No Arabic abstract
This paper continues the work Y. Suhov, M. Kelbert. FK-DLR states of a quantum bose-gas, arXiv:1304.0782 [math-ph], and focuses on infinite-volume bosonic states for a quantum system (a quantum gas) in a plane. We work under similar assumptions upon the form of local Hamiltonians and the type of the (pair) interaction potential as in the reference above. The result of the paper is that any infinite-volume FK-DLR functional corresponding to the Hamiltonians is shift-invariant, regardless of whether this functional is unique or not.
The generating functional of the cyclic representation of the CCR (Canonical Commutation Relations) representation for the thermodynamic limit of the grand canonical ensemble of the free Bose gas with attractive boundary conditions is rigorously computed. We use it to study the condensate localization as a function of the homothety point for the thermodynamic limit using a sequence of growing convex containers. The Kac function is explicitly obtained proving non-equivalence of ensembles in the condensate region in spite of the condensate density being zero locally.
We study the entropy of pure shift-invariant states on a quantum spin chain. Unlike the classical case, the local restrictions to intervals of length $N$ are typically mixed and have therefore a non-zero entropy $S_N$ which is, moreover, monotonically increasing in $N$. We are interested in the asymptotics of the total entropy. We investigate in detail a class of states derived from quasi-free states on a CAR algebra. These are characterised by a measurable subset of the unit interval. As the entropy density is known to vanishes, $S_N$ is sublinear in $N$. For states corresponding to unions of finitely many intervals, $S_N$ is shown to grow slower than $(log N)^2$. Numerical calculations suggest a $log N$ behaviour. For the case with infinitely many intervals, we present a class of states for which the entropy $S_N$ increases as $N^alpha$ where $alpha$ can take any value in $(0,1)$.
The Embedded-Atom Model (EAM) provides a phenomenological description of atomic arrangements in metallic systems. It consists of a configurational energy depending on atomic positions and featuring the interplay of two-body atomic interactions and nonlocal effects due to the corresponding electronic clouds. The purpose of this paper is to mathematically investigate the minimization of the EAM energy among lattices in two and three dimensions. We present a suite of analytical and numerical results under different reference choices for the underlying interaction potentials. In particular, Gaussian, inverse-power, and Lennard-Jones-type interactions are addressed.
The superstable Weakly Imperfect Bose Gas {(WIBG)} was originally derived to solve the inconsistency of the Bogoliubov theory of superfluidity. Its grand-canonical thermodynamics was recently solved but not at {point of} the {(first order)} phase transition. This paper proposes to close this gap by using the large deviations formalism and in particular the analysis of the Kac distribution function. It turns out that, as a function of the chemical potential, the discontinuity of the Bose condensate density at the phase transition {point} disappears as a function of the particle density. Indeed, the Bose condensate continuously starts at the first critical particle density and progressively grows but the free-energy per particle stays constant until the second critical density is reached. At higher particle densities, the Bose condensate density as well as the free-energy per particle both increase {monotonously}.
Models of quantum and classical particles on the d-dimensional cubic lattice with pair interparticle interactions are considered. The classical model is obtained from the corresponding quantum one when the reduced physical mass of the particle tends to infinity. For these models, it is proposed to define the convergence of the Euclidean Gibbs states, when the reduced mass tends to infinity, by the weak convergence of the corresponding Gibbs specifications, determined by conditional Gibbs measures. In fact it is proved that all conditional Gibbs measures of the quantum model weakly converge to the conditional Gibbs measures of the classical model. A similar convergence of the periodic Gibbs measures and, as a result, of the order parameters, for such models with the pair interactions possessing the translation invariance, has also been proven.