Fluctuation-dissipation relation and adiabatic noise-induced escape rate for nonequilibrium open systems

We consider the motion of an overdamped particle in a force field in presence of an external, adiabatic noise. We propose a new fluctuation-dissipation relation for this open nonequilibrium system and calculate the adiabatic noiseinduced rate of escape of the particle over a barrier. PACS number(s) : 05.40.-a,02.50.Ey Typeset using REVTEX 1 The motion of a Brownian particle in a fluid was first explained by Einstein in terms of fast thermal motion of fluid molecules striking the Brownian particle and causing it to undergo a random walk. One essential requirement of the theory is that the noise is of internal origin. This implies that the dissipative force which the Brownian particle experiences in course of its motion in the fluid and the stochastic force acting on the particle as a result of random impact of molecules arise from a common mechanism. From a microscopic point of view the system-reservoir Hamiltonian description developed over the last few decades [1,2] suggests that the coupling of the system and the reservoir co-ordinates determines both the noise and the dissipative terms of the Langevin equation describing the motion of the particle. It is therefore not difficult to anticipate that these two entities get related through the celebrated fluctuation-dissipation theorem (Einstein’s relation for diffusion and mobility is the first of its kind). The spiritual root of fluctuation-dissipation relation lies at the dynamical balance between inward flow of energy due to fluctuation of the reservoir into the system and the outward flow of energy from the system to the reservoir due to dissipation of the system. This dynamical balance is essential for attainment of a steady state of the nonequilibrium system. In their treatise on nonequilibrium statistical mechanics Lindenberg and West [2] have classified these systems as thermodynamically closed. However, there are quite a large number of physical situations, a comprehensive account of which has been given in [2], where the system is thermodynamically open, i.e., when the system is driven by an external noise which is independent of system’s characteristic damping. These are important for describing the effects of pump fluctuations on the emission of a dye laser [3], effects of fluctuating rate constants on a chemical reaction [4], effects of noise on electronic parametric oscillators [5] etc. The dynamics is still governed by a Langevin equation. The question is whether one can realize a relation between the fluctuation and the dissipation in such a situation. To the best of our knowledge this question has remained unanswered till date [2]. In the present letter we specifically address this issue and show that a relation of this kind can indeed be conceived for the steady state of the thermodynamically open system where the fluctuation is adiabatically slow such that it is characterized by a very 2 long correlation time τc where β ≪ ∆t ≪ τc . (1a) Here ∆t refers to the timescale over which we look for the average motion of the system. The latter inequality implies that 1 β , i.e., the inverse of damping constant defines the shortest timescale in the dynamics in contrast to the case of standard Brownian dynamics which obeys β ≫ ∆t ≫ τc . (1b) Our analysis is based on the following wellknown equation of motion of a particle of unit mass in one dimension when it is acted upon by an external field of force corresponding to a potential V (x) and an external, adiabatic stochastic force ξ(t), ẋ = − 1 β V (x) + 1 β ξ(t) . (2) where β and the correlation time τc of ξ(t) satisfy the inequality (1a). Also note that by virtue of this we have considered the overdamped limit. In a preceding paper [6] the equation of motion for probability density distribution function P (x, t) in phase space corresponding to the Langevin description (2) was derived. It has been shown that P (x, t) obeys the differential equation of motion which contains third order terms (beyond the usual FokkerPlanck terms) giving rise to third order noise. The appearance of these terms is generic for the stochastic process pertaining to the separation of timescales (1a) we consider here. The general expression for time evolution of probability density function is given by [6] ∂P (x, t) ∂t = 1 β ∂ ∂x [V (x)P (x, t)] + c01 β2 ∂P (x, t) ∂x2 − c2 β3 ∂ ∂x3 [V (x)P (x, t)] . (3) c0, c1 and c2 in Eq.(3) are as follows : c01 = c0 − c1 , c0 = ∫∞ 0 〈ξ(t) ξ(t− τ)〉 dτ c1 = ∫∞ 0 τ 〈ξ(t) dξ(t) dt ∣