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Physical Review

Transport Phenomena in a Bose-Einstein Gas

William Band

Published Jun 15, 1949• 4 citations
Publisher

Abstract

This paper develops the formal first order perturbation theory of transport phenomena in an ideal Bose-Einstein gas.The perturbation relaxation time is an unknown parameter in the theory. The ratio of thermal conductivity to viscosity is independent of this relaxation time; it increases discontinuously by a factor almost 2 at the Bose-Einstein condensation temperature ${T}_{\ensuremath{\lambda}}$. The isothermal Knudsen heat of transport through a small hole, classically equal to $2kT$, drops rapidly to zero as $T$ falls to ${T}_{\ensuremath{\lambda}}$: the Bose-Einstein gas flowing isothermally through a small hole carries no heat when its temperature is below ${T}_{\ensuremath{\lambda}}$ and it exhibits corresponding thermo-mechanical effects.The mathematical analysis is carried out in terms of an approximation in which the lowest energy states of the gas are treated discretely while above an arbitrary level they are treated by an integration approximation. The treatment is capable of generalization to models in which there are anomalies in the lowest level spacings.

DOI
10.1103/physrev.75.1937

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This paper develops the formal first order perturbation theory of transport phenomena in an ideal Bose-Einstein gas.The perturbation relaxation time is an unknown parameter in the theory. The ratio of thermal conductivity to viscosity is independent of this relaxation time; it increases discontinuously by a factor almost 2 at the Bose-Einstein condensation temperature T\ensuremathλ{T}_{\ensuremath{\lambda}}T\ensuremathλ​. The isothermal Knudsen heat of transport through a small hole, classically equal to 2kT2kT2kT, drops rapidly to zero as TTT falls to T\ensuremathλ{T}_{\ensuremath{\lambda}}T\ensuremathλ​: the Bose-Einstein gas flowing isothermally through a small hole carries no heat when its temperature is below T\ensuremathλ{T}_{\ensuremath{\lambda}}T\ensuremathλ​ and it exhibits corresponding thermo-mechanical effects.The mathematical analysis is carried out in terms of an approximation in which the lowest energy states of the gas are treated discretely while above an arbitrary level they are treated by an integration approximation. The treatment is capable of generalization to models in which there are anomalies in the lowest level spacings.
The theory of gases in nonstationary states, given by Lorentz and Enskog, is generalized for the quantum statistics to give the hydrodynamical equations and the distribution function in first and second approximation, and formal expressions for the viscosity and heat conductivity coefficients. These results are valid for all statistics and for all degrees of degeneration. Two essential points contribute to this generality: (a) Exact expressions independent of statistics and degeneration can be given for the coordinate and time derivatives of the coefficient AAA of the equilibrium distribution function in terms of the pressure and temperature gradients and time derivatives, though a closed expression for this coefficient as a function of vvv and TTT is known only in limiting cases; (b) The function WWW of the general equation of state for ideal gases in all statistics pv=(RTM)W(v23T)pv=(\frac{\mathrm{RT}}{M})W({v}^{\frac{2}{3}}T)pv=(MRT​)W(v32​T) is adiabatically invariant. Numerical values of the viscosity and heat conductivity coefficients, which should come out of the formal theory on the introduction of special assumptions about the molecular forces, have not yet been obtained. It is our hope that these results, when found, may furnish an experimental test of the existence of Einstein-Bose statistics in real gases, as is required by theory.
The thermal fluctuations of the order parameter affect the transport of Bose gas above the superfluid transition. Using the time-dependent Ginzburg-Landau theory in the classical region, we calculate the fluctuation corrections to the coefficients of thermal conductivity and shear viscosity. Above the transition temperature T c , the fluctuation contribution to the thermal conductivity diverges as ( T − T c ) -2+ d /2 for dimensions d . The viscosity drops near T c since the fluctuation contributes a correction which varies as − ( T − T c ) −2+ d /2 . The calculations are conducted under the condition that the fluctuation corrections are small compared with the normal part of the transport coefficients.
Expressions for the bulk viscosity, shear viscosity, and thermal conductivity of a quantum degenerate Bose gas above the critical temperature for Bose-Einstein condensation are derived using the Uehling-Uhlenbeck kinetic equation. For contact potentials and hard sphere interactions, the eigenvalues (relaxation rates) of the Uehling-Uhlenbeck collision operator have an upper cutoff. This cutoff requires summation over all discrete eigenvalues and eigenvectors of the collision operator when computing transport coefficients. We numerically compute the shear viscosity and thermal conductivity for any boson gas that interacts via a contact potential. We find that the bulk viscosity of the degenerate boson gas remains identically zero, as it is for the classical gas.
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∈/ k , where Δ∈ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermo dynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced, and from this we determine the density profile and the velocity distribution of particles. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. The final two sections are devoted to effects not captured by the simplest version of the semi-classical approximation: corrections to the transition temperature due to a finite particle number (Sec. 2.5), and thermodynamic properties of gases in lower dimensions (Sec. 2.6).
We derive the Boltzmann equation for relaxation rate, viscosity and thermal conductivity in ultra cold normal dipolar bose gas. We use anisotropic dipole- dipole interaction. We obtain the temperature dependence of the relaxation rate and give formal expressions for the shear viscosity and thermal conductivity coefficients. These coefficients are proportional to . The different form of potentials in calculations show themselves in the coefficients of the viscosity and thermal conductivity. The ratio of the viscosity coefficient for dipole-dipole interaction to contact potential is three order and this ratio for thermal conductivity is two order.
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.