) of identical, beating flagella of a biflagellate alga is known to be captured well by a minimal model describing the dynamics of coupled, limit-cycle, noisy oscillators (known as the noisy Kuramoto model). As demonstrated experimentally, the amplitudes of the noise terms therein, which stem from fluctuations of the rotary motors, depend on the flagella length. Here we address the conceptually important question which kind of synchrony occurs if the two flagella have different lengths such that the noises acting on each of them have different amplitudes. On the basis of a minimal model, too, we show that a different kind of synchrony emerges, and here it is mediated by a current carrying, steady-state; it manifests itself via correlated "drifts" of phases. We quantify such a synchronization mechanism in terms of appropriate order parameters $Q$ and $Q_{\cal S}$ - for an ensemble of trajectories and for a single realization of noises of duration ${\cal S}$, respectively. Via numerical simulations we show that both approaches become identical for long observation times ${\cal S}$. This reveals an ergodic behavior and implies that a single-realization order parameter $Q_{\cal S}$ is suitable for experimental analysis for which ensemble averaging is not always possible., Comment: 10 pages, 2 figures

Time-dependent processes are often analysed using the power spectral density (PSD), calculated by taking an appropriate Fourier transform of individual trajectories and finding the associated ensemble-average. Frequently, the available experimental data sets are too small for such ensemble averages, and hence it is of a great conceptual and practical importance to understand to which extent relevant information can be gained from $S(f,T)$, the PSD of a single trajectory. Here we focus on the behavior of this random, realization-dependent variable, parametrized by frequency $f$ and observation-time $T$, for a broad family of anomalous diffusions---fractional Brownian motion (fBm) with Hurst-index $H$---and derive exactly its probability density function. We show that $S(f,T)$ is proportional---up to a random numerical factor whose universal distribution we determine---to the ensemble-averaged PSD. For subdiffusion ($H<1/2$) we find that $S(f,T)\sim A/f^{2H+1}$ with random-amplitude $A$. In sharp contrast, for superdiffusion $(H>1/2)$ $S(f,T)\sim BT^{2H-1}/f^2$ with random amplitude $B$. Remarkably, for $H>1/2$ the PSD exhibits the same frequency-dependence as Brownian motion, a deceptive property that may lead to false conclusions when interpreting experimental data. Notably, for $H>1/2$ the PSD is ageing and is dependent on $T$. Our predictions for both sub- and superdiffusion are confirmed by experiments in live cells and in agarose hydrogels, and by extensive simulations., Comment: 13 pages, 5 figures, Supplemental Material can be found at https://journals.aps.org/prx/supplemental/10.1103/PhysRevX.9.011019/prx_SM_final.pdf

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. We also present a survey of 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., Comment: 32 pages, 4 figures, Journal of Physics: Condensed Matter, to appear

We study an outcome of a vote in a population of voters exposed to an externally applied bias in favour of one of two potential candidates. The population consists of ordinary individuals, that are in majority and tend to align their opinion with the external bias, and some number of contrarians --- individuals who are always hostile to the bias but are not in a conflict with ordinary voters. The voters interact among themselves, all with all, trying to find an opinion reached by the community as a whole. We demonstrate that for a sufficiently weak external bias, the opinion of ordinary individuals is always decisive and the outcome of the vote is in favour of the preferential candidate. On the contrary, for an excessively strong bias, the contrarians dominate in the population's opinion, producing overall a negative response to the imposed bias. We also show that for sufficiently strong interactions within the community, either of two subgroups can abruptly change an opinion of the other group., Comment: 11 pages, 6 figures

We study the degree of success of a single predator hunting a herd of prey on a two dimensional square lattice landscape. We explicitly consider the self volume of the prey restraining their dynamics on the lattice. The movement of both predator and prey is chosen to include an intelligent, decision making step based on their respective sighting ranges, the radius in which they can detect the other species (prey cannot recognise each other besides the self volume interaction): after spotting each other the motion of prey and predator turns from a nearest neighbour random walk into direct escape or chase, respectively. We consider a large range of prey densities and sighting ranges and compute the mean first passage time for a predator to catch a prey as well as characterise the effective dynamics of the hunted prey. We find that the prey's sighting range dominates their life expectancy and the predator profits more from a bad eyesight of the prey than from his own good eye sight. We characterise the dynamics in terms of the mean distance between the predator and the nearest prey. It turns out that effectively the dynamics of this distance coordinate can be captured in terms of a simple Ornstein-Uhlenbeck picture. Reducing the many-body problem to a simple two-body problem by imagining predator and nearest prey to be connected by a Hookean bond, all features of the model such as prey density and sighting ranges merge into the effective binding constant., Comment: 21 pages, 10 figures, IOPLaTeX

We analyze the joint distributions and temporal correlations between the partial maximum $m$ and the global maximum $M$ achieved by a Brownian Bridge on the subinterval $[0,t_1]$ and on the entire interval $[0,t]$, respectively. We determine three probability distribution functions: The joint distribution $P(m,M)$ of both maxima; the distribution $P(m)$ of the partial maximum; and the distribution $\Pi(G)$ of the gap between the maxima, $G = M-m$. We present exact results for the moments of these distributions and quantify the temporal correlations between $m$ and $M$ by calculating the Pearson correlation coefficient., Comment: 13 pages, 7 figures

We study a mesoscopic model of a chemically active colloidal particle which on certain parts of its surface promotes chemical reactions in the surrounding solution. For reasons of simplicity and conceptual clarity, we focus on the case in which only electrically neutral species are present in the solution and on chemical reactions which are described by first order kinetics. Within a self-consistent approach we explicitly determine the steady state product and reactant number density fields around the colloid as functionals of the interaction potentials of the various molecular species in solution with the colloid. By using Teubner's reciprocal theorem, this allows us to compute and to interpret -- in a transparent way in terms of the classical Smoluchowski theory of chemical kinetics -- the external force needed to keep such a catalytically active colloid at rest (\textit{stall} force) or, equivalently, the corresponding velocity of the colloid \textit{if} it is free to move. We use the particular case of triangular-well interaction potentials as a benchmark example for applying the general theoretical framework developed here. For this latter case, we derive explicit expressions for the dependences of the quantities of interest on the diffusion coefficients of the chemical species, the reaction rate constant, the coverage by catalyst, the size of the colloid, as well as on the parameters of the interaction potentials. These expressions provide a detailed picture of the phenomenology associated with catalytically-active colloids and self-diffusiophoresis., Comment: 63 pages, 5 figures

We study the correlations between the maxima $m$ and $M$ of a Brownian motion (BM) on the time intervals $[0,t_1]$ and $[0,t_2]$, with $t_2>t_1$. We determine exact forms of the distribution functions $P(m,M)$ and $P(G = M - m)$, and calculate the moments $\mathbb{E}\{\left(M - m\right)^k\}$ and the cross-moments $\mathbb{E}\{m^l M^k\}$ with arbitrary integers $l$ and $k$. We show that correlations between $m$ and $M$ decay as $\sqrt{t_1/t_2}$ when $t_2/t_1 \to \infty$, revealing strong memory effects in the statistics of the BM maxima. We also compute the Pearson correlation coefficient $\rho(m,M)$, the power spectrum of $M_t$, and we discuss a possibility of extracting the ensemble-averaged diffusion coefficient in single-trajectory experiments using a single realization of the maximum process., Comment: 5 pages, 5 figures

We study analytically the dynamics and the micro-structural changes of a host medium caused by a driven tracer particle moving in a confined, quiescent molecular crowding environment. Imitating typical settings of active micro-rheology experiments, we consider here a minimal model comprising a geometrically confined lattice system -- a two-dimensional strip-like or a three-dimensional capillary-like -- populated by two types of hard-core particles with stochastic dynamics -- a tracer particle driven by a constant external force and bath particles moving completely at random. Resorting to a decoupling scheme, which permits us to go beyond the linear-response approximation (Stokes regime) for arbitrary densities of the lattice gas particles, we determine the force-velocity relation for the tracer particle and the stationary density profiles of the host medium particles around it. These results are validated a posteriori by extensive numerical simulations for a wide range of parameters. Our theoretical analysis reveals two striking features: a) We show that, under certain conditions, the terminal velocity of the driven tracer particle is a nonmonotonic function of the force, so that in some parameter range the differential mobility becomes negative, and b) the biased particle drives the whole system into a nonequilibrium steady-state with a stationary particle density profile past the tracer, which decays exponentially, in sharp contrast with the behavior observed for unbounded lattices, where an algebraic decay is known to take place., Comment: 14 pages, 8 figures

We study the dynamics of a tracer particle (TP) on a comb lattice populated by randomly moving hard-core particles in the dense limit. We first consider the case where the TP is constrained to move on the backbone of the comb only, and, in the limit of high density of particles, we present exact analytical results for the cumulants of the TP position, showing a subdiffusive behavior $\sim t^{3/4}$. At longer times, a second regime is observed, where standard diffusion is recovered, with a surprising non analytical dependence of the diffusion coefficient on the particle density. When the TP is allowed to visit the teeth of the comb, based on a mean-field-like Continuous Time Random Walk description, we unveil a rich and complex scenario, with several successive subdiffusive regimes, resulting from the coupling between the inhomogeneous comb geometry and particle interactions. Remarkably, the presence of hard-core interactions speeds up the TP motion along the backbone of the structure in all regimes., Comment: 5 pages, 3 figures + supplemental material

We study the behavior of the stationary velocity of a driven particle in an environment of mobile hard-core obstacles. Based on a lattice gas model, we demonstrate analytically that the drift velocity can exhibit a nonmonotonic dependence on the applied force, and show quantitatively that such negative differential mobility (NDM), observed in various physical contexts, is controlled by both the density and diffusion time scale of obstacles. Our study unifies recent numerical and analytical results obtained in specific regimes, and makes it possible to determine analytically the region of the full parameter space where NDM occurs. These results suggest that NDM could be a generic feature of biased (or active) transport in crowded environments., Comment: 5 pages, 2 figures + supplemental material

We study the statistics of the first passage of a random walker to absorbing subsets of the boundary of compact domains in different spatial dimensions. We describe a novel diagnostic method to quantify the trajectory-to-trajectory fluctuations of the first passage, based on the distribution of the so-called uniformity index $\omega$, measuring the similarity of the first passage times of two independent walkers starting at the same location. We show that the characteristic shape of $P(\omega)$ exhibits a transition from unimodal to bimodal, depending on the starting point of the trajectories. From the study of different geometries in one, two and three dimensions, we conclude that this transition is a generic property of first passage phenomena in bounded domains. Our results show that, in general, the Mean First Passage Time (MFPT) is a meaningful characteristic measure of the first passage behaviour only when the Brownian walkers start sufficiently far from the absorbing boundary. Strikingly, in the opposite case, the first passage statistics exhibit large trajectory-to-trajectory fluctuations and the MFPT is not representative of the actual behaviour., Comment: 23 pages and 6 figures. To appear in the special volume "First-Passage Phenomena and Their Applications", Eds. R. Metzler, G. Oshanin, S. Redner. World Scientific (2013). arXiv admin note: text overlap with arXiv:1206.1003

We consider a driven tracer particle (TP) in a bath of hard-core particles undergoing continuous exchanges with a reservoir. We develop an analytical framework which allows us to go beyond the standard force-velocity relation used for this minimal model of active microrheology and quantitatively analyze, for any density of the bath particles, the fluctuations of the TP position and their correlations with the occupation number of the bath particles. We obtain an exact Einstein-type relation which links these fluctuations in the absence of a driving force and the bath particles density profiles in the linear driving regime. For the one-dimensional case we also provide an approximate but very accurate explicit expression for the variance of the TP position and show that it can be a non-monotoneous function of the bath particles density: counter-intuitively, an increase of the density may increase the dispersion of the TP position. We show that this non trivial behavior, which could in principle be observed in active microrheology experiments, is induced by subtle cross-correlations quantified by our approach.

We study a minimal model of active transport in crowded single-file environments which generalises 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 $\propto\sqrt{n}$ 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.

We focus on two models of nearest-neighbour random walks on d-dimensional regular hyper-cubic lattices that are usually assumed to be identical - the discrete-time Polya walk, in which the walker steps at each integer moment of time, and the Montroll-Weiss continuous-time random walk in which the time intervals between successive steps are independent, exponentially and identically distributed random variables with mean 1. We show that while for symmetric random walks both models indeed lead to identical behaviour in the long time limit, when there is an external bias they lead to markedly different behaviour., Comment: 5 pages, 1 figure

We analyze the form of the probability distribution function P_{n}^{(\beta)}(w) of the Schmidt-like random variable w = x_1^2/(\sum_{j=1}^n x^{2}_j/n), where x_j are the eigenvalues of a given n \times n \beta-Gaussian random matrix, \beta being the Dyson symmetry index. This variable, by definition, can be considered as a measure of how any individual eigenvalue deviates from the arithmetic mean value of all eigenvalues of a given random matrix, and its distribution is calculated with respect to the ensemble of such \beta-Gaussian random matrices. We show that in the asymptotic limit n \to \infty and for arbitrary \beta the distribution P_{n}^{(\beta)}(w) converges to the Mar\v{c}enko-Pastur form, i.e., is defined as P_{n}^{(\beta)}(w) \sim \sqrt{(4 - w)/w} for w \in [0,4] and equals zero outside of the support. Furthermore, for Gaussian unitary (\beta = 2) ensembles we present exact explicit expressions for P_{n}^{(\beta=2)}(w) which are valid for arbitrary n and analyze their behavior., Comment: 10 pages, 1 figure, submitted to JSTAT

The spontaneous spreading of non-volatile liquid droplets on solid substrates poses a classic problem in the context of wetting phenomena. It is well known that the spreading of a macroscopic droplet is in many cases accompanied by a thin film of macroscopic lateral extent, the so-called precursor film, which emanates from the three-phase contact line region and spreads ahead of the latter with a much higher speed. Such films have been usually associated with liquid-on-solid systems, but in the last decade similar films have been reported to occur in solid-on-solid systems. While the situations in which the thickness of such films is of mesoscopic size are rather well understood, an intriguing and yet to be fully understood aspect is the spreading of microscopic, i.e., molecularly thin films. Here we review the available experimental observations of such films in various liquid-on-solid and solid-on-solid systems, as well as the corresponding theoretical models and studies aimed at understanding their formation and spreading dynamics. Recent developments and perspectives for future research are discussed., Comment: 51 pages, 10 figures; small typos corrected

We study the distribution function $P(\omega)$ of the random variable $\omega = \tau_1/(\tau_1 + ... + \tau_N)$, where $\tau_k$'s are the partial Wigner delay times for chaotic scattering in a disordered system with $N$ independent, statistically equivalent channels. In this case, $\tau_k$'s are i.i.d. random variables with a distribution $\Psi(\tau)$ characterized by a "fat" power-law intermediate tail $\sim 1/\tau^{1 + \mu}$, truncated by an exponential (or a log-normal) function of $\tau$. For $N = 2$ and N=3, we observe a surprisingly rich behavior of $P(\omega)$ revealing a breakdown of the symmetry between identical independent channels. For N=2, numerical simulations of the quasi one-dimensional Anderson model confirm our findings., Comment: 4 pages, 5 figures

We study the distribution P(\omega) of the random variable \omega = x_1/(x_1 + x_2), where x_1 and x_2 are the wealths of two individuals selected at random from the same tempered Paretian ensemble characterized by the distribution \Psi(x) \sim \phi(x)/x^{1 + \alpha}, where \alpha > 0 is the Pareto index and $\phi(x)$ is the cut-off function. We consider two forms of \phi(x): a bounded function \phi(x) = 1 for L \leq x \leq H, and zero otherwise, and a smooth exponential function \phi(x) = \exp(-L/x - x/H). In both cases \Psi(x) has moments of arbitrary order. We show that, for \alpha > 1, P(\omega) always has a unimodal form and is peaked at \omega = 1/2, so that most probably x_1 \approx x_2. For 0 < \alpha < 1 we observe a more complicated behavior which depends on the value of \delta = L/H. In particular, for \delta < \delta_c - a certain threshold value - P(\omega) has a three-modal (for a bounded \phi(x)) and a bimodal M-shape (for an exponential \phi(x)) form which signifies that in such ensembles the wealths x_1 and x_2 are disproportionately different., Comment: 9 pages, 8 figures, to appear in Physica A

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 \{\tau_k\}, where \tau_k is the time of the first passage of the k-th searcher to the location of the target. Clearly, \tau_k's are independent, identically distributed random variables with the same distribution function \Psi(\tau). We evaluate then the distribution P(\omega) of the random variable \omega \sim \tau_1/bar{\tau}, where bar{\tau} = N^{-1} \sum_{k=1}^N \tau_k is the ensemble-averaged realization-dependent first passage time. We show that P(\omega) exhibits quite a non-trivial and sometimes a counterintuitive behaviour. We demonstrate that in some well-studied cases e.g., Brownian motion in finite d-dimensional domains) the \textit{mean} first passage time is not a robust measure of the search efficiency, despite the fact that \Psi(\tau) has moments of arbitrary order. This implies, in particular, that even in this simplest case (not saying about complex systems and/or anomalous diffusion) first passage data extracted from a single particle tracking should be regarded with an appropriate caution because of the significant sample-to-sample fluctuations., Comment: 35 pages, 18 figures, to appear in JSTAT

We analyze the mean time t_{app} that a randomly moving particle spends in a bounded domain (sphere) before it escapes through a small window in the domain's boundary. A particle is assumed to diffuse freely in the bulk until it approaches the surface of the domain where it becomes weakly adsorbed, and then wanders diffusively along the boundary for a random time until it desorbs back to the bulk, and etc. Using a mean-field approximation, we define t_{app} analytically as a function of the bulk and surface diffusion coefficients, the mean time it spends in the bulk between two consecutive arrivals to the surface and the mean time it wanders on the surface within a single round of the surface diffusion., Comment: 8 pages, 1 figure, submitted to JCP

The quasistationary spreading of a circular liquid drop on a solid substrate typically obeys the so-called Tanner law, with the instantaneous base radius R(t) growing with time as R ~ t^{1/10} -- an effect of the dominant role of capillary forces for a small-sized droplet. However, for droplets of nematic liquid crystals, a faster spreading law sets in at long times, so that R ~ t^alpha with alpha significantly larger than the Tanner exponent 1/10. In the framework of the thin film model (or lubrication approximation), we describe this "acceleration" as a transition to a qualitatively different spreading regime driven by a strong substrate-liquid interaction specific to nematics (antagonistic anchoring at the interfaces). The numerical solution of the thin film equation agrees well with the available experimental data for nematics, even though the non-Newtonian rheology has yet to be taken into account. Thus we complement the theory of spreading with a post-Tanner stage, noting that the spreading process can be expected to cross over from the usual capillarity-dominated stage to a regime where the whole reservoir becomes a diffusive film in the sense of Derjaguin., Comment: 15 pages, 4 figures, accepted in JPCM special issue

We study the survival of a prey that is hunted by N predators. The predators perform independent random walks on a square lattice with V sites and start a direct chase whenever the prey appears within their sighting range. The prey is caught when a predator jumps to the site occupied by the prey. We analyze the efficacy of a lazy, minimal-effort evasion strategy according to which the prey tries to avoid encounters with the predators by making a hop only when any of the predators appears within its sighting range; otherwise the prey stays still. We show that if the sighting range of such a lazy prey is equal to 1 lattice spacing, at least 3 predators are needed in order to catch the prey on a square lattice. In this situation, we establish a simple asymptotic relation ln(Pev)(t) \sim (N/V)2ln(Pimm(t)) between the survival probabilities of an evasive and an immobile prey. Hence, when the density of the predators is low N/V<<1, the lazy evasion strategy leads to the spectacular increase of the survival probability. We also argue that a short-sighting prey (its sighting range is smaller than the sighting range of the predators) undergoes an effective superdiffusive motion, as a result of its encounters with the predators, whereas a far-sighting prey performs a diffusive-type motion., Comment: 6 pages, 5 figures

We compare an efficiency of a deterministic "lawnmower" and random search strategies for finding a prescribed sequence of letters (a password) of length M in which all letters are taken from the same Q-ary alphabet. We show that at best a random search takes two times longer than a "lawnmower" search., Comment: To appear in J. Phys. A, special issue on "Random Search Problem: Trends and Perspectives", eds.: MEG da Luz, E Raposo, GM Viswanathan and A Grosberg

We study theoretically the effects of spatial confinement on the phoretic motion of a dissolved particle driven by composition gradients generated by chemical reactions of its solvent, which are active only on certain parts of the particle surface. We show that the presence of confining walls increases in a similar way both the composition gradients and the viscous friction, and the overall result of these competing effects is an increase in the phoretic velocity of the particle. For the case of steric repulsion only between the particle and the product molecules of the chemical reactions, the absolute value of the velocity remains nonetheless rather small., Comment: 18 pages, 4 figures, J. Chem. Phys. (in print; full bibliographic info and DOI to be added once available)

We study direct energy transfer by multipolar or exchange interactions between diffusive excited donor and diffusive unexcited acceptors. Extending over the case of long-range transfer of an excitation energy a non-perturbative approach by Bray and Blythe [Phys. Rev. Lett. 89, 150601 (2002)], originally developed for contact diffusion-controlled reactions, we determine exactly long-time asymptotics of the donor decay function in one-dimensional systems., Comment: 16 pages

We determine the probability that a partially melted heteropolymer at the melting temperature will either melt completely or return to a helix state. This system is equivalent to the splitting probability for a diffusing particle on a finite interval that moves according to the Sinai model. When the initial fraction of melted polymer is f, the melting probability fluctuates between different realizations of monomer sequences on the polymer. For a fixed value of f, the melting probability distribution changes from unimodal to a bimodal as the strength of the disorder is increased., Comment: 4 pages, 5 figures

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 sub-diffusively, such that the mean square displacement of each trap grows as $t^{\gamma}$ with $0\leq \gamma \leq 1$. Starting from a continuous time random walk (CTRW) description of the motion of the particle and of the traps, we derive lower and upper bounds for $P(t)$ and show that for $\gamma \leq 2/(d+2)$ 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 $d\leq 2$ 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.

We study the long-time behavior of the probability density Q_t of the first exit time from a bounded interval [-L,L] for a stochastic non-Markovian process h(t) describing fluctuations at a given point of a two-dimensional, infinite in both directions Gaussian interface. We show that Q_t decays when t \to \infty as a power-law $^{-1 - \alpha}, where \alpha is non-universal and proportional to the ratio of the thermal energy and the elastic energy of a fluctuation of size L. The fact that \alpha appears to be dependent on L, which is rather unusual, implies that the number of existing moments of Q_t depends on the size of the window [-L,L]. A moment of an arbitrary order n, as a function of L, exists for sufficiently small L, diverges when L approaches a certain threshold value L_n, and does not exist for L > L_n. For L > L_1, the probability density Q_t is normalizable but does not have moments., Comment: 10 pages

We study the long-time asymptotical behavior of the survival probability P_t of a tagged monomer of an infinitely long Rouse chain in presence of two fixed absorbing boundaries, placed at x = \pm L. Mean-square displacement of a tagged monomer obeys \bar{X^2(t)} \sim t^{1/2} at all times, which signifies that its dynamics is an anomalous diffusion process. Constructing lower and upper bounds on P_t, which have the same time-dependence but slightly differ by numerical factors in the definition of the characteristic relaxation time, we show that P_t is a stretched-exponential function of time, \ln(P_t) \sim - t^{1/2}/L^2. This implies that the distribution function of the first exit time from a fixed interval [-L,L] for such an anomalous diffusion has all moments., Comment: 6 pages, submitted to EPL

We study the long-time asymptotics of the probability P_t that the Riemann-Liouville fractional Brownian motion with Hurst index H does not escape from a fixed interval [-L,L] up to time t. We show that for any H \in ]0,1], for both subdiffusion and superdiffusion regimes, this probability obeys \ln(P_t) \sim - t^{2 H}/L^2, i.e. may decay slower than exponential (subdiffusion) or faster than exponential (superdiffusion). This implies that survival probability S_t of particles undergoing fractional Brownian motion in a one-dimensional system with randomly placed traps follows \ln(S_t) \sim - n^{2/3} t^{2H/3} as t \to \infty, where n is the mean density of traps., Comment: 13 pages, submitted to J.Phys.A

We study the effect of spatial confinement on the strength of propulsive diffusiophoretic forces acting on a particle that generates density gradients by exploiting the chemical free energy of its environment. Using a recently proposed simple model of a self-propelling device driven by chemical reactions taking place on some parts of its surface, we demonstrate that the force significantly increases in the presence of confining walls. We also show that such effects become even more pronounced in two-dimensional systems., Comment: 6 pages, 2 figures, uses epl2.cls

Within the framework of a semi-microscopic interface displacement model we analyze the linear stability of sessile ridges and drops of a non-volatile liquid on a homogeneous, partially wet substrate, for both signs and arbitrary amplitudes of the three-phase contact line tension. Focusing on perturbations which correspond to deformations of the three-phase contact line, we find that drops are generally stable while ridges are subject only to the long-wavelength Rayleigh-Plateau instability leading to a breakup into droplets, in contrast to the predictions of capillary models which take line tension into account. We argue that the short-wavelength instabilities predicted within the framework of the latter macroscopic capillary theory occur outside its range of validity and thus are spurious., Comment: 6 pages, 1 figure

We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time $n$, whose moves to the right or to the left are induced by the rise-and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site $X$ at time $n$, obtain the probability measure of different excursions and define the asymptotic distribution of the number of "U-turns" of the trajectories - permutation "peaks" and "through". In the second part, we focus on some statistical properties of surfaces obtained by randomly placing natural numbers $1,2,3, >...,L$ on sites of a 1d or 2d square lattices containing $L$ sites. We calculate the distribution function of the number of local "peaks" - sites the number at which is larger than the numbers appearing at nearest-neighboring sites - and discuss some surprising collective behavior emerging in this model., Comment: 16 pages, 5 figures; submitted to European Physical Journal, proceedings of the conference "Stochastic and Complex Systems: New Trends and Expectations" Santander, Spain

We study the search kinetics of an immobile target by a concentration of randomly moving searchers. The object of the study is to optimize the probability of detection within the constraints of our model. The target is hidden on a one-dimensional lattice in the sense that searchers have no a priori information about where it is, and may detect it only upon encounter. The searchers perform random walks in discrete time n=0,1,2, ..., N, where N is the maximal time the search process is allowed to run. With probability \alpha the searchers step on a nearest-neighbour, and with probability (1-\alpha) they leave the lattice and stay off until they land back on the lattice at a fixed distance L away from the departure point. The random walk is thus intermittent. We calculate the probability P_N that the target remains undetected up to the maximal search time N, and seek to minimize this probability. We find that P_N is a non-monotonic function of \alpha, and show that there is an optimal choice \alpha_{opt}(N) of \alpha well within the intermittent regime, 0 < \alpha_{opt}(N) < 1, whereby P_N can be orders of magnitude smaller compared to the "pure" random walk cases \alpha =0 and \alpha = 1., Comment: 19 pages, 5 figures; submitted to Journal of Physics: Condensed Matter; special issue on Chemical Kinetics Beyond the Textbook: Fluctuations, Many-Particle Effects and Anomalous Dynamics, eds. K.Lindenberg, G.Oshanin and M.Tachiya

We study the equilibrium properties of a model for a binary mixture of catalytically-reactive monomers adsorbed on a two-dimensional substrate decorated by randomly placed catalytic bonds. The interacting $A$ and $B$ monomer species undergo continuous exchanges with particle reservoirs and react ($A + B \to \emptyset$) as soon as a pair of unlike particles appears on sites connected by a catalytic bond. For the case of annealed disorder in the placement of the catalytic bonds this model can be mapped onto a classical spin model with spin values $S = -1,0,+1$, with effective couplings dependent on the temperature and on the mean density $q$ of catalytic bonds. This allows us to exploit the mean-field theory developed for the latter to determine the phase diagram as a function of $q$ in the (symmetric) case in which the chemical potentials of the particle reservoirs, as well as the $A-A$ and $B-B$ interactions are equal., Comment: 12 pages, 4 figures

We study kinetics of diffusion-limited catalytically-activated $A + B \to B$ reactions taking place in three dimensional systems, in which an annihilation of diffusive $A$ particles by diffusive traps $B$ may happen only if the encounter of an $A$ with any of the $B$s happens within a special catalytic subvolumen, these subvolumens being immobile and uniformly distributed within the reaction bath. Suitably extending the classical approach of Wilemski and Fixman (G. Wilemski and M. Fixman, J. Chem. Phys. \textbf{58}:4009, 1973) to such three-molecular diffusion-limited reactions, we calculate analytically an effective reaction constant and show that it comprises several terms associated with the residence and joint residence times of Brownian paths in finite domains. The effective reaction constant exhibits a non-trivial dependence on the reaction radii, the mean density of catalytic subvolumens and particles' diffusion coefficients. Finally, we discuss the fluctuation-induced kinetic behavior in such systems., Comment: To appear in J. Chem. Phys

We calculate the mean joint residence time of two Brownian particles in a sphere, for very general initial conditions. In particular, we focus on the dependence of this residence time as a function of the diffusion coefficients of the two particles. Our results can be useful for describing kinetics of bimolecular diffusion controlled reactions activated by catalytic sites.

Dynamics of a classical particle in a one-dimensional, randomly driven potential is analysed both analytically and numerically. The potential considered here is composed of two identical spatially-periodic saw-tooth-like components, one of which is externally driven by a random force. We show that under certain conditions the particle may travel against the averaged external force performing a saltatory unidirectional drift with a constant velocity. Such a behavior persists also in situations when the external force averages out to zero. We demonstrate that the physics behind this phenomenon stems from a particular behavior of fluctuations in random force: upon reaching a certain level, random fluctuations exercise a locking function creating points of irreversibility which the particle can not overpass. Repeated (randomly) in each cycle, this results in a saltatory unidirectional drift. This mechanism resembles the work of an escapement-type device in watches. Considering the overdamped limit, we propose simple analytical estimates for the particle's terminal velocity., Comment: 14 pages, 6 figures; appearing in Journal of Physics: Condensed Matter, special issue on Molecular Motors and Friction

We study here a standard next-nearest-neighbor (NNN) model of ballistic growth on one- and two-dimensional substrates focusing our analysis on the probability distribution function $P(M,L)$ of the number $M$ of maximal points (i.e., local ``peaks'') of growing surfaces. Our analysis is based on two central results: (i) the proof (presented here) of the fact that uniform one--dimensional ballistic growth process in the steady state can be mapped onto ''rise-and-descent'' sequences in the ensemble of random permutation matrices; and (ii) the fact, established in Ref. \cite{ov}, that different characteristics of ``rise-and-descent'' patterns in random permutations can be interpreted in terms of a certain continuous--space Hammersley--type process. For one--dimensional system we compute $P(M,L)$ exactly and also present explicit results for the correlation function characterizing the enveloping surface. For surfaces grown on 2d substrates, we pursue similar approach considering the ensemble of permutation matrices with long--ranged correlations. Determining exactly the first three cumulants of the corresponding distribution function, we define it in the scaling limit using an expansion in the Edgeworth series, and show that it converges to a Gaussian function as $L \to \infty$., Comment: 25 pages, 12 figures

The use of ultra-thin, i.e., monolayer films plays an important role for the emerging field of nano-fluidics. Since the dynamics of such films is governed by the interplay between substrate-fluid and fluid-fluid interactions, the transport of matter in nanoscale devices may be eventually efficiently controlled by substrate engineering. For such films, the dynamics is expected to be captured by two-dimensional lattice-gas models with interacting particles. Using a lattice gas model and the non-linear diffusion equation derived from the microscopic dynamics in the continuum limit, we study two problems of relevance in the context of nano-fluidics. The first one is the case in which along the spreading direction of a monolayer a mesoscopic-sized obstacle is present, with a particular focus on the relaxation of the fluid density profile upon encountering and passing the obstacle. The second one is the mixing of two monolayers of different particle species which spread side by side following the merger of two chemical lanes, here defined as domains of high affinity for fluid adsorption surrounded by domains of low affinity for fluid adsorption., Comment: 12 pages, 3 figures

We present a microscopic model of a charge carrier transfer under an action of a constant electric field in a complex medium. Generalizing previous theoretical approaches, we model the dynamical environment hindering the carrier motion by dynamic percolation, i.e., as a medium comprising particles which move randomly on a simple cubic lattice, constrained by hard-core exclusion, and may spontaneously annihilate and re-appear at some prescribed rates. We determine analytically the density profiles of the "environment" particles, as seen from the stationary moving charge carrier, and calculate its terminal velocity as the function of the applied field and other system parameters. We realize that for sufficiently small external fields the force exerted on the carrier by the "environment" particles shows a viscous-like behavior and define an analog of the Stokes formula for such dynamic percolative environments. The corresponding friction coefficient is also derived., Comment: appearing in Chem. Phys. Special Issue on Molecular Charge Transfer in Condensed Media - from Physics and Chemistry to Biology and Nano-Engineering, edited by A.Kornyshev (Imperial College London), M.Newton (Brookhaven Natl Lab) and J.Ulstrup (Technical University of Denmark)