Award Abstract #0072666
Homogenizations Continuum Limits and Kinetic Limits for Stochastic Models

NSF Org:	DMS Division of Mathematical Sciences
Initial Amendment Date:	July 20, 2000
Latest Amendment Date:	March 26, 2002
Award Number:	0072666
Award Instrument:	Continuing grant
Program Manager:	Dean M Evasius DMS Division of Mathematical Sciences MPS Directorate for Mathematical & Physical Sciences
Start Date:	July 15, 2000
Expires:	June 30, 2004 (Estimated)
Awarded Amount to Date:	$90000
Investigator(s):	Fraydoun Rezakhanlou rezakhan@math.berkeley.edu (Principal Investigator)
Sponsor:	University of California-Berkeley Sponsored Projects Office BERKELEY, CA 94704 510/642-8109
NSF Program(s):	PROBABILITY
Field Application(s):	0000099 Other Applications NEC
Program Reference Code(s):	OTHR,0000
Program Element Code(s):	1263

ABSTRACT

An outstanding and long studied problem in statistical mechanics is to establish the connection between the microscopic world and its macroscopic behavior. The investigator's research concerns stochastic models associated with the evolution of dilute gases and the formation of solids. As the first step, one derives a partial differential equation for the macroscopic evolution of such stochastic models. Roughly speaking, one shows that after a suitable scaling, the particle density of a dilute gas (respectively, the boundary surface of a solid) converges to a solution of the Boltzmann equation (respectively, Hamilton-Jacobi equation). Probabilistically, such a convergence is a law of large numbers and its corresponding central limit theorem provides us with some vital information about the microscopic model under the study.

Solids form through growth processes which take place at the surface. Imagine an already formed nucleus to which further material sticks from the ambient atmosphere. The process of the attachment is a function of a huge variety of growth mechanisms depending on the materials involved, their temperature, composition, etc. Following the tradition of statistical mechanics, one studies simplified models which nevertheless captures some of the essential physics. Such simplified models are proved to be useful in understanding the intricate formation of solids. It turns out that these models describe other phenomena such as the spread of infected cells in a tissue, the effect of impurity in the evolution of a fluid, etc. The investigator's research concerns the interplay between the microscopic growth rules and the macroscopic shape of the surface of a solid.

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