Exact spatio-temporal dynamics of confined lattice random walks in arbitrary dimensions:a century after Smoluchowski and Polya

A lattice random walk is a mathematical representation of movement through random steps on a lattice at discrete times. It is commonly referred to as Pólya’s walk when the steps occur in either of the nearest-neighbor sites. Since Smoluchowski’s 1906 derivation of the spatiotemporal dependence of the walk occupation probability in an unbounded one-dimensional lattice, discrete random walks and their continuous counterpart, Brownian walks, have developed over the course of a century into a vast and versatile area of knowledge. Lattice random walks are now routinely employed to study stochastic processes across scales, dimensions, and disciplines, from the one-dimensional search of proteins along a DNA strand and the two-dimensional roaming of bacteria in a petri dish, to the three-dimensional motion of macromolecules inside cells and the spatial coverage of multiple robots in a disaster area. In these realistic scenarios, when the randomly moving object is constrained to remain within a finite domain, confined lattice random walks represent a powerful modeling tool. Somewhat surprisingly, and differently from Brownian walks, the spatiotemporal dependence of the confined lattice walk probability has been accessible mainly via computational techniques, and finding its analytic description has remained an open problem. Making use of a set of analytic combinatorics identities with Chebyshev polynomials, I develop a hierarchical dimensionality reduction to find the exact space and time dependence of the occupation probability for confined Pólya’s walks in arbitrary dimensions with reflective, periodic, absorbing, and mixed (reflective and absorbing) boundary conditions along each direction. The probability expressions allow one to construct the time dependence of derived quantities, explicitly in one dimension and via an integration in higher dimensions, such as the first-passage probability to a single target, return probability, average number of distinct sites visited, and absorption probability with imperfect traps. Exact mean first-passage time formulas to a single target in arbitrary dimensions are also presented. These formulas allow one to extend the so-called discrete pseudo-Green function formalism, employed to determine analytically mean first-passage time, with reflecting and periodic boundaries, and a wealth of other related quantities, to arbitrary dimensions. For multiple targets, I introduce a procedure to construct the time dependence of the first-passage probability to one of many targets. Reduction of the occupation probability expressions to the continuous time limit, the so-called continuous time random walk, and to the space-time continuous limit is also presented

Comparison of two models of tethered motion

We consider a random walker whose motion is tethered around a focal point. We use two models that exhibit the same spatial dependence in the steady state but widely different dynamics. In one case, the walker is subject to a deterministic bias towards the focal point, while in the other case, it resets its position to the focal point at random times. The deterministic tendency of the biased walker makes the forays away from the focal point more unlikely when compared to the random nature of the returns of the resetting walker. This difference has consequences on the spatio-temporal dynamics at intermediate times. To show the differences in the two models, we analyze their probability distribution and their dynamics in presence and absence of partially or fully absorbing traps. We derive analytically various quantities: (i) mean first-passage times to one target, where we recover results obtained earlier by a different technique, (ii) splitting probabilities to either of two targets as well as survival probabilities when one or either target is partially absorbing. The interplay between confinement, diffusion and absorbing traps produces interesting non-monotonic effects in various quantities, all potentially accessible in experiments. The formalism developed here may have a diverse range of applications, from study of animals roaming within home ranges and of electronic excitations moving in organic crystals to developing efficient search algorithms for locating targets in a crowded environment.Comment: 26 pages, 7 figure

An anti-symmetric exclusion process for two particles on an infinite 1D lattice

A system of two biased, mutually exclusive random walkers on an infinite 1D lattice is studied whereby the intrinsic bias of one particle is equal and opposite to that of the other. The propogator for this system is solved exactly and expressions for the mean displacement and mean square displacement (MSD) are found. Depending on the nature of the intrinsic bias, the system's behaviour displays two regimes, characterised by (i) the particles moving towards each other and (ii) away from each other, both qualitatively different from the case of no bias. The continuous-space limit of the propogator is found and is shown to solve a Fokker-Planck equation for two biased, mutually exclusive Brownian particles with equal and opposite drift velocity.Comment: 19 pages, 5 figure

Dynamics of lattice random walk within regions composed of different media and interfaces

We study the lattice random walk dynamics in a heterogeneous space of two media separated by an interface and having different diffusivity and bias. Depending on the position of the interface, there exist two exclusive ways to model the dynamics: (1) Type A dynamics whereby the interface is placed between two lattice points, and (2) Type B dynamics whereby the interface is placed on a lattice point. For both types, we obtain exact results for the one-dimensional generating function of the Green's function or propagator for the composite system in unbounded domain as well as domains confined with reflecting, absorbing, and mixed boundaries. For the case with reflecting confinement in the absence of bias, the steady-state probability shows a step-like behavior for the Type A dynamics, while it is uniform for the Type B dynamics. We also derive explicit expressions for the first-passage probability and the mean first-passage time, and compare the hitting time dependence to a single target. Finally, considering the continuous-space continuous-time limit of the propagator, we obtain the boundary conditions at the interface. At the interface, while the flux is the same, the probability density is discontinuous for Type A and is continuous for Type B. For the latter we derive a generalized version of the so-called leather boundary condition in the appropriate limit.Comment: Submitted version: 30 pages, 8 figures. Accepted for publication in J. Stat. Mech.: Theory Ex

Closed-form solutions to the dynamics of confined biased lattice random walks in arbitrary dimensions

Biased lattice random walks (BLRW) are used to model random motion with drift in a variety of empirical situations in engineering and natural systems such as phototaxis, chemotaxis or gravitaxis. When motion is also affected by the presence of external borders resulting from natural barriers or experimental apparatuses, modelling biased random movement in confinement becomes necessary. To study these scenarios, confined BLRW models have been employed but so far only through computational techniques due to the lack of an analytic framework. Here, we lay the groundwork for such an analytical approach by deriving the Green's functions, or propagators, for the confined BLRW in arbitrary dimensions and arbitrary boundary conditions. By using these propagators we construct explicitly the time dependent first-passage probability in one dimension for reflecting and periodic domains, while in higher dimensions we are able to find its generating function. The latter is used to find the mean first-passage passage time for a $d$-dimensional box, $d$-dimensional torus or a combination of both. We show the appearance of surprising characteristics such as the presence of saddles in the spatio-temporal dynamics of the propagator with reflecting boundaries, bimodal features in the first-passage probability in periodic domains and the minimisation of the mean first-return time for a bias of intermediate strength in rectangular domains. Furthermore, we quantify how in a multi-target environment with the presence of a bias shorter mean first-passage times can be achieved by placing fewer targets close to boundaries in contrast to many targets away from them

Extreme Value Statistics and Arcsine Laws of Brownian Motion in the Presence of a Permeable Barrier

The Arcsine laws of Brownian motion are a collection of results describing three different statistical quantities of one-dimensional Brownian motion: the time at which the process reaches its maximum position, the total time the process spends in the positive half-space and the time at which the process crosses the origin for the last time. Remarkably the cumulative probabilities of these three observables all follows the same distribution, the Arcsine distribution. But in real systems, space is often heterogeneous, and these laws are likely to hold no longer. In this paper we explore such a scenario and study how the presence of a spatial heterogeneity alters these Arcsine laws. Specifically we consider the case of a thin permeable barrier, which is often employed to represent diffusion impeding heterogeneities in physical and biological systems such as multilayer electrodes, electrical gap junctions, cell membranes and fragmentation in the landscape for dispersing animals. Using the Feynman-Kac formalism and path decomposition techniques we are able to find the exact time-dependence of the probability distribution of the three statistical quantities of interest. We show that a permeable barrier has a large impact on these distributions at short times, but this impact is less influential as time becomes long. In particular, the presence of a barrier means that the three distributions are no longer identical with symmetry about their means being broken. We also study a closely related statistical quantity, namely, the distribution of the maximum displacement of a Brownian particle and show that it deviates significantly from the usual half-Gaussian form

Particle-Environment Interactions In Arbitrary Dimensions: A Unifying Analytic Framework To Model Diffusion With Inert Spatial Heterogeneities

Abridged abstract: Inert interactions between randomly moving entities and spatial disorder play a crucial role in quantifying the diffusive properties of a system. These interactions affect only the movement of the entities, and examples range from molecules advancing along dendritic spines to anti-predator displacements of animals due to sparse vegetation. Despite the prevalence of such systems, a general framework to model the movement explicitly in the presence of spatial heterogeneities is missing. Here, we tackle this challenge and develop an analytic theory to model inert particle-environment interactions in domains of arbitrary shape and dimensions. We use a discrete space formulation which allows us to model the interactions between an agent and the environment as perturbed dynamics between lattice sites. Interactions from spatial disorder, such as impenetrable and permeable obstacles or regions of increased or decreased diffusivity, as well as many others, can be modelled using our framework. We provide exact expressions for the generating function of the occupation probability of the diffusing particle and related transport quantities such as first-passage, return and exit probabilities and their respective means. We uncover a surprising property, the disorder indifference phenomenon of the mean first-passage time in the presence of a permeable barrier in quasi-1D systems. We demonstrate the widespread applicability of our formalism by considering three examples that span across scales and disciplines. (1) We explore an enhancement strategy of transdermal drug delivery. (2) We associate the disorder with a decision-making process of an animal to study thigmotaxis. (3) We illustrate the use of spatial heterogeneities to model inert interactions between particles by modelling the search for a promoter region on the DNA by transcription factors during gene transcription

Localization transition induced by learning in random searches

We solve an adaptive search model where a random walker or L\'evy flight stochastically resets to previously visited sites on a $d$-dimensional lattice containing one trapping site. Due to reinforcement, a phase transition occurs when the resetting rate crosses a threshold above which non-diffusive stationary states emerge, localized around the inhomogeneity. The threshold depends on the trapping strength and on the walker's return probability in the memoryless case. The transition belongs to the same class as the self-consistent theory of Anderson localization. These results show that similarly to many living organisms and unlike the well-studied Markovian walks, non-Markov movement processes can allow agents to learn about their environment and promise to bring adaptive solutions in search tasks.Comment: 5 pages, 5 figures + 4 pages of Supplemental Information. Accepted in Physical Review Letter

Bumblebees can discriminate between scent-marks deposited by conspecifics

Bumblebees secrete a substance from their tarsi wherever they land, which can be detected by conspecifics. These secretions are referred to as scent-marks, which bumblebees are able to use as social cues. Although it has been found that bumblebees can detect and associate scent-marks with rewarding or unrewarding flowers, their ability at discriminating between scent-marks from bumblebees of differing relatedness is unknown. We performed three separate experiments with bumblebees (Bombus terrestris), where they were repeatedly exposed to rewarding and unrewarding artificial flowers simultaneously. Each flower type carried scent-marks from conspecifics of differing relatedness or were unmarked. We found that bumblebees are able to distinguish between 1. Unmarked flowers and flowers that they themselves had scent-marked, 2. Flowers scent-marked by themselves and flowers scent-marked by others in their nest (nestmates), and 3. Flowers scent-marked by their nestmates and flowers scent-marked by non-nestmates. The bumblebees found it more difficult to discriminate between each of the flower types when both flower types were scent-marked. Our findings show that bumblebees have the ability to discriminate between scent-marks of conspecifics, which are potentially very similar in their chemical composition, and they can use this ability to improve their foraging success