We consider random walkers that deform the medium as they move, enabling a faster motion in regions which have been recently visited. This induces an effective attraction between walkers mediated by the medium, which can be regarded as a space metric, giving rise to a statistical mechanics toy model either for gravity, motion through deformable matter or adaptable geometry. In the strong-deformability regime, we find that diffusion is initially described by the porous medium equation, thus yielding subdiffusive behavior of an initially localized cloud of particles. Indeed, while the average width of a single cloud will sustain a $σ\sim t^{1/2}$ growth, the combined width of the whole ensemble will grow like $σ\sim t^{1/3}$ in a certain time regime. This difference can be accounted for by the strong correlations between the particles, which we explore indirectly through the fluctuations of the center of mass of the cloud and the expected value of the experienced density, defined as the average density measured by the particles themselves.
Directed manifolds (domain walls, interfaces, vortex lines) in a deformable medium can exist in a correlated state in which the manifold is self-localized by its own strain field. Depending on the temperature, manifold/medium dimensionalities, and the strength of the coupling with the medium, the degree of localization of the ground state can vary both continuously and discontinuously; there can be phase transitions from self- localized to the free-manifold state.
We consider the penetration length $l$ of random walkers diffusing in a medium of perfect or imperfect absorbers of number density $ρ$. We solve this problem on a lattice and in the continuum in all dimensions $D$, by means of a mean-field renormalization group. For a homogeneous system in $D>2$, we find that $l\sim \max(ξ,ρ^{-1/2})$, where $ξ$ is the absorber density correlation length. The cases of D=1 and D=2 are also treated. In the presence of long-range correlations, we estimate the temporal decay of the density of random walkers not yet absorbed. These results are illustrated by exactly…
We study chemotaxis in a porous medium using as a model a biased ("hungry") random walk on a percolating cluster. In close resemblance to the 1980s arcade game Pac-Man, the hungry random walker consumes food, which is initially distributed in the maze, and biases its movement towards food-filled sites. We observe that, on the percolating cluster, the mean-squared displacement of the pacman process shows anomalous dynamics, which follow a power law with a dynamical exponent different from both that of a self avoiding random walk as well as that of an unbiased random walk. The change in dynamics…
A relationship of the random walks on one-dimensional periodic lattice and the correlation functions of the XX Heisenberg spin chain is investigated. The operator averages taken over the ferromagnetic state play a role of generating functions of the number of paths made by the so-called "vicious" random walkers (the vicious walkers annihilate each other provided they arrive at the same lattice site). It is shown that the two-point correlation function of spins, calculated over eigen-states of the XX magnet, can be interpreted as the generating function of paths made by a single walker in a med…
We consider the continuous time random walk model (CTRW) of tracer's motion in porous medium flows based on the experimentally determined distributions of pore velocity and pore size reported in Holzner et al. Phys. Rev. E 92, 013015 (2015). The particle's passing through one channel is modelled as one step of the walk. The step's (channel) length is random and the walker's velocity at consecutive steps of the walk is conserved with finite probability mimicking that at the turning point there could be no abrupt change of velocity. We provide the Laplace transform of the characteristic function…
Consider $N$ points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory $μ$, leaves from the leftmost point and moves, at each discrete time step, to the nearest point, which has not been visited in the preceding $μ$ steps. We have obtained analytically the probability $P_N(μ) = (1 - 2^{-μ})^{N - μ- 1}$ that all $N$ points are visited in this open system, with $N \gg μ\gg 1$. The expression for $P_N(μ)$ evaluated in the mentioned limit is valid even for sm…
We study the extreme statistics of N non-intersecting Brownian motions (vicious walkers) over a unit time interval in one dimension. Using path-integral techniques we compute exactly the joint distribution of the maximum M and of the time τ_M at which this maximum is reached. We focus in particular on non-intersecting Brownian bridges ("watermelons without wall") and non-intersecting Brownian excursions ("watermelons with a wall"). We discuss in detail the relationships between such vicious walkers models in watermelons configurations and stochastic growth models in curved geometry on the one …
Optimization of multisite reactions in complex compartmentalized media
Laboratoire Ondes et Matière d’Aquitaine, CNRS/University of Bordeaux（阿基坦波与物质实验室，法国国家科学研究中心/波尔多大学）；Centre de physique théorique, CNRS / Aix-Marseille university（理论物理中心，法国国家科学研究中心/艾克斯-马赛大学）
The dense flow of air bubbles in a two-dimensional silo (through an aperture D) filled with a liquid is studied experimentally. A particle tracking technique has been used to bring out the main properties of the flow: displacements of the bubbles, transverse and axial velocities. The behavior of the air bubbles is observed to present similarities with non-deformable solid grains in a granular flow. Nevertheless, a correlation between the bubble velocities and their deformations has been evidenced. Moreover, a new discharge law (Beverloo-like) must be considered for such a system, where the flo…
We review recent studies demonstrating a nonuniversal (continuously variable) survival exponent for history-dependent random walks, and analyze a new example, the hard movable partial reflector. These processes serve as a simplified models of infection in a medium with a history-dependent susceptibility, and for spreading in systems with an infinite number of absorbing configurations. The memory may take the form of a history-dependent step length, or be the result of a partial reflector whose position marks the maximum distance the walker has ventured from the origin. In each case, a process …
Consider $N$ points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory $μ$, leaves from the leftmost point and moves, at each discrete time step, to the nearest point which has not been visited in the preceding $μ$ steps. Using open boundary conditions, we have calculated analytically the probability $P_N(μ) = (1 - 2^{-μ})^{N - μ- 1}$ that all $N$ points are visited, with $N \gg μ\gg 1$. This approximated expression for $P_N(μ)$ is reasonable even for sma…
The sticking of a soft polystyrene colloidal particle to a planar glass plate was studied by a microrheological technique using an optical tweezer to trap the particle and a piezoelectric-stage to position the plate and to sinusoidally drive it in-planeThe sticking process was well-described by a crossover of the mechanical coupling between the plate and the particle from being dominantly viscous to becoming dominantly elastic. By varying three parameters in the experiment, namely, the strength of the tweezer potential, the strength of the particle-plate interaction and the strength of the osc…
Levy walk (LW) process has been used as a simple model for describing anomalous diffusion in which the mean squared displacement of the walker grows non-linearly with time in contrast to the diffusive motion described by simple random walks or Brownian motion. In this paper we study a simple extension of the LW model in one dimension by introducing correlation among the velocities of the walker in different (flight) steps. Such correlation is absent in the LW model. The correlations are introduced by making the velocity at a step dependent on the velocity at the previous step in addition to th…
We study the first passage time properties of an integrated Brownian curve both in homogeneous and disordered environments. In a disordered medium we relate the scaling properties of this center of mass persistence of a random walker to the average persistence, the latter being the probability P_pr(t) that the expectation value <x(t)> of the walker's position after time t has not returned to the initial value. The average persistence is then connected to the statistics of extreme events of homogeneous random walks which can be computed exactly for moderate system sizes. As a result we obtain a…
We investigate the first passage time t_{j,N} to a given chemical or Euclidean distance of the first j of a set of N>>1 independent random walkers all initially placed on a site of a disordered medium. To solve this order-statistics problem we assume that, for short times, the survival probability (the probability that a single random walker is not absorbed by a hyperspherical surface during some time interval) decays for disordered media in the same way as for Euclidean and some class of deterministic fractal lattices. This conjecture is checked by simulation on the incipient percolation aggr…
We compute the joint probability density function (jpdf) P_N(M, τ_M) of the maximum M and its position τ_M for N non-intersecting Brownian excursions, on the unit time interval, in the large N limit. For N \to \infty, this jpdf is peaked around M = \sqrt{2N} and τ_M = 1/2, while the typical fluctuations behave for large N like M - \sqrt{2N} \propto s N^{-1/6} and τ_M - 1/2 \propto w N^{-1/3} where s and w are correlated random variables. One obtains an explicit expression of the limiting jpdf P(s,w) in terms of the Tracy-Widom distribution for the Gaussian Orthogonal Ensemble (GOE) of Random M…
We study the origin of buoyancy forces acting on a larger particle moving in a granular medium subject to horizontal shaking and its corrections before fluidization. In the fluid limit Archimedes' law is verified; before the limit memory effects counteract buoyancy, as also found experimentally. The origin of the friction is an excluded volume effect between active particles, which we study more exactly for a random walker in a random environment. The same excluded volume effect is also responsible for the mutual attraction between bodies moving in the granular medium. Our theoretical modeling…
Daily, are reported systems in nature that present anomalous diffusion phenomena due to irregularities of medium, traps or reactions process. In this scenario, the diffusion with traps or localised--reactions emerge through various investigations that include numerical, analytical and experimental techniques. In this work, we construct a model which involves a coupling of two diffusion equations to approach the random walkers in a medium with localised reaction point (or controlled diffusion). We present the exact analytical solutions to the model. In the following, we obtain the survival prob…
We study diffusion-limited pair annihilation $A+A\to 0$ on one-dimensional lattices with inhomogeneous nearest neighbour hopping in the limit of infinite reaction rate. We obtain a simple exact expression for the particle concentration $ρ_k(t)$ of the many-particle system in terms of the conditional probabilities $P(m;t|l;0)$ for a single random walker in a dual medium. For some disordered systems with an initially randomly filled lattice this leads asymptotically to $\bar{ρ(t)}=\bar{P(0;2t|0;0)}$ for the disorder-averaged particle density. We also obtain interesting exact relations for single…
Fundamental diagrams describing the relation between pedestrians speed and density are key points in understanding pedestrian dynamics. Experimental data evidence the onset of complex behaviors in which the velocity decreases with the density and different logistic regimes are identified. This paper addresses the issue of pedestrians transport and of fundamental diagrams for a scenario involving the motion of pedestrians escaping from an obscure tunnel. We capture the effects of the communication efficiency and the exit capacity by means of two thresholds controlling the rate at which particle…
Consider a medium characterized by N points whose coordinates are randomly generated by a uniform distribution along the edges of a unitary d-dimensional hypercube. A walker leaves from each point of this disordered medium and moves according to the deterministic rule to go to the nearest point which has not been visited in the preceding μsteps (deterministic tourist walk). Each trajectory generated by this dynamics has an initial non-periodic part of t steps (transient) and a final periodic part of p steps (attractor). The neighborhood rank probabilities are parameterized by the normalized in…
