First-passage times in complex scale-invariant media.

How long does it take a random walker to reach a given target point? This quantity, known as a first-passage time (FPT), has led to a growing number of theoretical investigations over the past decade. The importance of FPTs originates from the crucial role played by first encounter properties in various real situations, including transport in disordered media, neuron firing dynamics, spreading of diseases or target search processes. Most methods of determining FPT properties in confining domains have been limited to effectively one-dimensional geometries, or to higher spatial dimensions only in homogeneous media. Here we develop a general theory that allows accurate evaluation of the mean FPT in complex media. Our analytical approach provides a universal scaling dependence of the mean FPT on both the volume of the confining domain and the source-target distance. The analysis is applicable to a broad range of stochastic processes characterized by length-scale-invariant properties. Our theoretical predictions are confirmed by numerical simulations for several representative models of disordered media, fractals, anomalous diffusion and scale-free networks.

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.

Probing microscopic origins of confined subdiffusion by first-passage observables.

Subdiffusive motion of tracer particles in complex crowded environments, such as biological cells, has been shown to be widespread. This deviation from Brownian motion is usually characterized by a sublinear time dependence of the mean square displacement (MSD). However, subdiffusive behavior can stem from different microscopic scenarios that cannot be identified solely by the MSD data. In this article we present a theoretical framework that permits the analytical calculation of first-passage observables (mean first-passage times, splitting probabilities, and occupation times distributions) in disordered media in any dimensions. This analysis is applied to two representative microscopic models of subdiffusion: continuous-time random walks with heavy tailed waiting times and diffusion on fractals. Our results show that first-passage observables provide tools to unambiguously discriminate between the two possible microscopic scenarios of subdiffusion. Moreover, we suggest experiments based on first-passage observables that could help in determining the origin of subdiffusion in complex media, such as living cells, and discuss the implications of anomalous transport to reaction kinetics in cells.

Chemical and biological microstructures as probed by dynamic processes.

The dynamic process of electronic energy transfer is shown to be an important tool for probing the microstructure of molecular systems, particularly those in which donors and acceptors occupy specifically labeled sites of spatially confining host matrices. Special attention is given to analyzing the temporal behavior of the direct energy transfer reaction for systems in which the dipolar coupling is between a donor and randomly distributed acceptors. This dynamic process is dependent on two competing lengths when the donor and acceptor distribution is determined by the microstructure of the confining system: Rp, the dominant length characterizing the size of the confinement, and R0, which scales the strength of the dipolar coupling. When energy transfer processes are viewed in the context of these two competing lengths, a picture emerges of the microstructure of the confinement that is consistent with and corroborated by other structural probes.

Single-molecule pulling experiments: when the stiffness of the pulling device matters.

Using Langevin modeling, we investigate the role of the experimental setup on the unbinding forces measured in single-molecule pulling experiments. We demonstrate that the stiffness of the pulling device, K(eff), may influence the unbinding forces through its effect on the barrier heights for both unbinding and rebinding processes. Under realistic conditions the effect of K(eff) on the rebinding barrier is shown to play the most important role. This results in a significant increase of the mean unbinding force with the stiffness for a given loading rate. Thus, in contrast to the phenomenological Bell model, we find that the loading rate (the multiplicative value K(eff)V, V being the pulling velocity) is not the only control parameter that determines the mean unbinding force. If interested in intrinsic properties of a molecular system, we recommend probing the system in the parameter range corresponding to a weak spring and relatively high loading rates where rebinding is negligible.

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.

Survival probability of a particle in a sea of mobile traps: a tale of tails.

We study the long-time tails of the survival probability P(t) of an A particle diffusing in d-dimensional media in the presence of a concentration rho of traps B that move subdiffusively, such that the mean square displacement of each trap grows as tgamma with 0 < or = gamma < or =1. Starting from a continuous time random walk description of the motion of the particle and of the traps, we derive lower and upper bounds for P(t) and show that for gamma < or =2/(d+2) these bounds coincide asymptotically, thus determining asymptotically exact results. The asymptotic decay law in this regime is exactly that obtained for immobile traps. This means that for sufficiently subdiffusive traps, the moving A particle sees the traps as essentially immobile, and Lifshitz or trapping tails remain unchanged. For gamma >2/(d+2) and d< or =2 the upper and lower bounds again coincide, leading to a decay law equal to that of a stationary particle. Thus, in this regime the moving traps see the particle as essentially immobile. For d>2 , however, the upper and lower bounds in this gamma regime no longer coincide, and the decay law for the survival probability of the A particle remains ambiguous.

Saltatory drift in a randomly driven two-wave potential.

The dynamics of a classical particle in a one-dimensional, randomly driven potential is analysed both analytically and numerically. The potential considered here is composed of two identical spatially periodic saw-tooth-like components, one of which is externally driven by a random force. We show that under certain conditions the particle may travel against the averaged external force, performing a saltatory unidirectional drift with a constant velocity. Such a behaviour persists also in situations when the external force averages out to zero. We demonstrate that the physics behind this phenomenon stems from a particular behaviour of fluctuations in random force: upon reaching a certain level, random fluctuations exercise a locking function creating points of irreversibility which the particle cannot overpass. Repeated (randomly) in each cycle, this results in a saltatory unidirectional drift. This mechanism resembles the work of an escapement-type device in watches. Considering the overdamped limit, we propose simple analytical estimates for the particle's terminal velocity.

Actin-based motility: cooperative symmetry-breaking and phases of motion.

A microscopic model is proposed for the motility of a bead driven by the polymerization of actin filaments. The model exhibits a rich spectrum of behaviours similar to those observed in biomimetic experiments, which include spontaneous symmetry-breaking, various regimes of the bead's motion and correlations between the structure of the actin tail which propels the bead and the bead dynamics. The dependences of the dynamical properties (such as symmetry-breaking time, regimes of motion, mean velocity, and tail asymmetry) on the physical parameters (the bead radius and viscosity) agree well with the experimental observations. We find that most experimental observations can be reproduced taking into account only one type of filaments interacting with the bead: the detached filaments that push the bead. Our calculations suggest that the analysis of mean characteristics only (velocities, symmetry-breaking times, etc) does not always provide meaningful information about the mechanism of motility. The aim should be to obtain the corresponding distributions, which might be extremely broad and therefore not well represented by their mean only. Our findings suggest a simple coarse-grained description, which captures the main features obtained within the microscopic model.

Single stranded DNA translocation through a nanopore: a master equation approach.

We study voltage driven translocation of a single stranded DNA through a membrane channel. Our model, based on a master equation approach, investigates the probability density function of the translocation times, and shows that it can be either double peaked or mono peaked, depending on the system parameters. We show that the most probable translocation time is proportional to the polymer length, and inversely proportional to the first or second power of the voltage, depending on the initial conditions. The model recovers experimental observations on hetropolymers when using their properties inside the pore, such as stiffness and polymer-pore interaction.

Resonant activation in discrete systems.

The resonant activation phenomenon (RAP) in a discrete system is studied using the master equation formalism. We show that the RAP corresponds to a nonmonotonic behavior of the frequency dependent first passage time probability density function (PDF). An analytical expression for the resonant frequency is introduced, which, together with numerical results, helps understand the RAP behavior in the space spanned by the transition rates for the case of reflecting and absorbing boundary conditions. The limited range of system parameters for which the RAP occurs is discussed. We show that a minimum and a maximum in the mean first passage time can be obtained when both boundaries are absorbing. Relationships to some biological systems are suggested.

Manipulation of dynamical systems by symmetry breaking.

We propose an approach to manipulate and control transport in Hamiltonian systems which are characterized by a mixed phase space. The approach is based on symmetry breaking of the phase space structure by applying a zero-mean periodic force for a finite duration of time. This induces time and space reversal asymmetry, which modifies the internal dynamics of the system and leads to directed transport. It is shown that our strategy allows to perform manipulations both with individual particles and with statistical ensembles of particles.

Ballistic flights and random diffusion as building blocks for Hamiltonian kinetics.

We propose a kinetic approach to transport in Hamiltonian systems with a mixed phase space. The approach is based on the decomposition of the dynamical picture into two contributions: (a) ballistic flights, and (b) random diffusion. The kinetic scheme leads to a stochastic process with statistical properties which are similar to those produced by the original Hamiltonian. We show that our approach helps in obtaining an insight into several properties of Hamiltonian kinetics such as anomalous diffusion, chaos-assisted population exchange, and current rectification. In particular, the chaos-assisted exchange offers a classical counterpart for the recently reported chaos-assisted tunneling.

From deterministic dynamics to kinetic phenomena.

We investigate a one-dimensional Hamiltonian system that describes a system of particles interacting through short-range repulsive potentials. Depending on the particle mean energy epsilon the system demonstrates a spectrum of kinetic regimes, characterized by their transport properties ranging from ballistic motion to localized oscillations through anomalous diffusion regimes. We establish relationships between the observed kinetic regimes and the "thermodynamic" states of the system. The nature of heat conduction in the proposed model is discussed.

Do strange kinetics imply unusual thermodynamics?

We introduce a fractional Fokker-Planck equation (FFPE) for Lévy flights in the presence of an external field. The equation is derived within the framework of the subordination of random processes which leads to Lévy flights. It is shown that the coexistence of anomalous transport and a potential displays a regular exponential relaxation toward the Boltzmann equilibrium distribution. The properties of the Lévy-flight FFPE derived here are compared with earlier findings for a subdiffusive FFPE. The latter is characterized by a nonexponential Mittag-Leffler relaxation to the Boltzmann distribution. In both cases, which describe strange kinetics, the Boltzmann equilibrium is reached, and modifications of the Boltzmann thermodynamics are not required.

Catalytic reactions with bulk-mediated excursions: mixing fails to restore chemical equilibrium.

In this paper we analyze the effect of the bulk-mediated excursions (BME) of reactive species on the long-time behavior of the catalytic Langmuir-Hinshelwood-like A+B-->0 reactions in systems in which a catalytic plane (CP) is in contact with a liquid phase, containing concentrations of reactive particles. Such BME result from repeated particles desorption from the CP, subsequent diffusion in the liquid phase, and eventual readsorption on the CP away from the initial detachment point. This process leads to an effective superdiffusive transport along the CP. We consider both "batch" reactions, in which all particles of reactive species were initially adsorbed onto the CP, and reactions followed by a steady inflow of particles onto the CP. We show that for batch reactions the BME provide an effective mixing channel and here the mean-field-type behavior emerges. On the contrary, for reaction followed by a steady inflow of particles, we observe essential departures from the mean-field behavior and find that the mixing effect of the BME is insufficient to restore chemical equilibrium. We show that a steady state is established as t--> infinity, in which the limiting value of the mean coverage of the CP depends on the particles' diffusion coefficient in the bulk liquid phase, and that the spatial distributions of adsorbed particles are strongly correlated. Moreover, we show that the relaxation to such a steady state is a power-law function of time, in contrast to the exponential time dependence describing the approach to equilibrium in perfectly stirred systems.

Geometry-controlled kinetics.

It has long been appreciated that the transport properties of molecules can control reaction kinetics. This effect can be characterized by the time it takes a diffusing molecule to reach a target-the first-passage time (FPT). Determining the FPT distribution in realistic confined geometries has until now, however, seemed intractable. Here, we calculate this FPT distribution analytically and show that transport processes as varied as regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, fall into the same universality classes. Beyond the theoretical aspect, this result changes our views on standard reaction kinetics and we introduce the concept of 'geometry-controlled kinetics'. More precisely, we argue that geometry-and in particular the initial distance between reactants in 'compact' systems-can become a key parameter. These findings could help explain the crucial role that the spatial organization of genes has in transcription kinetics, and more generally the impact of geometry on diffusion-limited reactions.