Macroscopic Determinism in Interacting Systems Using Large Deviation Theory |
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Authors: | Brian R La Cour William C Schieve |
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Institution: | (1) Applied Research Laboratories, The University of Texas at Austin, P.O. Box 8029, Austin, Texas, 78713-8029;(2) Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems, Department of Physics, University of Texas at Austin, Austin, Texas, 78712 |
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Abstract: | We consider the quasi-deterministic behavior of systems with a large number, n, of deterministically interacting constituents. This work extends the results of a previous paper J. Statist. Phys.
99:1225–1249 (2000)] to include vector-valued observables on interacting systems. The approach used here, however, differs markedly in that a level-1 large deviation principle (LDP) on joint observables, rather than a level-2 LDP on empirical distributions, is employed. As before, we seek a mapping
t
on the set of (possibly vector-valued) macrostates such that, when the macrostate is given to be a
0 at time zero, the macrostate at time t is
t
(a
0) with a probability approaching one as n tends to infinity. We show that such a map exists and derives from a generalized dynamic free energy function, provided the latter is everywhere well defined, finite, and differentiable. We discuss some general properties of
t
relevant to issues of irreversibility and end with an example of a simple interacting lattice, for which an exact macroscopic solution is obtained. |
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Keywords: | determinism causality large deviation theory many-particle systems fluctuations nonequilibrium statistical mechanics cellular automata |
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