Abstract
We study N-particle systems in Rd whose interactions are governed by a hypersingular Riesz potential | x- y| -s, s> d, and subject to an external field. We provide both macroscopic results as well as microscopic results in the limit as N→ ∞ for random point configurations with respect to the associated Gibbs measure at scaled inverse temperature β. We show that a large deviation principle holds with a rate function of the form ‘β-Energy + Entropy’, yielding that the microscopic behavior (on the scale N-1/d) of such N-point systems is asymptotically determined by the minimizers of this rate function. In contrast to the asymptotic behavior in the integrable case s< d, where on the macroscopic scale N-point empirical measures have limiting density independent of β, the limiting density for s> d is strongly β-dependent.
Original language | English (US) |
---|---|
Pages (from-to) | 61-100 |
Number of pages | 40 |
Journal | Constructive Approximation |
Volume | 48 |
Issue number | 1 |
DOIs | |
State | Published - Aug 1 2018 |
Keywords
- Empirical measures
- Gibbs measure
- Large deviation principle
- Minimal energy
- Riesz gases
ASJC Scopus subject areas
- Analysis
- General Mathematics
- Computational Mathematics