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Progress of Theoretical Physics

Dynamics of Quantum Statistical Condensate

T. Usui

Published Aug 1, 1964• 8 citations
Publisher

Abstract

The equation of motion of the quantum statistical condensate, defined according to Penrose-Onsager and Yang, is derived. Oscillations of small amplitude are analyzed hereby, and it is shown that the equations of motion and of continuity of the superfluid and the kinetic equation of the excitation quanta are the appropriate equations in the cases of slight inhomogeneity. Weak interaction is assumed and its existence is taken into account in such an extent that it causes a modification of the excitation spectrum but not any scattering of the excitation quanta. Systems of Bose particles and of Fermi particles are treated.

DOI
10.1143/ptp.32.190

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The equation of motion of the quantum statistical condensate, defined according to Penrose-Onsager and Yang, is derived. Oscillations of small amplitude are analyzed hereby, and it is shown that the equations of motion and of continuity of the superfluid and the kinetic equation of the excitation quanta are the appropriate equations in the cases of slight inhomogeneity. Weak interaction is assumed and its existence is taken into account in such an extent that it causes a modification of the excitation spectrum but not any scattering of the excitation quanta. Systems of Bose particles and of Fermi particles are treated.
The equations of motion for the reduced density matrices of an interacting Bose system are treated using the concept of off-diagonal long-range order. The equations are decoupled on the one-particle level. With this approximation, the hydrodynamic equations for the densities and the velocities of the condensate and the depletion are derived. The interaction between the two fluid components depends on density variations and on the relative velocity. The latter interaction terms appear as kinetic pressure terms and a Magnus force term. Via the equation for the total momentum, a connection between the velocities of the condensate and the depletion, and the velocities in Landau's two-fluid model is derived. The excitation spectrum of the system is investigated in the phonon region. A comparison of the resulting velocities for the first and second sound with the experimentally determined ones shows that the functional dependence of the condensate density on temperature is similar to that of the superfluid component.
The evolution equation of a weakly interacting boson system is derived. The model includes the interaction between the atoms in the condensate and the surrounding gas of noncondensed particles. The Bogoliubov transformation is introduced in a full quantum context and the scattering kernel between dressed particles and the condensate phase is obtained. The final system is expressed by the Boltzmann evolution equation for noncondensed particles coupled to the Gross-Pitaevskii equation for the condensate. We consider an out-of-equilibrium situation that induces a fast production of condensed particles. We apply our model to study the condensation dynamics of positronium atoms by evaporation.
Starting from first principles of the statistical mechanics we construct\na dynamics of the condensed Bose gas at the nonzero\ntemperatures. In particular, we derive an equation of motion for\nthe condensate wavefunction and a quantum kinetic equation for\nthe distribution function for the excited atoms. Obtained generalized\nGross-Pitaevskii equation for condensate include effect of\ncollisions with thermal cloud atoms (non-condensate). The Boltzmann\nquantum kinetic equation for non-condensate was obtained\nusing Zubarev’s method of nonequilibrium statistical operator.\nTherefore was obtained a complete dynamics of the condensed\nBose gas at the nonzero temperatures.
We give an overview of the current theory of collective modes in trapped atomic gases at finite temperatures, when the dynamics of the condensate and non-condensate must both be considered. A simple introduction is given to the quantum field formulation of the dynamics of an interacting Bose-condensed system, based on equations of motion for the condensate wavefunction and single-particle Green's functions for the non-condensate atoms. We discuss the nature of excitations in the mean-field collisionless region, including the Beliaev second-order approximation for the self-energies. We also sketch the derivation of coupled two-fluid hydrodynamic equations using a simple kinetic equation which includes collisions between condensate and non-condensate atoms.
By expanding the grand partition function, we obtain the expression for partition function of two dimensional free particles which is seen to be of the same form as that of one dimensional harmonic oscillators. For such systems it is shown that the difference in statistics does not appear in thermal behavior. When the energy levels of the particles are modified as ε n 1 , n 2 = Δ + b ( n 1 2 + n 2 2 ), Δ >0, the Bose condensation occurs and, if we set Δ =2 b , the condensation temperature qualitatively agrees with that of Osborne. The speciffic heat of free Bosons in a narrow box is calculated as the function of temperature, its width d being taken 7.1 Å and 14.2 Å. The results show that the maximum of the specific heat occurs at higher temperatures than that of the case d =∞, and its height becomes lower. The weak excitation of a Fermi system can be treated as excitons which obey Bose statistics. And we can show, by the method analogous to that of Ward and Wilks and Dingle, that the sound wave in a Fermi gas should be considered as the second sound, or the propagation of fluctuation in exciton distribution. Its propagation velocity is shown to become p0/3mp_{0}/\sqrt{3}mp0​/3​m, where p 0 is the momentum at the Fermi surface.
A simplified derivation of the master-graph formulation of the quantum-statistical theory of an interacting quantum fluid is presented. A complete analysis of the self-energy structures arising from both particle statistics and particle interactions simplifies the master-graph expressions for the thermodynamic properties of the fluid. The xxx-ensemble formulation of Lee and Yang is used to treat the degenerate Bose fluid below its Bose-Einstein condensation temperature. The results for a Bose fluid above its Bose-Einstein condensation temperature and for a Fermi fluid are obtained as special cases.