Non-Markovian theory of activated rate processes. I. Formalism

Abstract

The escape of a particle from a potential well is treated using a generalized Langevin equation (GLE) in the low friction limit. The friction is represented by a memory kernel and the random noise is characterized by a finite correlation time. This non-Markovian stochastic equation is reduced to a Smoluchowski diffusion equation for the action variable of the particle and explicit expressions are obtained for the drift and diffusion terms in this equation in terms of the Fourier coefficients of the deterministic trajectory (associated with the motion without coupling to the heat bath) and of the Fourier transform of the friction kernel. The latter (frequency dependent friction) determines the rate of energy exchange with the heat bath. The resulting energy (or action) diffusion equation is used to determine the rate of achieving the critical (escape) energy. Explicit expressions are obtained for a Morse potential. These results for the escape rate agree with those from stochastic trajectories based on the original GLE. Non-Markovian effects are shown to have large effects on the rate of energy accumulation and relaxation within the well.

Modeling magnetosensitive ion channels in the viscoelastic environment of living cells

“…Then, our sensor would be really too slow to be functional. However, accounting for non-Markovian memory effects by a naive renormalization of the friction coefficient is patently wrong within the context of thermally activated dynamics, as it is well-recognized within nonMarkovian generalizations of Kramers rate theory [79][80][81][82][83]. Therefore, it is compelling to clarify the role of lowdimensional non-Markovian memory effects.…”

“…For φ ≥ φ max , a further increase of φ introduces a negative stiffness instability and the rod rotates to a new metastable minimum at φ min,2 ≈ 144. 81 o , where the channel opening probability becomes p(φ min,2 ) ≈ 0.926. At the first metastable minimum,…”

“…In fact, this asymptotic regime can become completely irrelevant for mesoscopic dynamics. This paradoxical fact also follows from the non-Markovian rate theory developed beyond the standard memoryless Kramers theory [78][79][80][81][82], see [83] for a review, and below. Namely this "much faster, not slower" [75,84,85], paradoxically due to subdiffusion, makes operating of such bistable magnetic sensors possible in viscoelastic cytosol.…”

Velocity dependence of friction and Kramers relaxation rates

“…The conventional means for describing this kind of problems are the Fokker-Planck equations (FPEs), or master equations, or the (generalized) Langevin equations for the reaction coordinate and the conjugated momentum of the particle in the external potential. Since the seminal work of Kramers, his approach has further been refined, and the influences of different dissipation (collision) mechanisms, 1,2,3,4,5 non-Poissonian collision statistics, 6,7,8 non-Markovian effects, 9,10,11,12 and position-dependent friction 13,14,15 on the reaction rates have been investigated (see also reviews 16,17,18,19 ).…”

Quantum phase-space function formulation of reactive flux theory

“…Over the decades the field has grown in various new directions, e.g. , extension of Kramers results to non-Markovian regime 4,5,6 , generalizations to higher dimensions 7,8 , inclusion of complex potentials 9,10 , generalization to open systems 11,12,13 , analysis of semiclassical and quantum effects 14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29 , thermal ratchet 30 and molecular motors 31 etc. These developments have been the subject of several reviews and monographs.We refer to 15,16,17,22 .…”