Phase transitions noise induced

The major aim of the present chapter is to utilize the theoretical tools of Chapter I to face a subject which is becoming of increasing interest the problem of noise-induced phase transitions. Hie theoretical starting point to investigate these phenomena is usually typified by stochastic differential equations such as... 

Henceforth this will be referred to as the first threshold. This effect has often been termed noise-induced phase transition. Figure 4 shows how Os,(x) is... 

This type of problem has recently become of great interest in the field of physics, being directly related to basic problems such as noise-induced phase transitions and the dependence of the relaxation time of the variable of interest on the intensity of external noise terms. As in Chapter X, the reduced equation of motion itself can be assumed to be rehable only when this relaxation time decreases with increasing noise intensity. Further problems of interest are whether or not the threshold of a noise-induced phase transition is characterized by a slowing dowrf " and what is the analytical form of the long-time decay. ... 

Thus, we have found that the mechanisms of escape from a nonhyperbolic attractor and a quasihyperbolic (Lorenz) attractor are quite different, and that the prehistory of the escape trajectories reflects the different structure of their chaotic attractors. The escape process for the nonhyperbolic attractor is realized via several steps, which include transitions between low-period saddle-cycles coexisting in the system phase space. The escape from the Lorenz attractor consist of two qualitatively different stages the first is defined by the stable and unstable manifolds of the saddle center point, and lies on the attractor the second is the escape itself, crossing the saddle boundary cycle surrounding the stable point attractor. Finally, we should like to point out that our main results were obtained via an experimental definition of optimal paths, confirming our experimental approach as a powerful instrument for investigating noise-induced escape from complex attractors. 

Bulsan, A. R., Schieve, W. C. Gran, R. F. (1978). Phase transitions induced by white noise in bistable optical system. Phys. Letters, 68A, 294-8. 

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. 

The mechanism of the one-way transition process is easily deduced from an examination of Fig. 1 [13]. Suppose the system is initially in the limit cycle state L2. Note that the basin boundary Bi intersects L2 thus, if a noise event occurs while the phase point is on L2 and to the left of Bi, it will find itself in the basin of FI and tend to evolve to FI in the absence of other noise events. Alternatively, if it is to the right of Bi when the noise acts, it will evolve to LI. The transition occurs by a shift of the basin boundary under the action of the parametric noise process (moving boundary mechanism) [13]. A similar argument shows that once a phase point is near FI or F2 it can never escape since both of these fixed points lie to the left of Bi and B2. Hence, under the stochastic dynamics the fixed point region will appear fuzzy as a result of noise-induced hops between FI and F2 likewise there will be noise-induced hops between LI and L2 leading to a fuzzy limit cycle. 

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