arXiv subjects

Paul C Bressloff

Publications and source records attributed to Paul C Bressloff.

At least 19 recordsLinked to original sources

arXiv2026

Global resetting and emergent correlations: exit statistics in an interval

There is considerable current interest in the emergence of statistical correlations within a population of otherwise non-interacting Brownian particles subject to a common fluctuating environment or drive. Examples include global stochastic resetting, switching confining potentials, fluctuating diffusivities, and stochastically gated boundaries. Most studies have focused on the analytical structure of the stationary joint probability density (assuming it exists). In this paper, we extend previous work on the exit statistics of multiple particles in stochastically gated domains to the case of global resetting in an interval with absorbing boundaries at both ends. First, we use a generalised Itô's lemma to derive a hierarchy of boundary value problems (BVPs) for the joint splitting probability that all particles exit from the same end of the interval. The BVPs form a nested sequence with respect to the initial number of particles $M$. We explicitly solve the BVP for a pair of particles ($M=2$) and use this to illustrate the emergence of pairwise correlations. Second, we show how the BVP for the splitting probability of $M$ Brownian particles can be mapped onto the $M$th order moment equation of a stochastic diffusion equation with resetting. We thus establish a general mathematical framework to study exit problems for globally-driven particle systems.

cond-mat.stat-mech↗

arXiv2026

Renewal theory for a run-and-tumble particle with stochastic resetting and a sticky boundary

We consider a run-and-tumble particle (RTP) with stochastic resetting confined to the half line $[0,\infty)$ with a sticky boundary at $x=0$. In the bulk the RTP tumbles at a constant rate $α>0$ between velocity states $\pm v$ with $v>0$ and randomly resets to its initial position and orientation $(x_0,k_0)\in(\mathbb{R}^+,\pm)$. When the RTP reaches the target at $x=0$ it attaches to the boundary for a random waiting time before either detaching and continuing to navigate the bulk domain or (permanently) entering the target. These events are the analogs of adsorption, desorption, and absorption of a particle by a partially reactive surface in physical chemistry. We use renewal theory to characterize the particle trajectory in terms of successive binding events under two distinct desorption protocols: via resetting to $(x_0,k_0)$ and via continuous movement from $x=0$ with velocity $+v$. First we derive the nonequilibrum stationary state (NESS) in the case of no absorption and characterize the accumulation at the boundary. Second, we compute the mean first passage time (MFPT) statistics. In addition to observing the usual unimodal dependence of the MFPT on bulk resetting, both the NESS and MFPT strongly depend on the initial orientation $k_0$ and the desorption protocol. For instance, if the initial orientation is toward the boundary, we find that the desorption-induced resetting protocol can reduce the MFPT more effectively than the non-resetting desorption protocol. We also show how matching the desorption kinetics with the bulk resetting or tumbling rate introduces a trade-off between minimizing the adsorption and absorption times. In this setting we find that the desorption protocol which minimizes the absorption MFPT for a given set of parameters is almost always the opposite of that favored when desorption and bulk kinetics are not the same.

cond-mat.stat-mech↗

arXiv2026

Renewal theory for Brownian motion with stochastically gated targets

There are a wide range of first passage time (FPT) problems in the physical and life sciences that can be modelled in terms of a Brownian particle binding to a reactive surface (absorption). However, prior to absorption, the particle may undergo several rounds of surface attachment (adsorption), detachment (desorption) and diffusion. Alternatively, the surface may be stochastically gated so that absorption can only occur when the gate is open. In both cases one can view each return to the surface as a renewal event. In this paper we develop a renewal theory for stochastically gated FPT problems along analogous lines to previous work on adsorption/desorption processes. We proceed by constructing a renewal equation that relates the joint probability density for particle position and the state of a gate (or multiple gates) to the probability density and FPT density for a totally absorbing (non-gated) boundary. This essentially decomposes sample paths into an alternating sequence of bulk diffusion and instantaneous adsorption/desorption events, which is terminated when adsorption coincides with an open gate. Through a variety of examples, we show how renewal theory provides a general mathematical framework for incorporating stochastic gating into FPT problems.

cond-mat.stat-mech↗

arXiv2026

Renewal theory for Brownian motion across a stochastically gated interface

Stochastically gated interfaces play an important role in a variety of cellular diffusion processes. Examples include intracellular transport via stochastically gated ion channels and pores in the plasma membrane of a cell, intercellular transport between cells coupled by stochastically gated gap junctions, and oxygen transport in insect respiration. Most studies of stochastically-gated interfaces are based on macroscopic models that track the particle concentration averaged with respect to different realisations of the gate dynamics. In this paper we use renewal theory to develop a probabilistic model of single-particle Brownian motion (BM) through a stochastically gated interface. We proceed by constructing a renewal equation for 1D BM with an interface at the origin, which effectively sews together a sequence of BMs on the half-line with a totally absorbing boundary at $x=0$. Each time the particle is absorbed, the stochastic process is immediately restarted according to the following rule: if the gate is closed then BM restarts on the same side of the interface, whereas if the gate is open then BM restarts on either side of the interface with equal probability. In order to ensure that diffusion restarts in a state that avoids immediate re-absorption. we assume that whenever the particle reaches the interface it is instantaneously shifted a distance $ε$ from the origin. We explicitly solve the renewal equation for $ε>0$ and show how the solution of a corresponding forward Kolmogorov equation is recovered in the limit $ε\rightarrow 0$. However, the renewal equation provides a more general mathematical framework by explicitly separating the first passage time problem of detecting the gated interface (absorption) and the subsequent rule for restarting BM. We conclude by extending the theory to higher-dimensional interfaces.

cond-mat.stat-mech↗

arXiv2025

Run-and-tumble particle with diffusion: boundary local times and the zero-diffusion limit

Incorporating boundary conditions into stochastic models of passive or active particle motion is usually implemented at the level of the associated forward or backward Kolmogorov equation, whose solution determines the probability distribution of sample paths. In order to write down the corresponding stochastic differential equation (SDE) that generates the individual sample paths, it is necessary to introduce a Brownian functional that keeps track of the boundary-particle contact time. We previously constructed the SDE for a (non-diffusing) run-and-tumble particle (RTP) on the half-line with either a reflecting or sticky wall at $x=0$. In this paper we extend the theory to include the effects of diffusion. One of the non-trivial consequences of combining drift-diffusion with tumbling is that the zero diffusion limit $D\rightarrow 0$ is singular in the sense that the number of boundary conditions is doubled when $D>0$. We use stochastic calculus to derive the forward Kolmogorov equation for two distinct boundary conditions that reduce, respectively, to non-sticky and sticky boundary conditions in the zero-diffusion limit. In the latter case, it is necessary to include a boundary layer in a neighbourhood of the wall and use singular perturbation theory. We also treat the wall as partially absorbing by assuming that the particle is absorbed when the amount of boundary-particle contact time (discrete or continuous local time) exceeds a quenched random threshold. Finally, we analyse the survival probability and corresponding FPT density for absorption at a non-sticky wall by deriving the corresponding backward Kolmogorov equation

cond-mat.stat-mech↗

arXiv2025

Stochastic 1D search-and-capture as a G/M/c queueing model

We study the accumulation of resources within a target due to the interplay between continual delivery, driven by 1D stochastic search processes, and sequential consumption. The assumption of sequential consumption is key because it changes the commonly used $G/M/\infty$ queue to a $G/M/c$ queue. Combining the theory of $G/M/c$ queues with the theory of first-passage times, we derive general conditions for the search process to ensure that the number of resources within the queue converges to a steady state and compute explicit expressions for the mean and variance of the number of resources within the queue at steady state. We then compare the performance of the $G/M/c$ queue with that of the $G/M/\infty$ queue for an increasing number of servers. We extend the model to consider two competing targets and show that, under specific scenarios, an additional target is beneficial to the original target. Finally, we study the effects of multiple searchers. Using renewal theory, we numerically compute the inter-arrival time density for $M$ searchers in the Laplace space, which allows us to exploit the explicit expressions for the steady-state statistics of the number of resources within $G/M/1$ and $G/M/\infty$ queues, and compare their behaviour with different numbers of searchers. Overall, the $G/M/c$ queue shows a tighter dependence on the configuration of the search process than the $G/M/\infty$ queue does.

cond-mat.stat-mech↗

arXiv2025

Macroscopic fluctuation theory and the absorption of Brownian particles by partially reactive targets

We use macroscopic fluctuation theory (MFT) to analyse current fluctuations in a non-interacting Brownian gas with one or more partially absorbing targets within a bounded domain $Ω\subset \R^d$. We proceed by coarse-graining a generalised Dean-Kawasaki equation with Robin boundary conditions at the target surfaces. The exterior surface $\partial Ω$ is maintained at a constant density $\owp$. We first derive MFT equations for the optimal noise-induced path for a single target under a saddle-point approximation of the associated path integral action. We then obtain the Gaussian distribution characterising small current fluctuations by linearising the MFT equations about the corresponding deterministic or noise-averaged system and solving the resulting stationary equations. The Robin boundary conditions are handled using the spectrum of a Dirichlet-to-Neumann operator defined on the target surface. We illustrate the theory by considering the finite interval and a circular annulus. In both cases we determine how the variance of the current depends on the rate of absorption $κ$. Finally, we extend our analysis to multiple partially absorbing targets. First, we obtain the general result that, in the case of partially absorbing targets ($0<κ<\infty$), the covariance matrix for current fluctuations supports cross correlations even in the absence of particle interactions. (These cross-correlations vanish in the totally absorbing limit $κ\rightarrow \infty$.) We then explicitly calculate the covariance matrix for circular targets in a 2D domain by assuming that the targets are much smaller than the characteristic size $L$ of the domain $Ω$ and applying methods from singular perturbation theory.

cond-mat.stat-mech↗

arXiv2025

Asymptotic analysis of the 2D narrow-capture problem for partially accessible targets

In this paper we use singular perturbation theory to solve the 2D narrow capture problem for a set of partially accessible targets $\calU_k$, $k=1,\ldots,N$, in a bounded domain $Ω\subset \R^2$. In contrast to previous models of narrow capture, we assume that when a searcher finds a target by attaching to the partially adsorbing surface $\partial \calU_k$ it does not have immediate access to the resources within the target interior. Instead, the searcher remains attached to the surface for a random waiting time $τ$, after which it either gains access to the resources within ({\em surface absorption}) or detaches and continues its search process ({\em surface desorption)}. We also consider two distinct desorption scenarios -- either the particle continues its search from the point of desorption or rapidly returns to its initial search position. We formulate the narrow capture problem in terms of a set of renewal equations that relate the probability density and target flux densities for absorption to the corresponding quantities for irreversible adsorption. The renewal equations, which effectively sew together successive rounds of adsorption and desorption prior to the final absorption event, provide a general probabilistic framework for incorporating non-Markovian models of desorption/absorption and different search scenarios following desorption. We solve the general renewal equations in two stages. First, we calculate the Laplace transformed target fluxes for irreversible adsorption by solving a Robbin boundary value problem (BVP) in the small-target limit using matched asymptotic analysis. We then use the inner solution of the BVP to solve the corresponding Laplace transformed renewal equations for non-Markovian desorption/absorption, which leads to explicit Neumann series expansions of the corresponding target fluxes.

cond-mat.stat-mech↗

arXiv2025

Slow-fast systems with stochastic resetting

In this paper we explore the effects of instantaneous stochastic resetting on a planar slow-fast dynamical system of the form $\dot{x}=f(x)-y$ and $\dot{y}=ε(x-y)$ with $0<ε\ll 1$. We assume that only the fast variable $x(t)$ resets to its initial state $x_0$ at a random sequence of times generated from a Poisson process of rate $r$. Fixing the slow variable, we determine the parameterized probability density $p(x,t|y)$, which is the solution to a modified Liouville equation. We then show how for $r\gg ε$ the slow dynamics can be approximated by the averaged equation $dy/dτ=\E[x|y]-y$ where $τ=εt$, $\E[x|y]=\int x p^*(x|y)dx$ and $p^*(x|y)=\lim_{t\rightarrow \infty}p(x,t|y)$. We illustrate the theory for $f(x)$ given by the cubic function of the FitzHugh-Nagumo equation. We find that the slow variable typically converges to an $r$-dependent fixed point $y^*$ that is a solution of the equation $y^*=\E[x|y^*]$. Finally, we numerically explore deviations from averaging theory when $r=O(ε)$.

nlin.AO↗

arXiv2025

Encounter-based model of a run-and-tumble particle with stochastic resetting

In this paper we analyze the effects of stochastic resetting on an encounter-based model of an unbiased run-and-tumble particle (RTP) confined to the half-line $[0,\infty)$ with a partially absorbing wall at $x=0$. The RTP tumbles at a constant rate $α$ between the velocity states $\pm v$ with $v>0$. Absorption occurs when the number of collisions with the wall (discrete local time) exceeds a randomly generated threshold $\widehat{\ell}$ with probability distribution $Ψ(\ell)$. The extended RTP model has three state variables, namely, particle position $X(t)\in [0,\infty)$, the velocity direction $σ(t)\in\{-1, 1\}$, and the discrete local time $L(t)\in {\mathbb N}$. We initially assume that only $X(t)$ and $σ(t)$ reset at a Poisson rate $r$, whereas $L(t)$ is not changed. This implies that resetting is not governed by a renewal process. We use the stochastic calculus of jump processes to derive an evolution equation for the joint probability distribution of the triplet $(X(t),σ(t),L(t))$. This is then used to calculate the mean first passage time (MFPT) by performing a discrete Laplace transform of the evolution equation with respect to the local time. We thus find that the MFPT's only dependence on the distribution $Ψ$ is via the mean local time threshold. We also identify parameter regimes in which the MFPT is a unimodal function of both the resetting and tumbling rates. Finally, we consider conditions under which resetting is given by a renewal process and show how the MFPT in the presence of local time resetting depends on the full statistics of $Ψ$.

cond-mat.stat-mech↗

arXiv2025

Diffusion-mediated adsorption versus absorption at partially reactive targets: a renewal approach

Renewal theory is finding increasing applications in non-equilibrium statistical physics. One example relates the probability density and survival probability of a Brownian particle or an active run-and-tumble particle with stochastic resetting to the corresponding quantities without resetting. A second example is so-called snapping out Brownian motion, which sews together diffusions on either side of an impermeable interface to obtain the corresponding stochastic dynamics across a semi-permeable interface. A third example relates diffusion-mediated surface adsorption-desorption (reversible adsorption) to the case of irreversible adsorption. In this paper we apply renewal theory to diffusion-mediated adsorption processes in which an adsorbed particle may be permanently removed (absorbed) prior to desorption. We construct a pair of renewal equations that relate the probability density and first passage time (FPT) density for absorption to the corresponding quantities for irreversible adsorption. We first consider the example of diffusion in a finite interval with a partially reactive target at one end. We use the renewal equations together with an encounter-based formalism to explore the effects of non-Markovian adsorption/desorption on the moments and long-time behaviour of the FPT density for absorption. We then analyse the corresponding renewal equations for a partially reactive semi-infinite trap and show how the solutions can be expressed in terms of a Neumann series expansion. Finally, we construct higher-dimensional versions of the renewal equations and derive general expression for the FPT density using spectral decompositions.

cond-mat.stat-mech↗

arXiv2024

Kuramoto model with stochastic resetting and coupling through an external medium

Most studies of collective phenomena in oscillator networks focus on directly coupled systems as exemplified by the classical Kuramoto model. However, there are growing number of examples in which oscillators interact indirectly via a common external medium, including bacterial quorum sensing (QS) networks, pedestrians walking on a bridge, and centrally coupled lasers. In this paper we analyze the effects of stochastic phase resetting on a Kuramoto model with indirect coupling. All the phases are simultaneously reset to their initial values at a random sequence of times generated from a Poisson process. On the other hand, the external environmental state is not reset. We first derive a continuity equation for the population density in the presence of resetting and show how the resulting density equation is itself subject to stochastic resetting. We then use an Ott-Antonsen (OA) ansatz to reduce the infinite-dimensional system to a four-dimensional piecewise deterministic system with subsystem resetting. The latter is used to explore how synchronization depends on a cell density parameter. (In bacterial QS this represents the ratio of the population cell volume and the extracellular volume.) At high densities we recover the OA dynamics of the classical Kuramoto model with global resetting. On the other hand, at low densities, we show how subsystem resetting has a major effect on collective synchronization, ranging from noise-induced transitions to slow/fast dynamics.

cond-mat.stat-mech↗

arXiv2024

Stochastic calculus of run-and-tumble motion: an applied perspective

The run-and-tumble particle (RTP) is one of the simplest examples of an active particle in which the direction of constant motion randomly switches. In the one-dimensional (1D) case this means switching between rightward and leftward velocities. Most theoretical studies of RTPs are based on the analysis of the Chapman-Kolmogorov (CK) differential equation describing the evolution of the joint probability densities for particle position and velocity state. In this paper we develop an alternative, probabilistic framework of 1D RTP motion based on the stochastic calculus of Poisson and diffusion processes. In particular, we show how a generalisation of Itô's lemma provides a direct link between sample paths of an RTP and the underlying CK equation. This allows us to incorporate various non-trivial extensions in a systematic fashion, including stochastic resetting and partially absorbing sticky boundaries. The velocity switching process and resetting process are represented by a pair of independent Poisson processes, whereas a sticky boundary is modelled using a boundary layer. We then use the probabilistic formulation to calculate stochastic entropy production along individual trajectories of an RTP, and show how the corresponding Gibbs-Shannon entropy is recovered by averaging over the ensemble of sample paths. Finally, we extend the probabilistic framework to a population of RTPs and use this to explore the effects of global resetting.

cond-mat.stat-mech↗

arXiv2024

Cellular diffusion processes in singularly perturbed domains

There are many processes in cell biology that can be modeled in terms of particles diffusing in a two-dimensional (2D) or three-dimensional (3D) bounded domain $Ω\subset \R^d$ containing a set of small subdomains or interior compartments $\calU_j$, $j=1,\ldots,N$ (singularly-perturbed diffusion problems). The domain $Ω$ could represent the cell membrane, the cell cytoplasm, the cell nucleus or the extracellular volume, while an individual compartment could represent a synapse, a membrane protein cluster, a biological condensate, or a quorum sensing bacterial cell. In this review we use a combination of matched asymptotic analysis and Green's function methods to solve a general type of singular boundary value problems (BVP) in 2D and 3D, in which an inhomogeneous Robin condition is imposed on each interior boundary $\partial \calU_j$. This allows us to incorporate a variety of previous studies of singularly perturbed diffusion problems into a single mathematical modeling framework. We mainly focus on steady-state solutions and the approach to steady-state, but also highlight some of the current challenges in dealing with time-dependent solutions and randomly switching processes

math.AP↗

arXiv2024

Entropy production for diffusion processes across a semipermeable interface

The emerging field of stochastic thermodynamics extends classical ideas of entropy, heat and work to non-equilibrium systems. One notable finding is that the second law of thermodynamics typically only holds after taking appropriate averages with respect to an ensemble of stochastic trajectories. The resulting average rate of entropy production then quantifies the degree of departure from thermodynamic equilibrium. In this paper we investigate how the presence of a semipermeable interface increases the average entropy production of a single diffusing particle. Starting from the Gibbs-Shannon entropy for the particle probability density, we show that a semipermeable interface or membrane $\calS$ increases the average rate of entropy production by an amount that is equal to the product of the flux through the interface and the logarithm of the ratio of the probability density on either side of the interface, integrated along $\calS$. The entropy production rate thus vanishes at thermodynamic equilibrium, but can be nonzero during the relaxation to equilibrium, or if there exists a nonzero stationary equilibrium state (NESS). We illustrate the latter using the example of diffusion with stochastic resetting on a circle, and show that the average rate of interfacial entropy production is a nonmonotonic function of the resetting rate and the permeability. Finally, we give a probabilistic interpretation of the interfacial entropy production rate using so-called snapping out Brownian motion. This also allows us to construct a stochastic version of entropy production.

cond-mat.stat-mech↗

arXiv2024

Global density equations for interacting particle systems with stochastic resetting: from overdamped Brownian motion to phase synchronization

A wide range of phenomena in the natural and social sciences involve large systems of interacting particles, including plasmas, collections of galaxies, coupled oscillators, cell aggregations, and economic ``agents'. Kinetic methods for reducing the complexity of such systems typically involve the derivation of nonlinear partial differential equations for the corresponding global densities. In recent years there has been considerable interest in the mean field limit of interacting particle systems with long range interactions. Two major examples are interacting Brownian particles in the overdamped regime and the Kuramoto model of coupled phase oscillators. In this paper we analyze these systems in the presence of local or global stochastic resetting, where the position or phase of each particle independently or simultaneously resets to its original value at a random sequence of times generated by a Poisson process. In each case we derive the Dean-Kawasaki (DK) equation describing hydrodynamic fluctuations of the global density, and then use a mean field ansatz to obtain the corresponding nonlinear McKean-Vlasov (MV) equation in the thermodynamic limit. In particular, we show how the MV equation for global resetting is driven by a Poisson shot noise process, reflecting the fact that resetting is common to all of the particles and thus induces correlations that cannot be eliminated by taking a mean field limit. We then investigate the effects of local and global resetting on nonequilibrium stationary solutions of the macroscopic dynamics and, in the case of the Kuramoto model, the reduced dynamics on the Ott-Antonsen manifold.

cond-mat.stat-mech↗

arXiv2023

A generalized Dean-Kawasaki equation for an interacting Brownian gas in a partially absorbing medium

The Dean-Kawasaki (DK) equation is a stochastic partial differential equation (SPDE) for the global density $ρ$ of a gas of $N$ over-damped Brownian particles. In the thermodynamic limit $N\rightarrow \infty$ with weak pairwise interactions, the expectation ${\mathbb E}[ρ]$ converges in distribution to the solution of a McKean-Vlasov (MV) equation. In this paper we derive a generalized DK equation for an interacting Brownian gas in a partially absorbing one-dimensional medium. In the case of the half-line with a totally reflecting boundary at $x=0$, the generalized DK equation is an SPDE for the joint global density $ρ(x,\ell,t)=N^{-1}\sum_{j=1}^Nδ(x-X_j(t))δ(\ell-L_j(t))$, where $X_j(t)$ and $L_j(t)$ denote the position and local time of the $j$th particle, respectively. Assuming the DK equation has a well-defined mean field limit, we derive the MV equation on the half-line with a reflecting boundary, and analyze stationary solutions for a Curie-Weiss (quadratic) interaction potential. We then use an encounter-based approach to develop the analogous theory for a partially absorbing boundary at $x=0$. Each particle is independently absorbed when its local time $L_j(t)$ exceeds a random threshold $\widehat{\ell}_j$ with probability distribution $Ψ(\ell)=¶[\widehat{\ell}_j>\ell]$. The joint global density is now summed over the set of particles that have not yet been absorbed, and expectations are taken with respect to the Gaussian noise and the random thresholds $\widehat{\ell}_j$. Extensions to finite intervals and partially absorbing traps are also considered.

cond-mat.stat-mech↗

arXiv2023

Asymptotic analysis of particle cluster formation in the presence of anchoring sites

The aggregation or clustering of proteins and other macromolecules plays an important role in the formation of large-scale molecular assemblies within cell membranes. Examples include lipid rafts and postsynaptic domains (PSDs). PSDs are rich in scaffolding proteins that transiently trap transmembrane neurotransmitter receptors, thus localizing them at specific spatial positions. Hence, PSDs play a key role in determining the strength of synaptic connections and their regulation during learning and memory. Recently, a 2D diffusion-mediated aggregation model of PSD formation has been developed in which the spatial locations of the clusters are determined by a set of fixed anchoring sites. The system is kept out of equilibrium by the recycling of particles between the cell membrane and interior. This results in a stationary distribution consisting of multiple clusters, whose average size can be determined using an effective mean-field description of the particle concentration around each anchored cluster. In this paper, we derive corrections to mean-field theory using asymptotic methods, assuming that the clusters are well separated and much smaller than the membrane surface. The analysis proceeds by matching inner solutions around each cluster with and an outer region where mean-field interactions occur. This generates an asymptotic expansion of the particle concentration that sums over all logarithmic terms involving the small parameter $ν=-1/\ln ε$, where $ε$ specifies the relative size of the clusters. This is then used to derive an expression for the radius of each cluster, which includes higher-order corrections to mean-field theory that depend on the positions of the clusters and the boundary of the domain. Finally, motivated by a recent study of light-activated protein oligomerization in cells, we develop the analogous theory for cluster formation in a 2D domain.

cond-mat.soft↗