Noise intensity

Noise. Technical differences exist between personal noise dosimeters and high accuracy sound level meters and these may alter the usual preference for personal monitors. But it is exposure to noise rather than general room noise that must be estimated for comparison with noise exposure criteria, the logarithmic expression and alternative means of summation (3 vs 5 db doubling) compHcate statistics. Exposure criteria for both dose and peak exposure must be evaluated, and space and time variabiUty of noise intensity can be immense. 

Noise is characterized by the time dependence of noise amplitude A. The measured value of A (the instantaneous value of potential or current) depends to some extent on the time resolution of the measuring device (its frequency bandwidth A/). Since noise always is a signal of alternating sign, its intensity is characterized in terms of the mean square of amplitude, A, over the frequency range A/, and is called (somewhat unfortunately) noise power. The Fourier transform of the experimental time dependence of noise intensity leads to the frequency dependence of noise intensity. In the literature these curves became known as PSD (power spectral density) plots. 

In Fig. 41 we plot the minority phase volume fraction, fm, versus the Euler characteristic density for a large number of simulation runs performed at different quench conditions. For the symmetric blends (< )0 = 0.5), fm = 0.5 and is independent of time and XEuier/ is always negative. For the asymmetric blends, fm decreases with time and xEu cr/F may change the sign. We have not observed the bicontinuous morphology for fm < 0.29, nor have we observed the droplet morphology for fm > 0.31. This observation suggests that the percolation occurs at fm = 0.3 0.01 and that the percolation threshold is not very sensitive to the quench conditions (noise intensity). 

The first paper that was devoted to the escape problem in the context of the kinetics of chemical reactions and that presented approximate, but complete, analytic results was the paper by Kramers [11]. Kramers considered the mechanism of the transition process as noise-assisted reaction and used the Fokker-Planck equation for the probability density of Brownian particles to obtain several approximate expressions for the desired transition rates. The main approach of the Kramers method is the assumption that the probability current over a potential barrier is small and thus constant. This condition is valid only if a potential barrier is sufficiently high in comparison with the noise intensity. For obtaining exact timescales and probability densities, it is necessary to solve the Fokker-Planck equation, which is the main difficulty of the problem of investigating diffusion transition processes. 

For a small noise intensity, the double integral may be evaluated analytically and finally we get the following expression for the escape time (inverse of the eigenvalue yj) of the considered bistable potential ... 

The first approach to obtain exact time characteristics of Markov processes with nonlinear drift coefficients was proposed in 1933 by Pontryagin, Andronov, and Vitt [19]. This approach allows one to obtain exact values of moments of the first passage time for arbitrary time constant potentials and arbitrary noise intensity moreover, the diffusion coefficient may be nonlinear function of coordinate. The only disadvantage of this method is that it requires an artificial introducing of absorbing boundaries, which change the process of diffusion in real smooth potentials. 

It is known that when the transition of an overdamped Brownian particle occurs over a potential barrier high enough in comparison with noise intensity A<[>/kT 1, time evolution of many observables (e.g., the probability of... 

In this section we will analyze the validity of exponential approximation of observables in wide range of noise intensity [88,89,91,92]. 

Probably, a similar procedure was previously used (see Refs. 1 and 93-95) for summation of the set of moments of the first passage time, when exponential distribution of the first passage time probability density was demonstrated for the case of a high potential barrier in comparison with noise intensity. 

Figure 10. Evolution of the survival probability for the potential Tf.v) — ax2 - for3 for different values of noise intensity the dashed curve denoted as MFPT (mean first passage time) represents exponential approximation with MFPT substituted into the factor of exponent.

Figure 11. Evolution of the survival probability for the potential

Finally, we have considered an example of metastable state without potential barrier exponential approximation with the MFPT of the point d for kT = 1 (Fig. 13). It is seen that even for such an example the exponential approximation [with the mean decay time (6.5)] gives an adequate description of the probability evolution and that this approximation works better for larger noise intensity. 

The second considered example is described by the monostable potential of the fourth order (x) = ax4/4. In this nonlinear case the applicability of exponential approximation significantly depends on the location of initial distribution and the noise intensity. Nevertheless, the exponential approximation of time evolution of the mean gives qualitatively correct results and may be used as first estimation in wide range of noise intensity (see Fig. 14, a = 1). Moreover, if we will increase noise intensity further, we will see that the error of our approximation decreases and for kT = 50 we obtain that the exponential approximation and the results of computer simulation coincide (see Fig. 15, plotted in the logarithmic scale, a = 1, xo = 3). From this plot we can conclude that the nonlinear system is linearized by a strong noise, an effect which is qualitatively obvious but which should be investigated further by the analysis of variance and higher cumulants. 

Figure 16. Evolution of the mean coordinate in the potential (x) — x4 f 4 — x2 for different values of noise intensity with initial distribution located in the potential minimum.

The exponential approximation may lead to a significant error in the case when the noise intensity is small, the potential is tilted, and the barrier is absent (purely dynamical motion slightly modulated by noise perturbations). But, to the contrary, as it has been observed for all considered examples, the single exponential approximation is more adequate for a noise-assisted process either (a) a noise-induced escape over a barrier or (b) motion under intensive fluctuations. 

Figure 19. The mean decay time as a function of frequency of the driving signal for different values of noise intensity, kT — 0.5,0.1,0.05, A — 1. The phase is equal to zero. Solid lines represent results of computer simulation, and dashed lines represent an adiabatic approximation (6.15).

The corresponding curves derived from Eqs. (6.15) and (6.16) are reported in Figs. 18 and 19 as dashed lines, from which we see that there is a good agreement between the modified adiabatic approximation and the numerical results up to go 1. Moreover, the approximation improves with the increase of the noise intensity. This could be due to the fact that the adiabatic approximation [96,108] is based on the concept of instantaneous escape, and for higher noise intensity the escape becomes faster. 

In the frame of the present review, we discussed different approaches for description of an overdamped Brownian motion based on the notion of integral relaxation time. As we have demonstrated, these approaches allow one to analytically derive exact time characteristics of one-dimensional Brownian diffusion for the case of time constant drift and diffusion coefficients in arbitrary potentials and for arbitrary noise intensity. The advantage of the use of integral relaxation times is that on one hand they may be calculated for a wide variety of desirable characteristics, such as transition probabilities, correlation functions, and different averages, and, on the other hand, they are naturally accessible from experiments. 

Maximum perturbed value of time delays Weighting parameters White plant disturbance intensity White sensor noise intensity Initial conditions... 

Stable states can be found, for example, by graphical solution of the equation 1 /x(4> — 4>o) = 7(potential minima [42,65], and it can be shown immediately that OB arises only if the system is biased by a sufficiently strong external field, that is, when it is far away from thermal equilibrium. If the noise intensity is weak, the system, when placed initially in an arbitrary state, will, with an overwhelming probability, approach the nearest potential minimum and will fluctuate near this minimum. Both the fluctuations and relaxation... 

For small noise intensities the distribution has sharp maxima near the stable states and their populations w 2 are described by the balance equations... 

The model (1)—(5) describes stochastic motion in a general OB system for white Gaussian noise in the low noise intensity limit. We now apply this model to the description of some experimental results on fluctuations and fluctuational transitions in some particular OB devices. 

It follows from the above discussion that an indicator of applicability of the description of stochastic motion in an OB system is an activation dependence of the transition probabilities VFnm on the noise intensity. Using level-crossing measurements (shown to be independent on the level positions), we found in our previous experiments [108] that the activation law applies over the whole range of noise intensities that we are using. 

For small noise intensities the system spends most of the time fluctuating near the stable positions, and interwell transitions occur only occasionally. Q((o) can then be represented as the sum of partial contributions from vibrations about the equilibrium positions x weighted with the populations of the corresponding stable states wn, and from interwell transitions. The intrawell contribution takes the form... 

Figure 3. Signal-to-noise-ratio (SNR) in the optical experiment for a signal at frequency fl = 3.9 Hz as a function of the internal noise intensity [115]. Inset the corresponding signal amplification.

Figure 4. Dependence of the signal-to-noise-ratio R dependence on the squared amplitude of the reference signal, measured [28] in an analog electronic experiment for noise intensities D = 0.015 (circles) and D = 0.14 (squares). The inset shows the dependence of R on the squared frequency coo for the same two noise intensities.

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