Yu. A. Makhnovskii

Wiener sausage volume moments

The statistical characteristics of a spatial region visited by a spherical Brownian particle during timet (Wiener sausage) are investigated. The expectation value and dispersion of this quantity are obtained for a space of arbitrary dimension. In the one-dimensional case the distribution of probability density and the moments of any order are determined for this quantity.

Time-dependent diffusion in tubes with periodic partitions

The presence of obstacles leads to a slowdown of diffusion. We study the slowdown when diffusion occurs in a tube, and obstacles are periodically spaced identical partitions with circular apertures of arbitrary radius in their centers. The mean squared displacement of a particle diffusing in such a system at large times is given by ⟨Δx2(t)⟩=2Defft, t→∞, where Deff is the effective diffusion coefficient, which is smaller than the particle diffusion coefficient in the tube with no partitions, D0. The latter characterizes the short-time behavior of the mean squared displacement, ⟨Δx2(t)⟩=2D0t, t→0. Thus, the particle diffusion coefficient decreases from D0 to Deff as time goes from zero to infinity. We derive analytical solutions for the Laplace transforms of the time-dependent diffusion coefficient and the mean squared displacement that show how these functions depend on the geometric parameters of the tube. To obtain these solutions we replace nonuniform partitions with apertures by effective partitions th...

Kinetics of diffusion-controlled reactions

Abstract We suggest a new approach to the problem of taking account of many-body effects in the kinetics of diffusion-controlled reactions of the type A+B→Pr for the survival probability of particles A which uniformly describes the process within the entire time interval is obtained. It is shown that the conventional expression which was derived without regarding many-body effects always underestimates the probability of survival of parti stage of the process. At the asymptotically large times when the conventional expression appreciably lowers the survival probability the fluctuation as

MANY-BODY EFFECTS IN DIFFUSION-LIMITED KINETICS

We review a novel approach to treating many-body effects in diffusion-limited kinetics. The derivation of the general expression for the survival probability of a Brownian particle in the presence of randomly distributed traps is given. The reduction of this expression to both the Smoluchowski solultion and the wellknown asymptotic behavior is demonstrated. It is shown that the Smoluchowski solution gives a lower bound for the particle survival probability. The correction to the Smoluchowski solution which takes into account the particle death slowdown in the initial process stage is described. The steady-state rate-constant concentration dependence and the reflection of many-body effects in it are discussed in detail.

Communication: Drift velocity of Brownian particle in a periodically tapered tube induced by a time-periodic force with zero mean: Dependence on the force period

We study the drift of a Brownian particle in a periodically tapered tube, induced by a longitudinal time-periodic force of amplitude ∣F∣ that alternates in sign every half-period. The focus is on the velocity dependence on the force period, which is usually considered not tractable analytically. For large ∣F∣ we derive an analytical solution that gives the velocity as a function of the amplitude and the period of the force as well as the geometric parameters of the tube. The solution shows how the velocity decreases from its maximum value to zero as the force period decreases from infinity (adiabatic regime) to zero. Our analytical results are in excellent agreement with those obtained from 3D Brownian dynamics simulations.

Driven Diffusion in a Periodically Compartmentalized Tube: Homogeneity versus Intermittency of Particle Motion

We study the effect of a driving force F on drift and diffusion of a point Brownian particle in a tube formed by identical cylindrical compartments, which create periodic entropy barriers for the particle motion along the tube axis. The particle transport exhibits striking features: the effective mobility monotonically decreases with increasing F, and the effective diffusivity diverges as F→∞, which indicates that the entropic effects in diffusive transport are enhanced by the driving force. Our consideration is based on two different scenarios of the particle motion at small and large F, homogeneous and intermittent, respectively. The scenarios are deduced from the careful analysis of statistics of the particle transition times between neighboring openings. From this qualitative picture, the limiting small-F and large-F behaviors of the effective mobility and diffusivity are derived analytically. Brownian dynamics simulations are used to find these quantities at intermediate values of the driving force for various compartment lengths and opening radii. This work shows that the driving force may lead to qualitatively different anomalous transport features, depending on the geometry design.

Smoluchowski-type theory of stochastically gated diffusion-influenced reactions

The Smoluchowski–Collins–Kimball theory of irreversible diffusion-influenced reactions with one of the reactants in excess is generalized to the case of stochastic gating when one of the reactants can be in one of M states. Distinct states are characterized by various efficiencies of the reaction of contacting partners. General expressions are derived for the rate constant and for the survival probability of the reactant which is in deficiency. We present these quantities in terms of the solution of the isolated pair problem. The difference between the cases when gating is due to the reactant, which is in excess, and one, which is in deficiency, is explicitly demonstrated. General relationships between the rate constants and the survival probabilities in the two cases are established. We show that in the former case the reaction goes faster compared to the latter one. To make the problem treatable analytically in the case when gating is due to the reactant which is in deficiency, a partial mean-field appr...

Directed transport of a Brownian particle in a periodically tapered tube

The problem of the motion of a Brownian particle in a periodically tapered tube induced by a time-periodic longitudinal force with zero mean is considered. Under the action of this force, the particle is shown to drift in a direction opposite to the constant load force applied to it. Analytical solutions for the drift velocity, the stopping force (the load causing the effect to disappear), and the efficiency of converting the energy introduced by perturbations into directed motion have been obtained at a large amplitude of the driving force, when the effect being discussed is maximal. In the range of its applicability extending from zero to asymptotically large force switching frequencies (proportional to the amplitude of the driving force), these solutions are in good agreement with the results of Brownian dynamics simulations.

Mutual influence of traps on the death of a Brownian particle

Abstract The role of many-body effects in the kinetics of diffusion-limited reactions has been studied in the simplest situation, i.e. in the case of death of a Brownian particle in two stationary traps. For this purpose, the survival probability of the particle was calculated using two methods: the traditional one, where the many-body effects are neglected, and a more precise one which allows one to take these effects into account. Comparison of the results obtained showed that neglecting many-body effects can lead to both over- and under-estimated values of the survival probability of the particle during a time interval t . The role of these effects in a more realistic situation when there are many traps is discussed.

Fluctuation slow-down of the death of Brownian particles in the case of movable traps

The influence of trap diffusion on the fluctuation slow-down of death of Brownian particles, discovered earlier in the case of stationary traps, is analysed. It is shown that fluctuation slow-down also takes place with movable traps if the diffusion is slow enough.