Level I theory of large deviations in the ideal gas

Abstract

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^dbe a convex Borel set. DenoteS _n=f(X₁)+f(X₂)+...+f(X_n). Using large-deviation 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 so-called discrete ideal gas, but we consider the continuous ideal gas.