Noise colored

Colored noise, transition state trajectory, 208-209 Configuration interaction (Cl), ab initio... 

Eigenvalues, transition state trajectory colored noise, 209... 

Friction interactions, multiparticle collision dynamics, single-particle friction and diffusion, 114—118 Friction kernel, transition state trajectory, colored noise, 209... 

Alternatively, the white-noise processes W(f) could be replaced by colored-noise processes. Since the latter have finite auto-correlation times, the resulting Lagrangian correlation functions for U and would be nonexponential. However, it would generally not be possible to describe the Lagrangian PDF by a Fokker-Planck equation. Thus, in order to simplify the comparison with Eulerian PDF methods, we will use white-noise processes throughout this section. 

Of course, equations (7.5) and (7.7) are still hard to solve and usually require approximate methods. Moreover it should be clear that this device does not solve the equation (7.1) for every given colored noise. It merely provides a model for investigating qualitatively the effect of positive rc. The conclusions obtained may... 

First, the Langevin equation receives in a separate chapter the attention merited by its popularity. In this chapter also non-Gaussian and colored noise are studied. Secondly, a chapter has been added to provide a more complete treatment of first-passage times and related topics. Finally, a new chapter was written about stochasticity in quantum systems, in which the origin of damping and fluctuations in quantum mechanics is discussed. Inevitably all this led to an increase in the volume of the book, but I hope that this is justified by the contents. 

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. 

Considering the highly processive mechanism of the protein degradation by the proteasome, a question naturally arises what is a mechanism behind such translocation rates Let us discuss one of the possible translocation mechanisms. In [52] we assume that the proteasome has a fluctuationally driven transport mechanism and we show that such a mechanism generally results in a nonmonotonous translocation rate. Since the proteasome has a symmetric structure, three ingredients are required for fluctuationally driven translocation the anisotropy of the proteasome-protein interaction potential, thermal noise in the interaction centers, and the energy input. Under the assumption that the protein potential is asymmetric and periodic, and that the energy input is modeled with a periodic force or colored noise, one can even obtain nonmonotonous translocation rates analytically [52]. Here we... 

The above approach gives good but slow response to time varying y. The algorithm can be further improved by treating C as a colored noise, implemented by... 

Dekker has studied multiplicative stochastic processes. In his work the stochastic Liouville equation was solved explicitly through first order in an expansion in terms of correlation times of the multiplicative Gaussian colored noise for a general multidimensional weakly non-Markovian process. He followed the suggestions of refs. 17 and 18 and applied, Novikov s theorem. In the general multidimensional case, however, he improved the earlier work by San Miguel and Sancho. ... 

Note that Eq. (118) is a colored noise representation of the dynamics expressed by Eq. (123). Even though the central moments of the solution to Eq. (123) scale in the present case in the same way as the moments did for the process in Section III, they are very different processes. The simplest way to see the difference is to note that the integral relation in Eq. (118) is linear so that the solution Fa(t) and the random fluctuations 2,(7) have the same statistics,... 

For colored noise sources the derivation of evolution equations for the probability densities is more difficult. In a Markovian embedding, i.e. if the Ornstein-Uhlenbeck process is defined via white noise (cf. chapter 1.3.2) and v t) is part of the phase space one again gets a Fokker-Planck equation for the density P x,y, Similarly, one finds in case of the telegraph... 

V. Beato and H. Engel. Coherence resonance phenomena in an excitable system driven by colored noise. Fluct. Noise Lett., 6 L85-L94, 2006. 

P. Hanggi and P. Jung. Colored noise in dynamical systems. Adv. Chem. Phys., 89 239, 1995. 

Lefever, R. J.W. Turner. 1986. Sensitivity of a Hopf bifurcation to multiplicative colored noise. Phys. Rev. Lett. 56 1631-4. 

If the mathematical model for the system of concern has too many uncertain parameters, the measurement will not provide sufficient mathematical constraints/equations to uniquely identify the uncertain parameters. However, experienced engineers can identify the critical substructures for monitoring. Then, a free body diagram can be drawn to focus on these critical substructures only. Note that the internal forces on the boundary of the substructures are unknown and difficult to measure, so they are treated as an uncertain input to the substructure. Furthermore, these internal forces share the dominant frequencies of the structure so they cannot be modeled arbitrarily as white noise or other prescribed colored noise. However, with the same idea as in Yuen and Katafygiotis [294], these interface forces can be treated as unknown inputs without assuming their time-frequency content [289]. This enables a large number of possible applications in structural health monitoring and also enhances the computational efficiency since one does not need to consider the whole system. 

We can think of the noise process in (7.40) as corresponding to a colored noise which has characteristics that directly depend on the stepsize, although the noise decorrelates in just a couple of timesteps. This is therefore no longer a Markov process, however it can be reformulated as such if one considers the appropriate extended space (or keeping the momenta/> ). 

Ceriotti, M., Bussi, G., Parrinello, M. Langevin equation with colored noise for constant-temperature molecular dynamics simulations. Phys. Rev. Lett. 102, 020,601 (2009). doi 10. 1103/PhysRevLett.l02.020601... 

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