The ninth chapter analyses interacting bosons on the far side of the λ-transition, showing that the partition function factorizes into the classical configuration integral and ideal non-local pure momentum permutation loops. Results include: condensation is into multiple multiply-occupied momentum states, the permutation loop entropy suppresses collisions, inviscid plug flow results from non-locality, and the predicted superfluid critical velocity quantitatively agrees with measured values. The fountain pressure between chambers of different temperature is quantitatively accounted for by the minimization of energy at constant subsystem entropy, which principle is a direct consequence of dissipationless superfluid flow. Superfluid equations of motion explain how collisions occur without dissipating the momentum of the condensed bosons.
The ninth chapter analyses interacting bosons on the far side of the λ-transition, showing that the partition function factorizes into the classical configuration integral and ideal non-local pure momentum permutation loops. Results include: condensation is into multiple multiply-occupied momentum states, the permutation loop entropy suppresses collisions, inviscid plug flow results from non-locality, and the predicted superfluid critical velocity quantitatively agrees with measured values. The fountain pressure between chambers of different temperature is quantitatively accounted for by the minimization of energy at constant subsystem entropy, which principle is a direct consequence of dissipationless superfluid flow. Superfluid equations of motion explain how collisions occur without dissipating the momentum of the condensed bosons.
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.
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 and the λ-transition are described in molecular detail for bosons interacting with a pair potential. New phenomena are identified that are absent in the usual ideal gas treatment. Monte Carlo simulations of Lennard-Jones helium-4 neglecting ground momentum state bosons give a diverging heat capacity approaching the transition. Pure permutation loops give continuous growth in the occupancy of the ground momentum state. Mixed ground and excited momentum state permutation loops give a discontinuous transition to the condensed phase. The consequent latent heat for the λ-transition is 3\% of the total energy. The predicted critical velocity for superfluid flow is within a factor of three of the measured values over three orders of magnitude in pore diameter.
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 phenomena of superfluidity and superconductivity are intimately connected with the existence of a condensate, a macroscopically occupied quantum state. Such condensates occur in a variety of different physical systems, as described in Chapter 1. The foundation for the description of superfluidity is a picture of the system as being comprised of a condensate and elementary excitations. In Chapter 8 we have seen how physical properties such as the energy and the density of a Bose–Einstein condensed system may be expressed in terms of a contribution from the condensate, plus one from the elementary excitations, and in this chapter we shall consider further developments of this basic idea to other situations. As a first application, we determine the critical velocity for creation of an excitation in a homogeneous system (Sec. 10.1). Following that, we show how to express the momentum density in terms of the velocity of the condensate and the distribution function for excitations. This provides the basis for a two-component description, the two components being the condensate and the thermal excitations (Sec. 10.2). In the past, this framework has proved to be very effective in describing the properties of superfluids and superconductors, and in Sec. 10.3 we apply it to dynamical processes. To describe the state of a superfluid, one must specify the condensate velocity, in addition to the variables needed to characterize the state of an ordinary fluid. As a consequence, the collective behaviour of a superfluid is richer than that of an ordinary one. Collective modes are most simply examined when excitations collide frequently enough that they are in local thermodynamic equilibrium.
A theory of superfluidity (S.F.) is developed from first principles using two novel concepts, (1) that of a 'superfluid Ensemble' (S.E.) i.e. a 'Restricted Ensemble' constructed from a 'Separable Phase Space' (S.P.S.) admitting independent configurations, at least one set of which are statistically equivalent. (2) The notion of 'Dynamical Equivalence' (D.E.), satisfied if and only if (i) all dynamical symmetries are rearranged (not broken) for two Lagrangian formulations of the same problem and if (ii) the expectation values of all the constants of motion are the same, even if their functional expressions are not. The dynamical variables (d.v.) of the S.P.S. are defined from the ('q' and 'c' number) fields of the most general 'Linear Coherent State Representation', more general than those of Glauber and Bogoljubov-Valatin combined. Three independent pairs of d.v. are obtained.
D.E. is proven for the Ideal Bose Gas and for a non-linear, interacting zero order Bose problem (I.Z.O.P.). An exact relation is obtained from the action principle, ensuring the cancellation of 'low and high order dangerous diagrams'. From this it follows that D.E. for the exact interacting problem must be demonstrable at infinite order of perturbation, in the finite volume limit. The I.Z.O.P. is posed in the Random Phase Approximation (R.P.A.), free from 'anomalous averages' and solved for the three branches of the excitation spectrum in a pure state description; the lowest branch is gapless, whilst the upper two coincide and show a gap. The standard strategy of linearization is found to be faulty.
The partition functions for both superfluid and non-superfluid ensembles are obtained for the I.Z.O.P. in the R.P.A. The coincident upper two branches (in a pure state description) split into a band in thermal equilibrium for the superfluid ensemble, in agreement with an upper band recently observed experimentally. O.D.L.R.O. is found in the second reduced density matrix, but ruled out in the first. Integral equations are obtained in thermal equilibrium - for the I.Z.O.P. in the R.P.A.; which differ, however, from those of existing approaches for the same problem. Most existing theories of S.F. are in fact shown not to predict superfluid behaviour. The present theory is applicable to arbitrary Bose or Fermi systems, whether superfluid or not. O.D.L.R.O. is found to be sufficient for SF. No a priori assumption is made as to the occurrence of Bose-Einstein condensation, its existence being here contingent on the solution of the integral equation; in any case, it is not to be associated with O.D.L.R.O. or S.F.