Kinetics of stochastically gated diffusion-limited reactions and geometry of random walk trajectories

Probing the modulation in facilitated diffusion guided by DNA-protein interactions in target search processes

Many fundamental biophysical processes involving gene regulation and gene editing rely, at the molecular level, on an intricate methodology of searching and locating the precise target base pair sequence on the genome by specific binding proteins. A unique mechanism, known as 'facilitated diffusion', which is a combination of 1D sliding along with 3D movement, is considered to be the key step for such events. This also explains the relatively much shorter timescale of the target searching process, compared to other diffusion-controlled biophysical processes. In this work, we aim to probe the modulation of target search dynamics of a protein moiety by estimating the rate of the target search process, and the statistics of the search rounds and timescales accomplished by the 1D and 3D motions, based on first passage time (FPT) calculations. This is studied with its characteristics getting influenced by various given conditions such as, when the DNA is rigid or flexible, and when the target is placed at different locations on the DNA. The current theoretical framework includes a Brownian dynamics simulation setup adopting a straightforward coarse-grained model for a diffusing protein on DNA. Moreover, this theoretical analysis provides insights into the complex target search dynamics by highlighting the significance of the chain dynamics in the mechanistic details of the facilitated diffusion process.

Diffusion-Controlled Reactions: An Overview

We review the milestones in the century-long development of the theory of diffusion-controlled reactions. Starting from the seminal work by von Smoluchowski, who recognized the importance of diffusion in chemical reactions, we discuss perfect and imperfect surface reactions, their microscopic origins, and the underlying mathematical framework. Single-molecule reaction schemes, anomalous bulk diffusions, reversible binding/unbinding kinetics, and many other extensions are presented. An alternative encounter-based approach to diffusion-controlled reactions is introduced, with emphasis on its advantages and potential applications. Some open problems and future perspectives are outlined.

Imperfect narrow escape problem

We consider the kinetics of the imperfect narrow escape problem, i.e., the time it takes for a particle diffusing in a confined medium of generic shape to reach and to be adsorbed by a small, imperfectly reactive patch embedded in the boundary of the domain, in two or three dimensions. Imperfect reactivity is modeled by an intrinsic surface reactivity κ of the patch, giving rise to Robin boundary conditions. We present a formalism to calculate the exact asymptotics of the mean reaction time in the limit of large volume of the confining domain. We obtain exact explicit results in the two limits of large and small reactivities of the reactive patch, and a semianalytical expression in the general case. Our approach reveals an anomalous scaling of the mean reaction time as the inverse square root of the reactivity in the large-reactivity limit, valid for an initial position near the extremity of the reactive patch. We compare our exact results with those obtained within the "constant flux approximation"; we show that this approximation turns out to give exactly the next-to-leading-order term of the small-reactivity limit, and provides a good approximation of the reaction time far from the reactive patch for all reactivities, but not in the vicinity of the boundary of the reactive patch due to the above-mentioned anomalous scaling. These results thus provide a general framework to quantify the mean reaction times for the imperfect narrow escape problem.

Theoretical insights into the full description of DNA target search by subdiffusing proteins

DNA binding proteins (DBPs) diffuse in the cytoplasm to recognise and bind with their respective target sites on the DNA to initiate several biologically important processes. The first passage time distributions (FPTDs) of DBPs are useful in quantifying the timescales of the most-probable search paths in addition to the mean value of the distribution which, strikingly, are decades of order apart in time. However, extremely crowded in vivo conditions or the viscoelasticity of the cellular medium among other factors causes biomolecules to exhibit anomalous diffusion which is usually overlooked in most theoretical studies. We have obtained approximate analytical expressions of a general FPTD and the two characteristic timescales that are valid for any single subdiffusing protein searching for its target in vivo. Our results can be applied to single-particle tracking experiments of target search.

Encounter-based approach to diffusion with resetting

An encounter-based approach consists in using the boundary local time as a proxy for the number of encounters between a diffusing particle and a target to implement various surface reaction mechanisms on that target. In this paper, we investigate the effects of stochastic resetting onto diffusion-controlled reactions in bounded confining domains. We first discuss the effect of position resetting onto the propagator and related quantities; in this way, we retrieve a number of earlier results but also provide complementary insights into them. Second, we introduce boundary local time resetting and investigate its impact. Curiously, we find that this type of resetting does not alter the conventional propagator governing the diffusive dynamics in the presence of a partially reactive target with a constant reactivity. In turn, the generalized propagator for other surface reaction mechanisms can be significantly affected. Our general results are illustrated for diffusion on an interval with reactive end points. Further perspectives and some open problems are discussed.

First-passage process in degree space for the time-dependent Erdős-Rényi and Watts-Strogatz models

In this work, we investigate the temporal evolution of the degree of a given vertex in a network by mapping the dynamics into a random walk problem in degree space. We analyze when the degree approximates a preestablished value through a parallel with the first-passage problem of random walks. The method is illustrated on the time-dependent versions of the Erdős-Rényi and Watts-Strogatz models, which were originally formulated as static networks. We have succeeded in obtaining an analytic form for the first and the second moments of the first-passage time and showing how they depend on the size of the network. The dominant contribution for large networks with N vertices indicates that these quantities scale on the ratio N/p, where p is the linking probability.

First-passage times to anisotropic partially reactive targets

We investigate restricted diffusion in a bounded domain towards a small partially reactive target in three- and higher-dimensional spaces. We propose a simple explicit approximation for the principal eigenvalue of the Laplace operator with mixed Robin-Neumann boundary conditions. This approximation involves the harmonic capacity and the surface area of the target, the volume of the confining domain, the diffusion coefficient, and the reactivity. The accuracy of the approximation is checked by using a finite-elements method. The proposed approximation determines also the mean first-reaction time, the long-time decay of the survival probability, and the overall reaction rate on that target. We identify the relevant lengthscale of the target, which determines its trapping capacity, and we investigate its relation to the target shape. In particular, we study the effect of target anisotropy on the principal eigenvalue by computing the harmonic capacity of prolate and oblate spheroids in various space dimensions. Some implications of these results in chemical physics and biophysics are briefly discussed.

Reversible target-binding kinetics of multiple impatient particles

Certain biochemical reactions can only be triggered after binding a sufficient number of particles to a specific target region such as an enzyme or a protein sensor. We investigate the distribution of the reaction time, i.e., the first instance when all independently diffusing particles are bound to the target. When each particle binds irreversibly, this is equivalent to the first-passage time of the slowest (last) particle. In turn, reversible binding to the target renders the problem much more challenging and drastically changes the distribution of the reaction time. We derive the exact solution of this problem and investigate the short-time and long-time asymptotic behaviors of the reaction time probability density. We also analyze how the mean reaction time depends on the unbinding rate and the number of particles. Our exact and asymptotic solutions are compared to Monte Carlo simulations.

Gated reactions in discrete time and space

How much time does it take for two molecules to react? If a reaction occurs upon contact, the answer to this question boils down to the classic first-passage time problem: find the time it takes for the two molecules to meet. However, this is not always the case as molecules switch stochastically between reactive and non-reactive states. The reaction is then said to be "gated" by the internal states of the molecules involved, which could have a dramatic influence on kinetics. A unified, continuous-time, approach to gated reactions on networks was presented in a recent paper [Scher and Reuveni, Phys. Rev. Lett. 127, 018301 (2021)]. Here, we build on this recent advancement and develop an analogous discrete-time version of the theory. Similar to continuous-time, we employ a renewal approach to show that the gated reaction time can always be expressed in terms of the corresponding ungated first-passage and return times, which yields formulas for the generating function of the gated reaction-time distribution and its corresponding mean and variance. In cases where the mean reaction time diverges, we show that the long-time asymptotics of the gated problem is inherited from its ungated counterpart. However, when molecules spend most of their time non-reactive, an interim regime of slower power-law decay emerges prior to the terminal asymptotics. The discretization of time also gives rise to resonances and anti-resonances, which were absent from the continuous-time picture. These features are illustrated using two case studies that also demonstrate how the general approach presented herein greatly simplifies the analysis of gated reactions.

Universal kinetics of imperfect reactions in confinement

Chemical reactions generically require that particles come into contact. In practice, reaction is often imperfect and can necessitate multiple random encounters between reactants. In confined geometries, despite notable recent advances, there is to date no general analytical treatment of such imperfect transport-limited reaction kinetics. Here, we determine the kinetics of imperfect reactions in confining domains for any diffusive or anomalously diffusive Markovian transport process, and for different models of imperfect reactivity. We show that the full distribution of reaction times is obtained in the large confining volume limit from the knowledge of the mean reaction time only, which we determine explicitly. This distribution for imperfect reactions is found to be identical to that of perfect reactions upon an appropriate rescaling of parameters, which highlights the robustness of our results. Strikingly, this holds true even in the regime of low reactivity where the mean reaction time is independent of the transport process, and can lead to large fluctuations of the reaction time - even in simple reaction schemes. We illustrate our results for normal diffusion in domains of generic shape, and for anomalous diffusion in complex environments, where our predictions are confirmed by numerical simulations.

Unified Approach to Gated Reactions on Networks

For two molecules to react they first have to meet. Yet, reaction times are rarely on par with the first-passage times that govern such molecular encounters. A prime reason for this discrepancy is stochastic transitions between reactive and nonreactive molecular states, which results in effective gating of product formation and altered reaction kinetics. To better understand this phenomenon we develop a unifying approach to gated reactions on networks. We first show that the mean and distribution of the gated reaction time can always be expressed in terms of ungated first-passage and return times. This relation between gated and ungated kinetics is then explored to reveal universal features of gated reactions. The latter are exemplified using a diverse set of case studies which are also used to expose the exotic kinetics that arises due to molecular gating.

Paradigm Shift in Diffusion-Mediated Surface Phenomena

Diffusion-mediated surface phenomena are crucial for human life and industry, with examples ranging from oxygen capture by lung alveolar surface to heterogeneous catalysis, gene regulation, membrane permeation, and filtration processes. Their current description via diffusion equations with mixed boundary conditions is limited to simple surface reactions with infinite or constant reactivity. In this Letter, we propose a probabilistic approach based on the concept of boundary local time to investigate the intricate dynamics of diffusing particles near a reactive surface. Reformulating surface-particle interactions in terms of stopping conditions, we obtain in a unified way major diffusion-reaction characteristics such as the propagator, the survival probability, the first-passage time distribution, and the reaction rate. This general formalism allows us to describe new surface reaction mechanisms such as for instance surface reactivity depending on the number of encounters with the diffusing particle that can model the effects of catalyst fooling or membrane degradation. The disentanglement of the geometric structure of the medium from surface reactivity opens far-reaching perspectives for modeling, optimization, and control of diffusion-mediated surface phenomena.

First Hitting Times to Intermittent Targets

In noisy environments such as the cell, many processes involve target sites that are often hidden or inactive, and thus not always available for reaction with diffusing entities. To understand reaction kinetics in these situations, we study the first hitting time statistics of a one-dimensional Brownian particle searching for a target site that switches stochastically between visible and hidden phases. At high crypticity, an unexpected rate limited power-law regime emerges for the first hitting time density, which markedly differs from the classic t^{-3/2} scaling for steady targets. Our problem admits an asymptotic mapping onto a mixed, or Robin, boundary condition. Similar results are obtained with non-Markov targets and particles diffusing anomalously.

Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains

How long does a diffusing molecule spend in a close vicinity of a confining boundary or a catalytic surface? This quantity is determined by the boundary local time, which plays thus a crucial role in the description of various surface-mediated phenomena, such as heterogeneous catalysis, permeation through semipermeable membranes, or surface relaxation in nuclear magnetic resonance. In this paper, we obtain the probability distribution of the boundary local time in terms of the spectral properties of the Dirichlet-to-Neumann operator. We investigate the short-time and long-time asymptotic behaviors of this random variable for both bounded and unbounded domains. This analysis provides complementary insights onto the dynamics of diffusing molecules near partially reactive boundaries.

Diffusion-influenced reactions on non-spherical partially absorbing axisymmetric surfaces

The calculation of the diffusion-controlled reaction rate for partially absorbing, non-spherical boundaries presents a formidable problem of broad relevance. In this paper we take the reference case of a spherical boundary and work out a perturbative approach to get a simple analytical formula for the first-order correction to the diffusive flux onto a non-spherical partially absorbing surface of revolution. To assess the range of validity of this formula, we derive exact and approximate expressions for the reaction rate in the case of partially absorbing prolate and oblate spheroids. We also present numerical solutions by a finite-element method that extend the validity analysis beyond spheroidal shapes. Our perturbative solution provides a handy way to quantify the effect of non-sphericity on the rate of capture in the general case of partial surface reactivity.

The escape problem for mortal walkers

We introduce and investigate the escape problem for random walkers that may eventually die, decay, bleach, or lose activity during their diffusion towards an escape or reactive region on the boundary of a confining domain. In the case of a first-order kinetics (i.e., exponentially distributed lifetimes), we study the effect of the associated death rate onto the survival probability, the exit probability, and the mean first passage time. We derive the upper and lower bounds and some approximations for these quantities. We reveal three asymptotic regimes of small, intermediate, and large death rates. General estimates and asymptotics are compared to several explicit solutions for simple domains and to numerical simulations. These results allow one to account for stochastic photobleaching of fluorescent tracers in bio-imaging, degradation of mRNA molecules in genetic translation mechanisms, or high mortality rates of spermatozoa in the fertilization process. Our findings provide a mathematical ground for optimizing storage containers and materials to reduce the risk of leakage of dangerous chemicals or nuclear wastes.

Effect of stochastic gating on channel-facilitated transport of non-interacting and strongly repelling solutes

Ligand- or voltage-driven stochastic gating-the structural rearrangements by which the channel switches between its open and closed states-is a fundamental property of biological membrane channels. Gating underlies the channel's ability to respond to different stimuli and, therefore, to be functionally regulated by the changing environment. The accepted understanding of the gating effect on the solute flux through the channel is that the mean flux is the product of the flux through the open channel and the probability of finding the channel in the open state. Here, using a diffusion model of channel-facilitated transport, we show that this is true only when the gating is much slower than the dynamics of solute translocation through the channel. If this condition breaks, the mean flux could differ from this simple estimate by orders of magnitude.

Diffusive escape through a narrow opening: new insights into a classic problem

We study the mean first exit time (T_ε) of a particle diffusing in a circular or a spherical micro-domain with an impenetrable confining boundary containing a small escape window (EW) of an angular size ε.

Diffusive flux in a model of stochastically gated oxygen transport in insect respiration

Oxygen delivery to insect tissues is controlled by transport through a branched tubular network that is connected to the atmosphere by valve-like gates, known as spiracles. In certain physiological regimes, the spiracles appear to be randomly switching between open and closed states. Quantitative analysis of this regime leads a reaction-diffusion problem with stochastically switching boundary condition. We derive an expression for the diffusive flux at long times in this problem. Our approach starts with the derivation of the passage probability for a single particle that diffuses between a stochastically gated boundary, which models the opening and closing spiracle, and the perfectly absorbing boundary, which models oxygen absorption by the tissue. This passage probability is then used to derive an expression giving the diffusive flux as a function of the geometric parameters of the tube and characteristic time scales of diffusion and gate dynamics.

Stochastically gated diffusion-limited reactions for a small target in a bounded domain

We calculate the reaction rate for stochastically gated ligands diffusing in a two-dimensional and a three-dimensional bounded domain with a single small target. Each ligand independently switches between an open and a closed state according to a two-state Markov process; a reaction between ligand and target can only occur when the former is an open state. In the large-time limit the reaction-rate is an exponentially decaying function of time, whose rate of decay is given by the principal eigenvalue of the Laplacian. We calculate the principal eigenvalue using matched asymptotics and determine the leading-order reduction in the reaction rate due to stochastic gating. We also develop a probabilistic interpretation of the reaction rate in terms of the first-passage time density to the target.

Active transport in dense diffusive single-file systems

We study a minimal model of active transport in crowded single-file environments which generalizes the emblematic model of single-file diffusion to the case when the tracer particle (TP) performs either an autonomous directed motion or is biased by an external force, while all other particles of the environment (bath) perform unbiased diffusions. We derive explicit expressions, valid in the limit of high density of bath particles, of the full distribution P((n))(X) of the TP position and of all its cumulants, for arbitrary values of the bias f and for any time n. Our analysis reveals striking features, such as the anomalous scaling [proportionality] √[n] of all cumulants, the equality of cumulants of the same parity characteristic of a Skellam distribution and a convergence to a Gaussian distribution in spite of asymmetric density profiles of bath particles. Altogether, our results provide the full statistics of the TP position and set the basis for a refined analysis of real trajectories of active particles in crowded single-file environments.

Exploration and trapping of mortal random walkers

Exploration and trapping properties of random walkers that may evanesce at any time as they walk have seen very little treatment in the literature, and yet a finite lifetime is a frequent occurrence, and its effects on a number of random walk properties may be profound. For instance, whereas the average number of distinct sites visited by an immortal walker grows with time without bound, that of a mortal walker may, depending on dimensionality and rate of evanescence, remain finite or keep growing with the passage of time. This number can in turn be used to calculate other classic quantities such as the survival probability of a target surrounded by diffusing traps. If the traps are immortal, the survival probability will vanish with increasing time. However, if the traps are evanescent, the target may be spared a certain death. We analytically calculate a number of basic and broadly used quantities for evanescent random walkers.

First passages in bounded domains: when is the mean first passage time meaningful?

We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability P(ω) distribution of the random variable ω=τ(1)/(τ(1)+τ(2)), which is a measure for how similar the first passage times τ(1) and τ(2) are of two independent realizations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether P(ω) represents a unimodal, bell-shaped form, or a bimodal, M-shaped behavior. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behavior, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of P(ω), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in two-dimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behavior even in bounded domains, in which all moments of the first passage distribution exist.

Intermittent pathways towards a dynamical target

In this paper, we investigate the quest for a single target, which remains fixed in a lattice, by a set of independent walkers. The target exhibits fluctuating behavior between a trap and an ordinary site of the lattice, whereas the walkers perform an intermittent kind of search strategy. Our searchers carry out their movements in one of two states, between which they switch randomly. One of these states (the exploratory phase) is a symmetric nearest-neighbor random walk and the other state (the relocating phase) is a symmetric next-nearest-neighbor random walk. By using the multistate continuous-time random-walk approach we are able to show that for dynamical targets, the intermittent strategy (despite the simplicity of the kinetics chosen for searching) improves detection, in comparison to displacements in a single state. We have obtained analytic results, which can be numerically evaluated, for the survival probability and for the lifetime of the target. Thus, we have studied the dependence of these quantities both in terms of the transition probability that describes the dynamics of the target and in terms of the parameter that characterizes the walkers' intermittency. In addition to our analytical approach, we have implemented Monte Carlo simulations, finding excellent agreement between the theoretical-numerical results and simulations.

Survival of an evasive prey

We study the survival of a prey that is hunted by N predators. The predators perform independent random walks on a square lattice with V sites and start a direct chase whenever the prey appears within their sighting range. The prey is caught when a predator jumps to the site occupied by the prey. We analyze the efficacy of a lazy, minimal-effort evasion strategy according to which the prey tries to avoid encounters with the predators by making a hop only when any of the predators appears within its sighting range; otherwise the prey stays still. We show that if the sighting range of such a lazy prey is equal to 1 lattice spacing, at least 3 predators are needed in order to catch the prey on a square lattice. In this situation, we establish a simple asymptotic relation ln P(ev)(t) approximately (N/V)(2)ln P(imm)(t) between the survival probabilities of an evasive and an immobile prey. Hence, when the density rho = N/V of the predators is low, rho << 1, the lazy evasion strategy leads to the spectacular increase of the survival probability. We also argue that a short-sighting prey (its sighting range is smaller than the sighting range of the predators) undergoes an effective superdiffusive motion, as a result of its encounters with the predators, whereas a far-sighting prey performs a diffusive-type motion.

Exact asymptotics for nonradiative migration-accelerated energy transfer in one-dimensional systems

We study direct energy transfer by multipolar or exchange interactions between diffusive excited donor and diffusive unexcited acceptors. Extending over the case of long-range transfer of an excitation energy a nonperturbative approach by Bray and Blythe [Phys. Rev. Lett. 89, 150601 (2002)], originally developed for contact diffusion-controlled reactions, we determine exactly long-time asymptotics of the donor decay function in one-dimensional systems.

Number of distinct sites visited by a subdiffusive random walker

The asymptotic mean number of distinct sites visited by a subdiffusive continuous-time random walker in two dimensions seems not to have been explicitly calculated anywhere in the literature. This number has been calculated for other dimensions for only one specific asymptotic behavior of the waiting time distribution between steps. We present an explicit derivation for two cases in all integer dimensions so as to formally complete a tableau of results. In this tableau we include the dominant as well as subdominant contributions in all integer dimensions. Other quantities that can be calculated from the mean number of distinct sites visited are also discussed.

Intermittent search process and teleportation

The authors study an intermittent search process combining diffusion and "teleportation" phases in a d-dimensional spherical continuous system and in a regular lattice. The searcher alternates diffusive phases, during which targets can be discovered, and fast phases (teleportation) which randomly relocate the searcher, but do not allow for target detection. The authors show that this alternation can be favorable for minimizing the time of first discovery, and that this time can be optimized by a convenient choice of the mean waiting times of each motion phase. The optimal search strategy is explicitly derived in the continuous case and in the lattice case. Arguments are given to show that much more general intermittent motions do provide optimal search strategies in d dimensions. These results can be useful in the context of heterogeneous catalysis or in various biological examples of transport through membrane pores.

Trapping reactions with subdiffusive traps and particles characterized by different anomalous diffusion exponents

A number of results for reactions involving subdiffusive species, all with the same anomalous exponent , have recently appeared in the literature and can often be understood in terms of a subordination principle whereby time in ordinary diffusion is replaced by tgamma. However, very few results are known for reactions involving different species characterized by different anomalous diffusion exponents. Here we study the reaction dynamics of a (sub)diffusive particle surrounded by a sea of (sub)diffusive traps in one dimension. We find rigorous results for the asymptotic survival probability of the particle in most cases, with the exception of the case of a particle that diffuses normally while the anomalous diffusion exponent of the traps is smaller than 2/3.

Kinetics of diffusion-limited catalytically activated reactions: An extension of the Wilemski–Fixman approach

We study the kinetics of diffusion-limited catalytically activated A+B-->B reactions taking place in three-dimensional systems, in which an annihilation of diffusive A particles by diffusive traps B may happen only if the encounter of an A with any of the Bs happens within a special catalytic subvolumen: these subvolumens being immobile and uniformly distributed within the reaction bath. Suitably extending the classical approach of Wilemski and Fixman [J. Chem. Phys. 58, 4009 (1973)] to such three-molecular diffusion-limited reactions, we calculate analytically an effective reaction constant and show that it comprises several terms associated with the residence and joint residence times of Brownian paths in finite domains. The effective reaction constant exhibits a nontrivial dependence on the reaction radii, the mean density of catalytic subvolumens, and particles' diffusion coefficients. Finally, we discuss the fluctuation-induced kinetic behavior in such systems.

Lattice theory of trapping reactions with mobile species

We present a stochastic lattice theory describing the kinetic behavior of trapping reactions A+B-->B, in which both the A and B particles perform an independent stochastic motion on a regular hypercubic lattice. Upon an encounter of an A particle with any of the B particles, A is annihilated with a finite probability; finite reaction rate is taken into account by introducing a set of two-state random variables--"gates," imposed on each B particle, such that an open (closed) gate corresponds to a reactive (passive) state. We evaluate here a formal expression describing the time evolution of the A particle survival probability, which generalizes our previous results. We prove that for quite a general class of random motion of the species involved in the reaction process, for infinite or finite number of traps, and for any time t, the A particle survival probability is always larger in the case when A stays immobile, than in situations when it moves.

Pascal principle for diffusion-controlled trapping reactions

In this paper, we analyze the long-time behavior of the survival probability P(A)(t) of an A particle, that performs lattice random walk in the presence of randomly moving traps B. We show that for both perfect and imperfect trapping reactions, for arbitrary spatial dimension d and for a rather general class of random walks, P(A)(t) is less than or equal to the survival probability of an immobile target A in the presence of randomly moving traps.

Trapping reactions with randomly moving traps: exact asymptotic results for compact exploration

In a recent paper, Bray and Blythe have shown that the survival probability P(A)(t) of an A particle diffusing with a diffusion coefficient D(A) in a one-dimensional system with diffusive traps B is independent of D(A) in the asymptotic limit t--> infinity and coincides with the survival probability of an immobile target in the presence of diffusive traps. Here, we show that this remarkable behavior has a more general range of validity and holds for systems of an arbitrary dimension d, integer or fractal, provided that the traps are "compactly exploring" the space, i.e., the "fractal" dimension d(w) of traps' trajectories is greater than d. For the marginal case when d(w)=d, as exemplified here by conventional diffusion in two-dimensional systems, the decay form is determined up to a numerical factor in the characteristic decay time.

Ultraslow vacancy-mediated tracer diffusion in two dimensions: the Einstein relation verified

We study the dynamics of a charged tracer particle (TP) on a two-dimensional lattice, all sites of which except one (a vacancy) are filled with identical neutral, hard-core particles. The particles move randomly by exchanging their positions with the vacancy, subject to the hard-core exclusion. In the case when the charged TP experiences a bias due to external electric field E (which favors its jumps in the preferential direction), we determine exactly the limiting probability distribution of the TP position in terms of appropriate scaling variables and the leading large-n (n being the discrete time) behavior of the TP mean displacement X(n); the latter is shown to obey an anomalous, logarithmic law /X(n)/=alpha(0)(/E/)ln(n). Comparing our results with earlier predictions by Brummelhuis and Hilhorst [J. Stat. Phys. 53, 249 (1988)] for the TP diffusivity D(n) in the unbiased case, we infer that the Einstein relation mu(n)=betaD(n) between the TP diffusivity and the mobility mu(n)=lim(/E/-->0)(/X(n)///E/n) holds in the leading n order, despite the fact that both D(n) and mu(n) are not constant but vanish as n--> infinity. We also generalize our approach to the situation with very small but finite vacancy concentration rho(v), in which case we find a ballistic-type law /X(n)/=pi(alpha)(0)(/E/)rho(v)n. We demonstrate that here, again, both D(n) and mu(n), calculated in the linear in rho(v) approximation, do obey the Einstein relation.

Uninfected random walkers in one dimension

We consider a system of unbiased diffusing walkers (A(Phi)<-->(Phi)A) in one dimension with random initial conditions. We investigate numerically the relation between the fraction of walkers U(t) which have never encountered another walker up to time t, calling such walkers "uninfected" and the fraction of sites P(t) which have never been visited by a diffusing particle. We extend our study to include the A+B--> Phi diffusion-limited reaction in one dimension, with equal initial densities of A and B particles distributed homogeneously at t=0. We find U(t) approximately [P(t)]gamma, with gamma approximately 1.39, in both models, though there is evidence that a smaller value of gamma is required for t-->infinity.

Trapping reaction with mobile traps

We present the Monte Carlo results for the two-species trapping reaction A+B-->B with diffusing A and B on lattices in one, two, and three dimensions. We use an algorithm that permits one to simulate the survival probabilities of A particles down to <10(-30) with high accuracy. The results for the survival probability agree much better with the exact asymptotic predictions of Bramson and Lebowitz [Phys. Rev. Lett. 61, 2397 (1988)] than with the heuristics of Kang and Redner [J. Phys. A 17, L451 (1984)]. But there are very large deviations from either, which show that even these simulations are far from asymptotia. This is supported by the rms displacement of A particles, which clearly shows that the asymptotic regime has not been reached, at least for d=2 and d=3.