Reaction rate theory: weak- to strong-friction turnover in Kramersʹ Fokker-Planck model

We present a unified treatment for both weak and strong friction — including the turnover regime — of Kramersʹ Fokker-Planck model for activated rate processes. No recourse to a specific microscopic model (à la Grabert et al.) or Langevin dynamics (à la Graham) will be made. Upon introducing some new theoretical concepts such as a constrained Gaussian transformation (CGT) and a dynamical extension of phase space (EPS), the analysis proceeds in terms of the unstable-mode energy and is systematic in a small parameter. We also introduce a nonlinear toy model for barrier recrossing for a simplified discussion of aspects like finite-barrier corrections, the boundary layer approximation, and exact solution in the absence of potential conditions.