Stochastic processes oscillator

In Section V the reorientation mechanism (A) was investigated in terms of the only (hat curved) potential well. Correspondingly, the only stochastic process characterized by the Debye relaxation time rD was discussed there. This restriction has led to a poor description of the submillimeter (10-100 cm-1) spectrum of water, since it is the second stochastic process which determines the frequency dependence (v) in this frequency range. The specific vibration mechanism (B) is applied for investigation of the submillimetre and the far-infrared spectrum in water. Here we shall demonstrate that if the harmonic oscillator model is applied, the small isotope shift of the R-band could be interpreted as a result of a small difference of the masses of the water isotopes. 

From the viewpoint of experimental workers, slow relaxations are abnormally (i.e. unexpected) slow transition processes. The time of a transition process is determined as that of the transition from the initial state to the limit (t -> oo) regime. The limit regime itself can be a steady state, a limit cycle (a self-oscillation process), a strange attractor (stochastic self-oscillation), etc. 

Both vibrational and rotovibrational relaxation can be described analyti-caDy as multiplicative stochastic processes. For these processes, RMT is equivalent to the stochastic Liouville equation of Kubo, with the added feature that RMT takes into account the back-reaction from the molecule imder consideration on the thermal bath. The stochastic Liouville equation has been used successfully to describe decoupling in the transient field-on condition and the effect of preparation on decay. When dealing with liquid-state molecular dynamics, RMT provides a rigorous justification for itinerant oscillator theory, widely applied to experimental data by Evans and coworkers. This implies analytically that decoupling effects should be exhibited in molecular liquids treated with strong fields. In the absence of experimental data, the computer runs described earlier amount to an independent means of verifying Grigolini s predictions. In this context note that the simulation of Oxtoby and coworkers are semistochastic and serve a similar purpose. 

These results were in contrast to the previous experiments described above, where nucleation was always observed. It is thought that as nucleation is a stochastic process, a single oscillation of a laser-induced bubble may not always lead to the nucleation. The probability of nucleation is increased by the continuously oscillating bubble produced by the standing wave system, and it is increased further by the multicavitation events produced by the ultrasonic horn. 

This section is a review of previous treatments of this subject. Subsequent sections are an application of the theory of stochastic processes to chemical rate phenomena the harmonic oscillator model of a diatomic molecule is used to obtain explicit results by the general formalism. 

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 ... 

Kuznetsov, P. I., Stratonovich, P. L. Tikhonov, V. I. (1965). The effect of electrical fluctuations on a valve oscillator. In Nonlinear transformations of stochastic processes, eds P. I. Kuznetsov, P. L. Stratonovich V. I. Tikhonov, pp. 223-Pergamon Press, Oxford. 

If extensive historical observations of the oscillation of the water levels of the basin are available, the evaluation of the probable maximum seiche should be based on a stochastic processing of the data. A stochastic processing of the data can only be done if records of these observations or measurements are available for the vicinity of the plant site and if all the forcing actions for which there is a potential in the basin are adequately represented in the data. The results of the stochastic evaluation should be verified by a simplified deterministic method. 

The original Langevin equation considers a Markovian stochastic process [6, 7] with simple constant friction H in the field of an external fluctuation force F f). For a harmonic oscillator this equation has the form ... 

Spontaneous Fluctuations at Chemical Instabilities via Stochastic SimulationT] Let us now consider an application of stochastic simulation methods to the Trimolecular reaction. Here we consider a few realizations of the underlying stochastic process in order to illustrate the main ideas. The initial investigation focuses on the size-dependence of fluctuations near the homogeneous transition to limit cycle oscillations. The deterministic system is defined according to Eqs. 14-17, with A = 1 and = 1... 

I/O data-based prediction model can be obtained in one step from collected past input and output data. However, thiCTe stiU exists a problem to be resolved. This prediction model does not require any stochastic observer to calculate the predicted output over one prediction horiajn. This feature can provide simplicity for control designer but in the pr ence of significant process or measurement noise, it can bring about too noise sensitive controller, i.e., file control input is also suppose to oscillate due to the noise of measursd output... 

Another possibility to quantify the response of a stochastic system to periodic signals is to generalize the notion of synchronization, which is known from deterministic nonlinear oscillators. We will pursue this idea in what follows. To this end we review in section 2.2 the notion of effective synchronization in stochastic systems. The mean number of synchronized system cycles turns out to be an appropriate quantity to characterize the synchronization properties of the system to the periodic signal. However the task remains to calculate this quantity. This calculation will be based on discrete renewal models for bistable and excitable dynamics. These discrete models are introduced in section 2.3. We first recapitulate the well known two state model for the stochastic dynamics of an overdamped particle in a doublewell system [10] and afterwards introduce a phenomenological discrete model for excitable dynamics. In section 2.4 a theory to calculate the mean frequency and effective diffusion coefficient in periodically driven renewal processes is presented. These two quantities allow to calculate the mean number of synchronized cycles. Finally in section 2.5 we apply this theory to investigate synchronization in bistable and excitable systems. 

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