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Physical Review A

Bose-Einstein Condensation for Charged Particles

Padmakar V. Panat, Hugh E. DeWitt, J. C. Garrison

Published Dec 1, 1973• 3 citations
Publisher

Abstract

The self-consistent random-phase approximation is used to derive an integral equation for the quasiparticle energy of a charged Bose gas near a critical point. An approximate analytical solution, valid at long wavelengths, is used to compute the critical exponents which are those of an ideal Bose gas with the single-particle spectrum $\ensuremath{\epsilon}(p)=A{p}^{2\ensuremath{-}\ensuremath{\eta}},\ensuremath{\eta}\ensuremath{\simeq}0.2$. The accuracy of the analytical solution is checked by an exact numerical solution which is also used in a calculation of the transition curve. A necessary condition for the correctness of the theory is found to be $\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\gg}0.01$.

DOI
10.1103/physreva.8.3154

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The self-consistent random-phase approximation is used to derive an integral equation for the quasiparticle energy of a charged Bose gas near a critical point. An approximate analytical solution, valid at long wavelengths, is used to compute the critical exponents which are those of an ideal Bose gas with the single-particle spectrum \ensuremathϵ(p)=Ap2\ensuremath−\ensuremathη,\ensuremathη\ensuremath≃0.2\ensuremath{\epsilon}(p)=A{p}^{2\ensuremath{-}\ensuremath{\eta}},\ensuremath{\eta}\ensuremath{\simeq}0.2\ensuremathϵ(p)=Ap2\ensuremath−\ensuremathη,\ensuremathη\ensuremath≃0.2. The accuracy of the analytical solution is checked by an exact numerical solution which is also used in a calculation of the transition curve. A necessary condition for the correctness of the theory is found to be (T\ensuremath−Tc)Tc\ensuremath≫0.01\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\gg}0.01Tc​(T\ensuremath−Tc​)​\ensuremath≫0.01.
Paper is published in J. Phys. A: Math. Theor. 43 (2010) 225001, doi:10.1088/1751-8113/43/22/225001. Exact analytical solution for the universal probability distribution of the order parameter fluctuations as well as for the universal statistical and thermodynamic functions of an ideal gas in the whole critical region of Bose-Einstein condensation is obtained. A universal constraint nonlinearity is found that is responsible for all nontrivial critical phenomena of the BEC phase transition. Simple analytical approximations, which describe the universal structure of the critical region in terms of confluent hypergeometric or parabolic cylinder functions, as well as asymptotics of the exact solution are derived. The results for the order parameter, all higher-order moments of BEC fluctuations, and thermodynamic quantities, including specific heat, perfectly match the known asymptotics outside critical region as well as the phenomenological renormalization-group ansatz with known critical exponents in the close vicinity of the critical point. Thus, a full analytical solution to a long-standing problem of finding a universal structure of the lambda-point for BEC in an ideal gas is found.
A closed form for the dielectric function of a charged Bose gas is found in the random-phase approximation. This dielectric constant is used to explore the quasiparticle energy spectrum (i.e., the allowed modes of oscillation of the gas), the damping of these quasiparticles, and the long-ranged form of the electrostatic potential around a test charge in the gas. We do this around the three temperature regions of interest; namely, T=0T=0T=0, $T=\ensuremath{\infty}$, and T=TcT={T}_{c}T=Tc​, where Tc{T}_{c}Tc​ is the transition temperature of the gas.
The ground state of a charged Bose gas is studied throughout a large range of densities in a self-consistent-field approximation formalism which includes the short-range correlations between bosons through a local-field correction. Numerical self-consistent calculations have been carried out in order to determine the static structure factor $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$. Hence, from $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ we have obtained the pair-correlation functions, the ground-state energy which essentially is the correlation energy, the pressure of the gas, and, from Feynman's relation, the energy spectrum of elementary excitations. The response of the system to a static impurity charge has also been investigated. In the high-density limit our calculations show the same results as those given by Bogoliubov's perturbation theory. In the intermediate-density region, corresponding to the strongly coupled system, our results are in very good agreement with recent calculations based on the hypernetted-chain integral equation as well as Monte Carlo computations.
We study the Bose-Einstein condensation (BEC) for a relativistic ideal gas of bosons. In the framework of canonical thermal field theory, we analyze the role of particles and anti-particles in the determination of BEC transition temperature. At the BEC transition point we obtain two universal curves, i.e. valid for any mass value: the scaled critical temperature as a function of the scaled charge density of the Bose system, and the density ratio of anti-particles versus the scaled critical temperature. Moreover, we numerically investigate charge densities and condensed fraction ranging from the non-relativistic to the ultra-relativistic temperature, where analytical results are obtained.
The analytical theory of Bose–Einstein condensation of an ideal gas in mesoscopic systems has been briefly reviewed in application to traps with arbitrary shapes and dimension. This theory describes the phases of the classical gas and the formed Bose–Einstein condensate, as well as the entire vicinity of the phase transition point. The statistics and thermodynamics of Bose–Einstein condensation have been studied in detail, including their self-similar structure in the critical region, transition to the thermodynamic limit, effect of boundary conditions on the properties of a system, and nonequivalence of the description of Bose–Einstein condensation in different statistical ensembles. The complete classification of universality classes of Bose–Einstein condensation has been given.