Thermal Origin of the Repulsive Logarithmic Schrödinger Free-Energy Term: Exact Occupancy Dependence, Neutron Bounds, and a Cold-Atom Test of the Counting Class
This paper derives the equilibrium thermodynamics of a populated collective occupation field under Maxwell–Boltzmann/Gibbs counting and identifies which of its natural occupancy potentials governs a deterministic collective field. The treatment is ensemble-exact: for cell weights W(n) = Gⁿ/n! the grand partition function is Ξ = exp(Gz), the mean occupation is n̄ = Gz, and the Legendre free energy of the mean is F_MB(n̄) = k_BT[n̄ ln(n̄/G) − n̄], giving μ_MB(n̄) = k_BT ln(n̄/G) for every n̄ > 0 — the logarithmic coefficient at the equilibrium free-energy level is exactly k_BT, with no Stirling approximation and no low-occupancy flattening. Three occupancy potentials that coincide at high occupation are shown to answer different questions: the ensemble chemical potential of the fluctuating mean (the object that governs a collective field), the discrete insertion cost ΔF(n) = k_BT ln[(n+1)/G] (exact for jump kinetics), and the sharp-occupation continuation k_BT[ψ₀(n+1) − ln G] (a derivative of an analytic continuation, with no ensemble carrier at low occupancy). The domain statement follows from the same ensemble: Var(n) = n̄, so relative fluctuations scale as n̄⁻¹ᐟ², and a deterministic collective field requires large occupation under a specified coarse graining. At occupation of order unity and below this paper predicts no nonlinear single-particle Schrödinger equation; the few-quantum sector must be derived from a discrete stochastic or Fock-space description. Neutron interferometry bounds therefore remain decisive against a universal one-particle logarithmic term, while the present mechanism is compatible with them as an explicitly collective effective closure. (This version supersedes v1.0, whose low-occupancy switch-off claim conflated the three potentials above; the complete revision record is Appendix B.) Monte Carlo verification (three independent seeds per temperature, energy bookkeeping validated against the full Hamiltonian; reproducibility details in Appendix B) gives interacting-chain slopes d ln⟨n⟩/dV = −2.2509 ± 0.0075, −1.0597 ± 0.0084, −0.5131 ± 0.0053 at T = 0.5, 1, 2, approaching the ideal −1/T as K/T decreases, and occupancy histograms with the distinct fingerprints R(n)(n+1) ≈ const (Maxwell–Boltzmann/Gibbs) versus R(n) ≈ const (uniform-configuration “Bose”) sampling. These tests validate the equilibrium counting implementation; they do not derive the promotion of the free-energy term into conservative phase dynamics. For Bose counting, μ_BE = k_BT ln[n̄/(n̄+G)] = k_BT ln[ν/(1+ν)], with ν = n̄/G, so the response agrees with the dilute Maxwell–Boltzmann form but saturates at high occupancy. A cold-atom search for a temperature-proportional ln ρ contribution therefore remains a conditional test of the counting class, subject to three explicit assumptions: a populated collective FDS field is present, its bath temperature is locked to the measured gas temperature, and the equilibrium free-energy derivative enters the effective phase dynamics. Failure of any one assumption changes the experimental interpretation.
This paper derives the equilibrium thermodynamics of a populated collective occupation field under Maxwell–Boltzmann/Gibbs counting and identifies which of its natural occupancy potentials governs a deterministic collective field. The treatment is ensemble-exact: for cell weights W(n) = Gⁿ/n! the grand partition function is Ξ = exp(Gz), the mean occupation is n̄ = Gz, and the Legendre free energy of the mean is F_MB(n̄) = k_BT[n̄ ln(n̄/G) − n̄], giving μ_MB(n̄) = k_BT ln(n̄/G) for every n̄ > 0 — the logarithmic coefficient at the equilibrium free-energy level is exactly k_BT, with no Stirling approximation and no low-occupancy flattening. Three occupancy potentials that coincide at high occupation are shown to answer different questions: the ensemble chemical potential of the fluctuating mean (the object that governs a collective field), the discrete insertion cost ΔF(n) = k_BT ln[(n+1)/G] (exact for jump kinetics), and the sharp-occupation continuation k_BT[ψ₀(n+1) − ln G] (a derivative of an analytic continuation, with no ensemble carrier at low occupancy). The domain statement follows from the same ensemble: Var(n) = n̄, so relative fluctuations scale as n̄⁻¹ᐟ², and a deterministic collective field requires large occupation under a specified coarse graining. At occupation of order unity and below this paper predicts no nonlinear single-particle Schrödinger equation; the few-quantum sector must be derived from a discrete stochastic or Fock-space description. Neutron interferometry bounds therefore remain decisive against a universal one-particle logarithmic term, while the present mechanism is compatible with them as an explicitly collective effective closure. (This version supersedes v1.0, whose low-occupancy switch-off claim conflated the three potentials above; the complete revision record is Appendix B.) Monte Carlo verification (three independent seeds per temperature, energy bookkeeping validated against the full Hamiltonian; reproducibility details in Appendix B) gives interacting-chain slopes d ln⟨n⟩/dV = −2.2509 ± 0.0075, −1.0597 ± 0.0084, −0.5131 ± 0.0053 at T = 0.5, 1, 2, approaching the ideal −1/T as K/T decreases, and occupancy histograms with the distinct fingerprints R(n)(n+1) ≈ const (Maxwell–Boltzmann/Gibbs) versus R(n) ≈ const (uniform-configuration “Bose”) sampling. These tests validate the equilibrium counting implementation; they do not derive the promotion of the free-energy term into conservative phase dynamics. For Bose counting, μ_BE = k_BT ln[n̄/(n̄+G)] = k_BT ln[ν/(1+ν)], with ν = n̄/G, so the response agrees with the dilute Maxwell–Boltzmann form but saturates at high occupancy. A cold-atom search for a temperature-proportional ln ρ contribution therefore remains a conditional test of the counting class, subject to three explicit assumptions: a populated collective FDS field is present, its bath temperature is locked to the measured gas temperature, and the equilibrium free-energy derivative enters the effective phase dynamics. Failure of any one assumption changes the experimental interpretation.
We discuss the exact particle number counting statistics of degenerate ideal Bose gases in the microcanonical, canonical, and grand-canonical ensemble, respectively, for various trapping potentials. We then invoke the Maxwell's Demon ensemble [Navez et el., Phys. Rev. Lett. (1997)] and show that for large total number of particles the root-mean-square fluctuation of the condensate occupation scales n0 / [T=Tc] r N s with scaling exponents r = 3=2, s = 1=2 for the3D harmonic oscillator trapping potential, and r = 1,s= 2=3 for the 3D box. We derive an explicit expression for r and s in terms of spatial dimension D and spectral index sigma of the single-particle energy spectrum. Our predictions also apply to systems where Bose-Einstein condensation does not occur. We point out that the condensate fluctuations in the microcanonical and canonical ensemble respect the principle of thermodynamic equivalence.
We present a new exact method to numerically compute the thermodynamical properties of an interacting Bose gas in the canonical ensemble. As in our previous paper (Carusotto I, Castin Y and Dalibard J 2001 Phys. Rev. A 63 023606), we write the density operator ρ as an average of Hartree dyadics | N :ϕ 1 ⟩⟨ N :ϕ 2 | and we find stochastic evolution equations for the wavefunctions ϕ 1,2 such that the exact imaginary-time evolution of ρ is recovered after averaging over noise. In this way, the thermal equilibrium density operator can be obtained for any temperature T . The method is then applied to study the thermodynamical properties of a homogeneous one-dimensional N -boson system: although Bose-Einstein condensation cannot occur in the thermodynamical limit, a macroscopic occupation of the lowest mode of a finite system is observed at sufficiently low temperatures. If k B T >>µ, the main effect of interactions is to suppress density fluctuations and to reduce their correlation length. Different effects such as a spatial antibunching of the atoms are predicted for the opposite k B T ⩽µ regime. Our exact stochastic calculations have been compared with existing approximate theories.
For ideal Bose atoms in an isotropic harmonic trap, we consider thermodynamic variables obtained from microcanonical, canonical, and grand canonical ensembles, each with certain variables specified and other variables fluctuating. For the first two of these ensembles, we derive recursion relations that link partition functions for different dimensions. We discuss fluctuations in general, and obtain expressions for variances of the atom number N, the chemical potential \ensuremath{\mu}, and the temperature T for small, Gaussian fluctuations in the grand canonical ensemble. Then from our recursion relations and others given elsewhere, we obtain probability distributions for N, for ground-state occupation number n0, for \ensuremath{\mu}, and for T. Below the critical temperature, the shape of the distributions for N, n0, and \ensuremath{\mu} are definitely not Gaussian for the grand canonical ensemble. For given temperature and small N, we find that the chemical potential values pertaining to the three ensembles differ. We compare the specific-heat function CN(T) for the three ensembles and propose to use the minimum of dCN/dT to define the critical temperature to facilitate comparisons with similar configurations of interacting atoms.
When a gas of bosonic atoms is cooled sufficiently near to absolute zero, it undergoes a startling transition from a collection of randomly-behaved individual atoms to a phase in which thousands, even millions, of atoms act in collective unison. In this new phase, known as a Bose-Einstein condensate (BEC), the atoms all occupy the exact same quantum mechanical ground state and can be collectively described by a single phase-coherent wavefunction of macroscopic size. But how exactly does this harmonious behaviour arise within a collection of incoherent atoms as their temperature decreases? Does the condensate wavefunction grow from a single central nucleus, or does it emerge simultaneously throughout the system? And how does the situation differ when the gas is cooled suddenly, away from thermodynamic equilibrium? Can non-equilibrium behaviour be inherited by the condensate that forms? In this thesis we aim to answer these and other questions by studying the formation dynamics of Bose-Einstein condensates and the development of phase coherence. The research described herein makes use of several theoretical tools, which we lay out on the workbench in Chapter 2: Semi-classical Hartree-Fock theory provides a mean-field description of a condensate and thermal cloud at equilibrium; Quantum kinetic theory is used to calculate the dynamics of condensate growth into an equilibrium state, from a non-equilibrium thermal cloud; The stochastic projected Gross-Pitaevskii equation (SPGPE) employs a classical-field description to encapsulate the non-equilibrium dynamics of the condensate and low-energy region of the atom cloud, coupled to an equilibrium thermal cloud. This thesis is composed of four principal studies, each constituting a separate chapter.In Chapter 3 we study the formation of a Bose-Einstein condensate in a cigar-shaped 3D harmonic trap, induced by the controlled addition of an attractive ``dimple'' potential along the weak axis in a recent experiment [Garrett et al., Phys. Rev. A 83, 013630 (2011)]. In this manner condensation is induced without cooling due to a localized increase in the phase space density. We perform a quantitative analysis of the thermodynamic transformation in both the sudden and adiabatic regimes for a range of dimple widths and depths. We find good agreement with equilibrium calculations based on self-consistent semi-classical Hartree-Fock theory describing the condensate and thermal cloud. We observe there is an optimal dimple depth that results in a maximum in the condensate fraction. We also study the non-equilibrium dynamics of condensate formation in the sudden turn-on regime, finding good agreement for the observed time dependence of the condensate fraction with calculations based on quantum kinetic theory. In Chapter 4 we study the onset of phase coherence during the formation of a BEC following a sudden quench to below the transition temperature. Using the SPGPE, we simulate a recent experiment performed at ANU in which a sudden RF quench is applied to a dilute gas of metastable Helium atoms confined in an elongated 3D harmonic magnetic trap. At regularly-spaced intervals following the quench, broadband RF pulses are applied to homogeneously outcouple small (<5%) pulses of atoms, which expand and fall to a microchannel plate detector. The second-order correlation function at the detector, postulated to provide a measure of phase coherence of the in-situ cloud from which they were outcoupled, is measured versus time and compared with the condensate growth curve. A delay was observed in the relaxation of the second-order correlation function, relative to the peak in the number of condensate atoms. In our simulations we incorporate the effects of periodic outcoupling both the condensate and thermal cloud, and monitor the time-dependence of first- and second-order correlation functions in both position and momentum space. We observe long-range phase coherence in the final equilibrium state, though mild persistent phase fluctuations are present. We also observe the curious phenomenon of ``superbunching'' - an enhancement of density fluctuations - at specific values of the momentum, which we relate to anomalous correlations of fluctuations. Despite obtaining a good quantitative fit to the experimentally-measured condensate growth curve in our SPGPE simulations, we do not observe any evidence of the delay in coherence observed in the experiment, which remains unexplained. In Chapter 5 we use the SPGPE to study the crossover from a true BEC with long-range phase coherence to a quasicondensate dominated by phase fluctuations, for the case of a Bose gas in an highly-elongated 3D harmonic trap. We use parameters corresponding to an experiment in which a sudden quench resulted in the formation of a non-equilibrium condensate, the dynamics of which we study in the chapter that follows. Sweeping through a range of temperatures and chemical potentials within the crossover region, we reveal the behaviour of phase fluctuations through analysis of correlation functions. We compare the ideal-gas approximation for the transition temperature with the more accurate value obtained via the Binder cumulant. By comparing the phase coherence length with the spatial extent across which density fluctuations are suppressed, we identify the BEC/quasicondensate crossover point. We compare directly with predictions of mean-field theory [Petrov et al., Phys. Rev. Lett. 87, 050404 (2001)] and find that phase fluctuations persist to lower temperatures in our SPGPE simulations. We calculate anomalous correlations of the fluctuation field and show that the resulting correction to the mean-field condensate self-interaction energy is significant within the quasicondensate regime. In Chapter 6 we study the formation of a BEC into a non-equilibrium state by simulating an experiment in which a dilute gas of Rubidium-87 atoms in a highly-elongated 3D harmonic magnetic trap is rapidly quenched to below the transition temperature. Following the quench, a condensate is observed to form with an excessively elongated shape, and subsequently undergo damped quadrupole shape oscillations [Shvarchuck et al., Phys. Rev. Lett. 89, 270404 (2002)]. We simulate this experiment by modifying the SPGPE to incorporate a hydrodynamic thermal cloud, parametrized by axially and temporally-dependent temperature and chemical potential. Comparison against simulations without this axial dependence suggests the hydrodynamic behaviour of the thermal cloud plays a crucial role in the formation of a non-equilibrium condensate. Analyses of the first-order correlation function and the condensate wavefunction reveal the spontaneous formation of quantum vortices, as well as strong phase fluctuations at equilibrium indicating quasicondensate behaviour.
Configuration Realism (CR) is a physical ontology consisting of a finite, timeless configuration space. Each configuration is a complete arrangement of fundamental degrees of freedom; none evolves; phase attaches additively to degrees of freedom and is defined relationally, relative to a stage of an observer chain. Information exists only as finite, lossy physical records. Each record defines a compatibility class, the configurations it cannot distinguish, and probability is defined as counting over these classes. The framework's derived core is proved and computed in three companion papers under named structural assumptions. The population of an outcome class equals N|α|²: the Born rule as a theorem, with the frequency reading consuming exactly one typicality principle. The CHSH quantity assembled from integer counts reaches 2√2 to O(1/N), the ontology giving up counterfactual definiteness in the plain Bell scenario—and the absoluteness of outcomes where Wigner's-friend extensions demand it—rather than locality or free settings. And equipped with a local move set, counted populations propagate as wavepackets: the effective mass is the inverse hopping rate of elementary refinement, the potential is a cell-dependent phase cost, and the counted dynamics reproduces Schrödinger phenomenology with the number of configurations conserved exactly at every stage. Around this core, closure laws order records by refinement. Observers are chains of record-bearing stages; change is difference read along a refinement chain; time is the emergent coordinate that labels it; and closure distance, accumulating in discrete quanta, measures the restructuring refinement requires. Direct consequences follow for entropy and the arrow of time, for the classical limit, and for objectivity as redundancy. The remainder is a labeled research program: causal structure, Lorentzian geometry, geodesic dynamics, and horizon thermodynamics as conjectured continuum limits of closure geometry, with the constructions that would establish them stated as open problems. A dedicated section collects the framework's distinctive predictions—the O(1/N) family separating a counting world from continuum quantum mechanics—each with its mechanism and in-principle test. Every claim in the paper is tabulated with its status: postulate, theorem, verified, argument, conjecture, or open.
In a finite, timeless configuration space where probability is counting and the population of an outcome class is proved to equal N|α|², we construct a refinement chain from local devices—nearest-neighbour beamsplitters and per-cell phase increments, all on the 3,600-value phase lattice of the companion models—and evolve wavepacket classes by literal enumeration. The small-angle brickwork realizes the lattice Hamiltonian H = −2η cos k + V(x): the effective mass m* = 1/(2η) is the inverse hopping rate—mass as resistance to record restructuring, obtained from the move set rather than posited—and the potential is a cell-dependent phase cost per closure step. Counted populations reproduce the Schrödinger spreading law σ²(t) = σ₀² + (t/2m*σ₀)², the group velocity to seven significant figures against the exact device chain, and harmonic oscillation at ω = √(2ηκ), with Σₓ nₓ = N exact at every stage—the counting form of unitarity, tested exactly. Two findings go beyond agreement. First, the counted packet has a literal edge: populations fall to one configuration and then to zero exactly where the Born value crosses 1/N, so a counting world's particle has finite support where the continuum Gaussian merely thins—the rational-probability prediction rendered in position space. Second, the ontology's record-relative definition of classes is load-bearing for dynamics: counting once at the measurement stage (the chain as one composite device) yields residues of O(1/N), measured log–log slope −0.94, whereas imputing an integer census to unrecorded intermediate stages—partitions where no records are—degrades the scaling to slope −0.62 through the amplitude granularity of few-configuration tail classes. A definitional clause changes dynamical accuracy by a factor of order √N. What is not shown is the formal continuum limit: given [CS] the discrete effective dynamics is derived, with the same conditional status as the companion theorems, and the passage to the continuum equation remains the framework's open problem. The cell structure is a named input [CS]—but not a foreign one: its three clauses are the discrete shadow of the framework's own emergent-space program, and we locate each, finding that a preferred partition is what the stability criterion already selects, that adjacency is what the elementary refinement move set already supplies, and that only the graph's large-scale regularity—together with an assumed uniformity of hopping rate, which is a flatness assumption—remains open. Reading the cells as emergent space has an immediate consequence: the construction's maximal group velocity is one instance of the framework's conjectured invariant speed, computed rather than posited, since a single elementary move defines both the refinement step and the spatial adjacency. Relaxing uniformity—a position-dependent hopping rate, hence a position-dependent effective mass—is the discrete form of curvature, and is the natural next model.