Large Deviations in the Superstable Weakly Imperfect Bose-Gas

The Superstable Weakly Imperfect Bose-Gas (Sup-WIBG) was originally proposed to solve some inconsistencies of the Bogoliubov theory based on the WIBG. The grand-canonical thermodynamics of the Sup-WIBG has been recently studied in details but only out of the point of the (first order) phase transition. The present paper closes this gap. The key technical tools are the Large Deviations (LD) formalism and in particular the analysis of the Kac distribution function. It turns out that the condensate fraction discontinuity as a function of the chemical potential (that occurs at the phase transition point) disappears if one considers it as a function of the total particle density. We prove that at this point the equilibrium state of the Sup-WIBG is a mixture of two (low- and high-density) pure phases related to two critical particle densities. Non-zero Bose-Einstein condensate starts at the smaller critical density and continuously grows (for a constant chemical potential) until the second critical density. For higher particle densities, the Bose condensate fraction as well as the chemical potential both increase monotonously.     

Bose-Einstein Condensate in a Linear Trap with a Dimple Potential

We study Bose-Einstein condensation in a linear trap with a dimple potential where we model dimple potentials by Dirac δ function. Attractive and repulsive dimple potentials are taken into account. This model allows simple, explicit numerical and analytical investigations of noninteracting gases. Thus, the Schrdinger equation is used instead of the Gross-Pitaevski equation. We calculate the atomic density, the chemical potential, the critical temperature and the condensate fraction. The role of the relative depth of the dimple potential with respect to the linear trap in large condensate formation at enhanced temperatures is clearly revealed. Moreover, we also present a semi-classical method for calculating various quantities such as entropy analytically. Moreover, we compare the results of this paper with the results of a previous paper in which the harmonic trap with a dimple potential in 1D is investigated.     

The thermodynamic limit for an imperfect Boson gas

We give a rigorous treatment in the infinite volume limit of a model Hamiltonian representing an imperfect Boson gas. In particular we obtain the exact expression for the mean particle density in the infinite volume limit as a function of the chemical potential, and show that the density function has a singularity at the critical density for Bose-Einstein condensation. We prove that, unlike the ideal Boson gas, the imperfect Boson gas has the same behaviour in the infinite volume limit for the grand canonical ensemble as for the canonical ensemble, and is moreover stable under small perturbations. We finally exhibit the possibility of ordinary condensation and prove that a system in an intermediate situation between two pure phases consists of a simple mixture of the two phases involved.     

Stability of Attractive Bose–Einstein Condensates

We propose the critical nonlinear Schrödinger equation with a harmonic potential as a model of attractive Bose–Einstein condensates. By an elaborate mathematical analysis we show that a sharp stability threshold exists with respect to the number of condensate particles. The value of the threshold agrees with the existing experimental data. Moreover with this threshold we prove that a ground state of the condensate exists and is orbital stable. We also evaluate the minimum of the condensate energy.     

Non-linear Stark effect and molecular localization

We consider a non-linear Stark effect as a model for localization and symmetry breaking of a molecule in a gas. By a comparison method with respect to the linear Stark effect, we prove the existence of level bifurcation and symmetry breaking at a critical value of the gas pressure exponentially small for large nuclear masses. Extending the Davies results, we confirm the predictions of Claverie-Jona Lasinio for pyramidal molecules as the ammonia one.     

Large Deviations for the Boundary Driven Symmetric Simple Exclusion Process

The large deviation properties of equilibrium (reversible) lattice gases are mathematically reasonably well understood. Much less is known in nonequilibrium, namely for nonreversible systems. In this paper we consider a simple example of a nonequilibrium situation, the symmetric simple exclusion process in which we let the system exchange particles with the boundaries at two different rates. We prove a dynamical large deviation principle for the empirical density which describes the probability of fluctuations from the solutions of the hydrodynamic equation. The so-called quasi potential, which measures the cost of a fluctuation from the stationary state, is then defined by a variational problem for the dynamical large deviation rate function. By characterizing the optimal path, we prove that the quasi potential can also be obtained from a static variational problem introduced by Derrida, Lebowitz, and Speer.     

Bose-Einstein condensate in a harmonic trap with an eccentric dimple potential

We investigate Bose-Einstein condensation of noninteracting gases in a harmonic trap with an offcenter dimple potential. We specifically consider the case of a tight and deep dimple potential, which is modeled by a point interaction. This point interaction is represented by a Dirac delta function. The atomic density, chemical potential, critical temperature and condensate fraction, and the role of the relative depth and the position of the dimple potential are analyzed by performing numerical calculations.     

Thermodynamics of chiral symmetry at low densities

The phase diagram of two-color QCD as a function of temperature and baryon chemical potential is considered. Using a low-energy chiral Lagrangian based on the symmetries of the microscopic theory, we determine, at the one-loop level, the temperature dependence of the critical chemical potential for diquark condensation and the temperature dependence of the diquark condensate and baryon density. The prediction for the temperature dependence of the critical chemical potential is consistent with the one obtained for a dilute Bose gas. The associated phase transition is shown to be of second order for low temperatures and first order at higher temperatures. The tricritical point at which the second order phase transition ends is determined. The results are carried over to QCD with quarks in the adjoint representation and to ordinary QCD at a non-zero chemical potential for isospin.     

Dynamics of a single vortex line in a Bose-Einstein condensate

We study experimentally the line of a single quantized vortex in a rotating prolate Bose-Einstein condensate confined by a harmonic potential. In agreement with predictions, we find that the vortex line is in most cases curved at the ends. We monitor the vortex line leaving the condensate. Its length is measured as a function of time and temperature. For a low temperature, the survival time can be as large as 10 sec. The length of the line and its deviation from the center of the trap are related to the angular momentum per particle along the condensate axis.     

Anderson localization of a Bose-Einstein condensate in a 3D random potential

We study the effect of Anderson localization on the expansion of a Bose-Einstein condensate, released from a harmonic trap, in a 3D random potential. We use scaling arguments and the self-consistent theory of localization to show that the long-time behavior of the condensate density is controlled by a single parameter equal to the ratio of the mobility edge and the chemical potential of the condensate. We find that the two critical exponents of the localization transition determine the evolution of the condensate density in time and space.     

Condensates in double-well potential with synthetic gauge potentials and vortex seeding

We demonstrate an enhancement in the vortex generation when artificial gauge potential is introduced to condensates confined in a double well potential. This is due to the lower energy required to create a vortex in the low condensate density region within the barrier. Furthermore, we study the transport of vortices between the two wells, and show that the traverse time for vortices is longer for the lower height of the well. We also show that the critical value of synthetic magnetic field to inject vortices into the bulk of the condensate is lower in the double-well potential compared to the harmonic confining potential.     

Existence of Weak Solutions for a Diffuse Interface Model for Viscous,Incompressible Fluids with General Densities

We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. In contrast to previous works, we study the general case that the fluids have different densities. This leads to an inhomogeneous Navier-Stokes system coupled to a Cahn-Hilliard system, where the density of the mixture depends on the concentration, the velocity field is no longer divergence free, and the pressure enters the equation for the chemical potential. We prove existence of weak solutions for the non-stationary system in two and three space dimensions.     

Second to tenth order susceptibilities of conserved charges within a modified Nambu-Jona-Lasinio model

We discuss the sign and energy dependence of second to tenth order susceptibilities of the baryon number,charge number, and strangeness for the analysis of critical conditions in heavy ion collisions in the LHC and RHIC by applying a modified Nambu-Jona-Lasinio model. This model is fitted to the quark condensate of the lattice QCD result at finite temperature and zero baryon chemical potential. The presence of a critical point made these susceptibilities deviate considerably from a Hadron-Resonance-Gas model that shows no criticality. The sign, magnitude, and energy dependence of these higher order fluctuations hint towards the existence and location of a critical point that could be discovered in future heavy ion collision experiments.     

Influence of Finite Chemical Potential on Critical Boson Mass in QED3

Using the coupled Dyson-Schwinger equation for the fermion propagator at finite chemical potential μ, we investigate the fermion chiral condensate when the gauge boson mass is nonzero in QED3. We show that the chiral symmetry restores when the boson mass is large enough, and the critical boson mass depends little on μ.     

General properties of polymer systems

We prove the existence of the thermodynamic limit for the pressure and show that the limit is a convex, continuous function of the chemical potential.The existence and analyticity properties of the thermodynamic limit for the correlation functions is then derived; we discuss in particular the Mayer Series and the virial expansion.In the special case of Monomer-Dimer systems it is established that no phase transition is possible; moreover it is shown that the Mayer Series for the density is a series of Stieltjes, which yields upper and lower bounds in terms of Padé approximants.Finally it is shown that the results obtained for polymer systems can be used to study classical lattice systems.Work presented in partial fullfilment of the Ph. D. Thesis.     

A perturbed mean field model of an interacting boson gas and the large deviation principle

This is a study of the equilibrium thermodynamics of a mean-field model of an interacting boson gas perturbed by a term quadratic in the occupation numbers of the free-gas energy-levels. We prove the existence of the pressure in the thermodynamic limit. We obtain also a variational formula for the pressure; this enables us to compare the effect of a smooth quadratic perturbation with that of a singular one (the Huang-Yang-Luttinger model). The proof uses a large deviation result for the occupation measure of the free boson gas which is of independent interest.     

Gluon condensates at finite baryon densities and temperature

We derive here the equation of state for quark matter with a nontrivial vacuum structure in QCD at finite temperature and baryon density. Using thermofield dynamics, the parameters of thermal vacuum and the gluon condensate function are determined through minimisation of the thermodynamic potential, along with a self-consistent determination of the effective gluon and quark masses. The scale parameter for the gluon condensates is related to the SVZ parameter in the context of QCD sum rules at zero temperature. With inclusion of quarks in the thermal vacuum the critical temperature at which the gluon condensate vanishes decreases as compared to that containing only gluons. At zero temperature, we similarly obtain the critical baryon density for the same to be about 0.36 fm^?3.     

Long Cycles in a Perturbed Mean Field Model of a Boson Gas

In this paper we give a precise mathematical formulation of the relation between Bose condensation and long cycles and prove its validity for the perturbed mean field model of a Bose gas. We decompose the total density ρ=ρ_short+ρ_long into the number density of particles belonging to cycles of finite length (ρ_short) and to infinitely long cycles (ρ_long) in the thermodynamic limit. For this model we prove that when there is Bose condensation, ρ_long is different from zero and identical to the condensate density. This is achieved through an application of the theory of large deviations. We discuss the possible equivalence of ρ_long≠ 0 with off-diagonal long range order and winding paths that occur in the path integral representation of the Bose gas     

Bose-Einstein Condensation of Photons and Photon Pairs

We investigate the Bose-Einstein condensation of photons and photon pairs in a two-dimension optical microcavity. We find that in the paraxial approximation, the mixed gas of photons and photon pairs is formally equivalent to a two dimension system of massive bosons with non-vanishing chemical potential, which implies the existence of two possible condensate phase. We also discuss the quantum phase transition of the system and obtain the critical point analytically. Moreover, we find that the quantum phase transition of the system can be interpreted as second harmonic generation.