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An example of the kind of question that interests my group is understanding why the weather is hard to forecast. Most people take for granted that the weather is unpredictable but what precisely makes it so? Is it because the radius of the earth is large compared to the depth of the atmosphere, so that the atmosphere consists of many uncorrelated regions of dynamical activity? Is it because the atmosphere is strongly driven out of equilibrium and so is turbulent? On what details does the forecasting time of about two weeks actually depend?
A similar question of interest is how fibrillation commences in a mammalian heart, i.e., how does the usual coherent periodic beating change to an incoherent nonperiodic dynamics? As one clue, the occurrence of fibrillation is known to be size-dependent, e.g., animals with small hearts such as mice or guinea pigs do not have heart attacks and fibrillation, if induced artificially in such hearts, spontaneously returns to a periodic dynamics. Why are big hearts more susceptible to fibrillation? Do whales, whose hearts can approach six feet in diameter, have heart attacks and, if not, how is fibrillation avoided? More generally, how does the dynamics of any nonequilibrium system depend on its size?
Rather than study such difficult questions directly, my group follows a theoretical physics tradition of finding simple idealized mathematical models that contain essential mechanisms of complex dynamics. We then study these models, usually by computer simulation, to answer some basic questions: what kinds of transitions (bifurcations) lead from simple to more complex dynamics? What kinds of spatiotemporal states are possible and how do they depend on parameters? How does the transport of energy and mass depend on the spatiotemporal structure of some medium?
A related part of my group's research is finding new or better ways to characterize the spatiotemporal disorder found in many nonequilibrium systems. What is really meant by the word "complex" and how does one compare one complex state with another, e.g., a computer simulation with experimental data?
Often our effort is directed towards understanding simpler experimental systems such as the dynamics of a shallow wide layer of convecting fluid (Rayleigh-Benard convection) since such experimental systems are much more completely characterized than the weather or a heart. E.g., scientists can perform convection experiments in which essentially all details are known and controlled to a high accuracy and for which we have high confidence that a complete quantitative description is available (the Boussinesq equations). This allows one to test quantitatively theory against experiment with an accuracy and confidence that is not possible in the fields of meteorology or physiology. Such quantitative comparisions of theory with experiment are essential if broad fundamental principles of nonequilibrium dynamics are to be discovered.
In carrying out this research, my group is grateful for support
from the National Science
Foundation and the Department