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Resonant activation in the presence of non-equilibrated baths

We study the generic problem of the escape of a classical particle over a fluctuating barrier under the influence of non-Gaussian noise mimicking the effects of not-fully equilibrated bath. Our attention focuses on the effect of the stable noises on the mean escape time and on the phenomenon of resonant activation (RA). Possible physical interpretation of the occurrence of L\'evy noises in the system of interest is discussed and the connection between the Tsallis statistics and the Fractional Fokker Planck Equation is addressed.


I. INTRODUCTION
In a classical Langevin approach the influence of the bath of surrounding molecules on a Brownian particle is described in terms of a mean-field, time-dependent stochastic force which is commonly assumed to be white Gaussian noise. That postulate is compatible with the assumption of a short correlation time of fluctuations, much shorter than the time-scale of the macroscopic motion and assumes that weak interactions with the bath lead to independent random variations of the parameter describing the motion. In more formal, mathematical terms Gaussianity of the state-variable fluctuations is a consequence of the Central Limit Theorem which states that normalized sum of independent and identically distributed random variables with finite variance converges to the Gaussian probability distribution. If, however, after random collisions jump lengths are ruled by broad distributions leading to the divergence of the second moment, the statistics of the process changes significantly. The existence of the limiting distribution is then guaranteed by the generalized Lévy-Gnedenko [1] limit theorem. According to the latter, normalized sums of independent, identically distributed (i.i.d) random variables with infinite variance converge in distribution to the Lévy statistics. At the level of the Langevin equation, Lévy noises are generalization of the Brownian motion and describe results of strong collisions between the test particle and the surrounding environment. In this sense, they lead to different models of the bath that go beyond a standard "close-toequilibrium" Gaussian description.
As documented elsewhere [2,3], not fully thermalized systems or systems driven away from the equilibrium can manifest interesting physical properties. In particular, such systems may exhibit large energy fluctuations with probabilities higher than those predicted by the Gaussian statistics. Nonequilibrated heat reservoir can be thus considered as a source of non-Gaussian noises. Formalisms that give physical background for this phenomena are based on the idea of nonextensive thermodynamics, established on a Tsallis statistics [4], and the Fractional Fokker Planck Equation (FFPE) [5]. The latter (FFPE) have been successfully applied [5,6,7,8,9] for describing anomalous diffusion processes, for which the nonlocal character brought about by the stable Lévy noise leads to the replacement of the local spatial derivatives in the diffusion term of the Fokker-Planck equation by a fractional derivative.
In contrast to the fractional calculus, the Tsallis statistics approach to stable, Lévy-type fluctuations is based on the introduction of a special entropy form [4,10] from which examined thermodynamical quantities can be derived [11]. The main property of the Tsallis entropy is its nonextensivity, i.e. the entropy of the system containing two noninteracting subsystems is different from the sum of the subsystems' entropies typical for equilibrium Gibssian ensembles and Gaussian measures.
Various examples of everyday life phenomena and a wide extent of experimental observations [2,12,13] show existence of long-range correlations, disorder, cooperativity and deviations from the Gaussian statistics, thus suggesting a strong need of study more general probability distribution than just Gaussian ones. In particular, α-stable distributions have been observed in anomalous dynamics and strange kinetics in amorphous semiconductors and glassy systems [8]. Lévy-flight models turn out to be adequate for the description of transport in heterogeneous catalysis, self-diffusion in micelle systems and analysis of geophysical data [14,15,16]. Related models have been also applied in financial modeling and analysis of economic time-series [17,18]. Among various aspects of the Lévy-type variables and processes, a problem of special interest is numerical generation of stable distributions and simulation of α-stable integrals and stochastic differential equations [19,20,21,22,23,24]. As broadly discussed in literature [20,22,24], crucial difficulties in solving stochastic differential equations with stable measures are caused by the noisy term, which allows for larger fluctuations with higher probabilities than Gaussian distributions. Numerical methods [17,20] for such equations are more sophisticated than for differential equations [25] and for stochastic differential equations with Gaussian noises [26]. In particular, the nonexistence of variance for stable variables makes the problem much more complicated to tackle both, numerically [17,20] and analytically [27,28]. It turns out, however, that with the use of suitable statistical estimation techniques, computer simulation procedures and numerical discretization methods it is possible to construct relevant approximations of stochastic in-tegrals with stable measures as integrators [20,21].
In this communication we study statistical properties of the generic system describing passage of a classical particle over a fluctuating potential barrier. The system is coupled to a non-Gaussian bath modeled by the Lévy stable noise. We present numerical results for the mean first passage time of the particle over the barrier, assuming a linear potential subject to Markovian dichotomous fluctuations. Our model belongs to the class of "on-off" models discussed in a paper by Doering and Gadoua [29] and further analyzed by Boguñá et al. [30]. A distinctive characteristics of this model is that part of the time the barrier is either switched off (i.e. it becomes flat), or the switching is performed between the barrier and a well, so that the particle can essentially roll rather than climb during these times. The main difference between the model considered here is a form of driving, additive fluctuations. Whereas in the previous papers mainly white Gaussian noises have been considered [29,31,32,33,34], here additive noises are assumed to be α-stable [21], that might arise from the contact with not fully equilibrated bath.
The problem of resonant activation (RA) [29] examined in this paper is an example of processes manifesting constructive role of noise [35]. The article deals with a modified version of the model proposed by Doering and Gadoua [29]. Section II presents the model and poses the problem to be studied. In Sections III and IV general considerations of Lévy noises in physical systems are addressed and the problem of underlying thermodynamic interpretation is discussed. Results of simulations with a short note on numerical methods applied for generating stable variables and integration of stochastic differential equations with stable measures are included in Section V. The paper is closed with the concluding remarks.

II. GENERIC MODEL SYSTEM
We consider an overdamped Brownian particle moving in a potential field between absorbing (x = 1) and reflecting (x = 0) boundaries (cf. Fig. 1), in the presence of noise that modulates the barrier height.
Time evolution of a state variable x(t) is described in terms of the Langevin equation where prime means differentiation over x, ζ(t) is a white Lévy process originating from the contact with not-fully equilibrated bath and η(t) stands for a Markovian dichotomous noise of intensity g taking one of two possible values ±1. Autocorrelation of the dichotomous noise is set to For simplicity, throughout the paper a particle mass, a friction coefficient and the Boltzman constant are all set to 1. The timedependent potential V ± (x) is assumed to be linear with the barrier switching between two configurations with an average FIG. 1: A model potential studied in the paper. The barrier height fluctuates dichotomously between the values H±. A particle starts its diffusive motion at a reflecting boundary x = 0 and continues until the absorption at x = 1.
rate γ (cf. Fig. 1) Both ζ and η noises are assumed to be statistically independent. The initial condition for Eq. (1) is i.e. initially particle is located at the reflecting boundary with equal choices of finding a potential barrier in (±) configurations The quantity of interest is the mean first passage time (MFPT) i.e., the average time which particle spends in the system before it becomes absorbed. Within the proposed approach, the MFPT is estimated as a first moment of the distribution of first passage times (FPT) obtained from the ensemble of simulated realizations of the stochastic process in question. Otherwise, for Lévy flights described by Eq. (1), the MFPT may be calculated after solving a relevant deterministic fractional Fokker-Planck equation [36] (FFPE) for the distribution function In the above FFPE p ± (x, t) are probability density functions (PDFs) for finding a particle at time t in the vicinity of x, while potential takes the value V ± (x). The fractional derivatives ∇ α is understood in the sense of the Fourier Transform [37] with α = 2 corresponding to the standard Brownian diffusion. The coefficient D denotes the generalized diffusion coefficient with the dimension [36] [D] = cm α sec −1 and can be related to the parameter σ characterizing the width of the PDF (see below).

III. LÉVY TYPE VARIABLES
The α-stable variables are random variables for which the the sum of random variables is distributed according to the same distribution as each variable, i.e.
where d = denotes equality in a distribution sense. Real constants c, d in Eq. (8) allow for rescaling and shifting of the initial probability distribution.
The characteristic function of the probability distribution that fulfills (8) can be parameterized in various ways [24]. In the usually chosen L α,β (ζ; σ, µ) [17,21,22] parameterization, a characteristic function of the Lévy type variables is given by Parameter α is called the stability index, β describes skewness of the distribution, σ is responsible for its scaling of and µ is a location parameter. The above parameterization (9) is continuous, in the sense that for every σ and k.

IV. LÉVY NOISES AND TSALLIS THERMODYNAMICS
Classical equilibrium thermodynamics can be derived by use of the variational principle for the Boltzman-Gibbs-Shannon entropy [38] where p(x) is a probability distribution of a given quantity. For example, x may be associated with a single jump length of a free Brownian particle [10,39] so that p(x) stands for a single jump length distribution. Optimizing S[p(x)] with additional constraints imposed on p(x) and ∞ −∞ leads to a Gaussian distribution p(x) used in the description of a normal Brownian diffusion [5] p(x) = exp(−bx 2 )/Z, Here Z = ∞ −∞ exp(−bx 2 ) = π/b is a partition function and b is a Lagrange multiplier, b = 1/(2σ 2 ).
As an extension of the above, a generalized q-entropy suggested by Tsallis [4] with a supplementary condition leads to [10] In the limit of q → 1, p q (x) reproduces Eq. (18) and the norm-constraint can be satisfied only for q < 3. N -folded distribution p q (x, N ) approximating probability distribution for macroscopic diffusion is determined by the general central limit theorem. Consequently, for q < 5/3, the asymptotic (N → ∞) distribution is Gaussian whereas for q > 5/3 it approaches the Lévy statistics [4,10] with the stability index α given by The variational principle can be also applied to the energy distribution [40] which subject to the generalized condition results in Alternatively, expression Eq. (25) can be derived by assuming subordination of the Gibssian exponential energy statistics to the statistics of fluctuations in temperature T (and hence in the In particular, a gamma-distributed temperature leads to a Lévy distribution p q (ε) = L α (bε) with the stability index given by the relative variance of fluctuations in b [41]. This observation shows a possible link of the Lévy distributions to models of nonequilibrated baths.

V. PHYSICAL INTERPRETATION
By definition, in the case of static barrier height and with the noise ζ uncorrelated at different times and obeying the Lévy statistics, the overdamped Langevin equation Eq. (1) describes Lévy flights [36,37] in a constant force field. For long times, the trajectory x(t) behaves as and yields the Lévy stable distribution [24,37] in the position of the particle. In consequence, if the first moment exists, i.e. for 1 < α ≤ 2, mean value of x(t) grows linearly with time, x(t) = H 0 t whereas the mean square displacement becomes [x(t) − x(t) ] 2 = 2D = 2σ 2 only for α = 2 when the finite second moment of L(ζ) exists. Thus the generalized Einstein relation connecting the first moment in the presence of constant force H 0 to the second moment in the absence of force x(t) H0 = H 0 x 2 (t) 0 is recovered only in the Brownian limit α = 2 with the noise amplitude σ related to the diffusion coefficient σ = D 1/2 . Obviously, for ζ noises with diverging mean square displacement, the classical fluctuation dissipation theorem is violated and the Einstein relation does not longer hold [9,37,42]. When analyzed from the perspective of the continuous time random walks (CTRW), Lévy flights characterize walks with a Poisson waiting time and a Lévy distribution of the jump length. The scaling nature of the jump length PDF leads then to a clustering of the Lévy flights visible via interruption of the local motion by occasional long sojourns on all length scales [5]. This fractal character of the Lévy flight-trajectory can be contrasted with subdiffusive CTRW [5,9] that fills the two-dimensional space completely and features no clusters. In such a case, however, the time intervals between consecutive steps are governed by the power-law waiting time distributions that lead to a sublinear dependence x(t) ≈ t ν with ν denoting the power-law index of the waiting time PDF. Therefore, subdiffusive CTRW in a constant external force acting along the x direction perfectly satisfies the fluctuationdissipation theorem [5,9] which holds also in the corresponding fractional Fokker-Planck equation framework.
Diverging mean-square displacement cannot be valid for a particle with non-diverging mass. In fact, for massive particles, a finite velocity of propagation exists making very long instantaneous jumps impossible. For that reason, the dilemma of diverging mean-square displacement in the Lévy flight can be overcome by CTRW version of Lévy walks with a suitable time cost penalizing long jumps. Nevertheless, in many physical systems of interest diffusion of state variable x(t) with diverging second moment does not violate physical principles and is a legitimate way of physical modeling [14,15,43,44].
In the above equations V and W are independent random variables, such that V is uniformly distributed in the interval (− π 2 , π 2 ) and W is exponentially distributed with a unit mean [17,21,22].
The problem described by Eq. (1) and corresponding Eq. (28) for the stability index α = 2 is a well known case of resonant activation resolved in the series of papers [29,33,34]. Accordingly, for α = 2 integration in Eq. (28) is performed with respect to a Gaussian measure and standard numerical methods can be applied [45] to solve the Langevin equation under study. Otherwise, the ordinary forward (or backward) in time Fokker Planck Equation [46,47,48] for the PDF can be derived in a closed form, from which the MFPT can be easily calculated. Also, for the Gaussian fluctuations theoretical background is provided by the standard thermodynamics [38] where parameter σ can be related to the intensity of the thermal fluctuations imposed on a physical mode x.
The choice of the σ, which scales the distribution width and varies the intensity of fluctuations, corresponds, although not in a self-transparent way, to the change of system temperature. However, the problem of definition of the system temperature is more subtle here than in a presence of a standard bath [10,41,49,50,51] because the system is not in the state of equilibrium.
Lévy type variables have been generated by use of recipes given by Eq. (30) and (34). The trajectory x(t) has been generated according to (28) for various noise realizations. Its evaluation proceeded till the time t ′ when x(t ′ ) ≥ 1 for the first time. The resulting distribution of first passage times (FPT) has been further used to evaluate MFPT.
Asymptotic lines plotted in Fig. 2 have been calculated numerically by use of the Monte Carlo method [52]. As expected, typical behavior [29,30,33] has been recovered for γ → 0 and for γ → ∞ Fig. 2 presents results of simulations averaged over 10 3 realizations with a time-step ∆t = 10 −5 . For calculation of the asymptotic lines (Eqs. (36) and (37) ) ∆t = 10 −5 and averaging over 10 4 realizations has been performed. For the barrier switching between H ± = ±8, the phenomenon of resonant activation is clearly visible cf. Fig. 2 (lower panel). By comparison to the results obtained for a motion of a particle subject to the additive white Gaussian noise [29,30,31,32,33], the value of the MFPT is smaller and resonant activation is observed at lower frequencies γ. Moreover, for H ± = ±8, asymptotic values of MFPT estimated for γ → ∞ and γ → 0 are higher (lower) than in the corresponding white Gaussian noise case [29,30,33].
For other cases under consideration, i.e. for H − = 0, H + = 8 and H − = 4, H + = 8, the resonant activation has not been observed (cf. Fig. 2 (upper and middle  panel)). However, values of reported MFPTs for the system driven by non-Gaussian Lévy noises are always significantly smaller than in the case of Gaussian-noise driving. It is caused    (30) and (34). In order to gain sufficient statistics averaging of Eq. (28) over 10 3 realizations has been performed. Asymptotic lines have been evaluated for reference static potential profiles with the time step dt = 10 −5 and averaged over 10 4 realizations. Error bars have been estimated by use of standard techniques for the MC data analysis [52]. They represent standard deviation from the mean and remain within the symbol size. Lines have been drawn to guide the eye. by the fact that the Cauchy and Lévy-Smirnoff noises allow for higher values of driving fluctuations in the Langevin equation (1).
As it can be inferred from Fig. 2, values of the MFPT for the Cauchy noise are higher than the corresponding times for the Lévy-Smirnoff noise, compare the left and the right panel of the Fig. 2. This observation can be explained by the difference between both statistics: for the given sets of parameters, the Lévy-Smirnoff distribution is more "heavily tailed" than the Cauchy distribution.
The procedure applied for numerical integration of (28) is valid for every allowed value of α. In particular, the case with α = 2 has been investigated along these lines in order to test the numerical approach adopted in simulation of Eq. (1). The test concluded a perfect agreement of numerically simulated results with formerly solved Gaussian cases of RA [29,32,33,34].

VII. SUMMARY
We have considered a thermally activated process that occurs in a system coupled to a non-Gaussian noise source introduced by a not fully equilibrated thermal bath. Another external stochastic process is assumed to be responsible for dichotomous fluctuations of the potential barrier which has been modeled by the linear function with a varying slope.
In comparison to the Gaussian case [34], when the RA phenomenon has been observed for all barrier setups (H ± = ±8, H + = 8, H − = 0, H + = 8, H − = 4) analyzed in this study, non-Gaussian additive stable noises produce resonant activation observable only for the H ± = ±8 case. Resulting MFPTs are significantly smaller than those obtained for the Gaussian source of fluctuations. The effect is due to higher probabilities of the extreme events (large fluctuations) allowed by the Lévy statistics. Similarly to the Gaussian-bath case, a typical asymptotic behavior of the MFPT has been recovered for large and small frequencies of the dichotomous noise: for small γ MFPT tends to the average MFPTs for the both barrier configurations, while for large γ it becomes equal to the MFPT over the average potential barrier [29,31,32].
These observations have several implications in relation to chemical kinetics in conformationally varying media [55,56,57] where flipping barriers separating reactants' and products' basins may be due to a dynamic isomerization of the activated complex. The effects of fluctuations in force or potential on the diffusive process have been also extensively studied in the context of motor proteins [58]. Under typical conditions, i.e. with a Gaussian additive thermal noise and with a flipping barrier height, the RA phenomenon is registered in all those realms as a maximal flux of particles or a maximum reaction rate. In contrary to this finding, the analogous systems influenced by the presence of non-Gaussian additive noises exhibit RA only for potentials switching between the barrier and the well (Fig. 2, lower panel). Otherwise, the kinetics is fairly insensitive to the flipping rate of an erecting barrier (Fig. 2, upper panel) or becomes slowed down (Fig. 2, middle panel) when the frequency of switching between a high and a low barrier becomes large.
In addition, physical interpretation of the Lévy noisy term in equation (1) was discussed in some detail. In general, descriptions provided by the FFPE approach and the Tsallis statistics are not in a full agreement [5,36,37]. In particular, the FFPE description of Lévy flights in a harmonic potential [37,53] produces different stationary PDF than the one predicted by the entropy Eq. (19). Nevertheless, the Tsallis formalism of nonextensive statistical mechanics (or the "superstatistics" concept, in general [50,51]) offers an intriguing interpretation of non-Gibssian ensembles that can model not fully equilibrated baths. Especially, for nonequilbrium systems composed of regions that exhibit spatiotemporal fluctuations of an intensive quantity (like pressure, chemical potential, inverse temperature or the energy dissipation rate [50]) generalized statistics may emerge in consequence of a statistical subordination of the intensive variable to those fluctuations.
The situation in which resulting probability distributions are not Gaussian are frequent in various physical situations. The phenomena of special interest are random walks that may lead to Lévy distributions [5] and hence provide a possible realization of Lévy noise sources.
There are several candidates for such a noise origin to be taken into account. Obviously, all noise sources need to be infinite to allow, with a nonzero probability, infinite fluctuations. Among possible models of chaotic nonequilibrium baths perhaps the most natural is a model of the fluid-like bath [54] abruptly perturbed with a local heating to a very high temperature. Gradual spreading of energy progresses until a new equilibrium state is reached and for some finite periods of time, the probing system that experiences local change of the bath temperature will be governed by non-Boltzman statistics [41,51,54]. This and similar approaches validate then use of the Lévy-type statistics for description of not-fully equilibrated baths. As known from other similar studies [28], the Lévy-type structure of the noise imposed on a double-well potential can affect profoundly noise induced jumping between metastable states and result in stationary PDF deviating from the usual Gibbs distribution.