Yurii A. Makhnovskii

Kinetics of escape through a small hole

We study the time dependence of the survival probability of a Brownian particle that escapes from a cavity through a round hole. When the hole is small the escape is controlled by an entropy barrier and the survival probability decays as a single exponential. We argue that the rate constant is given by k=4Da/V, where a and V are the hole radius and the cavity volume and D is the diffusion constant of the particle. Brownian dynamics simulations for spherical and cubic cavities confirmed both the exponential decay of the survival probability and the expression for the rate constant for sufficiently small values of a.

Boundary homogenization for trapping by patchy surfaces

We analyze trapping of diffusing particles by nonoverlapping partially absorbing disks randomly located on a reflecting surface, the problem that arises in many branches of chemical and biological physics. We approach the problem by replacing the heterogeneous boundary condition on the patchy surface by the homogenized partially absorbing boundary condition, which is uniform over the surface. The latter can be used to analyze any problem (internal and external, steady state, and time dependent) in which diffusing particles are trapped by the surface. Our main result is an expression for the effective trapping rate of the homogenized boundary as a function of the fraction of the surface covered by the disks, the disk radius and trapping efficiency, and the particle diffusion constant. We demonstrate excellent accuracy of this expression by testing it against the results of Brownian dynamics simulations. (c) 2004 American Institute of Physics.

Communications: Drift and diffusion in a tube of periodically varying diameter. Driving force induced intermittency

We show that the effect of driving force F on the effective mobility and diffusion coefficient of a particle in a tube formed by identical compartments may be qualitatively different depending on the compartment shape. In tubes formed by cylindrical (spherical) compartments the mobility monotonically decreases (increases) with F and the diffusion coefficient diverges (remains finite) as F tends to infinity. In tubes formed by cylindrical compartments, at large F there is intermittency in the particle transitions between openings connecting neighboring compartments.

TRANSIENT DIFFUSION IN A TUBE WITH DEAD ENDS

A particle diffusing in a tube with dead ends, from time to time enters a dead end, spends some time in the dead end, and then comes back to the tube. As a result, the particle spends in the tube only a part of the entire observation time that leads to slowdown of its diffusion along the tube. We study the transient diffusion in a tube with periodic identical dead ends formed by cavities of volume V(cav) connected to the tube by cylindrical channels of length L and radius a, which is assumed to be much smaller than the tube radius R and the distance l between neighboring dead ends. Assuming that the particle initial position is uniformly distributed over the tube, we analyze the monotonic decrease of the particle diffusion coefficient D(t) from its initial value D(0)=D, which characterizes diffusion in the tube without dead ends, to its asymptotic long-time value D(infinity)=D(eff)<D. We derive an expression for the Laplace transform of D(t), denoted by D(s), where s is the Laplace parameter. Although the expression is too complicated to be inverted analytically, we use it to find the relaxation time of the process as a function of the geometric parameters of the system mentioned above. To check the accuracy of our results, we ran Brownian dynamics simulations and found the mean squared displacement of the particle as a function of time by averaging over 5x10(4) realizations of the particle trajectory. The time-dependent mean squared displacement found in simulations is compared with that obtained by numerically inverting the Laplace transform of the mean squared displacement predicted by the theory, which is given by 2D(s)/s. Comparison shows excellent agreement between the two time dependences that support the approximations used when developing the theory.

Communication: Turnover behavior of effective mobility in a tube with periodic entropy potential

Using Brownian dynamics simulations, we study the effective mobility and diffusion coefficient of a point particle in a tube formed from identical compartments of varying diameter, as functions of the driving force applied along the tube axis. Our primary focus is on how the driving force dependences of these transport coefficients are modified by the changes in the compartment shape. In addition to monotonically increasing or decreasing behavior of the effective mobility in periodic entropy potentials reported earlier, we now show that the effective mobility can even be nonmonotonic in the driving force.

Particle size effect on diffusion in tubes with dead ends: Nonmonotonic size dependence of effective diffusion constant

Diffusion of a spherical particle of radius r in a tube with identical periodic dead ends is analyzed. It is shown that the effective diffusion constant follows the Stokes-Einstein relation, D(eff)(r) proportional to 1r, only when r is larger or much smaller than the radius of the dead end entrance. In between, D(eff)(r) not only deviates from the 1r behavior but may also even become a nonmonotonic function, which increases with the particle radius for a certain range of r.

Force-dependent mobility and entropic rectification in tubes of periodically varying geometry.

We investigate transport of point Brownian particles in a tube formed by identical periodic compartments of varying diameter, focusing on the effects due to the compartment asymmetry. The paper contains two parts. First, we study the force-dependent mobility of the particle. The mobility is a symmetric non-monotonic function of the driving force, F, when the compartment is symmetric. Compartment asymmetry gives rise to an asymmetric force-dependent mobility, which remains non-monotonic when the compartment asymmetry is not too high. The F-dependence of the mobility becomes monotonic in tubes formed by highly asymmetric compartments. The transition of the F-dependence of the mobility from non-monotonic to monotonic behavior results in important consequences for the particle motion under the action of a time-periodic force with zero mean, which are discussed in the second part of the paper: In a tube formed by moderately asymmetric compartments, the particle under the action of such a force moves with an effective drift velocity that vanishes at small and large values of the force amplitude having a maximum in between. In a tube formed by highly asymmetric compartments, the effective drift velocity monotonically increases with the amplitude of the driving force and becomes unboundedly large as the amplitude tends to infinity.

Stochastic gating influence on the kinetics of diffusion-limited reactions

We study how the kinetics of diffusion-influenced reactions is modified when the reactivity of species fluctuates in time (stochastically gated) with emphasis on the many-particle aspect of the problem. Because of the fact that the dynamics of ligand binding to proteins originally motivated the problem, it is considered in that context. Recently, Zhou and Szabo [J. Phys. Chem. 100, 2597 (1996)] have demonstrated many-particle effects in the problem and found that the kinetics of reaction between a gated protein with a large number of ligands significantly differs from that between a protein and gated ligands. With our approach, the difference between the kinetics of ligand-gated and protein-gated reactions appears formally the same as the difference between the target and trapping problems despite the origin of the corresponding effects and their manifestations are distinctly different. A simple approximate method to treat the many-particle effects is proposed. The theory is applied to a particular two-st...

ROLE OF TRAP CLUSTERING IN THE TRAPPING KINETICS

The Smoluchowski theory describes the kinetics of trapping of Brownian particles by absorbers randomly placed without correlations between their positions. We generalize this theory to take trap correlations into account when traps occur in spherical clusters distributed in space in a noncorrelated manner. A cluster contains n traps uniformly distributed within the cluster. An effective medium treatment is used to handle trap-correlation effects. Explicit expressions are obtained for the time-dependent rate coefficient and the particle survival probability valid for the entire range of n and cluster radius R. We analyze how the trap clustering manifests itself in the kinetics. In particular, we show that there exists a domain of the parameters n and R, where the kinetics is well fitted by a stretched exponential function for more than 99% of the decay. Such behavior should be contrasted to the essentially exponential kinetics predicted by Smoluchowski theory for noncorrelated traps.

Net transport due to noise-induced internal reciprocating motion

We consider a system of two coupled Brownian particles fluctuating between two states. The fluctuations are produced by both equilibrium thermal and external nonthermal noise, the transition rates depending on the interparticle distance. An externally induced modulation of the transition rates acts on the internal degree of freedom (the interparticle distance) and generates reciprocating motion along this coordinate. The system moves unidirectionally due to rectification of the internal motion by asymmetric friction fluctuations and thus operates as a dimeric motor that converts input energy into net movement. The properties of the motor are primarily determined by the properties of the reciprocating engine, represented by the interparticle distance dynamics. Two main mechanisms are recognized by which the engine operates: energetic and informational. In the physically important cases where only one of the motion-inducing mechanisms is operative, exact solutions can be found for the model with linearly coupled particles. We focus on the informational mechanism, in which thermal noise is involved as a vital component and the reciprocating velocity exhibits a rich behavior as a function of the model parameters. An efficient rectification method for the reciprocating motion is also discussed.