Bose-Einstein condensation at low temperatures is driven by occupation entropy, since configurations of labeled particles with multiply-occupied states have more weight. An open quantum system collapses into decoherent states weighted by the reservoir entropy. Thus arises quantum statistical mechanics in classical phase space. The symmetrization function generalizes quantum state occupancy to the momentum continuum. Hamilton's classical equations of motion hold only for an open quantum system with negligible condensation. In the condensed regime the molecular equations of motion preserve the entropy, consistent with fountain pressure measurements of superfluid flow. Condensed bosons in the same momentum state experience non-local forces and move as a rigid body that does not dissipate momentum in classical collisions. High-temperature superconductivity is statistical mechanical in nature, requires bound fermion pairs of neighboring electrons, and is driven by the monotonic-oscillatory transition at high charge coupling.
Bose-Einstein condensation at low temperatures is driven by occupation entropy, since configurations of labeled particles with multiply-occupied states have more weight. An open quantum system collapses into decoherent states weighted by the reservoir entropy. Thus arises quantum statistical mechanics in classical phase space. The symmetrization function generalizes quantum state occupancy to the momentum continuum. Hamilton's classical equations of motion hold only for an open quantum system with negligible condensation. In the condensed regime the molecular equations of motion preserve the entropy, consistent with fountain pressure measurements of superfluid flow. Condensed bosons in the same momentum state experience non-local forces and move as a rigid body that does not dissipate momentum in classical collisions. High-temperature superconductivity is statistical mechanical in nature, requires bound fermion pairs of neighboring electrons, and is driven by the monotonic-oscillatory transition at high charge coupling.
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 exchange of conserved variables between a subsystem and a reservoir or environment entangles their wave functions. Their subsequent collapse, together with the uniform weight of wave space, gives rise to quantum statistical mechanics formulated as the trace of the probability and observable operators. A dual representation in position and momentum space allows a permutation loop expansion for particle symmetrization, and a second expansion accounting for the non-commutativity of the position and momentum operators. The leading term gives classical statistical mechanics, and higher-order terms give successive quantum corrections that become increasingly important for low temperatures, high densities and small masses. The probability density in phase space is stable under the stochastic dissipative equations of motion. Trapped Bose-Einstein condensates may be modeled as a system of non-interacting harmonic oscillators. Most bosons in the condensate are in the ground state, in contrast to the -transition in 4He where the fraction in the ground state is negligible.
We have seen in Chapter 3 , that as temperature is increased quantum statistics, both Bose-Einstein or Fermi-Dirac, tend to classical. High temperatures blur out quantum effects in a system, which can be safely described in classical terms. In this chapter, we will consider some systems which manifest quantum properties at macroscopic level, but at very low temperatures. We recall that helium is the only substance which remains liquid at low temperature, down to absolute zero (see Chapter 6 ), a phenomenon related to the quantum behavior of its constituent atoms. Liquid helium was instrumental to the discovery that some metals are able to sustain an electrical current for long periods of time without any driving force. This phenomenon is named superconductivity, and is also a macroscopic manifestation of the quantum behavior of the electrons in metals. Later on it was discovered that liquid helium itself shows a similar property, named superfluidity, consisting in its ability to flow through a capillary without viscosity. Hence, a persistent flow of liquid can be maintained without any driving force. While the quantum agents of superconductivity are electrons, which are spin- 1/2 fermions, those of superfluidity are 4 He atoms, which are spin- 0 bosons. There are other systems able to display similar behavior as liquid 3 He, atomic nuclei and nuclear matter. In fact, superfluidity and superconductivity are new phases of matter related to the bosonic character of 4 He atoms or of pairs of fermions (either electrons, 3 He atoms or nucleons), the same as Bose-Einstein condensation in ultracold atomic gases. The temperatures where these new phases manifest range from 165 K for superconductivity down to a few nano-Kelvin for Bose-Einstein condensation.
The two-fluid theory for superfluidity is not entirely consistent with the evidence for condensation into multiple highly occupied momentum states. On the far side of the -transition wave function symmetrisation is in the form of pure momentum loops and momentum state occupancy, the non-local nature of which gives inviscid plug flow, and the entropy of which gives the superfluid critical velocity. Equations of motion for condensed bosons conserve entropy. A quantum stochastic molecular dynamics algorithm based on a continuous form of occupancy gives the viscosity of liquid 4He, which is less than the classical calculation, increasingly so as the temperature is decreased through the condensation transition. Molecular trajectories show the suppression of collisions that transfer longitudinal momentum laterally in the superfluid regime. A second algorithm that treats condensed bosons in the same momentum state as a rigid body with non-local forces also yields a reduced viscosity but with greater uncertainty.
The modern theory of Bose-Einstein condensation, superfluidity, and superconductivity is reviewed. The thermodynamic principle for superfluid flow and the equation of motion for condensed bosons are given. Computer simulations of Lennard-Jones 4He give the λ-transition and the superfluid viscosity. The statistical mechanical theory of high-temperature superconductivity is presented. Critical comparison is made with older approaches, such as ground energy state condensation, irrotational superfluid flow, and the macroscopic wavefunction.
At finite temperatures below the phase transition point, the Bose-Einstein condensation, the macroscopic occupation of a single quantum state by particles of integer spin, is not complete. In the language of superfluid helium, this means that the superfluid coexists with the normal fluid. Our goal is to describe this coexistence in trapped, dilute atomic gases with repulsive interactions via mean field laws that account for a {\em spatially varying} particle interaction strength. By starting with the N-body Hamiltonian, N≫1, we formally derive a system of coupled, nonlinear evolution equations in 3+1 dimensions for the following quantities: (i) the wave function of the macroscopically occupied state; and (ii) the single-particle wave functions of thermally excited states. For stationary (bound) states and a scattering length with {\em periodic microstructure} of subscale ε, we heuristically extract effective equations of motion via periodic homogenization up to second order in ε.