Jakob Yngvason

Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional

The ground-state properties of interacting Host gases in external potentials, as considered in recent exptriments, are usually described by means of the Gross-Pitaevskii energy functional. We present here a rigorous proof of the asymptotic exactness of this approximation for the ground-state energy and particle density of a dilute Bose gas with a positive interaction.

A Rigorous Derivation¶of the Gross–Pitaevskii Energy Functional¶for a Two-dimensional Bose Gas

Abstract: We consider the ground state properties of an inhomogeneous two-dimensional Bose gas with a repulsive, short range pair interaction and an external confining potential. In the limit when the particle number N is large but ρ̅a2 is small, where ρ̅ is the average particle density and a the scattering length, the ground state energy and density are rigorously shown to be given to leading order by a Gross–Pitaevskii (GP) energy functional with a coupling constant g~1/|1n(ρ̅a2)|. In contrast to the 3D case the coupling constant depends on N through the mean density. The GP energy per particle depends only on Ng. In 2D this parameter is typically so large that the gradient term in the GP energy functional is negligible and the simpler description by a Thomas–Fermi type functional is adequate.

A FRESH LOOK AT ENTROPY AND THE SECOND LAW OF THERMODYNAMICS

In days long gone, the second law of thermodynamics (which predated the first law) was regarded as perhaps the most perfect and unassailable law in physics. It was even supposed to have philosophical import: It has been hailed for providing a proof of the existence of God (who started the universe off in a state of low entropy, from which it is constantly degenerating); conversely, it has been rejected as being incompatible with dialectical materialism and the perfectibility of the human condition.

String-localized quantum fields and modular localization

We study free, covariant, quantum (Bose) fields that are associated with irreducible representations of the Poincaré group and localized in semi-infinite strings extending to spacelike infinity. Among these are fields that generate the irreducible representations of mass zero and infinite spin that are known to be incompatible with point-like localized fields. For the massive representations and the massless representations of finite helicity, all string-localized free fields can be written as an integral, along the string, of point-localized tensor or spinor fields. As a special case we discuss the string-localized vector fields associated with the point-like electromagnetic field and their relation to the axial gauge condition in the usual setting.

Asymptotics of heavy atoms in high magnetic fields. II. Semiclassical regions

The ground state energy of an atom of nuclear chargeZe in a magnetic fieldB is exactly evaluated to leading order asZ→∞ in the following three regions:B≪Z4/3,B≈Z4/3 andZ4/3≪B≪Z3. In each case this is accomplished by a modified Thomas-Fermi (TF) type theory. We also analyze these TF theories in detail, one of their consequences being the nonintuitive fact that atoms are spherical (to leading order) despite the leading order change in energy due to theB field. This paper complements and completes our earlier analysis [1], which was primarily devoted to the regionsB≈Z3 andB≫Z3 in which a semiclassical TF analysis is numerically and conceptually wrong. There are two main mathematical results in this paper, needed for the proof of the exactitude of the TF theories. One is a generalization of the Lieb-Thirring inequality for sums of eigenvalues to include magnetic fields. The second is a semiclassical asymptotic formula for sums of eigenvalues that isuniform in the fieldB.

Bose-Einstein quantum phase transition in an optical lattice model

Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an external potential, such as that presented by an optical lattice. We present a model of this phenomenon which we are able to analyze rigorously. The system is a hard core lattice gas at half of the maximum density and the optical lattice is modeled by a periodic potential of strength λ. For small λ and temperature, BEC is proved to occur, while at large λ or temperature there is no BEC. At large λ the low-temperature states are in a Mott insulator phase with a characteristic gap that is absent in the BEC phase. The interparticle interaction is essential for this transition, which occurs even in the ground state. Surprisingly, the condensation is always into the p=0 mode in this model, although the density itself has the periodicity of the imposed potential.

The Ground State Energy of a Dilute Two-Dimensional Bose Gas

The ground state energy per particle of a dilute, homogeneous, two-dimensional Bose gas, in the thermodynamic limit is shown rigorously to be E0/N=(2πℏ2ρ/m)|ln(ρa2)|−1, to leading order, with a relative error at most O(|ln(ρa2)|−1/5). Here N is the number of particles, ρ=N/V is the particle density and a is the scattering length of the two-body potential. We assume that the two-body potential is short range and nonnegative. The amusing feature of this result is that, in contrast to the three-dimensional case, the energy, E0 is not simply N(N−1)/2 times the energy of two particles in a large box of volume (area, really) V. It is much larger.

One-dimensional bosons in three-dimensional traps.

Recent experimental and theoretical work has indicated conditions in which a trapped, low density Bose gas ought to behave like the 1D delta-function Bose gas solved by Lieb and Liniger. Up until now, the theoretical arguments have been based on variational/perturbative ideas or numerical investigations. There are four parameters: density, transverse and longitudinal dimensions, and scattering length. In this paper we explicate five parameter regions in which various types of 1D or 3D behavior occur in the ground state. Our treatment is based on a rigorous analysis of the many-body Schrödinger equation.

There are no causality problems for Fermi's two-atom system.

A repeatedly discussed gedanken expeirment, proposed by Fermi to check Einstein causality, is reconsidered. It is shown that, contrary to a recent statement made by Hegerfeldt, there appears no causality paradox in a proper theoretical description of the experiment.

A GUIDE TO ENTROPY AND THE SECOND LAW OF THERMODYNAMICS

This article is intended for readers who, like us, were told that the second law of thermodynamics is one of the major achievements of the nineteenth cenwry—that it is a logical, perfect, and unbreakable law—but who were unsatisfied with the “derivations” of the entropy principle as found in textbooks and in popular writings.