[en] 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 directed 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 an effective Hookean bond, all features of the model such as prey density and sighting ranges merge into the effective binding constant. (paper)

[en] Power-law Poissonian ensembles are Poisson processes that are defined on the positive half-line, and that are governed by power-law intensities. Power-law Poissonian ensembles are stochastic objects of fundamental significance; they uniquely display an array of fractal features and they uniquely generate a span of important applications. In this paper we apply three different methods—oligarchic analysis, Lorenzian analysis and heterogeneity analysis—to explore power-law Poissonian ensembles. The amalgamation of these analyses, combined with the topology of power-law Poissonian ensembles, establishes a detailed and multi-faceted picture of the statistical structure and the statistical phase transitions of these elemental ensembles. (paper)

[en] A standard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, $T\to <!--\; ???\; -->\mathrm{\infty <!--\; ???\; -->}$. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit $T\to <!--\; ???\; -->\mathrm{\infty <!--\; ???\; -->}$ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion. We demonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent. We also compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing single-trajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement. (paper)

[en] We study the position of a biased tracer particle (TP) in a bath of hardcore particles moving on a lattice of arbitrary dimension and in contact with a reservoir. Starting from the master equation satisfied by the joint probability of the TP position and the bath configuration and resorting to a mean-field-type approximation, we presented a computation of the fluctuations of the TP position in a previous publication (Bénichou et al 2013 Phys. Rev. E 87 032164). Counter-intuitively, on a one-dimensional lattice, the diffusion coefficient of the TP was shown to be a nonmonotonic function of the density of bath particles, and reaches a maximum for a nonzero value of the density. Here, we (i) give the details of this computation and offer a physical insight into the understanding of the nonmonotonicity of the diffusion coefficient; (ii) extend the mean-field-type approximation to decouple higher-order correlation functions, and obtain the evolution equation satisfied by the cumulant generating function of the position of the TP, valid in any space dimension. In the particular case of a one-dimensional lattice, we solve this equation and obtain the probability distribution of the TP position. We show that the position rescaled by its fluctuations is asymptotically distributed according to a Gaussian distribution in the long-time limit. (paper)

[en] We analyse the power spectral density (PSD) S _T(f) (with T being the observation time and  f  the frequency) of a fractional Brownian motion (fBm), with an arbitrary Hurst index , undergoing a stochastic resetting to the origin at a constant rate r—the resetting process introduced some time ago as an example of an efficient, optimisable search algorithm. To this end, we first derive an exact expression for the covariance function of an arbitrary (not necessarily a fBm) process with a reset, expressing it through the covariance function of the parental process without a reset, which yields the desired result for the fBm in a particular case. We then use this result to compute exactly the power spectral density for fBM for all frequency f. The asymptotic, large frequency f behaviour of the PSD turns out to be distinctly different for sub- and super-diffusive fBms. We show that for large f, the PSD has a power law tail: where the exponent for (sub-diffusive fBm), while for all . Thus, somewhat unexpectedly, the exponent in the superdiffusive case H  >  1/2 sticks to its Brownian value and does not depend on H. (paper)

[en] We study the probability distribution function P^(β)_n(w) of the Schmidt-like random variable w = x²₁/(∑_j=1ⁿx²_j/n), where x_j, (j = 1, 2, …, n), are unordered eigenvalues of a given n × n β-Gaussian random matrix, β being the Dyson symmetry index. This variable, by definition, can be considered as a measure of how any individual (randomly chosen) 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 β-Gaussian random matrices. We show that in the asymptotic limit n → ∞ and for arbitrary β the distribution P^(β)_n(w) converges to the Marčenko–Pastur form, i.e. is defined as P_n^(β)(w)∼√((4 - w)/w) for w ∈ [0, 4] and equals zero outside of the support, despite the fact that formally w is defined on the interval [0, n]. Furthermore, for Gaussian unitary ensembles (β = 2) we present exact explicit expressions for P^(β=2)_n(w) which are valid for arbitrary n and analyse their behaviour. (paper)

[en] We study equilibrium properties of catalytically-activated reactions taking place on a lattice of adsorption sites. The particles undergo continuous exchanges with a reservoir maintained at a constant chemical potential μ and react when they appear at the neighbouring sites, provided that some reactive conditions are fulfilled. We model the latter in two different ways: in the Model I some fraction p of the bonds connecting neighbouring sites possesses special catalytic properties such that any two As appearing on the sites connected by such a bond instantaneously react and desorb. In the Model II some fraction p of the adsorption sites possesses such properties and neighbouring particles react if at least one of them resides on a catalytic site. For the case of annealed disorder in the distribution of the catalyst, which is tantamount to the situation when the reaction may take place at any point on the lattice but happens with a finite probability p, we provide an exact solution for both models for the interior of an infinitely large Cayley tree—the so-called Bethe lattice. We show that both models exhibit a rich critical behaviour: for the annealed Model I it is characterised by a transition into an ordered state and a re-entrant transition into a disordered phase, which both are continuous. For the annealed Model II, which represents a rather exotic model of statistical mechanics in which interactions of any particle with its environment have a peculiar Boolean form, the transition to an ordered state is always continuous, while the re-entrant transition into the disordered phase may be either continuous or discontinuous, depending on the value of p. (paper: classical statistical mechanics, equilibrium and non-equilibrium)

[en] We study equilibrium properties of binary lattice-gases comprising A and B particles, which undergo continuous exchanges with their respective reservoirs, maintained at chemical potentials μ _A = μ _B = μ. The particles interact via on-site hard-core exclusion and also between the nearest-neighbours: there are a soft repulsion between AB pairs and also interactions of arbitrary strength J, positive or negative, for AA and BB pairs. For tree-like Bethe and Husimi lattices we determine the full phase diagram of such a ternary mixture of particles and voids. We show that for J being above a lattice-dependent threshold value, the critical behaviour is similar: the system undergoes a transition at μ = μ _c from a phase with equal mean densities of species into a phase with a spontaneously broken symmetry, in which the mean densities are no longer equal. Depending on the value of J, this transition can be either continuous or of the first order. For sufficiently large negative J, the behaviour on the two lattices becomes markedly different: on the Bethe lattice there exist two separate phases with different kinds of structural order, which are absent on the Husimi lattice, due to stronger frustration effects. (paper)

[en] We study the dynamics of a Brownian particle in a strongly correlated quenched random potential defined as a periodically–extended (with period L) finite trajectory of a fractional Brownian motion with arbitrary Hurst exponent H∈(0,1). While the periodicity ensures that the ultimate long–time behavior is diffusive, the generalized Sinai potential considered here leads to a strong logarithmic confinement of particle trajectories at intermediate times. These two competing trends lead to dynamical frustration and result in a rich statistical behavior of the diffusion coefficient D _L: although one has the typical value D_L^typ∼exp(−βL^H), we show via an exact analytical approach that the positive moments (k>0) scale like 〈D_L^k〉∼exp[−c^′(kβL^H)^1/(1+H)], and the negative ones as 〈D_L^−k〉∼exp(a^′(kβL^H)²), c^′ and a^′ being numerical constants and β the inverse temperature. These results demonstrate that D _L is strongly non-self-averaging. We further show that the probability distribution of D _L has a log–normal left tail and a highly singular, one–sided log–stable right tail reminiscent of a Lifshitz singularity. (fast track communication)

[en] The question how the extremal values of a stochastic process achieved on different time intervals are correlated to each other has been discussed within the last few years on examples of the running maximum of a Brownian motion, of a Brownian bridge and of a Slepian process. Here, we focus on the two-time correlations of the running range of Brownian motion (BM)—the maximal extent of a Brownian trajectory on a finite time interval. We calculate exactly the covariance function of the running range and analyse its asymptotic behaviour. Our analysis reveals non-trivial correlations between the value of the largest descent (rise) of a BM from the top to a bottom on some time interval, and the value of this property on a larger time interval. (paper)