Bose–Einstein condensation is an intensely studied quantum phenomenon that emerges at low temperatures. While preceding Bose–Einstein condensation models do not consider what statistics apply above the condensation temperature, we show that neglecting this question leads to inconsistencies. A mathematically rigorous calculation of Bose–Einstein condensation temperature requires evaluating the thermodynamic balance between coherent and incoherent particle populations. The first part of this work develops such an improved Bose–Einstein condensation temperature calculation, for both three-dimensional and two-dimensional scenarios. The progress over preceding Bose–Einstein condensation models is particularly apparent in the two-dimensional case, where preceding models run into mathematical divergence. In the Discussion section, we compare our mathematical model against experimental superconductivity data. A remarkable match is found between experimental data and the calculated Bose–Einstein condensation temperature formulas. Our mathematical model therefore appears applicable to superconductivity, and may facilitate a rational search for higher-temperature superconductors.
Bose–Einstein condensation is an intensely studied quantum phenomenon that emerges at low temperatures. While preceding Bose–Einstein condensation models do not consider what statistics apply above the condensation temperature, we show that neglecting this question leads to inconsistencies. A mathematically rigorous calculation of Bose–Einstein condensation temperature requires evaluating the thermodynamic balance between coherent and incoherent particle populations. The first part of this work develops such an improved Bose–Einstein condensation temperature calculation, for both three-dimensional and two-dimensional scenarios. The progress over preceding Bose–Einstein condensation models is particularly apparent in the two-dimensional case, where preceding models run into mathematical divergence. In the Discussion section, we compare our mathematical model against experimental superconductivity data. A remarkable match is found between experimental data and the calculated Bose–Einstein condensation temperature formulas. Our mathematical model therefore appears applicable to superconductivity, and may facilitate a rational search for higher-temperature superconductors.
This work is an up-to-date and partially improved analysis of the basics of the mathe- matical description of Bose-Einstein condensation in terms of first principles of Quantum Mechanics. The main aims are: 1) to provide a compact, yet coherent overview: - of the basic mathematical tools used to formalise Bose-Einstein condensation, - of the mathematical techniques for studying several features of this physical phenomenon, - and of how such means emerge as the natural ones in connection with their physical interpretation; 2) to discuss and to place in the above perspective some new contributions and improvements: - on equivalent characterizations of Bose-Einstein condensation, - on the strength of the convergence for some currently available asymptotic results, - and on the effects of interparticle correlations on the energy and the dynamics of the many-body condensate.
Bose-Einstein condensation, superfluidity, and superconductivity are quantum mechanics made visible. They mark the boundary between the classical and the quantum worlds, and they show the macroscopic role of quantum mechanics in condensed matter. This book presents these phenomena in terms of particles, their positions, and their momenta, giving a concrete visualisation and description that is not possible with traditional wave functions. A single approach that bridges the classical-quantum divide provides new insight into the role of particle interactions in condensation, the nature of collisions in superfluid flow, and the physical form of Cooper pairs in high-temperature superconductors. High-temperature superconductivity is explored with quantum statistical mechanics, which links it to Bose-Einstein condensation. Identifying a new mechanism for Cooper pairing, this explains the differences between the low- and high-temperature superconducting regimes and the role of the molecular structure of the conductor. The new perspective offered by this book on Bose-Einstein condensation, superfluidity, and high-temperature superconductivity gives particle-based explanations as well as mathematical and computational methods for these macroscopic quantum phenomena so that readers understand the role of particle interactions and structure in the physics of these phenomena. This book will appeal to undergraduate and graduate students, lecturers, academics, and scientific researchers in the fields of Bose-Einstein condensation and condensates, superfluidity, and superconductivity. It will also be of interest to those working with thermodynamics, statistical mechanics, statistical physics, quantum mechanics, molecular dynamics, materials science, condensed matter physics, and theoretical chemistry. Key Features: · Explores Bose-Einstein condensation with new evidence for multiple condensed states and novel Monte Carlo simulations for interacting bosons · Establishes the thermodynamic nature of condensed bosons from an analysis of fountain pressure measurements, including that they carry energy and entropy, and the thermodynamic principle of superfluid flow · Derives equations of motion for condensed bosons, and performs molecular dynamics simulations of the viscosity with molecular trajectories that give rise to superfluidity · Identifies the mechanism for electron pairing in high-temperature superconductivity
The purpose of this work is twofold.In the first part we describe superfluidity/superconductivity as an emergent phenomenon resulting from two-body correlations in presence of the Bose-Einstein condensation of particles.We briefly discuss the underlying mechanism for bosons as well as fermion pairs and illustrate various realizations of superfluidity emphasizing the recent examples.In the second part we study the glassy liquid of incoherent pairs which might exist above the transition temperature T c in the underdoped regime of cuprate superconductors.In particular, we explore the angular variation of pseudogap within two-dimensional version of the boson-fermion model using for a quantitative analysis the projective method.We find that above T c the pseudogap closes first near the nodal areas restoring there pieces (arcs) of the Fermi surface whereas remaining parts of the large Fermi surface around the antinodal points are still absent due to incoherent pairs.Upon increasing temperature the length of the Fermi arcs enlarges because the superconducting correlations are gradually suppressed.An intriguing death of Fermi surface can thus be closely related to the Bogoliubov quasiparticles whose existence in the pseudogap state has been predicted by us and confirmed recently by the angle resolved photoemission spectroscopy measurements on Bi 2 Sr 2 CaCu 2 O 8 and La 1.895 Sr 0.105 CuO 4 compounds.
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
Abstract The Bose-Einstein condensation of a gas is investigated. Starting from the well-known formulae for Bose statistics, the problem has been generalized to include a variety of potential fields in which the particles of the gas move, and the number w of dimensions has not been restricted to three. The energy levels are taken to be εi≡εs1,....,s10=constanth2ms1α−1a12+...+swαaw2(1≤α≤2) the quantum numbers being s1 ,w= 1, 2, ..., and a1, ..., aw being certain characteristic lengths. (For α = 2, the potential field is that of the w-dimensional rectangular box; for α = 1, we obtain the w-dimensional harmonic oscillator field.) A direct rigorous method is used similar to that proposed by Fowler & Jones (1938). It is shown that the number q = w/α determines the appearance of an Einstein transition temperature T0·For q≤ 1 there is no such point, while for q > 1 a transition point exists. For 1 < q≤ 2, the mean energy ϵ- per particle and the specific heat dϵ-/dT are continuous at T = T0· For q > 2, the specific heat is discontinuous at T = T0, giving rise to a A λ-point. A well-defined transition point only appears for a very large (theoretically infinite) number N of particles. T0 is finite only if the quantity v = N/(a1 .... aw)2/ α¯is finite. For a rectangular box, v is equal to the mean density of the gas. If v tends to zero or infinity as N→ ∞, then T0 likewise tends to zero or infinity. In the case q > 1, and at temperatures T < T 0' there is a finite fraction N0 /N of the particles, given by N 0/N = 1-(T/T0)q, in the lowest state. London’s formula (1938 b) for the three-dimensional box is an example of this equation. Some further results are also compared with those given by London’s continuous spectrum approximation.
Abstract In this work, the authors show an apparent connection in the behaviors of superfluid Helium in its two stable isotopes, where the lambda transition to superfluidity is seen to correspond to a theoretical Bose–Einstein condensation temperature of the ideal parts of the two isotopes. A statistical model for the physical and thermodynamic behaviors is developed based on their known properties, where the equations of the partition function, entropy, and specific heat are constructed in such a way where one term applies to the ideal part of the fluid and the second to its interacting part. The calculated temperatures for the formation of a theoretical Bose–Einstein condensation in the ideal parts of both isotopes almost completely matches their lambda transition temperatures. The models of the two isotopes are very different due to the fermionic nature of Helium-3, for which the model is formed based on the Bardeen–Cooper–Schrieffer (BCS) theory of Cooper pairs of fermions. Using this formulation, the Bose–Einstein condensation temperature for the pairs of Helium-3 atoms is calculated, as opposed to the condensation temperature for Helium-4, which is directly derived from the bosonic atoms, leading to its much lower value than Helium-4.