Effect of External Noise

In this section we explore the systematic effect of an external noise on the front velocity. Consider a reaction term F p, a) that depends not only on the density but also on a parameter a that fluctuates. Assuming small fluctuations around its mean value, we can write a x, t) = a — e where is the mean value, e is a 

A crucial feature of the multiplicative noise case is that the noise term in (4.97) has a nonzero mean value. Using Novikov s theorem [324] for Gaussian noise in the Stratonovich interpretation, we find that 

This rearrangement allows us to distinguish explicitly between the systematie contribution from the noise term and a residual stochastic one. Since the noise is white in time, the average of the noise term has no explicit time dependence. Writing p(x, t) = Pq(x, t) + Sp, where Pq(x, t) = (p(x, t)) and the perturbative fluctuation 

To illustrate our approach, we consider the Ginzburg-Landau reaction term Rip) = P(1 - p)(o + p), where a Then /(po) = 

Po(l-Po)( m+Po)andg(Po) = -Pod-Po)-hi the absence of noise, the velocity of the front propagating into the unstable state 0 is given by 

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Finally, neither the effect of external noise, which affects nonequilibrium transitions in chemical and biological systems (Lefever, 1981 Horsthemke Lefever, 1984 Lefever Turner, 1986), nor the stochastic aspects of these transitions (Nicolis, Baras Malek-Mansour, 1984) are considered - with the exception of the glycolytic system (chapter 2). Such a simplification, justified in the first approximation by the absence of systematic noise in the biological systems considered, permits us to avoid complicating from the outset the analysis of systems whose kinetics is already complex. 

Chemical systems, on which the effects of external noise have been observed, have been studied both experimentally and theoretically in the last few years. De Kepper Horsthemke (1978) studied the effect of fluctuating illumination on photochemical reactions (see also Micheau et ai, 1984). The effect of nonperfect mixing in continuous stirred tank reactor (CSTR) was... 

For the sake of clarity I will discuss the effect of external noise in as simple a case as possible. I will therefore assume that the system has the following three properties i) It is spatially homogeneous. This corresponds to the limit of fast transport, ii) The system is macroscopic and can be described by intensive variables. This corresponds to the thermodynamic limit and implies that any finite size effects, such as internal fluctuations, can be neglected, iii) The state of the system can be described by one variable. This is only a point of mathematical convenience explicit analytical results can be obtained for one variable systems. 

Effect of external noise sources on immissions at workplaces... 

The influence of external noise on nonequilibrium systems can advance or delay the onset of oscillatory behaviour. Without drawing much attention, this phenomenon was apparently first described in the field of radio engineering [1-3]. KUTZNETSOV et al. [ 1] in a paper on the valve oscillator remark that the amplitude of the oscillations tends to zero if the intensity of the noise exceeds a certain threshold. Conversely, studying numerically the effect of substrate input noise on an oscillatory enzymatic reaction, HAHN et al. [4] found that it may induce quasi-periodic behaviour under conditions where oscillations do not occur according to the deterministic equations. 

The large oscillations starting at around 3.8 ms and 14 ms (Fig. 4-a) are attributed to the passage of two successive clouds of bubbles. The sudden variations (spikes) are the effect of external electrical noise. The amplitude of the second harmonic oscillation is 60 mV (estimated using a BPF, not shown here). 

Cryogenics. At room temperature, thermal excitation of charge carriers within the Ge(Li) detector produces unacceptable levels of electrical noise in the external counting circuits. To overcome this effect it... 

P. Kuban and P.C. Hauser, Effects of the cell geometry and operating parameters on the performance of an external contactless conductivity detector for microchip electrophoresis, Lab Chip, 5 (2005) 407-415. J.G.A. Brito-Neto, J.A.F. da Silva, L. Blanes and C.L. do Lago, Understanding capacitively coupled contactless conductivity detection in capillary and microchip electrophoresis. Part 2. Peak shape, stray capacitance, noise, and actual electronics, Electroanalysis, 17 (2005) 1207-1214. 

A more reahstic and more general treatment would presumably lead to a set of equations like Eqs. (52), with the potential V(x) fluctuating as a consequence of couplings with nonreactive modes (see Section III). For the sake of simplicity, we study separately the two different aspects. While Section III was devoted to pointing out the role of multiphcative fluctuations (derived from nonlinear microscopic Liouvillians) in the presence of additive white noise, this subsection is focused on the effects of a non-Markovian fluctuation-dissipation process (with a time convolution term provided by a rigorous derivation from a hypothetical microscopic Liouvillian) in the presence of a time-independent external potential. 

The advancement of nano-technologies allows, on the one hand, to realize precision devices, which can not effectively operate without precision control of external and internal electric noise and, on the other hand, to create precise detectors, which can control this noise. A new quantum effect found recently [1,2] allows to create a device based on superconductor nano-structures which will be able to detect very weak noise down to the equilibrium one. 

Most, if not all, QSAR methods require selection of relevant or informative descriptors before modeling is actually performed. This is necessary because the method could otherwise be more susceptible to the effects of noise. The a priori selection of descriptors, however, carries with it the additional risk of selection bias [73], when the descriptors are selected before the dataset is divided into the training and test sets (Figure 6.6A). Because of selection bias, both external validation and cross validation could significantly overstate pre-... 

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