Stochastic noise additive

In several respects the finite inertia of the system of Eq. (1.7) produces effects similar to those of a weak additive stochastic force. In particular, as a consequence of finite inertia, the escape from the well is also possible in the absence of the additive stochastic noise. These effects are not taken into... 

One cannot expect that such an agreement covers also the large-Q region. Two main reasons have already beoi singled out inotia and weak residual stochastic forces of additive kind. In particular, as shown in Figs. 6 and 7, the additive stochastic noise forces to increase when the large-0 re-... 

But in real radar applications many different noise and clutter background signal situations can occur. The target echo signal practically always appears before a background signal, which is filled with point, area or even extended clutter and additional superimposed noise. Furthermore the location of this background clutter varies in time, position and intensity. Clutter is, in real applications, a complicated time and space variant stochastic process. 

This Ansatz is the essential step. The -expansion is not just one out of a plethora of approximation schemes, to be judged by comparison with experimental or numerical results 0. It is a systematic expansion in and is the basis for the existence of a macroscopic deterministic description of systems that are intrinsically stochastic. It justifies as a first approximation the standard treatment in terms of a deterministic equation with noise added, as in the Langevin approach. It will appear that in the lowest approximation the noise is Gaussian, as is commonly postulated. In addition, however, it opens up the possibility of adding higher approximations. 

It should be emphasized that this way of including fluctuations has no other justification than that it is convenient and bypasses a description of the noise sources, compare IX.4. It may provide some qualitative insight into the effect of noise, but does not describe its actual mechanism. For instance, fluctuations in the pumping should give rise to randomness in the coefficient a, rather than to an additive term. Yet the equation (7.6) has been the subject of extensive study and it is famous in statistical mechanics under the name of generalized Ginzburg-Landau equation. It may well serve us as an illustration for a stochastic process.510... 

Let us now consider stochastic motion in an OB system. In general, noise in an OB system may result from fluctuations of the incident field, or from thermal and quantum fluctuations in the system itself. We shall consider the former. The fluctuations of the intensities of the input or reference signals give rise respectively to either multiplicative or additive noise driving the phase. Both types of fluctuations can be considered within the same approach [108], Here we discuss only the effects of zero-mean white Gaussian noise in the reference signal ... 

The advantage of separating out the additive stochastic component is that the character of noise-like sounds is not modified with the time scale in particular, the noise may be stretched without the tonality that occurs in very large stretching of sine waves. On the other hand, the timbre of transient aharmonic sounds may be altered. In addition, component separation may suffer from a lack of fusion, unlike sine-wave modification which models all components similarly. One approach to improve fusion of the two components is to exploit the property that for many sounds the stochastic component is in synchrony with the deterministic component. In speech, for exam-... 

Analytical results are possible if we assume collective oscillations of the peptide elements, e.g., F(t) = Acos(interaction center undergoes local thermal fluctuations, represented by mutually uncorrelated white noise of intensity a2 fi(t) = where = u2[Pg.379]

After the kinetic model for the network is defined, a simulation method needs to be chosen, given the systemic phenomenon of interest. The phenomenon might be spatial. Then it has to be decided whether in addition stochasticity plays a role or not. In the former case the kinetic model should be described with a reaction-diffusion master equation [81], whereas in the latter case partial differential equations should suffice. If the phenomenon does not involve a spatial organization, the dynamics can be simulated either using ordinary differential equations [47] or master equations [82-84]. In the latter case but not in the former, stochasticity is considered of importance. A first-order estimate of the magnitude of stochastic fluctuations can be obtained using the linear noise approximation, given only the ordinary differential equation description of the kinetic model [83-85, 87]. 

SA of SODEs describing chemically reacting systems was introduced early on, in the case of white noise added to an ODE (Dacol and Rabitz, 1984). In addition to expected values (time or ensemble average quantities), SA of variances or other correlation functions, or even the entire pdf, may also be of interest. In other words, in stochastic or multiscale systems one may also be interested in identifying model parameters that mostly affect the variance of different responses. In many experimental systems, the noise is due to multiple sources as a result, comparison with model-based SA for parameter estimation needs identification of the sources of experimental noise for meaningful conclusions. 

The additive stochastic force fa(t), assumed to be a white Gaussian noise, is defined by... 

To recover the ideal case of Eq. (1.1) we would have to assume that (u ), vanishes. The analog simulation of Section III, however, will involve additive stochastic forces, which are an unavoidable characteristic of any electric circuit. It is therefore convenient to regard as a parameter the value of which will be determined so as to fit the experimental results. In the absence of the coupling with the variable Eq. (1.7) would describe the standard motion of a Brownian particle in an external potential field G(x). This potential is modulated by a fluctuating field The stochastic motion of in turn, is driven by the last equation of the set of Eq. (1.7), which is a standard Langevin equation with a white Gaussian noise defined by... 

Figures 12c and b show the dependence of on Q for some values of i as obtained by using the experimental method of Section III. We would like to stress again that a great deal of attention has been devoted to limiting the effects of spurious additive noise from the circuit and that if the well is not exactly symmetric the multiplicative noise can itself produce a spread of the variable x. In spite of our efforts, a weak additive stochastic force proves to be present in our electrical circuit.

We develop a stochastic model for the superlattice approximating the random fluctuations of the current densities by additive Gaussian white noise m t) with... 

We have concluded from these analyses that not only are the measured irregularities in rotation and elongation purely random white noise, but that they are, in addition, not correlated. We would expect that on a microscopic scale, all growth processes would be stochastic, but what is surprising is that the macroscopic growth kinetics of a Phycomyces sporanglophore is also stochastic. 

In this section, we consider the combination of stochastic perturbation with a deterministic thermostat. Methods constructed in this way can be ergodic for the canonical distribution while also providing flexibility in way equilibrium is achieved. We distinguish in (6.16) between multiplicative noise, where B = B(z) varies with z and additive noise, where B is constant. In our treatment of this topic we will only consider additive noise. The presence of multiplicative noise may complicate discretization. As we shall see, the reliance on additive noise improves the performance of discretization schemes. 

The equation above is the conventional linear Fokker-Planck equation for stochastic processes with additive noise and the noise strength measured in terms of the temperature T. 

A Tuned Mass Damper (TMD) is one of the simplest and the most rehable passive device for vibration control in a wide range of applications, and for this reason many optimization criteria have been proposed for this specific device. Essentially, a TMD consists in an additional mass connected to a main system by a spring and a damper. The main system, excited by a base acceleration, is modelled as a stochastic stationary coloured noise and introducing the global space state vector ... 

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