Stochastic forces intensity

The fundamental reason why excitation has an influence on the decay process is the breakdown of condition (24). This has profound consequences on relaxation when the time scale of the stochastic force i (r) is not well separated from that of the variable of interest A, that is, when memory effects are important. The deexdtation effect is a direct consequence of the invalidation of Eq. (24) by intense external force fields and was first investigated by Grigolini and coworkers following the computer simulations described in Section II. 

Stochastic resonance is a kinetic effect universally inherent to bi- or multistable dynamic systems exposed to either white or color noise. Its main manifestation is the appearance of a maximum on the noise intensity dependencies of the signal-to-noise ratio in a system subject to a weak driving force. Essentially, this behavior is due to the presence of an exponential Kramers time x cx exp(AU/3>) of the system switching between energy minima here AU is the effective height of the energy barrier separating the potential wells and 3> is the noise intensity. 

One of the difficulties in performing SA of stochastic or more generally multiscale models is that a closed form equation does not often exist. As a result, brute force SA has so far been the method of choice, which, while possible, is computationally intensive. As suggested in Raimondeau et al. (2003), since the response obtained is noisy, one has to introduce relatively large perturbations to ensure that the responses are reliable, so that meaningful SA results are obtained. For most complex systems, local SA may not be feasible. However, I do not see this being an impediment since SA is typically used to rank-order the... 

We are thus led to assume that V and the fiuctuation-dissipation process driving the virtual body depends on the variable 17 so as to simulate the efiects of the H-bond dynamics. For example, a strong interaction potential V accompanied by friction and stochastic torques (forces) of weak intensity simulates the solidlike properties of the environment when the tt ed molecule is characterized by four hydrogen bonds. In this case a reasonable approximation is to simulate such an environment by ice Ih. In the opposite limit, a weak potential V with strong friction and stochastic torques (forces) simulates the difiusional properties of the unbounded molecule (liquid water at high temperature or in very dilute nonpolar solution). 

On a molecular scale liquid surfaces are not flat, but subject to Jluctuations. These irregularities have a stochastic nature, meaning that no external force is needed to create them, that they cannot be used to perform work and are devoid of order. Their properties can only be described by statistical means as explained in sec. 1.3.7. Surface fluctuations are also known as thermal ripples, or thermal waves, in distinction to mechanically created waves that will be discussed in detail in sec. 3.6. Except near the critical point, the amplitudes of these fluctuations are small, in the order of 1 nm, but they can, in principle, be measured by the scattering of optical light. X-ray and neutron beams. From the scattered intensity the root mean square amplitude can be derived and this quantity can, in turn, be related to the surface tension because this tension opposes the fluctuations ). 

As a result of the mechanical action of mixing tools in high intensity mixers (see Section 7.4.2) an aerated, turbulent particulate matter system with stochastic particle movement develops. Similar conditions exist if the particles are suspended in a fluidized bed. The main difference between the two methods is that in the mixers particle movement is caused by mechanical forces while in fluidized beds drag forces, that are induced by a flow of gas, are the main reason for the movement of the particulate matter. Therefore, fluidized beds are not only used as excellent environments in which gas efficiently and intimately contacts particles but also for dry mixing of particulate solids and coalescence of particles which, in the presence of binding mechanisms, causes agglomeration. 

In [4] further studies are presented on fluctuations near limit cycles, on the basis of approximate solutions of the master equation (rather than the Fokker-Planck equation). In [5] there is an analysis of fluctuations (the stochastic potential) for a periodically forced limit cycle, with references to earlier work. Both these articles are intensive mathematical treatments. 

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