Turnover of activationless escape rate for collisional dynamics

We consider the Bhatnagar–Gross–Krook (BGK) model for the dynamics of a particle in the phase space. Namely, the particle follows Newtonian trajectories that are randomly interrupted by collisions which thermalize its velocity. For this collisional model, we analyze the activationless escape of a free particle from a unit interval as a function of the collision frequency, γ. Approximate analytic expressions, which compare favorably with simulations, are derived for the effective and asymptotic rate constants, k and Γ, that describe the escape kinetics. Both rate constants show a turnover behavior as functions of γ similar to the rate constants found when the particle motion is governed by the Langevin dynamics. It is found that as γ→0, k∼1/ln(1/γ) (with an amplitude 13 times smaller than in Langevin dynamics) and Γ∼γ (rather than Γ∼γ1/3 in Langevin dynamics) while when γ→∞ both rate constants vanish as γ−1 like in the Langevin dynamics.