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). Simulated results indicate good agreement with the theory.