Α system of charged weakly interacting bosons with «d" symmetry is considered.The interaction U(d) = U (s) f (0) contains the angle dependence f(θ) .Υ2,2 2 consistent with «d" symmetry.This system resembles much more free bosons than the system with «s" symmetry (interaction U(s)).For example at Τ = 0, n n, where n, n are the densities of particles and condensate respectively.
Α system of charged weakly interacting bosons with «d" symmetry is considered.The interaction U(d) = U (s) f (0) contains the angle dependence f(θ) .Υ2,2 2 consistent with «d" symmetry.This system resembles much more free bosons than the system with «s" symmetry (interaction U(s)).For example at Τ = 0, n n, where n, n are the densities of particles and condensate respectively.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01
The condensation and accumulation phenomena of charged Bose-Einstein particles in a magnetic field are examined, neglecting electrostatic interactions. For weak fields a single condensation occurs at the same temperature as that for uncharged particles; for strong fields there is a separate lower accumulation temperature, indicating the accumulation of particles in the lowest translational state. The total energy, free energy, and polarization are calculated, and applied to the properties of a gas of electron pairs, and a solution of deuterons in a metal, these being systems whose low temperature behavior might show a Bose-Einstein condensation. It is shown that the Bose-Einstein condensation of free electron pairs cannot provide a theory of superconductivity.
A new approach developed in Ref.1 for investigating the spatial boson correlations at low temperatures is considered in the particular case of the Coulomb interaction potential. This approach is based on integro–differential equations for the radial distribution function and can be used beyond the weak coupling regime. PACS: 05.30.Jp, 67.90.+z Description of the charged Bose liquid, or, in other words, of the strongly correlated many–boson system with the Coulomb interparticle potential, is a long–standing problem of the condensed matter physics. Recently, interest in this problem has been renewed in the connection with the noteworthy results of applying the Bose and Bose–Fermi liquid models [2] to the explanation of various phenomena inherent in the superconductors with high critical temperatures. These models are based on the Bogoliubov approach [3] implying the weak coupling regime. For the system of charged bosons the weak coupling approximation can be used at small values of the Brueckner parameter rS ∼ 0.1. However, questions requiring detailed investigations often concern the materials with relatively low densities of charge carriers which correspond to rS−values even larger than those of the electron gas at 1 the metallic densities. For example, there are the results [4] testifying to the possibility of forming the so–called large bipolaron (e.g. the two–electron cluster with a large radius of electron–electron separation, see papers in Ref. [5]) inside the polaron environment at the electron densities ∼ 10 19 cm −3 close to the carrier densities of the heavily doped degenerate polar semiconductors. To investigate the possible phase transition of the charge carrier system from the polaron state to the bipolaron phase at these densities, one should operate with the charged Bose liquid of large bipolarons for rS ≫ 1. Indeed, the ”large–bipolaron“ Bohr radius can be expressed [4] as abip =
We show, within mean-field theory, that the normal state of compressible quasi-two-dimensional bose gas with arbitrarily weak attractive interactions becomes unstable with respect to pair condensation at a temperature Tc of order T0, the bose degeneracy temperature. Below Tc the ground-state many-particle wave function is the boson analog of the Bardeen-Cooper-Schrieffer wave function and the charged gas has the principal electromagnetic properties of a conventional superconductor.
Abstract Chapter 20 examines effects of weak repulsive interactions in a Bose–Einstein condensate and the transition from Bose–Einstein condensate to superfluid phase transition. Renormalization group methods are used and a universal amplitude is calculated by non–perturbative methods. After the discovery of the predicted Bose–Einstein condensation, which is a property of free bosons, an interesting issue was the effects of weak repulsive interactions. In this chapter, it is shown that, near the transition temperature, the initial non–relativistic field theory can be replaced by a relativistic effective Euclidean field theory known to describe a superfluid phase transition (a dimensional reduction). These theoretical considerations are illustrated by an evaluation of the universal variation of the transition temperature at weak coupling. For this purpose, the O(2) symmetry of the model is generalized to O(N) symmetry, and large N techniques are used.
Abstract 1. Gordon has proved that the scattering of particles by an inverse square field is the same on the wave mechanics as on the classical theory. Mott has however shown, from consideration of the symmetry of the wave functions, that the scattering is quite different from the classical when the scattering and scattered particles are identical. The scattering of alpha particles by helium is typical of the case of the collision of identical particles that have no pin and obey the Einstein-Bose statistics. For this case, the expression given by Mott for the number of alpha particles scattered between angles θ, and θ +dθ, in going a distance dr in a gas containing n atoms per unit volume, is N dθ dr = 4πnN0e4/m2v4 {cosec4 θ + sec4 θ + 2 cosec2 θ sec2 θ cos u} dθ dr, (1) where u = 8/137 c/v log cot θ, (2) and N0 is the total number of incident particles, v, m and e, their initial velocity, mass and charge and c the velocity of light.