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Analytical and numerical investigation of escape rate for a noise driven bath

We consider a system-reservoir model where the reservoir is modulated by an external noise. Both the internal noise of the reservoir and the external noise are stationary, Gaussian and are characterized by arbitrary decaying correlation functions. Based on a relation between the dissipation of the system and the response function of the reservoir driven by external noise we numerically examine the model using a full bistable potential to show that one can recover the turn-over features of the usual Kramers' dynamics when the external noise modulates the reservoir rather than the system directly. We derive the generalized Kramers' rate for this nonequilibrium open system. The theoretical results are verified by numerical simulation.


I. INTRODUCTION
More than half a century ago Kramers proposed a diffusion model for chemical reactions in terms of the theory of Brownian motion in phase space [1]. Since then the model and several of its variants have been ubiquitous in many areas of physics, chemistry and biology for understanding the nature of activated processes in classical [2][3][4][5][6][7], quantum and semiclassical [8][9][10][11] systems, in general. These have become the subject of several reviews [12][13][14] and monograph [15] in the recent past.
In the majority of these treatments one is essentially concerned with an equilibrium thermal bath at a finite temperature which stimulates the reaction co-ordinate to cross the activation energy barrier. The inherent noise of the medium is of internal origin. This implies that the dissipative force which the system experiences in course of its motion in the medium and the stochastic force acting on the system as a result of random impact from the constituents of the medium arise from a common mechanism. From a microscopic point of view the system-reservoir Hamiltonian description [16][17][18][19] developed over the decades suggests that the coupling of the system and the reservoir co-ordinates determines both the noise and the dissipative terms in the Langevin equation describing the motion of the system. It is therefore not difficult to anticipate that these two entities get related through a fluctuation-dissipation relation [20] ( these systems are sometimes classified as thermodynamically closed system in contrast to the systems driven by external noise in nonequilibrium statistical mechanics [21] ). However, when the reservoir is modulated by an external noise it is likely that this relation gets affected in a significant way. Since the modulation of the reservoir crucially depend on its response function, one can further envisage a connection between the dissipation of the system and the response function of the reservoir due to the external noise from a microscopic point of view.
In the present paper we explore this connection in the context of activated rate processes when the reservoir is modulated by an external noise. Specifically our object here is twofold: (i) to explore the role of reservoir response as a function of external noise on the system dynamics in contrast to direct driving of the system by external noise, (ii) to calculate the generalized Kramers' rate for the steady state of this nonequilibrium open system taking full care of thermodynamic consistency condition. Both the internal and the external noises are Gaussian, stationary and are characterized by arbitrary decaying correlation functions.
While the internal noise of the reservoir is thermal, the external noise may be of thermal or non-thermal type. We consider the stochastic motion to be spatial diffusion limited and calculate the rate of escape in the intermediate to strong damping regime. It is worth to mention that the externally generated nonequilibrium fluctuations can bias the Brownian motion of a particle in an anisotropic medium and may be used for designing molecular motors and pumps [22]. We further mention that nonequilibrium, non-thermal systems have also been investigated phenomenologically by a number of workers in several other contexts, e.g., for examining the role of color noise in stationary probabilities [23], properties of nonlinear systems [24], nature of crossover [25], rate of diffusion limited coagulation processes [26], effect of monochromatic noise [27], etc. While these treatments concern direct driving of the system by an external noise the present consideration is based on modulation of the bath. A number of different situations depicting the modulation of the bath by an external noise may be physically relevant. As, for example, we consider a simple unimolecular conversion (say, an isomerization reaction) from A −→ B. The reaction is carried out in a photochemically active solvent under the influence of external fluctuating light intensity. Since the fluctuations in the light intensity result in the fluctuations in the polarization of the solvent molecules, the effective reaction field around the reactant system gets modified. Provided the required stationarity of this nonequilibrium open system is maintained (which is not difficult in view of the experiments performed in the studies of external noise-induced transitions in photochemical systems [28]) the dynamics of barrier crossing becomes amenable to the present theoretical analysis that follows.
The remaining part of this paper is organized as follows: In Sec. II we discuss a systemreservoir model where the later is modulated by an external noise and establish an interesting connection between the dissipation of the system and the response function of the reservoir due to external noise. The stochastic motion in a linearized potential field is described in terms of a Fokker-Planck equation in Sec. III. Based on the traditional flux over population method [29] we derive in Sec. IV the generalized expression for the Kramers' rate of escape from a metastable well. In Sec. V we numerically analyze the model and the bath modulated dynamics for the full potential and verify the theoretical rate with numerical simulation. The paper is concluded in Sec. VI.

AN EXTERNAL NOISE
We consider a classical particle of mass M is linearly coupled to a heat bath of N harmonic oscillators driven by an external noise. The total Hamiltonian is given by [18] In Eq.(1), x and p are the co-ordinate and momentum of the system particle; (q i , p i ) are the variables associated with the i-th oscillator and ω i and m i are the corresponding frequency and mass, respectively. g i x measures the interaction between the particle and the bath.
V (x) is the potential energy of the particle. H int is assumed to be of the form The coupling function κ i measures the strength of interaction and ǫ(t) is the external noise which we assume to be stationary and Gaussian with zero mean, i.e., ǫ(t) e = 0 and is characterized by an arbitrary correlation function as follows; Here . . . e implies the averaging over the external noise.
We then eliminate the bath degrees of freedom in the usual way [16,18,19,30] to obtain the following generalized Langevin equation [ while constructing Eq.(3) we have set M and m i equal to unity ] where f (t) is the internal forcing function generated through the coupling between the system and the heat bath and is given by In Eq. (3), π(t) is a fluctuating force term due to the external noise ǫ(t) and is given by where The form of Eq.(3) therefore suggests that the system is driven by two forcing functions f (t) and π(t). f (t) depends on the initial conditions of the bath oscillators for a fixed choice of the initial condition of the system degrees of freedom. To define the statistical properties of f (t), we assume that the initial distribution is one in which the bath is equilibrated at t = 0 in the presence of the system but in the absence of the external noise agency such that Now, at t = 0 + , the external noise agency is switched on and the bath is modulated by ǫ(t). The system is governed by Eq. (3), where, apart from the internal noise f (t), another fluctuating force π(t) appears, that depends on the external noise ǫ(t). Therefore, one can define an effective noise ξ(t)(= f (t) + π(t)) whose correlation is given by along with ξ(t) = 0, where . . . means we have taken two averages independently. It should be emphasized that the above relation (8) is not a fluctuation-dissipation relation due to the appearance of the external noise intensity. Rather it serves as a thermodynamic consistency condition.
Let us now digress a little bit about π(t). The statistical properties of π(t) are determined by the normal mode density of the bath frequencies, the coupling of the system with the bath, the coupling of the bath with the external noise and the external noise itself. Eq. (6) is reminiscent of the familiar linear relation between the polarization and the external field where π and ǫ play the role of the former and the later, respectively. ϕ(t) can then be interpreted as a response function of the reservoir due to external noise ǫ(t). The very structure of π(t) suggests that this forcing function although originating from an external force is different from a direct driving force acting on the system. The distinction lies at the very nature of the bath characteristics (rather than system characteristics) as reflected in the relation (6) and (7).
With the coupling coefficients g(ω) = g 0 / √ τ c ω and κ(ω) = √ τ c ωκ 0 , in the continuum limit [30] γ(t) and ϕ(t) reduce to the following forms and where g 0 and κ 0 are constants and τ −1 c is the cutoff frequency of the oscillator. D(ω) is the density of modes of the heat bath.
From the above two relations, we obtain Eq. (11) is an important content of the present model. This expresses how the dissipative kernel γ(t) depends on the response function ϕ(t) of the medium due to external noise ǫ(t) We now consider the system to be a harmonically bound particle of unit mass and of frequency ω 0 . Then because of Eq.(6) the Langevin equation (3) becomeṡ The Laplace transform of Eq. (12) allows us to write a formal solution for the displacement of the form where we have made use of the relation (11) explicitly. Here with x(0) and v(0) being the initial position and initial velocity of the oscillator, respectively, which are nonrandom and The kernel h(t) is the Laplace inversion of , is the Laplace transform of the friction kernel γ(t), and The time derivative of Eq. (13) yields where Next we calculate the variances. From the formal solution of x(t) and v(t), the explicit expressions for the variances are obtained which are given below; and where we have assumed that the noises f (t) and ǫ(t) are symmetric with respect to the time argument and have made use the fact that f (t) and ǫ(t) are uncorrelated.
Due to the Gaussian property of the noises f (t) and ǫ(t) and the linearity of the Langevin equation (12), we see that the joint probability density p(x, v, t) of the oscillator must be Gaussian. The joint characteristic function associated with the density is Using the method of characteristic function [31,32] and the above expression (24) we find the general Fokker-Planck equation associated with the probability density function p(x, v, t) for the process (12); and The functions φ(t) and ψ(t) are defined by where the covariances are to be calculated for a particular given noise process.
For the internal noise processes it had been shown earlier that for several models the various time dependent parametersω 2 0 (t),γ(t), etc. do exist asymptotically as t → ∞. The above consideration shows that h(t), h 1 (t), etc. do not depend on the nature of the noise but depend only on the relaxationγ(t).
We now discuss the asymptotic properties of φ(t) and ψ(t), which in turn are dependent on the variances σ 2 xx (t) and σ 2 vv (t), as t → ∞ since they play a significant role in our further analysis that follows.
From Eqs. (21) and (22), we may write where 'i' denotes the part corresponding to internal noise f (t) and 'e' corresponds to the external noise ǫ(t). Since the average velocity of the oscillator is zero as t → ∞ we see from Eq. (19) that h(t) and h 1 (t) must be zero as t → ∞. Also from Eq. (14) we observe that the function χ x (t) must decay to zero for long times. Hence, from Eq.(15) we see that the Now, σ 2(i) xx (t) and σ 2(i) vv (t) of Eqs. (21) and (22) can be written in the form and From the above two expressions [ Eqs. (28) and (29) ] we see that It is important to note that these stationary values are not related to the intensity and correlation time of the internal noise.
We next consider the parts, σ 2(e) xx (t) and σ 2(e) vv (t), due to the presence of the external noise.
The Laplace transform of Eq.(13) yields the expressioñ where From the above equation (31) we can calculate the variance σ 2 xx in the Laplace-transformed space which can be identified as the Laplace transform of Eq. (21). Thus, for the part σ 2(e) xx (t) we observe that,σ 2(e) xx (s) contains terms like κ 0 g 0 τ c ω 2 0h (s) 2 ǫ 2 (s) e . Since, we have assumed the stationarity of the noise ǫ(t), we conclude that ifC(0) exists [ where C(t−t ′ ) = ǫ(t)ǫ(t ′ ) e ], then the stationary value of σ 2(e) xx (t) exists and becomes a constant that depends on the correlation time and the strength of the noise. Similar argument is also valid for σ 2(e) vv (t).
Summarizing the above discussions we note that, (i) the internal noise-driven parts of σ 2 xx (t) and σ 2 vv (t), i.e., σ 2(i) xx and σ 2(i) vv approach the fixed values which are independent of the noise correlation and the intensity as t → ∞, Hence we conclude, following the Ref. (33) and our preceding discussions that even in presence of an external noise the above terms do exist asymptotically and we write the steady state Fokker-Planck equation for the asymptotic values of the parameters as, where,ω 2 0 ,γ, φ(∞), ψ(∞), etc. are to be calculated from the general definition (26) for the steady state. As an explicit example we consider the case of a δ-correlated external noise and Ornstein-Uhlenbeck internal noise for which we provide the expressions for variances σ 2 xx (t), σ 2 vv (t) and σ 2 xv (t) and the relaxation function h(t) given in the Appendix-A.
The general steady state solution of the above equation (33) is where and Z is the normalization constant. The solution (34) can be verified by direct substitution.
The distribution (34) is not an equilibrium distribution. This stationary distribution for the nonequilibrium open system plays the role of an equilibrium distribution of the closed system which may, however, be recovered in the absence of external noise term.

IV. KRAMERS' ESCAPE RATE
We now turn to the problem of decay of a metastable state. In Kramers approach [1], the particle coordinate x corresponds to the reaction coordinate and its values at the minima of the potential V (x) denotes the reactant and product states.
Linearizing the motion around barrier top at x = x b the Langevin equation (3) can be written down asẏ where, y = x − x b and the barrier frequency ω 2 b is defined by Correspondingly the motion of the particle is governed by the Fokker-Planck equation (25) ∂p where, the suffix 'b' indicates that all the coefficients are to be calculated using the general definition (26) for the barrier top region.
It is apparent from Eqs. (33) and (38) that since the dynamics is non-Markovian and the system is thermodynamically open one has to deal with the renormalized frequencies ω 0 andω b near the bottom or top of the well, respectively. We make the ansatz that the nonequilibrium, steady state probability p b , generating a nonvanishing diffusion current j, across the barrier is given by where V (x) is the renormalized linear potential as After making use of the appropriate transformations and boundary conditions for reduced distribution functions [7] we obtain the barrier crossing rate k given by Here E 0 is the activation energy, . Since the temperature due to internal thermal noise, the strength of the external noise and the damping constant are buried in the parameters D 0 , D b , ψ 0 , ψ 0 and Λ the generalized expression look somewhat cumbersome.
We point out that the subscripts '0' and 'b' in D and ψ refer to the well or barrier top region, respectively. Eq.(43) is one of the key results of this paper. We note here that The dependence of the rate on the parameters can be exposed explicitly once we consider the typical cases.  system are compared to that for the standard Kramers' turnover, i.e., when the dynamics is calculated in absence of the external noise ( dashed curve ). As expected the rate in the driven system ( be it directly or through bath modulation ) is always higher that in the undriven one. Fig.(1) also shows that although in the low damping region the direct driving causes a much higher rate, it is, however, the bath modulation which becomes more effective in inducing activated barrier crossing in the high friction regime. It is thus interesting to note that the barrier crossing dynamics of the system in contact with an external noise modulated bath captures the essential turnover features of the Kramers' dynamics of the closed system. We therefore realize that although open, the nature of the response function of the reservoir as well as the thermodynamic consistency condition (8), make the open system feel like a closed system.
In Fig.(2) we show the variation of inverse of the mean first passage time as a function of the strength of the external noise D, keeping all other parameter same as before but for g 2 0 = 2.0. It is apparent that the barrier crossing is more facilitated by modulating the bath than driving the system directly for higher values of external noise strength.
B. The rate : Theoretical results vs. numerical simulation So far we have considered the full potential V (x) = 1 4 x 4 − 1 2 x 2 and vanishingly small correlation times for external and internal noises for numerical computation. We now turn to our basic theoretical result Eq.(43) which is a generalization of Kramers' rate for bath modulated dynamics for intermediate to strong damping regime. To examine its validity we calculate the rate (43) as a function of the damping constant g 2 0 ( g 0 is related to both γ(t) and ϕ(t) in Eqs. (ii) Many of the earlier treatments of the rate concern direct (phenomenological) driving (iii) Based on numerical simulation of the full model potential we show that one can recover the turnover features of the Kramers' dynamics when the external noise modulates the reservoir rather than the system directly. This recovery, we believe, is an offshoot of the derived thermodynamic consistency condition (8).
(iv) Provided the long time limit of the moments for the stochastic processes pertaining to the external and internal noises characterized by arbitrary decaying correlation functions exist, the expression for generalized Kramers' rate of barrier crossing for the open system we derive here is fairly general. We have shown that it agrees reasonably well with numerical simulation using full potential for this model. open systems and may serve as a basis for studying processes occurring within irreversibly driven environments [6,35] and for thermal ratchet problems [22].  We consider a particular case as an example where the external noise ǫ(t) is δ-correlated and the internal noise is an Ornstein-Uhlenbeck process, i.e., Consequently, from the fluctuation-dissipation relation we derive the dissipative kernel as, It should be noted that for τ c → 0, the above noise process become δ-correlated.