White stochastic force

I, y, k denotes a circular permutation / Ij, are the molecular moments of inertia and F is an external potential depending on molecular orientation and /(I) white stochastic forces defined by... 

In Eq. (13), the vector q denotes a set of mass-weighted coordinates in a configuration space of arbitrary dimension N, U(q) is the potential of mean force governing the reaction, T is a symmetric positive-definite friction matrix, and , (/) is a stochastic force that is assumed to represent white noise that is Gaussian distributed with zero mean. The subscript a in Eq. (13) is used to label a particular noise sequence For any given a, there are infinitely many... 

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... 

The study of a Brownian particle suspended in a fluid lead also to the introduction of the exponentially correlated Ornstein-Uhlenbeck process [48], the only Markovian Gaussian non-white stochastic process [19, 22]. We present here the Langevin approach to this problem, hence we analyze the forces that act on a single Brownian particle. We suppose the particle having a mass m equal to unity, and we assume the force due to the hits with thermal activated molecules of the fluid to be a stochastic variable. Moreover, due to the viscosity of the fluid, a friction force proportional to the velocity of the particle has to be considered. All this yields the following equation... 

In Equation 3.1, is the friction constant for the motion along x. Ax t) is the random displacement due to the stochastic force and in the presence of white noise, it can be written as... 

The evolution of the general Fourier BK>de, x ip,t) is therefore determined by the intramolecular elastic forces and a stochastic contribution, due to other chains or solvent, whose statistical properties must be specified Adopting the traditional Langevin specification of the stochastic force we will say that i (p t) is Gaussian of zero mean, white and stationary... 

The dWi are Gaussian white noise processes, and their strength a is related to the kinetic friction y through the fluctuation-dissipation relation.72 When deriving integrators for these methods, one has to be careful to take into account the special character of the random forces employed in these simulations.73 A variant of the velocity Verlet method, including a stochastic dynamics treatment of constraints, can be found in Ref. 74. The stochastic... 

Alternatively one may postulate that all higher cumulants are zero. This specifies all stochastic properties of L(t) in terms of the single parameter F. The L(t) defined in this way is called Gaussian white noise. From the mathematical point of view it does not really exist as a stochastic function (no more than the delta function exists as a function) and in physics it never really occurs but serves as a model for any rapidly fluctuating force. 

External noise denotes fluctuations created in an otherwise deterministic system by the application of a random force, whose stochastic properties are supposed to be known. Examples are a noise generator inserted into an electric circuit, a random signal fed into a transmission line, the growth of a species under influence of the weather, random loading of a bridge, and most other stochastic problems that occur in engineering. In all these cases clearly (4.5) holds if one inserts for A(y) the deterministic equation of motion for the isolated system, while L(t) is approximately but never completely white. Thus for external noise the Stratonovich result (4.8) and (4.9) applies, in which A(y) represents the dynamics of the system with the noise turned off. 

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. 

This is an equation written, for example, for a Brownian particle of mass m moving along the x axis under the influence of a potential V x). Here (t) is a white-noise driving force (a stochastic variable) coming from the Brownian movement of the surroundings, and fx(l) is the systematic friction force. This equation can be solved exactly only in the known special cases V=0 and V=yx, where y is a coeflScient independent of time. Equation (3) is the Langevin equation equivalent to the Kramers equation ... 

Equation (8.28) implies that all frequencies are equally presented in this random force spectrum. A stochastic process of this type is called a white noise. 

As discussed in Section 8.2.1, the Langevin equation (8.13) describes a Markovian stochastic process The evolution of the stochastic system variable x(Z) is determined by the state of the system and the bath at the same time t. The instantaneous response of the bath is expressed by the appearance of a constant damping coefficient y and by the white-noise character of the random force 7 (Z). 

Most of the research on CR was focused on the case of white noise. White noise is a good approximation as long a.s the intrinsic time scale.s of the deterministic system are much larger than the correlation time of the external fiuctnatioiis. This is the case for example in neuronal dynamics where a neuron can be exteniallj forced by another randomly bursting neuron. In general, when deterministic and stochastic time scales are not well separated from each other, not only the amplitude but also the temporal correlation is expected to influence iioise-induced phenomena as CR. In... 

As noted before, the Brownian force n t) may be modeled as a white noise stochastic process. White noise is a zero mean Gaussian random process with a constant power spectrum given in (72). Thus,... 

The statistical properties of the random force f(0 are modeled with an extreme economy of assumptions f(t) is assumed to be a stationary and Gaussian stochastic process, with zero mean (f(0 = 0), uncorrelated with the initial value v(t = 0) of the velocity fluctuations, and delta-correlated with itself, f(0f(t ) = f25(t -1 ) (i.e it is a purely random, or white, noise). The stationarity condition is in reality equivalent to the fluctuation-dissipation relation between the random and the dissipative forces in Equation 1.1, which essentially fixes the value of y. In fact, from Equation 1.1 and the assumed properties of f(t), we can derive the expression y(f)v(t) = exp [v(0)v(0) -+ ylM °, where Xg = In equilibrium, the long-time asymptotic value y/M must coincide with the equilibrium average (vv) = (k TIM)t given by the equipartition theorem (with I being the 3 X 3 Cartesian unit tensor), and this fixes the value of y to y=... 

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