Condensed-Matter Physics (1986) / Chapter Skim
Currently Skimming:
11 Nonlinear Dynamics, Instabilities, and Chaos
11 Nonlinear Dynamics, Instabilities, and Chaos
Pages 215-235
The Chapter Skim interface presents what we've algorithmically identified as the most significant single chunk of text within every page in the chapter.
Select key terms on the right to highlight them within pages of the chapter.
From page 215... ...
and turbulence, subjects of both fundamental interest and practical importance because of the simplicity of fluids and their ubiquity in nature. However, chaotic dynamics are increasingly being found in other areas of condensed-matter physics, such as superconducting devices and electronic conduction in semiconductors.
|
From page 216... ...
The eventual impact of these new mathematical tools will probably be greater than that of the specific physical systems to which they are currently being applied. Although the discussion in this chapter emphasizes nonlinear dynamics in the context of instabilities and chaotic motion in fluids and other condensed-matter systems, a much broader approach could be taken, in which the methods of nonlinear dynamics are applied to a wide range of interdisciplinary problems in plasma physics, astrophysics, and even accelerator technology.
|
From page 217... ...
For example, closed loops in state space describe periodic oscillations; attractors having the topology of a torus describe more complex oscillations termed quasi-periodic; and forms known as strange attractors describe chaotic motion. A strange attractor is a set on which trajectories wander erratically.
|
From page 218... ...
No qualitative change separated laminar and turbulent motion in this view. In 1971 a suggestion was made that turbulent flows could actually be modeled by strange attractors having only a few degrees of freedom.
|
From page 219... ...
3. Intermittency, in which laminar and chaotic phases alternate in These basic routes to chaotic motion were found both theoretically in simple models and experimentally in a limited class of fluid systems where the spatial pattern of the flow is fixed and the transition occurs continuously.
|
From page 220... ...
The discovery that the routes to chaos have universal properties is a particularly exciting development and accounts in part for the speed with which nonlinear dynamics (applied especially but not exclusively to fluid motion) has been accepted as a part of condensed-matter physics.
|
From page 221... ...
Whereas earlier work emphasized the use of frequency spectra, one can now use geometrical methods to study the forms of strange attractors in phase space and to measure their properties. An example of the shape of trajectories in phase space near the onset of chaotic fluid flow is shown in Figure li.3.
|
From page 222... ...
this is not the case: time dependence is accompanied by large variations in the spatial pattern of the flow. One widely applicable theoretical approach that has proven successful in describing pattern changes is known as stability theory.
|
From page 223... ...
Various types of random motions can similarly be described in terms of objects with fractional dimension and can probably be understood using the language of dynamical systems. For example, the invasive percolation of a two-phase fluid in a porous medium, various aggregation processes, and the behavior of electrons lopping in a random material all show dynamically produced structures of fractional dimensionality.
|
From page 224... ...
show chaotic dynamics, a fact that may be important in device applications. Cooled extrinsic germanium photoconductors exhibit a series of progressively more complex instabilities with an increasing applied dc electric field.
|
From page 225... ...
be observably altered by a small perturbation and therefore could not be predictable. Early in its history, the probabilistic nature of statistical mechanics was realized not to be obviously consistent with an integrable and predictable Newtonian dynamics.
|
From page 226... ...
If finite accuracy measurements of an arbitrary observable of the system, made at regular time intervals from the beginning of time to the present, do not permit precise prediction of the next measurement, the system has positive metric entropy, i.e., information is gained from each measurement. Another concept recently adapted from information theory is that of algorithmic complexity, which measures the unpredictability of individual orbits.
|
From page 227... ...
instabilities and chaotic motion are widespread phenomena with common features; and (c) the growing understanding of the origins of chaotic behavior and the limits of predictability contribute to the foundations of physics.
|
From page 228... ...
continue to provide insight into behavior just beyond the onset of instabilities and will play an important role in understanding pattern selection. However, the difficulty of extending nonlinear stability theory beyond the second instability is clear.
|
From page 229... ...
Experimental Methods Now that the usefulness of phase-space methods has been demonstrated, we anticipate that a great deal of attention will be given to the development of reliable ways of measuring such quantities as the dimensionality, Lyapunov exponents, and metric entropy of strange attractors. One important goal of this work is to understand chaotic behavior significantly above its onset and to begin to understand the relationship between chaotic time dependence and spatial variations.
|
From page 230... ...
The transition from complete integrability to full chaos involves a divided phase space exhibiting chaos surrounding integrable islands of stability. Do these islands decrease in size but never disappear as the energy of the system increases'?
|
From page 231... ...
At present, there is a growing amount of convincing evidence indicating that the quantum mechanics of finite bounded systems contains no chaotic time dependence. This leads to the deep suspicion that the quantum description of a classical chaotic system does not tend to the proper classical limit; much worse, it may be that quantum mechanics was tacitly constructed on the notion of intergrability.
|
From page 232... ...
There is a great need for precise measurements of morphological response to varying growth conditions in simple, carefully characterized model systems. In the case of phase transitions involving nucleation or spinodal decomposition, the major opportunity now is to use neutrons, synchrotron radiation, and other modern methods to make real-time observations of nonequilibrium processes in alloy solids, polymers, and similar materials.
|
From page 233... ...
Research on networks is closely related to advances in computer science, where efforts are being made to apply nonlinear dynamics to the behavior of computing machines. Phenomena in which fractal structures occur in real space rather than in phase space are being actively studied.
|
From page 235... ...
Appendixes
|
Key Terms
This material may be derived from roughly machine-read images, and so is provided only to facilitate research.
More
information on Chapter Skim is available.