Generalization of the Classical Kramers Rate for Non-markovian Open Systems out of Equilibrium I Introduction: Metastability and Fluctuations Ii Generalizing the Langevin Approach Iii Our Fokker–planck Equation

We analyze the behavior of a Brownian particle moving in a double-well potential. The escape probability of this particle over the potential barrier from a metastable state toward another state is known as the Kramers problem. In this work we generalize Kramers' rate theory to the case of an environment always out of thermodynamic equilibrium reckoning with non-Markovian effects. We consider a Brownian particle immersed in an environment (e.g., a fluid) under the influence of an external potential. Due to the environmental fluctuations the escape rate of this particle over the barrier separating two metastable states –in a double-well potential, for instance– is known as the Kramers problem [1, 2], even though the rate theory has been already tackled by van't Hoff and Arrhenius as late as 1880 [2]. For many years this phenomenon has had various applications in physical, chemical, astronomical , and biological systems [1, 2, 3, 4]. Originally, Kramers [1] investigated the Brownian movement in a reservoir at thermodynamic equilibrium taking into account only Markovian effects. He also worked out a method for calculating the escape probability from a Fokker–Planck equation (nowadays known as the Kramers equation) associated with a given set of Langevin equations. Even then, several generalizations of this Kramers' pioneering work have arisen in the literature with experimental verifications [2, 3, 5, 6], e.g., in Josephson junction measuring the decay of the supercurrent. In the theory of escape rate non-Markov and/or nonequilibrium features are commonly introduced through memory effects contained in the friction kernel present in generalized Langevin equations and using either the Fokker– Planck equation found by Adelman and Mazo [7] or a non-Markovian Smolu-chowski equation [8], or yet using the Fokker–Planck equation in energy picture [9]. The equilibrium Kramers rate using only the non-Markovian generalized Langevin equation is investigated in [10]. In nonequilibrium situations the Kramers theory has been also studied in Markovian open systems with oscillating barriers [11], as well as in periodically driven stochastic systems [12]. It should be remarked that a feature common to all above approaches is that the mean value of the stochastic term present in the Langevin equations is zero. Following a diverse way, in the present paper we propose a generalization of the Langevin equations and construct the respective Fokker-Planck equation. In this context we evaluate the Kramers escape rate away from the equilibrium taking into account non-Markovian effects related to different time scales inherent …