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Low Temperature Physics

The Kramers problem with accommodative boundary conditions for quantum Fermi gases

Alexander A. Kostikov, A. V. Latyshev, Aleksandr Alekseevich Yushkanov

Published Sep 1, 2008• 3 citations
Publisher

Abstract

The Kramers problem of isothermal slip of a quantum Fermi gas with Cercignani boundary conditions is solved analytically. The velocity of isothermal slip is obtained as a function of the accommodation coefficient and the reduced chemical potential—the ratio of the chemical potential to the product of Boltzmann’s constant and the absolute temperature. The distribution function of the molecules is presented in explicit form.

DOI
10.1063/1.2973711

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International Journal of Modern Physics B•Mar 1, 2017•2 citations
The Kramers problem of isothermal slip of a quantum Fermi gas with Cercignani boundary conditions is solved analytically. The velocity of isothermal slip is obtained as a function of the accommodation coefficient and the reduced chemical potential—the ratio of the chemical potential to the product of Boltzmann’s constant and the absolute temperature. The distribution function of the molecules is presented in explicit form.
A solution is obtained for Kramers’ problem of isothermal slip of a quantum Bose gas along a flat solid surface. The reflection of molecules from the wall is assumed to be specular-diffuse. A model kinetic equation with a collision integral in the τ-model form and collision frequency proportional to the molecular velocity is used. The dependence of the slip velocity on the value of a parameter equal to the ratio of the chemical potential to the product of Boltzmann’s constant by the absolute temperature is analyzed.
Distribution of mass velocity of quantum Bose-gas in half-space is received. Far from half-space border the gradient of mass velocity is set. The mass velocity of Bose-gas directly at a wall is found also. All results are received on basis of the analytical solution of Kramer's problem. The analysis of dependence of velocity coefficients from parametre quantity is carried out. The parametre representing the relation of chemical potential to product of Boltzmann constant on absolute temperature. Graphic comparison of coefficients of velocity Bose- and Fermi-gases is resulted.
The effects of low dimensionality on the thermodynamics of a Fermi gas trapped by isotropic power-law potentials are analyzed. Particular attention is given to different characteristic temperatures that emerge, at low dimensionality, in the thermodynamic functions of state and in the thermodynamic susceptibilities (isothermal compressibility and specific heat). An energy-entropy argument that physically favors the relevance of one of these characteristic temperatures, namely, the nonvanishing temperature at which the chemical potential reaches the Fermi energy value, is presented. Such an argument allows interpreting the nonmonotonic dependence of the chemical potential on temperature, as an indicator of the appearance of a thermodynamic regime, where the equilibrium states of a trapped Fermi gas are characterized by larger fluctuations in energy and particle density as is revealed in the corresponding thermodynamics susceptibilities.