A. M. Berezhkovskii

Irreversible bimolecular reactions of Langevin particles

The reaction A+B→B is studied when the reactants diffuse in phase space, i.e., their dynamics is described by the Langevin equation. The steady-state rate constants are calculated for both the target (static A and mobile B’s) and trapping (mobile A and static B’s) problems when the reaction is assumed to occur at the first contact. For Brownian dynamics (i.e., ordinary diffusion), the rate constant for both problems is a monotonically decreasing function of the friction coefficient γ. For Langevin dynamics, however, we find that the steady-state rate constant exhibits a turnover behavior as a function of γ for the trapping problem but not for the target problem. This turnover is different from the familiar Kramers turnover of the rate constant for escape from a deep potential well because the reaction considered here is an activationless process.

THE INFLUENCE OF POLYMER ON THE DIFFUSION OF A SPHERICAL TRACER

We analyze how the addition of a small number of polymer molecules influences the diffusion constant of a spherical tracer, whose radius is small compared to the size of the polymer. We show that the polymer chain can be regarded as a two-dimensional object which is an impenetrable obstacle for the tracer. It is also shown that the diffusion constant of the tracer, in contrast to the solution viscosity, is independent of chain length, depending only on the monomer concentration.

Collision model for activated rate processes: turnover behavior of the rate constant

A theory of reaction rates is developed on the basis of the Bhatnagar–Gross–Krook model, which assumes instantaneous Maxwellization of the particle velocity at each collision. This model may be regarded as an alternative to the Kramers model for reaction dynamics in the condensed phase. The main results are two expressions for the rate constant for single- and double-well potentials. These cover the entire range of collision frequency. These expressions predict a turnover of the rate constant as a function of the collision frequency, analogous to the Kramers–Mel’nikov–Meshkov solution for the rate constant in the Kramers model. In contrast to the prediction for the Kramers model, the maximal value of the rate constant is noticeably below the TST estimate even for so high a barrier as 30kBT. This is a consequence of two facts: (1) The rate constant grows slowly from zero at small collision frequencies. (2) In addition, the rate of growth increases weakly with the barrier height, ΔU, as ln(ΔU/kBT). Simulate...

Turnover in the mean survival time for a particle moving between two traps

There is presently no solution to the problem of characterizing the statistics of trapping of a one-dimensional Brownian particle that moves between two traps, when velocity is relevant. We analyze a simplified version of this problem in which the continuous velocity is replaced by three velocities, ±v and 0, allowing the particle to make random transitions between these states. This model is easily solved and exhibits turnover behavior as a function of the noise intensity (i.e., the jump rate).

Some generalizations of the trapping problem

We consider two effects on kinetics that have not generally been taken into account in analyses of classical trapping models for the reactions A + T → T and A + T → 0: the effects due to correlations in the initial positions of the As and those due to the state of mobility of each of the species. Our analysis is formulated in terms of three-dimensional Brownian motion. We give a heuristic treatment of the short-time regime based on statistical properties of the Wiener sausage. The effects of initially correlated positions are modelled in terms of a set of a multiplicity of A particles located at the origin.