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CRC Press eBooks

Ideal Bose Gas

Michael J.R. Hoch

Published Apr 19, 2016• 0 citations
Publisher

Abstract

At low temperatures, the number of particles in the ground state, with ε = 0, becomes large, and care must be exercised in the evaluation of the integral. e density of states given by Equation 4.14 has the form ρ(ε) ∝ ε1/2, which gives ρ(ε) = 0 for ε = 0. e ground state is therefore excluded in the integral. For fermions, this introduces a negligible error because the exclusion principle prevents multiple occupancy of a particular state but for bosons this is not the case. At very low temperatures, the number of bosons in the ground state may be expected to become macroscopically large.

DOI
10.1201/9781439850541-18

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At low temperatures, the number of particles in the ground state, with ε = 0, becomes large, and care must be exercised in the evaluation of the integral. e density of states given by Equation 4.14 has the form ρ(ε) ∝ ε1/2, which gives ρ(ε) = 0 for ε = 0. e ground state is therefore excluded in the integral. For fermions, this introduces a negligible error because the exclusion principle prevents multiple occupancy of a particular state but for bosons this is not the case. At very low temperatures, the number of bosons in the ground state may be expected to become macroscopically large.
To obtain an expression for the Fermi energy εF, deˆned in Section 12.3, consider a system of N fermions of spin 12 and mass m in a container of volume V at T = 0 K. In zero-applied magnetic ˆeld, the mS = ± 12 spin states are degenerate, and it is convenient to introduce a degeneracy factor in expressions that involve the density of single particle states. e fermions ˆll the lowest energy states up to the energy εF = µ. By equating the number of particles to the number of states in the range 0 to εF, an expression for εF is obtained. In general, for fermions, N n er r r= ∑ = ∑ +−1 1/( ).( )b e m Conversion of the sum to an integral gives N f= ∞∫2 0 r e e e( ) ( ) ,d (13.1) where ρ(ε) is the density of states and the factor 2 is the spin degeneracy factor. f ee b e m( ) = +−1 1/( )( ) is the Fermi function, which at T = 0 K has the form f(ε) = 1 for e eʺ F and f(ε) = 0 for e e> F. is may be seen in Figure 12.1. At T = 0 K, Equation 13.1 may therefore be written as N = ∫2 0 r e ee ( ) .dF (13.2) From Equation 4.14, the density of single particle states for particles in a box is r e p e( ) ( / )( / ) / /= V m4 22 2 3 2 1 2 , and substitution in Equation 13.2 gives N V m V m = ⎛ ⎝ ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ e pF = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 3 m N V .
The properties of the ideal Bose gas are calculated from the integral equations for the energy and the number of particles as a function of the temperature and chemical potential. It is shown that the integral equations break down below the Einstein temperature that corresponds to the transition to the low-temperature state. The lowest single-particle energy level must be treated explicitly to get the proper equations. With the inclusion of the lowest single-particle energy level, the low-temperature behavior is calculated. The occupation of the lowest level becomes comparable to the total number of particles in the system below the Einstein temperature, and equal to the total number of particles at zero temperature. A numerical solution to the properties of the Bose gas is discussed, and the detailed calculations are assigned to the problems at the end of the chapter.
At sufficiently low temperatures, bosons predominantly occupy the ground state with zero energy and momentum. In addition, both superconductivity in metals and superfluidity in liquid helium-3, at sufficiently low temperatures, are associated with pairing interactions of fermions that lead to bosonic behavior. In contrast, fermions in a Fermi gas at low T populate the lowest states up to the Fermi level as a result of the Pauli exclusion principle. Beautiful experiments have been carried out on vapors of alkali atoms such as rubidium-87 that are cooled to very low temperatures by means of sophisticated experimental techniques that involve laser cooling followed by evaporative cooling.
At low temperatures the De Broglie wavelength of the gas particles becomes on the order of their average separation, and the effects of their indistinguishability become important. In the absence of a phase transition in the gas, the quantum mechanical Wigner distribution function for a dilute gas of fermions or bosons satisfies the Uehling-Uhlenbeck equation. This equation satisfies an H- theorem with equilibrium solutions being ideal boson or ideal fermion distributions. Navier-Stokes equations can be derived by standard methods. A low temperature gas of weakly interacting bosons undergoes a Bose-Einstein condensation with a macroscopically occupied ground state. A different approach is required to describe the non-equilibrium processes in such a situation. A kinetic equation can be derived for the Bogoliubov excitations in the gas at very low temperatures. The associated hydrodynamic equations are the Landau-Khaltnikov, two fluid equations, and explicit expressions are obtained for the six associated transport coefficients.
Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, ρρρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is 2πℏ2aρ/m2π\hbar^2 a ρ/m2πℏ2aρ/m where aaa is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange N7/5N^{7/5}N7/5 law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.