Stochastic linear oscillator

The stochastic Bloch equation is a semiphenomenological equation with some elements of quantum mechanics in it. To understand better whether our results are quantum mechanical in origin, we analyze a classical model. Lorentz invented the theory of classical, linear interaction of light with matter. Here, we investigate a stochastic Lorentz oscillator model. We follow Allen and Eberley [108] who considered the deterministic model in detail. The classical model is also helpful because its physical interpretation is clear. We show that for weak laser intensity, the stochastic Bloch equations are equivalent to classical Lorentz approach. 

Since the first report of oscillation in 1965 (159), a variety of other nonlinear kinetic phenomena have been observed in this reaction, such as bi-stability, bi-rhythmicity, complex oscillations, quasi-periodicity, stochastic resonance, period-adding and period-doubling to chaos. Recently, the details and sub-systems of the PO reaction were surveyed and a critical assessment of earlier experiments was given by Scheeline and co-workers (160). This reaction is beyond the scope of this chapter and therefore, the mechanistic details will not be discussed here. Nevertheless, it is worthwhile to mention that many studies were designed to explore non-linear autoxidation phenomena in less complicated systems with an ultimate goal of understanding the PO reaction better. 

The main features of the copper catalyzed autoxidation of ascorbic acid were summarized in detail in Section III. Recently, Strizhak and coworkers demonstrated that in a continuously stirred tank reactor (CSTR) as well as in a batch reactor, the reaction shows various non-linear phenomena, such as bi-stability, oscillations and stochastic resonance (161). The results from the batch experiments can be suitably illustrated with a two-dimensional parameter diagram shown in Pig. 5. 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

The studies of Ertl and co-workers showed that the reason for self-oscillations [142, 145, 185-187] and hysteresis effects [143] in CO oxidation over Pt(100) in high vacuum ( 10 4 Torr) is the existence of spatio-temporal waves of the reversible surface phase transition hex - (1 x 1). The mathematical model [188] suggests that in each of the phases an adsorption mechanism with various parameters of CO and 02 adsorption/desorption and their interaction is realized, and the phase transition is modelled by a semi-empirical method via the introduction of discontinuous non-linearity. Later, an imitation model based on the stochastic automat was used [189] to study the qualitative characteristics for the dynamic behaviour of the surface. 

The model is based on the standard tight-binding Hamiltonian consisting of a donor, a number of bridge sites, and an acceptor, all coupled to form a linear chain. In addition, a single linearly coupled oscillator is included, representing a high-frequency vibrational coordinate coupled to the electron transfer. The lack of detailed information about this system makes it appropriate to treat the bath stochastically. Thus... 

In the classical limit (hcoj k T), the reaction coordinate X t) in each quantum state can be described as a Gaussian stochastic process [203]. It is Gaussian because of the assumed linear response. As follows from the discussion in Section II.A, if the collective solvent polarization follows the linear response, the ET system can be effectively represented by two sets of harmonic oscillators with the same frequencies but different equilibrium positions corresponding to the initial and final electronic states [26, 203]. The reaction coordinate, defined as the energy difference between the reactant and the product states, is a linear combination of the oscillator coordinates, that is, it is a linear combination of harmonic functions and is, therefore, Gaussian. The mean value is = — , for state 1 and = , for state 2, respectively. We can represent Xi(r) and X2 t) in terms of a single Gaussian stochastic process x(t) with zero mean as follows ... 

In the literature and the previously mentioned identification methods, the input is either measured or modeled as a prescribed parametric stochastic model (even though the parameters may be unknown). This seems to be a necessary condition for model identification purpose. For example, consider a linear single-degree-of-freedom system. In the frequency domain, the response X is equal to the input 7, magnified by the transfer function of the oscillator Jf ... 

A method for accurately estimating the second-order statistical responses required for robust design optimization in case of nonlinear response. This is accomplished by a modified method of stochastic equivalent linearization, especially purported to take into account the actual non Gaussian behavior of hysteretic oscillators proposed by the author in (Hurtado and Barbat 1996 Hurtado and Barbat 2000). 

Figure 2 Comparison of Gaussian (dashed line) and non-Gaussian stochastic equivalent linearization (solid line) for an hysteretic oscillator using Monte Carlo results (dotted line).

The second-order quantities needed by robust optimization can be obtained with the method of stochastic equivalent linearization, which is the only nonlinear random vibration technique useful for large structures. To this end it is necessary to apply non Gaussian approaches due to the saturation of the restoring forces about the strength values of hysteretic oscillators. 

In this section we find the FPT distribution using the ME approach, by using discrete stochastic growth models for y and the ME framework (see Section II.B.l). In the continuum limit of y, the results for the FPT distributions are essentially unchanged. As previously noted, for the purpose of the present discussion we assume that cell size growth is either linear of exponential. (Also see phase oscillator model in the following text for a different interpretation of the linear growth model.)... 

Au SK, Beck JL (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 16 263-277 Caughey TK, Payne HJ (1967) On the response of a class of self-excited oscillators to stochastic excitation, hit J Non Linear Mech 2 125-151... 

Stochastic Analysis of Linear Systems, Fig. 2 NGSM functions, lj,uu(f) (/ = 0, 1, 2) and bandwidth parameter, of the transient response of an oscillator... 

Stochastic Analysis of Linear Systems, Fig. 5 NGSMs of the response of an oscillator with coq = 27t rad/sec, = 0.05 for the Hsu and Bernard (1978) model (solid line) and for the Spanos and Solomos (1983) model (dashed line)... 

The very restrictive conditions that make available exact solutions for the stochastic dynamical systems motivated the development of approximate solution techniques. Methods such as the equivalent linearization have been developed to generate first-order approximate solutions. These techniques can also be applied in some cases to single-degree-of-freedom nonlinear oscillator with hysteretic behavior. 

From the spectral density of R t) we can find the spectral density of stochastic observables that are related to R via linear Langevin equations. For example, consider the Langevin equation (8.13) with V(x) = /2)mo (the so called Brownian harmonic oscillator)... 

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