Level I theory of large deviations in the ideal gas
 T. Lehtonen,
 E. Nummelin
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
LetX _{1},X _{2},... be i.i.d. random elements (the states of the particles 1,2,...). Letf be an ℝ^{d}valued, measurable function (an observable) and letB ⊂R^{d}be a convex Borel set. DenoteS _{n}=f(X_{1})+f(X_{2})+...+f(X_{n}). Using largedeviation theory, it may be shown that, under certain regularity conditions, there exists a point υ_{B} (the dominating point of B) so that, givenS _{n}/nε B, actually S_{n}/n→υ_{ B } in probability as n→∞. Having this conditional weak law of large numbers as our starting point, we consider physical systems of independent particles, especially the ideal gas. Given an observed energy level, we derive convergence results for empirical means, empirical distributions, and microcanonical distributions. Results are obtained for a closed system with a fixed number of particles as well as for an open particle system in the space (a Poisson random field). Our approach is elementary in the sense that we need not refer to the abstract “level II” theory of large deviations. However, the treatment is not restricted to the socalled discrete ideal gas, but we consider the continuous ideal gas.
 Aizenman, M., Goldstein, S., and Lebowitz, J. L. (1978).Communications in Mathematical Physics,62, 279–302.
 Csiszar, I. (1984).Annals of Probability,12, 768–793.
 Dobrushin, R. L., and Tirozzi, B. (1977).Communications in Mathematical Physics,54, 173–192.
 Ellis, R. S. (1985).Entropy, Large Deviations and Statistical Mechanics, Springer, New York.
 Lanford, O. E. (1973). Entropy and equilibrium states in classical statistical mechanics, inStatistical Mechanics and Mathematical Problems, Springer, Berlin, pp. 1–113.
 MartinLöf, A. (1979).Statistical Mechanics and the Foundations of Thermodynamics, Springer, Berlin.
 Ney, P. (1983).Annals of Probability 11, 158–167.
 Nummelin, E. (1987). A conditional weak law of large numbers, in Proceedings of the Seminar on Stability Problems for Stochastic Models, Suhumi, USSR. Lecture Notes in Mathematics,1142, 259–262.
 Rockafellar, R. T. (1970).Convex Analysis, Princeton University Press, Princeton, New Jersey.
 Title
 Level I theory of large deviations in the ideal gas
 Journal

International Journal of Theoretical Physics
Volume 29, Issue 6 , pp 621635
 Cover Date
 19900601
 DOI
 10.1007/BF00672036
 Print ISSN
 00207748
 Online ISSN
 15729575
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Industry Sectors
 Authors

 T. Lehtonen ^{(1)}
 E. Nummelin ^{(2)}
 Author Affiliations

 1. Department of Mathematics and Statistics, Helsinki School of Economics, Runeberginkatu 2224, 00100, Helsinki, Finland
 2. Department of Mathematics, University of Helsinki, Hallituskatu 15, 00100, Helsinki, Finland