2007/07/17 by Paul Federbush, Federbush, Paul · 1 citation
Engineering · #Advanced Materials and Mechanics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.0707.2525
openalex publication_date 2007/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider tiles of some fixed size, with an associated weighting on the shapes of tile, of total mass 1. We study the pressure, p, of tilings with those tiles; the pressure, one over the volume times the logarithm of the partition function. (The quantity we define as "pressure" could, perhaps equally harmoniously with physics notation, be called "entropy per volume", neither nomenclature is "correct".) We let p0 (easy to compute) be the pressure in the limit of absolute smoothness (the weighting function is constant). Then as smoothness, suitably defined, increases, p converges to p0, uniformly in the volume. It is the uniformity requirement that makes the result non-trivial. This seems like a very basic result in the theory of pressure of tilings. Though at the same time, perhaps non-glamorous, being bereft of geometry and not very difficult. The problem arose for us out of study of a problem in mathematical physics, associated to a model of ferromagnetism.