Royce Zia uses a checkerboard and a computer game to illustrate theoretical condensed matter physics, a discipline devoted to understanding the cooperative behavior in systems with large numbers of constituent particles.
Zia studies the statistical mechanics of driven diffusive systems, a special class of physical systems far from thermal equilibrium. Typical systems consist of too many constituents to be described in detail, such as air molecules in a room. Nevertheless, thanks to statistical mechanics, many overall properties can be predicted.
The goal is to understand how a variety of macroscopic states emerges from an ensemble of simple microscopic constituents.
“Though ice, water, and steam have very different properties, all are results of just H20 molecules subjected to very simple interactions,” Zia says. Another example is a collection of carbon atoms, which can form ash, graphite, diamond, and, more recently, Buckyballs.”
The impact of cooperative behavior can be appreciated from examples far from physics. Zia likes to use the analogy of a collection of children. “Even if the interaction between two children leads to well-behaved play, we can imagine the result of having 30 of them in a living room instead. On the other hand, a dramatically different state will ensue if we put the same set of kids on a football field.”
The examples of H20 molecules and carbon atoms noted above represent cooperative behavior in thermal equilibrium. By contrast, under conditions far from equilibrium, the same collection of particles can produce an even greater variety of states. The key difference lies in the presence of some form of energy flowing through non-equilibrium states. Examples range from physical patterns like snowflakes to the full biological gamut. Zia’s research focuses on the question common in all these cases: How do complex macroscopic patterns emerge from a few simple dynamical rules governing the evolution of microscopic constituents? In search of an answer, he looks for the bare essentials needed to produce complex behavior, through the study of simple model systems that display rich and surprising phenomena.
Zia uses a checkerboard to illustrate one such model. Given a few of the red and black checkers arranged on the board and trying to move against each other, Zia imposes a few simple rules: A piece moves randomly forwards or sideways by single steps (provided the target square is empty), and, when pushed off the board, they can “wrap around” to come back up on the other side. If only a few checkers are present, they can manipulate their way around their opponents, he demonstrates, but when more and more pieces are added, a new pattern emerges. They lose options for movement, form clusters and, finally, crash into a giant stalemate. The essential ingredients for the transition, in this case, are inherent randomness and the two preferred directions of movement. An analogous phenomena in familiar daily life is the traffic jam. With fast cars and slow trucks on a multilane highway, jams occur when too many vehicles are present.
To illustrate the importance of randomness, Zia shows the fragility of intricate patterns in the game of ‘Life’, which relies on deterministic rules of evolution. The addition or removal of a single individual can destroy a self-replicating colony. “On the other hand, stochastic rules pick out robust patterns instead of fragile ones,” he says. “Most of the steady states around us are stable against small, random perturbations. To build simple models for these stable patterns, it is crucial to incorporate stochastic rules.”
Although the main goal of Zia’s work — the search for overarching principles that underlie non-equilibrium statistical mechanics — is general and abstract, the models studied have potential applications in many areas. Apart from traffic jams, other examples may be found in biology (e.g., electrophoresis — a method commonly used in DNA analysis, rates of translation in protein synthesis, biological motors) or materials science (e.g., super-ionic conductors, polymer dynamics, surface growth, granular flow).
The tools of his trade range from sophisticated mathematical techniques (stochastic differential equations, path integrals, renormalization group) to down-to-earth computer simulations of pieces jumping around on a checkerboard. While advanced mathematics is needed in the final understanding of how complex patterns emerge, it is simulations that facilitate the discovery of these surprising phenomena.
Dr. Zia is professor of physics.
— Written by Sally Harris
