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ScienceWeek
THEORETICAL PHYSICS: ON DISCRETE BREATHERS
The following points are made by David K. Campbell (Nature 2004 432:455):
1) Stanislaw Ulam (1909-1986), the celebrated Polish mathematician and godfather of the field now known as nonlinear science, remarked that using the term "nonlinear science" was like "calling the bulk of zoology the study of non-elephants". He meant that linear processes are the exception rather than the rule; that most phenomena are inherently nonlinear; and that the effects of nonlinearity are apparent everywhere in nature, from the synchronized flashing of fireflies through clear-air turbulence to tornadoes and tsunamis.
2) Perhaps Ulam should have carried the metaphor of a nonlinear "zoo" a bit further, for the remarkable taxonomy of "non-elephants" observed in recent physics experiments has been truly breathtaking. One of the newest species to be added to the menagerie is the "discrete breather", or intrinsic localized mode. Briefly characterized, discrete breathers are spatially localized, time-periodic, stable excitations that exist and propagate in spatially extended, perfectly periodic, discrete systems.
3) Sato and Sievers[1] have recently reported the sighting of a particularly elusive form of discrete breather that exists at the atomic scale in a magnetic solid. Coupled with other recent observations in systems ranging from Josephson-junction arrays[2,3], through micromechanical systems[4], to photonic crystals[5] and optical-switching waveguide arrays, this new observation underscores the ubiquity of these nonlinear excitations and the importance of understanding their role in determining the properties of these widely different physical systems.
4) Discreteness and nonlinearity lead to the existence of stable discrete breathers. The discreteness of the system means that there is a finite range of frequencies in which linear excitations can exist. If a large-amplitude (and hence nonlinear) excitation is created -- a putative discrete breather -- its frequency can be shifted out of the allowed band of linear excitations; for nonlinear systems such as the plane pendulum (with so-called soft nonlinearities), this frequency lies below the lowest allowed linear frequency. For highly discrete systems, the allowed linear band can be very narrow, as the highest allowed linear frequency decreases with increasing discreteness. This means that harmonics of the discrete-breather frequency can all lie above the linear band, so that the discrete breather cannot couple to linear excitations and is therefore stable against decaying into them. Although there are additional subtleties, these simple arguments capture the essence of the phenomenon.
5) Given the ubiquity of such breathers in discrete nonlinear physical systems (which exist on essentially all length scales), these nonlinear excitations are likely to be important in many physical phenomena, including melting, fracture, and the buckling and folding of biopolymers. They may also prove useful in technologies ranging from "smart" materials with tunable collective responses to light-induced, all-optical switches and networks.
References (abridged):
1. Sato, M. & Sievers, A. J. Nature 432, 486-488 (2004)
2. Trias, E., Mazo, J. J. & Orlando, T. P. Phys. Rev. Lett. 84, 741-744 (2000)
3. Binder, P., Abraimov, D., Ustinov, A. V., Flach, S. & Zolotaryuk, Y. Phys. Rev. Lett. 84, 745-748 (2000)
4. Sato, M., Hubbard, B. E., Sievers, A. J., Ilic, B. & Craighead, H. G. Europhys. Lett. 66, 318-323 (2004)
5. Fleischer, J. W., Segev, M., Efredmidis, N. K. & Christodoulides, D. N. Nature 422, 147-150 (2003)
Nature http://www.nature.com/nature
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Related Material:
THEORETICAL PHYSICS: CHAOS AND NONLINEAR DYNAMICS
Notes by ScienceWeek:
In general, a nonlinear dynamical system is a system described by time-dependent differential equations such that the rates of change of one or more dependent variables of the system depend in a nonlinear fashion on the variables themselves. Certain nonlinear dynamical systems, some of which are of great scientific interest, exhibit "chaotic dynamics"
In this context, the term "chaos" refers to unpredictable behavior arising in a system that obeys deterministic laws but exhibits unpredictability. The essential idea is that in certain systems small perturbations may produce a cascade of larger perturbations, so that eventually the behavior of such systems cannot be predicted from prior states no matter if the systems appear simple and obey deterministic laws. Examples of chaotic nonlinear dynamical systems are the weather and populations of organisms, and instances of chaotic dynamics have now been documented in most scientific disciplines.
Because the differential equations for many nonlinear systems are often intractable (i.e., no explicit quantitative solutions are possible), a focus of theoretical research on nonlinear systems has been on analysis of the qualitative behavior of such systems, in particular on analysis of the "phase space" and "trajectories" in the phase spaces of such systems. The idea is essentially as follows: If the state of a system depends upon N variables, the instantaneous state of the system can be viewed as a point (phase point) in an N-dimensional space (phase space; system hyperspace), and as the state of the system changes, its phase point can be viewed as describing a trajectory in its phase space. Qualitative analysis of the possible families of solutions of nonlinear differential equations can provide information about such phase space trajectories, and there are certain real systems for which qualitative analysis of the phase space trajectories of the system has revealed significant properties of the system otherwise difficult to delineate.
The following points are made by J.P. Gollub and M.C. Cross (Nature 2000 404:710):
1) The techniques of nonlinear dynamics are well-developed, but the impact of this field has been largely confined to phenomena in which there are only a few important time-dependent quantities. Unfortunately, this excludes a vast range of important problems in which the behavior of one point in space can be quite different (though statistically similar) to that at another location. A particular example is convective behavior.
2) The traditional approach to studying nonlinear dynamical behavior is to plot the dynamical variables of the system as a multidimensional phase space graph indicating how the behavior changes over time. For example, a simplified model of the Solar System consisting of the Sun and 9 planets would require a phase space with as many as 60 dimensions (3 position and 3 momentum coordinates for each body). In the case of a convecting fluid, a complete description of the flow pattern requires knowledge of the velocity and temperature at a very large number of locations, so the number of dimensions of the phase plot are enormous (from thousands to millions, depending on the desired spatial resolution). As a result, the methods of nonlinear dynamics are cumbersome and progress has been slow, even though many interesting examples of spatiotemporal chaos have been explored both experimentally and numerically.
3) Recent research (D.A. Egolf et al: Nature 404:733 2000) involving numerical studies of an accepted model of thermal convection indicates that the origin of unpredictable motion in chaotic thermal convective systems, at least in one particular form of spatiotemporal chaos, lies in what occurs in small regions of space and over short time-scales. These local changes in the organization of the flow affect the surrounding regions in such a way that the entire future evolution is affected. The authors state: "This is something akin to Ed Lorenz's famous remark [E.N. Lorenz: J. Atmos. Sci. 20:130 1963] that the localized flapping of a butterfly's wings might change the weather dramatically over the entire world a few weeks later." Although such sensitivity to localized fluctuations has never been confirmed as the source of the unpredictability of the weather, it is apparently the origin of chaotic dynamics in thermal convection.
4) The authors conclude: "The methods used by Egolf et al should apply to many other forms of chaos in spatially extended systems (physical, chemical, and biological) for which reliable model equations are available, so that the key processes leading to the complex dynamics can be identified. Applications to areas as diverse as cardiology and atmospheric dynamics might be expected eventually. Moreover, it is not unreasonable to imagine that insight into the processes leading to unpredictability will also lead to progress in modifying or controlling the dynamics of these systems."
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