Bistable system

Consider the bistable one-level system with symmetrical potential profile cp(x) = cp(—x) with respect to the origin, where the potential barrier of the height (3 is located (Fig. 9). According to the principle of conformity and (5.119) the relaxation time 0 of this bistable system (the decision interval is [—oo, 0]) is equal to... 

Expression (5.120) means that the relaxation time in an arbitrary symmetrical one-level bistable system is two times smaller than the MFPT T(—d, +d)—that is, two times smaller than the decay time of the metastable state—shown with the dashed line taken into account in Fig. 9. 

This is concerned with the fact that in the case of the relaxation time, roughly speaking only half of all Brownian particles should leave the initial potential minimum to reach the equilibrium state, while for the profile of the decay time case all particles should leave the initial minimum. Expression (5.120), of course, is true only in the case of the sufficiently large potential barrier, separating the stable states of the bistable system, when the inverse probability current from the second minimum to the initial one may be neglected (see Ref. 33). 

Figure 9. A potential profile of type I representing a one-level bistable system. The dashed line shows the absorbing wall.

Fig. 34. The macroscopic rate equation for a bistable system and the domains of attraction.

All information about the evolution is, of course, contained in the M-equation, but explicit solutions for bistable systems are rare510. Yet the one-step process and the one-dimensional Fokker-Planck equation have three features that can be found without solving the equation. 

B. Fluctuations and Fluctuational Transitions in an OB (Optically Bistable) System... 

One of the most important general features of fluctuations in a bistable system is the onset of a narrow zero-frequency spectral peak for parameter values lying in the range of the kinetic phase transition. This peak arises from... 

It will be apparent from the above discussion that the double-cavity membrane system is ideally suited to investigations of fluctuations and fluctuational transition phenomena. Stochastic resonance and huge noise-induced amplification of a heterodyne signal have been observed. We would emphasize that noise-protected heterodyning is a general phenomenon that may occur in bistable systems of various sorts, and that it may therefore be of interest for applications in engineering. 

In principle any molecule able to exist in two reversible, switchable states can represents a molecular switch (bistable device) with potential to form part of molecular circuitry or act as molecular memory. An excellent component for switchable molecular devices is the 1,2-dithienylethene system, which has been exploited ingeniously by Lehn in a number of bistable systems.54 The core switching element is the transformation of the dithienylethene unit between two stable states as a function of the wavelength of incident radiation (Scheme 11.8). 

In a more formal way, a single-domain particle is a rotationally bistable system with the potentials... 

The approach developed above is very convenient for obtaining a comprehensive account of the frequency and temperature behavior of the phase shift in the system in question. The problem has special interest since those dependencies for bistable systems have been put under discussion with contradicting conclusions [101,102]. Some particular numerical simulations on the same subject reported in Ref. 103, although interesting, are insufficient to draw out a final clarification. 

We show that application of a constant force (bias held) results in shifting the position of the ordinary SR peak together with the anticipated reduction of its height and sharpness. For the quadratic SR the situation is more complicated. There, the joint action of the thermal noise and constant bias leads to formation of a mountain-like surface over the plane of those parameters. In other words, for each given value of the bias held there exists a unique value of the noise strength that maximizes SNR and vice versa. The discovered effect can be useful, for example, for evaluation of the parameters of bistable systems through susceptibility measurements. In addition, it has to be taken into account when designing any devices where the nonlinear SR is employed. 

The preparation of chiral magnetic materials is not limited to those that show paramagnetism and magnetic ordering. Another important phenomenon is that of spin crossover, as it can lead to bistable systems, that in addition can be switched by light [213,214]. A homochiral spin crossover... 

The simplest situation arises for bistable systems in which the basic pattern is a front, i.e. an interfacial region that connects the two locally stable stationary states and propagates in space.10 Thereby one of the two states expands on the expense of the other one. Two experimental examples of potential fronts propagating along the... 

Fig. 32. Illustration of front motion in a bistable system due to the interplay of homogeneous dynamics and migration coupling (see text).

Landauer studied the behavior of a bistable system consisting of a tunnel diode and focused his attention on the activation jump between the two states resulting from thermal fluctuations. Later, Matsuno ° analyzed the power spectrum of the noise affecting the electric conductivity of a Gunn diode. Diode oscillations were proven to be accompanied by modulation of both amplitude and frequency. The noise spectrum was shown to exhibit typical 1// characteristics, and fluctuations were found to appear in the region close to the critical threshold with an increasing relaxation time as the threshold was approached. This behavior is largely reminiscent of that ex-... 

In practice this system was put to work by electrochemical action [100]. Oxidation of the starting Cu (4j yielded the Cu (4) species with its typical 670-nm band, as discussed for the previous bistable system. The rearrangements to Cu (5) and to Cu (6) could not be resolved, even if the overall process was rather slow (minutes or hours), and was strongly affected by the presence of coordinating solvents or ions, as previously discussed. The final point of the oxidation reaction, Cu (6), was clearly characterized by spectro- and electrochemical properties. Interestingly, the oxidation step from Cu (4), through the intermediate Cu (5), to could also... 

The magneto-elastic system is representative of a wide class of driven bistable systems. An easier system to visualize is a damped particle in a double-well potential (Figure 12.5.2). Here the two wells correspond to the two buckled states of the beam, separated by the hump at. r = 0. 

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