Biomedical Engineering Reference
In-Depth Information
removed except by drifting off the boundary of the system or annihilating with
another defect of opposite topological sign (a right-handed, spiral). The dynam-
ics of the system may be described as a changing pattern of defect locations.
Since the behavior of dissipative nonlinear dynamical systems with many de-
grees of freedom resembles in several ways the behavior of systems treated by
statistical mechanics, the language of statistical mechanics sometimes creeps
into discussions of deterministic nonlinear dynamical systems. In particular,
bifurcations are sometimes referred to as
phase transitions
.
The statistical approach is very tempting to physicists. The tools of statisti-
cal mechanics have been spectacularly successful in analyzing phase transitions
in equilibrium systems, and they become more and more accurate as system size
increases. They are based on the idea that the details of how a system moves in
its very high-dimensional state space are unimportant for making statistical pre-
dictions about the states it is likely to be found in. Unfortunately, the fundamen-
tal assumptions of statistical mechanics are strongly violated in the driven,
dissipative systems of biomedical interest. At present, there is increasing evi-
dence that statistical mechanics can account for the behavior of a number of
deterministic systems far from equilibrium. Examples include coupled discrete
maps that undergo bifurcation that are quantitatively similar to equilibrium
phase transitions (18), and sheared granular materials like sand in which effec-
tive temperatures can be defined for describing the wanderings of individual
grains in space and time (16). These applications of statistical mechanics are as
yet poorly understood, however, and in the absence of a fundamental theory of
nonequilibrium pattern formation, theoretical insight comes largely from nu-
merical simulation of model equations and analysis of the generic types of be-
havior on a case by case basis.
Construction of an appropriate model for a spatially extended dissipative
system involves a healthy dose of intuition as well as a few constructive princi-
ples. One generally begins with the selection of the simplest PDE that incorpo-
rates the symmetries and general features of the physical system. This may be a
known set of equations that is selected either because its solutions seem to
match the observed behavior qualitatively or because the underlying physics of
the system is expected to be in the same universality class. One then simulates
the system numerically and attempts to find a regime in parameter space where
the spatiotemporal dynamics is roughly reminiscent of the real system. Analysis
of the model can then lead to hypotheses about the effects of varying parameters
in the real system. To obtain more accurate predictions, one then adds terms to
the model that alter the detailed behavior without changing the big picture.
Given the complexity of types of bifurcations that can occur in large systems,
however, one often discovers new and unexpected attractors, or one finds that
detailed models just don't work and aspects of the physics that were thought to
be irrelevant actually must be included in order to obtain reasonable representa-
tions of the true dynamical attractors. A good example of progress of this sort is
