Physics, chaotic processes

At the end of this chapter we will now briefly discuss a theoretical approach to the description of chaotic processes encountered in chemical kinetics, see Section 1.3 and Sections 6.2.2.4, 6.3.2.4. In Section 6.2.2.4 we described the method of generation and physical meaning of a chaotic state of the Belousov-Zhabotinskii reaction carried out in a flow reactor. 

Although chaotic fluctuations could theoretically be purely deterministic and be described using simple ordinary differential equations, real physical processes usually contain a stochastic (or noise) conq)onent 11, S3, 54). Note that a combination of many deterministic-chaotic processes could lead to an apparent randomness of experimentally observed data and a transition to highdimensional chaos (with a correlation dimension more than S [5/]), which is difficult to distinguish from randomness. 

Perikinetic motion of small particles (known as colloids ) in a liquid is easily observed under the optical microscope or in a shaft of sunlight through a dusty room - the particles moving in a somewhat jerky and chaotic manner known as the random walk caused by particle bombardment by the fluid molecules reflecting their thermal energy. Einstein propounded the essential physics of perikinetic or Brownian motion (Furth, 1956). Brownian motion is stochastic in the sense that any earlier movements do not affect each successive displacement. This is thus a type of Markov process and the trajectory is an archetypal fractal object of dimension 2 (Mandlebroot, 1982). 

From a physical point of view, the rhythmic phenomena are related to the fact that biological systems are maintained under far-from-equilibrium conditions through a continuous dissipation of energy [23]. However, non-equilibrium conditions can also give rise to more complicated behaviors. Chaotic dynamics, for instance, can arise either as a regular rhythmic process is destabilized and develops through a cascade of period-doubling bifurcations [24], by torus destruction in connection with the interaction of two or more rhythms, or via different types of intermittency... 

None of the classically chaotic quantum systems so far investigated in the atomic and molecular physics literature exhibits type III quantum chaos. On the other hand, atomic and molecular physics systems provide excellent examples for quantized chaos, the topic of this section. The attractive feature of the term quantized chaos is that it does not imply anything about what happens to the classical chaos when it is quantized. Usually, especially in bounded time independent quantum systems, classical chaos does not survive the quantization process. The quantized system does not exhibit any instabilities, or sensitivity to initial conditions, e.g. sensitivity to small variations in the wave function at time t = 0. 

The data shown in Fig. 6.9 and Fig. 6.10 confirm our suspicion that for weak microwave fields no chaos mechanisms have to be invoked for an adequate physical understanding of microwave ionization data. The situation, however, is quite different in the case of strong microwave fields. In this case the ionization routes are very comphcated, and the multiphoton pictmre loses its attractiveness. It has to be replaced by a picture based on chaos. Chaos provides a simpler description of the ionization process and consequently a better physical insight. The discussion of the chaotic strong-field regime is the topic of the following section. 

In Section 2.3 we studied the tent map, a schematic model for ionization that was able to produce fractal structures as a result of ionization. An important question is therefore whether the results presented in Section 2.3 are only of academic interest, or whether fractal structures can appear as a result of ionization in physical systems. In order to answer this question we return to the microwave-driven one-dimensional hydrogen atom. As we know from the previous chapter, this model is ionizing and realistic enough to qualitatively reproduce measured ionization data. Therefore this model is expected to be a fair representative for a large class of chaotic ionization processes. 

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