Noise induced transitions

I.I. Gichman and A.W. Skorochod, Stochastische Differentialgleichungen (Akademie-Verlag, Berlin 1971) ch. 5 W. Horsthemke and R. Lefever, Noise-Induced Transitions (Springer, Berlin 1984) sect. 5.5. 

In the presence of weak noise there is a finite probability of noise-induced transitions between the chaotic attractor and the stable limit cycle. In Fig. 14 the filled circles show the intersections of one of the real escape trajectories with the given Poincare section. The following intuitive escape scenario can be expected in the Hamiltonian formalism. Let us consider first the escape of the system from the basin of attraction of a stable limit cycle that is bounded by an saddle cycle. In general, escape occurs along a single optimal trajectory qovt(t) connecting the two limit cycles. 

W. Horsthemke and R. Lefever, Noise-Induced Transitions. Springer Verlag, Berlin, 1984. 

U. Erdmann, W. Ebeling, and A. S. Mikhailov. Noise-induced transition from translational to rotational motion of swarms. Phys. Rev. E, 71 051904 (2005). 

Horsthemke, W. R. Lefever. 1984. Noise-Induced Transitions. Theory and Applications in Physics, Chemistry and Biology. Springer, Berlin. 

Lefever, R. 1981. Noise-induced transitions in biological systems. In Stochastic Nonlinear Systems. R. Lefever L. Arnold, eds. Springer, Berlin, pp. 127-36. 

Horsthemke, W. Lefever, R. (2006). Noise-induced Transitions Applications to Physics, Chemistry and Biology. New York Springer. 

The theory of noise-induced transition (Horsthemke Lefever, 1984a) emphasise that fluctuations (noise) superimposed on deterministic motions might have a crucial role in forming ordered structures i.e. they may operate as organising forces . Noise may destroy the stability of the deterministic attractors and stochastic models might exhibit completely different qualitative properties from those of the deterministic model. 

According to the traditional point of view fluctuations are averaged out. It was clearly demonstrated that noise can support the transition of a system from a stable state to another stable regime. Since stochastic models might exhibit qualitatively different behaviour than their deterministic counterparts, external noise can support transitions to states which are not available (or even do not exist) in a deterministic framework. (The theory of noise-induced transition, as well as its applications are discussed in the book of Horsthemke Lefever (1984b). 

In Subsection 5.8.2 we give a short introduction to the mathematical formalism of external noise. In Subsection 5.8.3 specific models are given for illustrating the existence of noise-induced transitions in chemical systems. Further remarks in connection with the development of the theory will be given in Subsection 5.8.4. 

In many applications of the theory of noise-induced transition it is assumed that fo is linear in A, i.e. 

Noise-induced transition an example for white noise idealisation... 

Noise-induced transition the effect of coloured noise... 

To investigate the robustness of white-noise-induced transitions a nonzero but short correlation time has been used (see, for example. Chapter 8 of Horsthemke Lefever (1984b)). 

Erdi, P. Barna, G. (1985). Self-organization of neural networks noise induced transition. Phys. Letters, 107A, 287-90. 

Schimansky-Geier, L., Tolstopjatenko, A. V. Ebeling, W. (1985). Noise induced transitions due to external additive noise. Phys. Letters, 108A, 329-32. 

The organization of this paper is as follows First I will discuss the modeling of nonlineat systems coupled to a noisy environment. Then I will highlight the main theoretical results and I will conclude by mentioning problems currently under study and experimental results on noise-induced transitions. More details on most of the aspects of noise-induced phenomena discussed here can be found in the recent monograph [5]. ... 

This gives further support to our identification of with the "phases" of the system. ii) multiplicative noise, g(x) const. In this case the effect of the external noise is modulated by the state of the system. For small noise intensities the term g g in (33) may be neglected and we find that x % x. However, if becomes larger and if g g is sufficiently nonlinear compared to h and g, then the extrema Xj can be very different, in number and location, from the deterministic steady states. Since this change in the behavior of the system arises without any changes in the systemic parameters but simply by varying the noise intensity, we have called this phenomenon a noise-induced transition. 

Noise-induced transitions have been studied theoretically in quite a few physical and chemical systems, namely the optical bistability [12,13,5], the Freedricksz transition in nematics [14,15,16,5], the superfluid turbulence in helium II [17], the dye laser [18,19], in photochemical reactions [20], the van der Pol-Duffing oscillator [21] and other nonlinear oscillators [22]. Here I will present a very simple model which exhibits a noise-induced critical point. The so-called genetic model was first discussed in [4]. I will not describe its application to population genetics in this paper, see [5] for this aspect, but use a chemical model reaction scheme ... 

So far we have discussed noise-induced transitions only in the white noise limit. Naturally, the question arises if these noise-induced phenomena are robust. In other words, are essentially the same phenomena observed for cor small, but nonvanishing The answer is positive. Noise-induced transitions are HO-t an artifact of white noise they occur also for colored noise, i.e. for environments with nonzero correlation times. This has been established by various techniques, namely the wide band perturbation expansion [8,5], the approximate Fokker-Planck operator techniques [24], and approximate renormalized equations of evolution for Gaussian noise [25]. These methods are perturbation expansions and are limited to small T or order to ex-... 

Let me conclude by listing the experimental studies on noise-induced transitions and by expressing the hope that their number will increase in the future. At present the theoretical part of the field of noise-induced transitions is more developed than its experimental side. There is a clear need for more experimental... 

Stochastically driven systems exhibit a variety of interesting nonequilibrium effects. These have been recently reviewed by HORSTHEMKE and LEFEVER [1] and also addressed by other authors in this workshop. In this contribution we focus our attention on the role played by the internal fluctuations of a system driven by an external noise [2,3,4]. External noise effects are usually studied in the thermodynamic limit in which internal fluctuations become negligible. This procedure assumes that the external driving noise completely dominates the fluctuations in the system. Nevertheless, a framework in which internal and external fluctuations are simultaneously considered is necessary to calculate finite size effects. Within such a framework a better understanding of the physical contents of "noise induced transition" phenomena [1] is obtained by investigating how changes in a stationary distribution induced by external noise are smoothed out by internal fluctuations. A major novel outcome of the unified theory of internal and external fluctuations presented here is the existence of "crossed-fluctuation" contributions which couple the two independent sources of randomness in the system. 

More recently, while the interest in noise induced transitions is increasing (see e.g. [5-7] and the references cited therein), similar observations have been made by KABASHIMA et al. [8,9] and by DE KEPPER, In the first... 

We have shown using a Galerkin type of resolution scheme that when the kinetic constant of the trimolecular step in (1) fluctuates, new pure noise induced transitions become possible. For the sake of clarity let us recall for example that when A = 2, the extrema of the probability density behave as represented in figure 1. u is plotted as a function of a and for the values of e indicated. The curves labelled 1 correspond to the situation typical above the Hopf bifurcation. For small intensities (curve in the lower left corner) one sees that the noise suppresses the extremum corresponding to the usual limit cycle this is the same behavior as in 2.2. Its amplitude which is equal to one (in normalizing with respect to the deterministic limit cycle, i.e. 

W. Horsthemke, R. Lefever Noise induced transitions. Theory and applications... 

The observed fluctuations are very sensitive to stirring. The pronounced decrease of the oscillation period with stirring rate is accompanied by a marked increase of the fluctuation amplitude. Figure 2 illustrates our interpretation in terms of the contraction of the limit cycle from its deterministic limit 1-2-3-4 to the stochastic limit cycle l -2 -3 -4. The latter is dominated by noise-induced transitions. The interaction between local (stochastic) and global dynamics is seen to be profound. 

Noise Induced Transitions (plenary lecture). By W. Horsthemke. 150... 

Reactions taking place far from equilibrium show a number of unusual features, even at the macroscopic level. Consequently, the aim of much of the research in this area is rather different, and has concentrated on characterizing and unraveling the kinetics giving rise to behavior like oscillations, chaos and bistability. So the microscopic details of the reactive event, which are presumably like those in close-to-equilibrium systems, are not studied, but rather the complex structure of the macroscopic temporal and spatial evolution is the focus of attention. However, there is a class of phenomena which is analogous in some respects to the rate problem in systems close to equilibrium these are external noise-induced transitions between bistable states. The purpose of this article is to describe these phenomena and point out how some of the theoretical tools developed for equilibrium systems can be exploited to investigate these new rate processes. 

Under noisy dynamics, during part of its trajectory, the system may enter a regime where bistability ceases to exist, and noise-induced transitions occur. This mechanism is the one-dimensional analog of the tangent mechanism described in [13]. 

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