arXiv subjects

Paul C. Bressloff

Publications and source records attributed to Paul C. Bressloff.

At least 19 recordsLinked to original sources

arXiv2026

Universality in the target arrival statistics of non-conservative search processes

Stochastic search processes in which agents are continuously inserted into and removed from a target search domain are fundamental to a wide class of physical and artificial systems. The theory of such non-conservative search processes is, however, much less developed than for search processes with a fixed number of agents. Recent work on the non-conservative case has characterized the fastest first-passage time (FPT) to a target, after which the search process is terminated. Here we consider the complementary problem of continuous arrival statistics to a target. In particular, we derive the full time-dependent distribution of target arrivals, which we use to show that the inter-arrival time distribution is exactly exponential in steady-state regardless of the underlying search process. Thus, counterintuitively, the arrival statistics of non-conservative search are both universal and much simpler than sequential search-and-capture processes involving a fixed number of agents. Our methods also yield an exact expression for the fastest FPT distribution with time-dependent insertion, thereby generalizing previous work on fixed-rate insertion for non-conservative search. Altogether these results have major implications for target resource accumulation, wherein the delivery of resources is counter-balanced by their downstream consumption.

cond-mat.stat-mech↗

arXiv2026

2D target search with boundary-induced resetting

Most studies of stochastic search processes with resetting focus on spontaneous resetting events that occur at a random sequence of times typically generated by a Poisson process. One of the major consequences of spontaneous resetting is that it yields a finite mean first passage time (MFPT) in unbounded domains, resulting in an optimal resetting rate that minimises the MFPT. An optimal resetting rate can also occur in bounded domains provided that the reset point is not too far from the target. Recently, there has been growing interest in event-based rather than spontaneous resetting, where resetting is triggered when a specific threshold is reached. Identifying the threshold with a physical non-target boundary then leads to so-called boundary-induced resetting. In this paper, we explore the effects of boundary-induced resetting on single-particle diffusive search in a two-dimensional (2D) bounded domain $Ω$ containing one or more small interior targets. The interior target boundaries are totally absorbing, whereas the exterior boundary $\partial Ω$ is sticky. That is, whenever the particle reaches a point on $\partial Ω$ it remains attached for a random waiting time $τ$ after which it immediately resets to a fixed interior point $\x_0\in Ω$. Using renewal theory, matched asymptotic analysis and Green's functions we calculate the unconditional MFPT for absorption by any of the targets and compare this with the corresponding MFPT of a search process on the same domain with a reflecting surface $\partial Ω$ and no resetting. We show that boundary-induced resetting only reduces the MFPT relative to the no resetting case if the searcher's reset point $\x_0$ is sufficiently close to one of the targets.

cond-mat.stat-mech↗

arXiv2026

Slow-fast dynamics of the McKean model with stochastic resetting and diffusion

In this paper we investigate the combined effects of stochastic resetting and diffusion on a slow--fast dynamical system given by the piecewise-linear McKean model. That is, the fast variable $v$ is subject to Gaussian white noise with effective diffusivity $D$ and is reset to a fixed value $v_r$ at a random sequence of times generated from a Poisson process with rate $r$. Assuming that resetting occurs on the fast timescale, we freeze the slow variable $w$ under an adiabatic approximation and determine the resulting non-equilibrium stationary state (NESS) of the fast variable as the solution of a modified Fokker--Planck equation. We show that the NESS can be expressed in terms of parabolic cylinder functions, whose asymptotic behavior allows us to recover the corresponding NESS without diffusion in the small-diffusion limit. The NESS is used to derive an averaged equation for the slow dynamics, whose solution converges to a stable fixed point $w^*$ that depends on $D$, $r$ and $v_r$. This fixed point effectively determines the long-time behaviour of the full system. Finally, we analyze the regime in which resetting occurs on the same timescale as the slow variable. We show how the slow variable now undergoes noisy oscillations due to resetting-induced switching between branches of the fast nullcline and derive the corresponding NESS for $w$ in the non-diffusive case.

cond-mat.stat-mech↗

arXiv2026

Approach to Steady-State in Nested Resetting Processes

We characterise the approach to nonequilibrium steady state in nested resetting processes by deriving accumulation times for the system, which quantify the effective first-passage time for the local establishment of steady state. For equal resetting rates, we obtain closed-form expressions showing that relaxation propagates as a wavefront with a delay that increases linearly along the resetting chain. We then extend this analysis to heterogeneous resetting rates, deriving exact steady-state distributions, spatial moments and accumulation times for both degenerate and non-degenerate resetting rates. We show that relaxation to steady state is ultimately governed by the minimum resetting rate in the system and, in degenerate systems, its multiplicity. These results provide a comprehensive analytical description of first-passage to nonequilibrium steady state in hierarchically coupled stochastic resetting systems, with potential applications to relaxation and transport processes in biological systems.

cond-mat.stat-mech↗

arXiv2026

Persistence, resetting, and first-passage times of an active Ornstein--Uhlenbeck particle

We study the first-passage time statistics of an active Ornstein--Uhlenbeck particle on a finite one-dimensional interval with an absorbing boundary at one end and a reflecting boundary at the other, subject to stochastic resetting. In the weak-activity regime, we develop a perturbative solution of the Fokker--Planck equation and use a renewal framework to obtain analytical expressions for the first-passage-time distribution and its low-order moments. In the absence of resetting, activity can either increase or decrease the mean first-passage time, depending on the relation between the persistence time and the diffusive time to absorption. When resetting is introduced, we determine both the onset of beneficial resetting and the finite range of resetting rates for which resetting lowers the mean first-passage time of the active system. We further show that the resetting transition depends on the velocity resetting protocol. Finally, by comparing the active particle with resetting to the passive Brownian particle without resetting, we construct a phase diagram that identifies the regions of parameter space in which activity and resetting together reduce the mean first-passage time.

cond-mat.stat-mech↗

arXiv2026

Resetting-induced instability in queues fed by a search process in an interval

Proper management of resources whose arrival and consumption are subject to environmental randomness is an intrinsic process in both natural and artificial systems. This phenomenon can be modeled as a queuing process whose arrival distribution is determined by a search process with stochastic resetting. When the queuing system has a limited number of servers and the search process occurs within a bounded domain, the dynamics of expediting or delaying the search through stochastic resetting interact with the long-term dynamics of the number of resources in the queue. We combine results from queuing theory with those from search processes with stochastic resetting in a bounded domain to obtain regions of the parameter space of the search process that ensure convergence of the number of resources in the queue to a steady state. Furthermore, we find a threshold resetting rate at which the effects of stochastic resetting shift from reducing convergence regions to expanding them. Finally, we demonstrate that this threshold value grows exponentially with the number of servers, making it harder for stochastic resetting to improve the convergence of the queueing system.

cond-mat.stat-mech↗

arXiv2025

Random search with stochastic resetting: when finding the target is not enough

In this paper we consider a random search process with stochastic resetting and a partially accessible target $\calU$. That is, when the searcher finds the target by attaching to its surface $\partial \calU$ it does not have immediate access to the resources within the target interior. After a random waiting time, the searcher either gains access to the resources within or detaches and continues its search process. We also assume that the searcher requires an alternating sequence of periods of bulk diffusion interspersed with local surface interactions before being able to attach to the surface. The attachment, detachment and target entry events are the analogs of adsorption, desorption and absorption of a particle by a partially reactive surface in physical chemistry. In applications to animal foraging, the resources could represent food or shelter while resetting corresponds to an animal returning to its home base. We begin by considering a Brownian particle on the half-line with a partially accessible target at the origin $x=0$. We calculate the non-equilibrium stationary state (NESS) in the case of reversible adsorption and obtain the corresponding first passage time (FPT) density for absorption when adsorption is only partially reversible. We then reformulate the stochastic process in terms of a pair of renewal equations that relate the probability density and FPT density for absorption in terms of the corresponding quantities for irreversible adsorption. The renewal equations allow us to incorporate non-Markovian models of absorption and desorption. They also provide a useful decomposition of quantities such as the mean FPT (MFPT) in terms of the number of desorption events and the statistics of the waiting time density. Finally, we consider various extensions of the theory, including higher-dimensional search processes and an encounter-based model of absorption.

cond-mat.stat-mech↗

arXiv2023

Global density equations for a population of actively switching particles

There are many processes in cell biology that can be modelled in terms of an actively switching particle. The continuous degrees of freedom evolve according to a hybrid stochastic differential equation (hSDE) whose drift term depends on a discrete internal or environmental state that switches according to a continuous time Markov chain. In this paper we derive global density equations for a population of non-interacting actively switching particles, either independently switching or subject to a common randomly switching environment. In the case of a random environment, we show that the global particle density evolves according to a hybrid stochastic partial differential equation (hSPDE). Averaging with respect to the Gaussian noise processes yields a hybrid partial differential equation (hPDE) for the one-particle density. We use the corresponding functional Chapman-Kolmogorov equation to derive moment equations for the one-particle density and show how a randomly switching environment induces statistical correlations. We also discuss the effects of particle interactions, which generate moment closure problems at both the hSPDE and hPDE levels. The former can be handled by taking a mean field limit, but the resulting hPDE is now a nonlinear functional of the one-particle density. We then develop the analogous constructions for independently switching particles. We introduce a discrete set of global densities that are indexed by the single-particle internal states. We derive an SPDE for the densities by taking expectations with respect to the switching process, and then use a slow/fast analysis to reduce the SPDE to a scalar stochastic Fokker-Planck equation in the fast-switching limit. We end by deriving path integrals for the global densities in the absence of interactions and relate this to recent studies of Brownian gases and run-and-tumble particles.

cond-mat.stat-mech↗

arXiv2023

Encounter-based model of a run-and-tumble particle II: absorption at sticky boundaries

In this paper we develop an encounter-based model of a run-and-tumble particle (RTP) confined to a finite interval $[0,L]$ with partially absorbing, sticky boundaries at both ends. We assume that the particle switches between two constant velocity states $\pm v$ at a rate $α$. Whenever the particle hits a boundary, it becomes stuck by pushing on the boundary until either a tumble event reverses the swimming direction or it is permanently absorbed. We formulate the absorption process by identifying the first passage time (FPT) for absorption with the event that the time $A(t)$ spent attached to either wall up to time $t$ (the occupation time) crosses some random threshold $\hat{A}$. Taking $Ψ({a})\equiv\mathbb{P}[\hat{A}\gt{a}]$ to be an exponential distribution, $Ψ[{a}]=e^{-κ_0a}$, we show that the joint probability density for particle position $X(t)$ and velocity state $σ(t)=\pm $ satisfies a well-defined boundary value problem (BVP) with $κ_0$ representing a constant absorption rate. The solution of this BVP determines the so-called occupation time propagator, which is the joint probability density for the triplet $(X(t),A(t),σ(t))$. The propagator is then used to incorporate more general models of absorption based on non-exponential (non-Markovian) distributions $Ψ(a)$. We illustrate the theory by calculating the mean FPT (MFPT) and splitting probabilities for absorption. We also show how our previous results for partially absorbing, non-sticky boundaries can be recovered in an appropriate limit. Absorption now depends on the number of collisions $\ell(t)$ of the RTP with the boundary. Finally, we extend the theory by taking absorption to depend on the individual occupation times at the two ends.

cond-mat.stat-mech↗

arXiv2023

Renewal equations for single-particle diffusion in multi-layered media

In this paper we develop a probabilistic model of single-particle diffusion in 1D multi-layered media by constructing a multi-layered version of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of reflected BM, each of which is restricted to a single layer. Each round of reflected BM is killed when the local time at one end of the layer exceeds an independent, exponentially distributed random variable. (The local time specifies the amount of time a reflected Brownian particle spends in a neighborhood of a boundary.) The particle then immediately resumes reflected BM in the same layer or the layer on the other side of the boundary with equal probability, and the process is iterated We proceed by constructing a last renewal equation for multi-layered snapping out BM that relates the full probability density to the probability densities of partially reflected BM in each layer. We then show how transfer matrices can be used to solve the Laplace transformed renewal equation, and prove that the renewal equation and corresponding multi-layer diffusion equation are equivalent. We illustrate the theory by analyzing the first passage time (FPT) problem for escape at the exterior boundaries of the domain. Finally, we use the renewal approach to incorporate a generalization of snapping out BM based on the encounter-based method for surface absorption; each round of reflected BM is now killed according to a non-exponential distribution for each local time threshold. This is achieved by considering a corresponding first renewal equation that relates the full probability density to the FPT densities for killing each round of reflected BM. We show that for certain configurations, non-exponential killing leads to an effective time-dependent permeability that is normalizable but heavy-tailed.

cond-mat.stat-mech↗

arXiv2022

Stochastically switching diffusion with partially reactive surfaces

In this paper we develop a hybrid version of the encounter-based approach to diffusion-mediated absorption at a reactive surface, which takes into account stochastic switching of a diffusing particle's conformational state. For simplicity, we consider a two-state model in which the probability of surface absorption depends on the current particle state and the amount of time the particle has spent in a neighborhood of the surface in each state. The latter is determined by a pair of local times $\ell_{n,t}$, $n=0,1$, which are Brownian functionals that keep track of particle-surface encounters over the time interval $[0,t]$. We proceed by constructing a differential Chapman-Kolmogorov equation for a pair of generalized propagators $P_n(\x,\ell_0,\ell_1,t)$, where $P_n$ is the joint probability density for the set $(\X_t,\ell_{0,t},\ell_{1,t})$ when $N_t=n$, where $\X_t$ denotes the particle position and $N_t$ is the corresponding conformational state. Performing a double Laplace transform with respect to $\ell_0,\ell_1$ yields an effective system of equations describing diffusion in a bounded domain $Ω$, in which there is switching between two Robin boundary conditions on $\partial Ω$. The corresponding constant reactivities are $κ_j=D z_j$, $j=0,1$, where $z_j$ is the Laplace variable corresponding to $\ell_j$ and $D$ is the diffusivity. Given the solution for the propagators in Laplace space, we construct a corresponding probabilistic model for partial absorption, which requires finding the inverse Laplace transform with respect to $z_0,z_1$. We illustrate the theory by considering diffusion of a particle on the half-line with the boundary at $x=0$ effectively switching between a totally reflecting and a partially absorbing state. Finally, we indicate how to extend the analysis to higher spatial dimensions using the spectral theory of Dirichlet-to-Neumann operators.

cond-mat.stat-mech↗

arXiv2022

Threshold surface reactions and local time resetting

In this paper we consider a threshold surface absorption mechanism for a particle diffusing in a domain containing a single target $\calU $. The target boundary $\partial \calU$ is taken to be a reactive surface that modifies an internal state $Z_t$ of the particle when in contact with the surface at time $t$, with $Z_0=h$. The state $Z_t$ is taken to be a monotonically decreasing function of the so-called boundary local time, and absorption occurs as soon as $Z_t$ reaches zero. (The boundary local time is a Brownian functional that determines the amount of time that the particle spends in a neighborhood of $\partial \calU$.) We first show how to analyze threshold surface absorption in terms of the joint probability density or generalized propagator $P_0(\x,\ell,t|\x_0)$ for the pair $(\X_t,\ell_t)$ in the case of a perfectly reflecting surface, where $\X_t$ and $\ell_t$ denote the particle position and local time at time $t$, respectively, and $\x_0$ is the initial position. We then introduce a generalized stochastic resetting protocol in which both the position $\X_t$ and the internal state $Z_t$ are reset to their initial values, $\X_t\rightarrow \x_0$ and $Z_t\rightarrow h$, at a Poisson rate $r$. The latter is mathematically equivalent to resetting the boundary local time, $\ell_t\rightarrow 0$. Since resetting is governed by a renewal process, the survival probability with resetting can be expressed in terms of the survival probability without resetting, which means that the statistics of absorption can be determined by calculating the Laplace transform of $P_0(\x,\ell,t|\x_0)$ with respect to $t$. We contrast this with the case where only particle position is reset, which is not governed by a renewal process. We illustrate the theory using the simple examples of diffusion on the half-line and a spherical target in a spherical domain.

cond-mat.stat-mech↗

arXiv2022

Diffusion-mediated surface reactions and stochastic resetting

In this paper, we investigate the effects of stochastic resetting on diffusion in $\R^d\backslash \calU$, where $\calU$ is a bounded obstacle with a partially absorbing surface $\partial \calU$. We begin by considering a Robin boundary condition with a constant reactivity $κ_0$, and show how previous results are recovered in the limits $κ_0\rightarrow 0,\infty$. We then generalize the Robin boundary condition to a more general probabilistic model of diffusion-mediated surface reactions using an encounter-based approach. The latter considers the joint probability density or propagator $P(\x,\ell,t|\x_0)$ for the pair $(\X_t,\ell_t)$ in the case of a perfectly reflecting surface, where $\X_t$ and $\ell_t$ denote the particle position and local time, respectively. The local time determines the amount of time that a Brownian particle spends in a neighborhood of the boundary. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. We construct the boundary value problem (BVP) satisfied by the propagator in the presence of resetting, and use this to derive implicit equations for the marginal density of particle position and the survival probability. We highlight the fact that these equations are difficult to solve in the case of non-constant reactivities, since resetting is not governed by a renewal process. We then consider a simpler problem in which both the position and local time are reset. In this case, the survival probability with resetting can be expressed in terms of the survival probability without resetting, which allows us to explore the dependence of the MFPT on the resetting rate $r$ and the type of surface reactions. The theory is illustrated using the example of a spherically symmetric surface.

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arXiv2022

Local accumulation times in a diffusion-trapping model of synaptic receptor dynamics

The lateral diffusion and trapping of neurotransmitter receptors within the postsynaptic membrane of a neuron plays a key role in determining synaptic strength and plasticity. Trapping is mediated by the reversible binding of receptors to scaffolding proteins (slots) within a synapse. In this paper we introduce a new method for analyzing the transient dynamics of synapses in a diffusion-trapping model of receptor trafficking. Given a population of spatially distributed synapses, each of which has a fixed number of slots, we calculate the rate of relaxation to the steady-state distribution of bound slots (synaptic weights) in terms of a set of local accumulation times. Assuming that the rates of exocytosis and endocytosis are sufficiently slow, we show that the steady-state synaptic weights are independent of each other (purely local). On the other hand, the local accumulation time of a given synapse depends on the number of slots and the spatial location of all the synapses, indicating a form of transient heterosynaptic plasticity. This suggests that local accumulation time measurements could provide useful information regarding the distribution of synaptic weights within a dendrite.

q-bio.QM↗

arXiv2022

The narrow capture problem: an encounter-based approach to partially reactive targets

A general topic of current interest is the analysis of diffusion problems in singularly perturbed domains with small interior targets or traps (the narrow capture problem). One major application is to intracellular diffusion, where the targets typically represent some form of reactive biochemical substrate. Most studies of the narrow capture problem treat the target boundaries as totally absorbing. In this paper, we analyze the three-dimensional narrow capture problem in the more realistic case of partially reactive target boundaries. We begin by considering classical Robin boundary conditions. Matching inner and outer solutions of the single-particle probability density, we derive an asymptotic expansion of the Laplace transformed flux into each reactive surface in powers of $ε$, where $ερ$ is a given target size. In turn, the fluxes determine the splitting probabilities for target absorption. We then extend our analysis to more general types of reactive targets by combining matched asymptotic analysis with an encounter-based formulation of diffusion-mediated surface reactions. That is, we derive an asymptotic expansion of the joint probability density for particle position and the boundary local time. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. Finally, we illustrate the theory by exploring how the leading-order contributions to the splitting probabilities depend on the choice of surface reactions. In particular, we show that there is an effective renormalization of the target radius of the form $ρ\rightarrow ρ-\widetildeΨ(1/ρ)$, where $\widetildeΨ$ is the Laplace transform of the stopping local time distribution.

cond-mat.stat-mech↗

arXiv2022

Diffusion-mediated surface reactions, Brownian functionals and the Feynman-Kac formula

Many processes in cell biology involve diffusion in a domain $Ω$ that contains a target $\calU$ whose boundary $\partial \calU$ is a chemically reactive surface. Such a target could represent a single reactive molecule, an intracellular compartment or a whole cell. Recently, a probabilistic framework for studying diffusion-mediated surface reactions has been developed that considers the joint probability density or propagator for the particle position and the so-called boundary local time. The latter characterizes the amount of time that a Brownian particle spends in the neighborhood of a point on a totally reflecting boundary. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. In this paper we generalize the theory of diffusion-mediated surface reactions to cases where the whole interior target domain $\calU$ acts as a partial absorber rather than the target boundary $\partial \calU$. Now the particle can freely enter and exit $\calU$, and is only able to react (be absorbed) within $\calU$. The appropriate Brownian functional is then the occupation time (accumulated time that the particle spends within $\calU$) rather than the boundary local time. We show that both cases can be considered within a unified framework by using a Feynman-Kac formula to derive a boundary value problem (BVP) for the propagator of the corresponding Brownian functional, and introducing an associated stopping condition. We illustrate the theory by calculating the mean first passage time (MFPT) for a spherical target $\calU$ located at the center of a spherical domain $Ω$. This is achieved by solving the propagator BVP directly, rather than using spectral methods. We find that if the first moment of the stopping time density is infinite, then the MFPT is also infinite, that is, the spherical target is not sufficiently absorbing.

cond-mat.stat-mech↗

arXiv2022

Local accumulation time for diffusion in cells with gap junction coupling

In this paper we analyze the relaxation to steady-state of intracellular diffusion in a pair of cells with gap-junction coupling. Gap junctions are prevalent in most animal organs and tissues, providing a direct diffusion pathway for both electrical and chemical communication between cells. Most analytical models of gap junctions focus on the steady-state diffusive flux and the associated effective diffusivity. Here we investigate the relaxation to steady state in terms of the so-called local accumulation time. The latter is commonly used to estimate the time to form a protein concentration gradient during morphogenesis. The basic idea is to treat the fractional deviation from the steady-state concentration as a cumulative distribution for the local accumulation time. One of the useful features of the local accumulation time is that it takes into account the fact that different spatial regions can relax at different rates. We consider both static and dynamic gap junction models. The former treats the gap junction as a resistive channel with effective permeability $μ$, whereas the latter represents the gap junction as a stochastic gate that randomly switches between an open and closed state. The local accumulation time is calculated by solving the diffusion equation in Laplace space and then taking the small-$s$ limit. We show that the accumulation time is a monotonically increasing function of spatial position, with a jump discontinuity at the gap junction. This discontinuity vanishes in the limit $μ\rightarrow \infty$ for a static junction and $β\rightarrow 0$ for a stochastically-gated junction, where $β$ is the rate at which the gate closes. Finally, our results are generalized to the case of a linear array of cells with nearest neighbor gap junction coupling.

cond-mat.stat-mech↗

arXiv2021

Path integrals for stochastic hybrid reaction-diffusion processes

We construct path integrals for stochastic hybrid reaction-diffusion (RD) processes, in which the reaction terms depend on the discrete state of a randomly switching environment. We proceed by spatially discretizing a given RD system and using a spinor representation of the environmental states to derive a path integral for the lattice model. In the case of large molecular numbers, the corresponding continuum path integral action is expressed in terms of an effective Hamiltonian, which involves a concentration field $u(\x,t)$, $\x\in \R^d$, a conjugate field $v(\x,t)$, and $M$ auxiliary conjugate pairs $(c_m(t),ϕ_m(t))$, where $M$ is the number of discrete environmental states. The variable $c_m(t)$ determines the effective probability that a sample path is exposed to the $m$-th environmental state at time $t$, with $\sum_{m=1}^Mc_m(t)=1$. We then consider the semi-classical (adiabatic) limit $ε\rightarrow 0$, where $ε^{-1}$ determines the rate of switching between the environmental states. We show how the auxiliary variables can be eliminated to yield an action functional for the fields $u$ and $v$ alone. The associated Hamiltonian is the sum of a diffusion term and the Perron or principal eigenvalue of a functional linear operator involving the reaction terms and the matrix generator of the switching process. The reduced path integral is then used to derive a functional Hamilton-Jacobi equation for least action paths and to obtain a Gaussian noise approximation of the stochastic hybrid RD system in the adiabatic limit. Finally, the path integral in the case of low molecular numbers is constructed by considering a corresponding RD master equation. It is now necessary to take into account two sources of noise, one due to the switching environment and the other due to fluctuations in molecular numbers.

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