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The Journal of Chemical Physics

Distance Correlations and Bose-Einstein Condensation

F. London

Published Apr 1, 1943• 50 citations
Publisher

Abstract

The correlation effect on the molecular distance distribution in a Bose-Einstein gas is calculated as a function of volume and temperature, particularly for the case of degeneracy (condensation) of the gas. It is shown that, under certain conditions in the condensed state, the density as seen from any one molecule appears, over a rather extended region around it, to be considerably larger than the true density. This result seems to have direct relation to the fact that the low temperature modification of liquid helium has a negative coefficient of thermal expansion.

DOI
10.1063/1.1723828

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The correlation effect on the molecular distance distribution in a Bose-Einstein gas is calculated as a function of volume and temperature, particularly for the case of degeneracy (condensation) of the gas. It is shown that, under certain conditions in the condensed state, the density as seen from any one molecule appears, over a rather extended region around it, to be considerably larger than the true density. This result seems to have direct relation to the fact that the low temperature modification of liquid helium has a negative coefficient of thermal expansion.
We consider the properties of a gas of bosonic diatomic molecules in the limit when few of the molecules are dissociated. Taking into account the effects of dissociation and scattering among molecules and atoms, we calculate the dispersion relation for a molecule, and the thermal depletion of the condensate. We calculate the dependence of the Bose-Einstein-condensation (BEC) temperature of a uniform gas on the atom-atom scattering length, and conclude that, for a broad Feshbach resonance, the condensation temperature increases as the molecular state becomes less strongly bound, thereby giving rise to a maximum in the transition temperature in the BEC-BCS crossover. We also argue on general grounds that, for a gas in a harmonic trap and for a narrow Feshbach resonance, the condensation temperature will decrease with increasing scattering length.
The condition which has to be fulfilled in order that the Bose-Einstein condensation be accompanied by a discontinuity of the heat capacity at the condensation temperature is discussed. The nature of the transition depends on the distribution of the energy levels of the individual particles forming the gas. The condensation is studied for several types of individual energy spectra. It is shown that all those Bose-Einstein (B.E.) assemblies which admit a non-ambiguous definition of the statistical distribution function present, beside interesting analogies, a qualitatively different behavior from that observed in liquid helium. It appears that as long as the transition in momentum space of non-ideal B.E. systems is not understood, the possibility of connection between B.E. condensation and the transition observed in liquid helium remains an open problem.
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 L2L^2L2-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\textit{Full-text embargoed until: 2026-01-01}Full-text embargoed until: 2026-01-01
When an ideal Bose gas is enclosed in two boxes with different volumes connected by a narrow tube, and the temperature is below the transition temperature for Bose-Einstein condensation, it is expected that the particles will be localized in the larger box, because the ground state is localized here. In this paper, we consider the effect of the interaction between particles on the localization using Gross-Pitaevskii theory. Our consideration is limited to the case of positive scattering length and zero temperature. We determine a condition for the localization with an approximate method. It is shown that the system must be extremely dilute or extremely small for localization to occur. The localization condition is confirmed by a numerical calculation.
The pair-correlation function is one of the basic quantities to characterize the coherence properties of a Bose-Einstein condensate. We calculate this function in the experimentally important case of a zero temperature Bose-Einstein condensate in a metastable triplet helium state using the variational method with a pair-excitation ansatz. We compare our result with a pair-correlation function obtained for the hard-sphere potential with the same scattering length. Both functions are practically indistinguishable for distances greater than the scattering length. At smaller distances, due to interatomic interactions, the helium condensate shows strong correlations.