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Proceedings of the Physical Society

Charged Boltzmann Gas at Absolute Zero

M. J. Stephen

Published May 1, 1962• 21 citations
Publisher

Abstract

The correlation energy of a charged Boltzmann gas at absolute zero is calculated. The result is exact at high density in the same sense as the result of Gell-Mann and Brueckner for the Fermi gas. Investigation of the magnetic properties reveals that at absolute zero this system exhibits a Meissner effect of the London type. At any finite temperature the system is diamagnetic. Consideration of the effect of the Coulomb interaction on the Meissner effect leads to the conjecture that the London constant is independent of the particle interactions. The results are also applicable to a charged boson gas at absolute zero.

DOI
10.1088/0370-1328/79/5/311

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The correlation energy of a charged Boltzmann gas at absolute zero is calculated. The result is exact at high density in the same sense as the result of Gell-Mann and Brueckner for the Fermi gas. Investigation of the magnetic properties reveals that at absolute zero this system exhibits a Meissner effect of the London type. At any finite temperature the system is diamagnetic. Consideration of the effect of the Coulomb interaction on the Meissner effect leads to the conjecture that the London constant is independent of the particle interactions. The results are also applicable to a charged boson gas at absolute zero.
The ground state energy and elementary excitations of a charged gas of bosons at high densities are examined by use of the method developed by Bogoliubov for boson gases. It is conjectured, but not herein established, that this method yields exact results in the high-density limit analogous to those obtained by Gell-Mann and Brueckner, and by Sawada, in the corresponding case of a charged fermion gas. The ground state energy is essentially correlation energy, and is therefore negative, and its magnitude varies as the one-fourth power of the density at high densities. The elementary excitations have for low momenta the energy appropriate to plasma waves, and for high momenta the energy appropriate to single-particle excitation. There is therefore an energy gap, suggesting that the gas is both a superfluid and a superconductor at low temperatures. At low densities the behavior of a charged gas is independent of statistics; hence, such a gap must disappear as the system is expanded.
Green-function techniques are applied to the charged Bose gas at the absolute zero of temperature. The procedure permits some general remarks to be made concerning the expansions of the ground-state energy and depletion parameter in terms of rS{\mathcal{r}}_{S}rS​, the ratio of the interparticle spacing to the Bohr radius. It is found that these series contain not only all integer powers of rS34{{\mathcal{r}}_{S}}^{\frac{3}{4}}rS​43​ as had been conjectured, but additional functions of rS{\mathcal{r}}_{S}rS​ as well. The first-two terms in the expansion of the ground-state energy are calculated exactly as are the first-four terms in the expansion of the depletion parameter. The former expansion agrees with the results of Lee and Feenberg. The latter, containing a noninteger power of rS34{{\mathcal{r}}_{S}}^{\frac{3}{4}}rS​43​, is new. It is shown that the ground-state energy diagrams which must be summed in the Bose gas are characterized by equal numbers of variables of integration. It is emphasized that these are not the most divergent diagrams as in the electron-gas calculation of Gell-Mann and Brueckner. This difference leads us to reformulate the diagrammatic procedure so as to emphasize its self-consistency. The underlying similarity of the charged Fermi and Bose-gas calculations is then made apparent.
Abstract We prove a rigorous upper bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of interaction potentials. Our result covers the Coulomb potential, and in this case we obtain the analogue of the Gell-Mann–Brueckner formula c1ρlog⁡(ρ)+c2ρc_{1}\rho \log \left( \rho \right) +c_{2}\rho c1​ρlog(ρ)+c2​ρ c 1 ρ log ρ + c 2 ρ in the high density limit. We do this by refining the analysis of our bosonization method to deal with singular potentials, and to capture the exchange contribution which is absent in the purely bosonic picture.
It is shown that an ideal gas of charged bosons exhibits the essential equilibrium features of a superconductor. The onset of Bose-Einstein condensation marks the transition temperature Tc{T}_{c}Tc​. Below Tc{T}_{c}Tc​ a Meissner-Ochsenfeld effect is exhibited which is described in a very good approximation by London's equation.The singular nature of the condensed ideal Bose gas exhibits itself in a space dependence of the London constant $\ensuremath{\lambda}$, determined by the boundary conditions on the wave function. It is shown that the electrostatic repulsion between the bosons compensates this effect and leads to a spatially constant $\ensuremath{\lambda}$, independently of the boundary conditions.The critical field Hc(T){H}_{c}(T)Hc​(T) is determined and found to be related to the penetration depth d(T)d(T)d(T) by ${H}_{c}=\frac{\ensuremath{\hbar}c}{2e{d}^{2}}$ (eee being the boson charge).The B(H)B(H)B(H) law is different from the one usually assumed for actual superconductors. Corresponding changes occur in the thermodynamical relation.A comparison with superconducting metals is made. The main conclusion is that if superconductivity in metals is due to the concurrence of bosons, then the number of these bosons must be strongly temperaturedependent below Tc{T}_{c}Tc​.
It is well-known that the 2-dimensional Bose gas does not condense, and that no transition occurs in its thermodynamic behavior. Despite this, we find that in a magnetic field a 2-dimensional gas of charged, ideal bosons has a nearly sharp transition temperature, below which it displays an "imperfect" Meissner-Ochsenfeld effect, practically indistinguishable from the perfect London one. This fact is relevant to the application of the Bose gas model of superconductivity to thin films.The main interest of the model is, however, that it exhibits an (essentially perfect) Meissner-Ochsenfeld effect without any irregularity in CV(T){C}_{V}(T)CV​(T).
We prove a rigorous lower bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of singular interactions, including the Coulomb potential. Combined with the upper bound obtained by Christiansen, Hainzl, and Nam [Comm. Math. Phys. 401 (2023), pp. 1469–1529], our result establishes an analogue of the Gell-Mann–Brueckner formula c 1 ρ log ⁡ ( ρ ) + c 2 ρ c_{1}\rho \log \left (\rho \right )+c_{2}\rho for the correlation energy of the electron gas in the high-density limit. Moreover, our analysis allows us to go beyond mean-field scaling while still covering the same class of potentials.