On the uniqueness of Gibbs states in the Pirogov-Sinai theory

We prove that, for low-temperature systems considered in the Pirogov-Sinai theory, uniqueness in the class of translation-periodic Gibbs states implies global uniqueness, i.e. the absence of any non-periodic Gibbs state. The approach to this infinite volume state is exponentially fast.Comment: 12 pages, Plain TeX, to appear in Communications in Mathematical Physic

Sound speed measurements in liquid oxygen-liquid nitrogen mixtures

The sound speed in liquid oxygen (LOX), liquid nitrogen (LN2), and five LOX-LN2 mixtures was measured by an ultrasonic pulse-echo technique at temperatures in the vicinity of -195.8C, the boiling point of N2 at a pressure of I atm. Under these conditions, the measurements yield the following relationship between sound speed in meters per second and LN2 content M in mole percent: c = 1009.05-1.8275M+0.0026507 M squared. The second speeds of 1009.05 m/sec plus or minus 0.25 percent for pure LOX and 852.8 m/sec plus or minus 0.32 percent for pure LN2 are compared with those reported by past investigators. Measurement of sound speed should prove an effective means for monitoring the contamination of LOX by Ln2

Invariant measures for Burgers equation with stochastic forcing

In this paper we study the following Burgers equation du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t) where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise in t. We prove the existence and uniqueness of an invariant measure by establishing a ``one force, one solution'' principle, namely that for almost every realization of the force, there is a unique distinguished solution that exists for the time interval (-infty, +infty) and this solution attracts all other solutions with the same forcing. This is done by studying the so-called one-sided minimizers. We also give a detailed description of the structure and regularity properties for the stationary solutions. In particular, we prove, under some non-degeneracy conditions on the forcing, that almost surely there is a unique main shock and a unique global minimizer for the stationary solutions. Furthermore the global minimizer is a hyperbolic trajectory of the underlying system of characteristics.Comment: 84 pages, published version, abstract added in migratio

Minimal Area of a Voronoi Cell in a Packing of Unit Circles

We present a new self-contained proof of the well-known fact that the minimal area of a Voronoi cell in a unit circle packing is equal to $2\sqrt{3}$, and the minimum is achieved only on a perfect hexagon. The proof is short and, in our opinion, instructive

The hard-core model on Z3\mathbb{Z}^3 and Kepler's conjecture

We study the hard-core model of statistical mechanics on a unit cubic lattice $\mathbb{Z}^3$, which is intrinsically related to the sphere-packing problem for spheres with centers in $\mathbb{Z}^3$. The model is defined by the sphere diameter $D>0$ which is interpreted as a Euclidean exclusion distance between point particles located at spheres centers. The second parameter of the underlying model is the particle fugacity $u$. For $u>1$ the ground states of the model are given by the dense-packings of the spheres. The identification of such dense-packings is a considerable challenge, and we solve it for $D^2=2, 3, 4, 5, 6, 8, 9, 10, 11, 12$ as well as for $D^2=2\ell^2$, where $\ell\in\mathbb{N}$. For the former family of values of $D^2$ our proofs are self-contained. For $D^2=2\ell^2$ our results are based on the proof of Kepler's conjecture. Depending on the value of $D^2$, we encounter three physically distinct situations: (i) finitely many periodic ground states, (ii) countably many layered periodic ground states and (iii) countably many not necessarily layered periodic ground states. For the first two cases we use the Pirogov-Sinai theory and identify the corresponding periodic Gibbs distributions for $D^2=2,3,5,8,9,10,12$ and $D^2=2\ell^2$, $\ell\in\mathbb{N}$, in a high-density regime $u>u_*(D^2)$, where the system is ordered and tends to fluctuate around some ground states. In particular, for $D^2=5$ only a finite number out of countably many layered periodic ground states generate pure phases