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arXiv:0906.5061 · Open Physics

Effect of Bohm potential on a charged gas

D. Mostacci, Vincenzo Molinari, Francesco Pizzio

Published Dec 4, 2009• 0 citations
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Abstract

Abstract Bohm’s interpretation of Quantum Mechanics leads to the derivation of a Quantum Kinetic Equation (QKE): in the present work, propagation of waves in charged quantum gases is investigated starting from this QKE. Dispersion relations are derived for fully and weakly degenerate fermions and bosons (for the latter above critical temperature) and the differences discussed. Use of a kinetic equation permits investigation of “Landau-type” damping: it is found that the presence of damping in fermion gases is dependent upon the degree of degeneracy, whereas it is always present in boson gases. In fully degenerate fermions a phenomenon appears that is akin to the “zero sound” propagation.

DOI
10.2478/s11534-009-0160-3

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Abstract Bohm’s interpretation of Quantum Mechanics leads to the derivation of a Quantum Kinetic Equation (QKE): in the present work, propagation of waves in charged quantum gases is investigated starting from this QKE. Dispersion relations are derived for fully and weakly degenerate fermions and bosons (for the latter above critical temperature) and the differences discussed. Use of a kinetic equation permits investigation of “Landau-type” damping: it is found that the presence of damping in fermion gases is dependent upon the degree of degeneracy, whereas it is always present in boson gases. In fully degenerate fermions a phenomenon appears that is akin to the “zero sound” propagation.
In Bohm's interpretation of Quantum Mechanics, quantum effects are governed by a “quantum potential” (known as Bohm potential), and particles follow definite trajectories. In the present work, Liouville's theorem is invoked, an appropriate Liouville equation is derived, and following the BBGKY method a quantum kinetic equation (QKE) is derived. To demonstrate the working of the QKE, two examples of application are presented: the thermal equilibrium of a quantum gas and the propagation of disturbances in a force free gas of non-interacting bosons. In contrast to the classical collisionless Boltzmann equation, waves are found to be possible in the absence of interaction or external forces, due only to Bohm potential (zero sound propagation).
Starting from Bohm’s point of view on Quantum Mechanics, adopting the Bohm potential description, a Quantum Kinetic Equation can be derived. When molecular interactions are described as a self-consistent field, a Quantum Vlasov Equation (QVE) can be written. With this QVE, the evolution of a small perturbation can be described, yielding a wave propagation with or without a damping coefficient, depending on conditions. With this approach applied to a liquid of either fermions or bosons, the propagation of waves in quantum liquids is described.
In this work the propagation of sound waves in a degenerate quantum gas is considered. We modify the Wang Chang and Uhlenbeck method for description of the sounds in a Maxwell gas for the case of a degenerate quantum gas. Using this approach, we constructed a dispersion relation for sound waves in a condensed Bose gas at finite temperatures, and calculate the velocities of first and second sounds in the first approximation. The possibility of the theoretical investigation for sound damping is discussed.
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.