We can imagine creating a particle in the condensate by inducing a transition into it. As we saw in Section 16.2, Bose enhancement makes the rate proportional to the existing boson density. Thus, the creation amplitude ψ∗(r) should be proportional to the square root of the density. We define |ψ(r, t)|2d3r as the number of condensate particles in the volume element d3r, so that the total number of particles in the condensate is ∫ d3r |ψ(r)|2 = N0 (20.1) In Chapter 18, we were able to give a complete mathematical description of the Bose-Einstein condensation in the ideal gas. Such a treatment is not possible when there is interaction between particles. Here, we shall describe it in Ginsburg-Landau theory, with the complex field ψ as order parameter. The Landau free energy is taken to be E[ψ, ψ∗] = ∫ d3r [ h¯2 2m |∇ψ |2 + (Uext − μ)ψ∗ψ + g2 (ψ ∗ψ)2 ] (20.2) where |∇ψ |2 = ∇ψ∗ · ∇ψ , m is the mass of the particles, Uext(r) is an external potential, and μ is the chemical potential that determines N0. There are equivalent choices for independent variables: ψ and its complex conjugate ψ∗, real and imaginary parts Re ψand Im ψ, modulus and phase angle √n0 and ϕ in the polar representation ψ(x) = √ n0(x) e iϕ(x) where n0 is the density of the condensate.
We can imagine creating a particle in the condensate by inducing a transition into it. As we saw in Section 16.2, Bose enhancement makes the rate proportional to the existing boson density. Thus, the creation amplitude ψ∗(r) should be proportional to the square root of the density. We define |ψ(r, t)|2d3r as the number of condensate particles in the volume element d3r, so that the total number of particles in the condensate is ∫ d3r |ψ(r)|2 = N0 (20.1) In Chapter 18, we were able to give a complete mathematical description of the Bose-Einstein condensation in the ideal gas. Such a treatment is not possible when there is interaction between particles. Here, we shall describe it in Ginsburg-Landau theory, with the complex field ψ as order parameter. The Landau free energy is taken to be E[ψ, ψ∗] = ∫ d3r [ h¯2 2m |∇ψ |2 + (Uext − μ)ψ∗ψ + g2 (ψ ∗ψ)2 ] (20.2) where |∇ψ |2 = ∇ψ∗ · ∇ψ , m is the mass of the particles, Uext(r) is an external potential, and μ is the chemical potential that determines N0. There are equivalent choices for independent variables: ψ and its complex conjugate ψ∗, real and imaginary parts Re ψand Im ψ, modulus and phase angle √n0 and ϕ in the polar representation ψ(x) = √ n0(x) e iϕ(x) where n0 is the density of the condensate.
Abstract Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function <ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G2 in the G1-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.
Bose–Einstein condensation (BEC) is discussed at the level of an advanced course of statistical thermodynamics, clarifying some formal and physical aspects that are usually not covered by the standard pedagogical literature. The non-conventional approach adopted starts by showing that the continuum limit, in certain cases, cancels out the crucial role of the bosonic ground level. If so, a correct treatment of the problem, including the ground level population N 0 by construction, leads to BEC in a straightforward way. For a density of states of the form G ( )∝ γ , the chemical potential µ is explicitly calculated as a function of the temperature T and of the number N of bosons, for various significant values of the positive exponent γ. In the thermodynamic limit, in which the boson number N diverges and BEC is a sharp process, the chemical potential µ is a singular function of T at the critical temperature T B , determined by an appropriate critical exponent. The condensate population N 0 is studied analytically and numerically as a function of the temperature, for various values of N and for different γ. This provides an accurate description of the way BEC approaches the character of a sharp phase transition. Some aspects of the real experiments on BEC, involving a finite number of bosons, are also illustrated.
In this paper, first of all, we proved if the ideal Bose gas with a finite volume and number of particles has a non-degenerate single-particle energy level n, the chemical potential μ can take the value μn = n and there is a phase transition temperature Tp,n, where n = 0,1,2 Taking 0 ≤ n Tp,n+1. When the temperature T > Tp,n or T ≤ Tp,n+1, μ # n and the most probable occupation number Nn = 0. In the temperature interval Tp,n ≥ T > Tp,n+1, μ = n and 0 ≤ Nn = N - NjsupNn, where Nj is the most probable occupation number in the degenerate level j. Thus, if the finite ideal Bose gas has some non-degenerate single-particle levels, there exists a characteristic temperature Tp = Tp,0. The chemical potential μ is quantized when T ≤ Tp, and this leads to the creation of a macroscopic quantum state (pure state) or Bose-Einstein condensation phase. Tp = Tp,0 is a first-order phase transition point, Tp,n#0 is a zero-order phase transition point. Next, we obtained a new expression of the most probable distribution of the finite ideal Bose gas. In this expression Nj is directly proportional to gj - 1, where gj and Nj are, respectively, the degeneracy and the most probable occupation number in the degenerate level j. This property agrees with what chemical potential can be quantized if there is a non-degenerate level for the finite ideal Bose gas. Finally, using this expression, we defined a micro-partition function M, obtained the statistical expressions of some thermodynamical quantities.
In dieser Arbeit erörtern wir ein eindimensionales Lieb Liniger Modell für Bosonen mit einem zusätzlichen Zufallspotenzial. Für dieses Modell von schwach wechselwirkendem Bosegas wurde von Seiringer, Yngvason und Zagrebnov gezeigt, dass für N nach unendlich, wobei N die Anzahl an Teilchen bezeichnet, die Gross Pitaevskii Energie zur quantenmechanischen Energie konvergiert. Weiters wurde gezeigt, dass für den Vielteilchenlimes vollständige Bose Einstein Kondensation (BEC) existiert. Für die Diskussion von Superfluidität, ist eine genauere Analyse der Gross Pitaevskii Dichte nötig und für bestimmte Limiten wird diese Analyse in dieser Arbeit durchgeführt. Im Speziellen können wir zeigen, dass für schwache Wechselwirkung keine Superfluidität im Limes existiert, auch wenn Bose Einstein Kondensation vorkommt. Andererseits können wir zeigen, dass es einen Parameterbereich gibt, in dem vollständige Superfluidität stattfindet. Daher fordert die Existenz von Superfluidität stärkere Voraussetzungen als die Existenz von BEC, in diesem eindimensionalen Modell. In derselben Arbeit von Seiringer, Yngvason und Zagrebnov wurde gezeigt, dass in einem speziellen Parameterbereich, ein deterministischer Limes für die Gross Pitaevskii Energie existiert welche von dem Zufallspotenzial abhängt. In dieser Arbeit erweitern wir diesen Beweis um Abschätzungen für die Konvergenz in Wahrscheinlichkeit. Weiters beweisen wir, dass es möglich ist, diese Konvergenz auch für ein Bernoulli Zufallspotenzial zu erreichen.