Nonstandard analysis and lattice statistical mechanics: a variational principle

Hurd, A. E.

doi:10.1090/S0002-9947-1981-0590413-2

Nonstandard analysis and lattice statistical mechanics: a variational principle
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by A. E. Hurd

Trans. Amer. Math. Soc. 263 (1981), 89-110

DOI: https://doi.org/10.1090/S0002-9947-1981-0590413-2

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Abstract:

Using nonstandard methods we construct a configuration space appropriate for the statistical mechanics of lattice systems with infinitely many particles and infinite volumes. Nonstandard representations of generalized equilibrium measures are obtained, yielding as a consequence a simple proof of the existence of standard equilibrium measures. As another application we establish an extension for generalized equilibrium measures of the basic variational principle of Landord and Ruelle. The same methods are applicable to continuous systems, and will be presented in a later paper.

References

Robert M. Anderson, A non-standard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), no. 1-2, 15–46. MR 464380, DOI 10.1007/BF02756559
Martin Davis, Applied nonstandard analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0505473
H. Föllmer and J. L. Snell, An “inner” variational principle for Markov fields on a graph, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 39 (1977), no. 3, 187–195. MR 445642, DOI 10.1007/BF00535471
L. L. Helms and P. A. Loeb, Applications of nonstandard analysis to spin models, J. Math. Anal. Appl. 69 (1979), no. 2, 341–352. MR 538222, DOI 10.1016/0022-247X(79)90147-1

Nonstandard models in nonequilibrium statistical mechanics

24

H. Jerome Keisler, An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. 48 (1984), no. 297, x+184. MR 732752, DOI 10.1090/memo/0297
Oscar E. Lanford III, Time evolution of large classical systems, Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974) Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975, pp. 1–111. MR 0479206
O. E. Lanford III and D. Ruelle, Observables at infinity and states with short range correlations in statistical mechanics, Comm. Math. Phys. 13 (1969), 194–215. MR 256687
A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I, Arch. Rational Mech. Anal. 59 (1975), no. 3, 219–239. MR 391830, DOI 10.1007/BF00251601
Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122. MR 390154, DOI 10.1090/S0002-9947-1975-0390154-8
R. A. Minlos, Lectures on statistical physics, Uspehi Mat. Nauk 23 (1968), no. 1 (139), 133–190 (Russian). MR 0223152
A. Ostebee, P. Gambardella, and M. Dresden, “Nonstandard” approach to the thermodynamic limit, Phys. Rev. A (3) 13 (1976), no. 2, 878–881. MR 395505, DOI 10.1103/PhysRevA.13.878
Chris Preston, Random fields, Lecture Notes in Mathematics, Vol. 534, Springer-Verlag, Berlin-New York, 1976. MR 0448630
Abraham Robinson, Non-standard analysis, Nederl. Akad. Wetensch. Proc. Ser. A 64 = Indag. Math. 23 (1961), 432–440. MR 0142464

Non-standard analysis

David Ruelle, Statistical mechanics: Rigorous results, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0289084
K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, Pure and Applied Mathematics, No. 72, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0491163