Project Details
Description
This grant is supported by the Divisions of Materials Research, Physics, and Mathematical Sciences. The research covers a broad program in statistical mechanics with an aim to better understand macroscopic phenomena originating in the collective behavior of its microscopic constituents. The methods used range from rigorous mathematical analysis to computer simulations. Topics to be studied include:
(i) Symmetry breaking transitions leading to the formation of spatial (temporal) patterns are fascinating and important examples of collective phenomena. They are paradigms of emergent behavior, with no counterpart in the properties of individual atoms or molecules. They occur in very diverse situations, ranging from crystallization in equilibrium systems, the development of rolls and cells in fluids heated from below to the formation of patterns in morphogenesis. For equilibrium systems the state observed with overwhelming probability is the one which maximizes the entropy: the logarithm of the number of microscopic states (with given constraints) corresponding to the macroscopic structure. This translates into minimization of the free energy at a fixed temperature, etc. For nonequilibrium systems there is no such general principle. We have however recently found a rigorous generalization of free energy to a model nonequilibrium system. Extending this work to more realistic systems will provide a framework fo rpattern formation in nonequilibrium with some of the generality now enjoyed by equilibrium.
(ii) The behavior of alloys and fluid mixtures following a quench from a uniform high temperature phase into the coexistence region continues to offer challenging problems. Of particular theoretical interest and practical importance is the case of alloys with elastic interactions where the kinetics determine many of the important physical properties. Our approach to these problems includes analytic derivations of macroscopic equations describing phase segregation from microscopic models; investigations of the solution of these highly nonlinear equations; and, computer simulations. An important simplification occurs when the interactions giving rise to the phase segregation are long range Kac potentials. These systems permit the investigation of detailed properties of the interface (soliton) between different phases.
(iii) Work on other aspects of nonequilibrium phenomena include dynamical systems approach to current carrying systems in which the time evolution is described by thermostated deterministic non-Hamiltonian dynamics; the study of very large, formally infinite, Hamiltonian systems which can be divided into a subsystem and reservoirs; microscopic derivation of macroscopic (hydrodynamics type) equations for realistic systems.
(iv) A surprisingly large number of macroscopic phenomena, such as boiling and freezing, can be treated as if the atomic world was (effectively) classical. This is no longer so at low temperatures or ano sizes. There one is in the world of the quantum where the phenomena is much richer and calculations much harder. Work will continue on both equilibrium and nonequilibrium systems which are intrinsically quantum mechanical. The Schroedinger time evolution of even the simplest model system is exceedingly complex and fascinating once one goes beyond perturbation theory.
(v) Biological systems are now at the frontier of science. There have already been many applications of statistical mechanical ideas to biology. These range from ecological and epidemiological systems to neural network models of the immune system and the brain. While only a few of these applications have really been on target, the methodology of statistical mechanics does seem to provide the right framework for describing how higher level patterns or behavior emerge from the activity of a multitude of interacting simpler entities. Research will be carried out on a number of these topics.
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This grant is supported by the Divisions of Materials Research, Physics, and Mathematical Sciences. The research covers a broad program in statistical mechanics with an aim to better understand macroscopic phenomena originating in the collective behavior of its microscopic constituents. The methods used range from rigorous mathematical analysis to computer simulations.
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Status | Finished |
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Effective start/end date | 1/1/02 → 3/31/05 |
Funding
- National Science Foundation: $487,500.00