David A. Egolf, Georgetown University

Most of my research involves trying to understand systems far-from-equilibrium (meaning that the energy input and the energy dissipated are not precisely balanced at all places and at all times). Unlike the situation for systems in equilibrium, researchers have so far been stymied in their efforts to develop a general, predictive theory of nonequilibrium systems. My research group uses the tools of nonlinear dynamics and statistical physics to try to establish the underpinnings for such a theory. Most of the work also involves the use of large-scale computation, mostly on my computer clusters of about 100 CPUs. The types of systems I study range from fluid and granular systems to electrochemical waves in heart tissue to idealized mathematical models. Some of the projects are described below. The work has been supported by the NSF, Research Corporation, NASA, and the Sloan Foundation.

Dynamical Events in Spiral Defect Chaos

The behaviors of systems far-from-equilibrium can often appear hopelessly complicated --- a muddled situation with a wide variety of length scales and time scales and often a chaotic dynamics that defies prediction. We have been using the mathematical tools of nonlinear dynamics to identify within this wild behavior particular events which determine the future evolution of the system. Using massively-parallel simulations of Rayleigh-Bénard convection, we found that the complex behavior of the Spiral Defect Chaos state of convection is actually largely determined by particular events (convection roll breaking) that are highly localized in space and time. It is only in the vicinity of these events that the system is particularly sensitive to small perturbations, so the rest of the system, although complex-looking, is to a large degree predictable.

D.A. Egolf, I.V. Melnikov, W. Pesch, and R.E. Ecke, Nature 404:733-736 (2000).

Fibrillating Heart Tissue

We have been applying these same techniques to a model of fibrillating heart tissue, with the hope of inspiring the development of targeted defibrillation in which tiny electrical (or chemical) pulses can be introduced in specified portions of the heart to lead the heart out of fibrillation in a gentle way. Again, we found that particular events within the complicated chaotic dynamics of fibrillation largely determine the future behavior of the fibrillating tissue. However, we found an additional wrinkle that some events which initially appear to be extremely sensitive to perturbations are quickly followed by periods of behavior that cancel the effect of the sensitive period to a large degree.

2D Granular Shear

We have also been using the same dynamical analysis techniques to study the behavior of a two-dimensional granular layer subjected to shear. This system has been studied extensively since Bob Behringer and collaborators discovered that the system goes through a glass-like jamming transition as the density of grains is varied. We have found intriguing connections between the Lyapunov vectors describing the most important dynamical modes and the grains that are rearranging cooperatively near the jamming transition. In addition, we have measured mathematically well-defined dynamical length scales and time scales that diverge near the jamming transition density.

Building Blocks of Spatiotemporal Chaos

Such complicated behavior arising from isolated events provides hope that we may be able to develop an understanding of these systems by characterizing the events and the parts of the system involved in the events (much like the way we can understand a great deal about gases by understanding atoms and the ways they interact when they get close to each other). The idea that spatiotemporal chaotic systems could be considered as a collection of weakly-interacting subsystems was originally argued by Ruelle and later expanded upon by Cross and Hohenberg and others. In a series of papers, we have explored this idea in hopes of identifying these building blocks on which to base a statistical mechanics of spatiotemporal chaos.

M.P. Fishman and D.A. Egolf, Phys. Rev. Lett. 96:054103 (2006).
D.A. Egolf, Phys. Rev. Lett. 81:4120-4123 (1998).
C.S. O'Hern, D.A. Egolf, and H.S. Greenside, Phys. Rev. E 53:3374-3386 (1996).
D.A. Egolf and H.S. Greenside, Phys. Rev. Lett. 74:1751-1754 (1995).
D.A. Egolf and H.S. Greenside, Nature 369:129-131 (1994).

Statistical Mechanics Far-from-equilibrium

One of the reasons researchers are hopeful that a statistical mechanics of spatiotemporal chaos might be developed is that the behavior of many of these systems resembles the phase transitions of equilibrium systems. In a paper in Science I showed that, remarkably, a coarse-graining of a simple spatiotemporal chaotic system is indistinguishable from an equilibrium system of the Ising universality class.

D.A. Egolf, Science 287:101-104 (2000).

More Granular Dynamics

Since arriving at Georgetown, I have developed a successful collaboration with Prof. Jeffrey Urbach to study the thermodynamics of driven granular systems. In a series of papers, we have described our efforts to understand the behavior of these granular systems. In early work, we explored the relationship between the way the system is driven and correlations within the granular media. In more recent work, we studied two phenomena that appear in our non-equilibrium granular systems and also in equilibrium systems --- two-phase coexistence and the depletion force --- and characterized the similarities and differences between the equilibrium and non-equilibrium situations. Our current work involves driving a granular system using both shaking and shearing to introduce multiple time-scales, and we're seeing some intriguing behavior.

P. Melby, A. Prevost, D.A. Egolf, and J.S. Urbach, Depletion force in a bidisperse granular layer, To appear in Phys. Rev. E.
P. Melby, F.V. Reyes, A. Prevost, R. Robertson, P. Kumar, D.A. Egolf, and J.S. Urbach, J. Phys. Cond. Matt. 17:S2689-S2704 (2005).
A. Prevost, P. Melby, D.A. Egolf, and J.S. Urbach, Phys. Rev. E 70:050301 (2004).
A. Prevost, D.A. Egolf, and J.S. Urbach, Phys. Rev. Lett. 89:084301 (2002).

Biophysics

My colleague Jeff Urbach and I have two students working on projects to study the behavior of biopolymers in solution and in networks. This work will hopefully give us insight into important mechanisms at work within cells.

QCD Calculations using Chiral Perturbation Theory

My scientific interests are quite broad, so I occasionally work on projects far afield from my usual research. I have been working with Professor Roxanne Springer at Duke to calculate various quantities using an effective theory of low energy quantum chromodynamics (QCD) --- heavy baryon chiral perturbation theory. In the more recent of our two joint papers, at the request of experimentalists at Fermilab we calculated the decay rates of doubly-heavy baryons into a variety of channels.

D.A. Egolf, R.P. Springer, and J. Urban, Phys. Rev. D 68:013003 (2003).
D.A. Egolf, I.V. Melnikov, and R.P. Springer, Phys. Lett. B 451:267-274 (1999).