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arXiv:1001.3111

KAPITSA RESISTANCE IN DEGENERATE QUANTUM GASES WITH BOGOLYUBOV ENERGY EXCITATIONS IN THE PRESENCE OF BOSE - EINSTEIN CONDENSATE

Anatoly V. Latyshev, Alexander A. Yushkanov

Published Jan 18, 2010• 0 citations
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Abstract

,Moscow State Regional University, 105005, Moscow, Radio st., 10–A(Dated: January18,2010)The linearized kinetic equation modelling behaviour of thedegenerate quantumbose - gas withthe frequency of collisions depending on momentum of elementary excitations is constructed.The general case of dependence of the elementary excitations energy on momentum according toBogolyubov formula is considered. The analytical solution of the half–space boundary problemon temperature jump on border of degenerate bose - gas in the presence of a Bose — Einsteincondensateisreceived. ExpressionforKapitsaresistanceisreceived.Keywords: degenerate quantum Bose gas, collision integral, Bose — Einstein condensate,Bogolyubovexcitations,temperaturejump,Kapitsaresistance.

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,Moscow State Regional University, 105005, Moscow, Radio st., 10–A(Dated: January18,2010)The linearized kinetic equation modelling behaviour of thedegenerate quantumbose - gas withthe frequency of collisions depending on momentum of elementary excitations is constructed.The general case of dependence of the elementary excitations energy on momentum according toBogolyubov formula is considered. The analytical solution of the half–space boundary problemon temperature jump on border of degenerate bose - gas in the presence of a Bose — Einsteincondensateisreceived. ExpressionforKapitsaresistanceisreceived.Keywords: degenerate quantum Bose gas, collision integral, Bose — Einstein condensate,Bogolyubovexcitations,temperaturejump,Kapitsaresistance.
The paper considers a model for Bose gases in the so-called 'high-temperature range' below the temperature Tc, where Bose-Einstein condensation sets in.The model is of non-linear two-component type, consisting of a kinetic equation with periodic boundary conditions for the distribution function of a gas of excitations interacting with a Bose condensate, which is described by a Gross-Pitaevskii equation. Results on well-posedness and long time behaviour are proved in a H1-setting close to equilibrium.
Abstract Bose–Einstein condensation in a Bose gas is studied analytically, in any positive dimensionality ( d > 0) for identical bosons with any energy-momentum positive-exponent ( s > 0) plus an energy gap Δ between the ground state energy ε 0 and the first excited state, i.e. ε = ε 0 for k = 0 and ε = ε 0 + Δ + c s k s , for k > 0, where ℏ k is the particle momentum and c s a constant with dimensions of energy multiplied by a length to the power s > 0. Explicit formula with arbitrary d / s and Δ are obtained and discussed for the critical temperature and the condensed fraction, as well as for the equation of state from where we deduce a generalized Δ independent thermal de Broglie wavelength. Also the internal energy is calculated from where we obtain the isochoric specific heat and its jump at T c . When Δ > 0, for any d > 0 exists a Bose–Einstein critical temperature T c ≠ 0 where the internal energy shows a peak and the specific heat shows a jump. Both the critical temperature and the specific heat jump increase as functions of the gap but they decrease as functions of d / s . At sufficiently high temperatures Δ-independent classical results are recovered while for temperatures below the critical one the gap effects are predominant. For Δ = 0 we recover previous reported results.
We review some modelling and numerical aspects in quantum kinetic theory for a gas of interacting bosons and we try to explain what makes Bose-Einstein condensation in a dilute gas mathematically interesting and numerically challenging. Particular care is devoted to the development of efficient numerical schemes for the quantum Boltzmann equation that preserve the main physical features of the continuous problem, namely conservation of mass and energy, the entropy inequality and generalized Bose-Einstein distributions as steady states. These properties are essential in order to develop numerical methods that are able to capture the challenging phenomenon of bosons condensation. We also show that the resulting schemes can be evaluated with the use of fast algorithms. In order to study the evolution of the condensate wave function the Gross-Pitaevskii equation is presented together with some schemes for its efficient numerical solution. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We derive the mean field kinetic equation for the momentum distribution of Bogoliubov excitations (bogolons) in a spatially uniform Bose-Einstein condensate (BEC), with a focus on the collision integrals. We use the method of Peletminksii and Yatsenko rather than the standard non-equilibrium Green's function formalism. This method produces three collision integrals G12{\cal G}^{12}G12, G22{\cal G}^{22}G22 and G31{\cal G}^{31}G31. Only G12{\cal G}^{12}G12 and G22{\cal G}^{22}G22 have been considered by previous authors. The third collision integral G31{\cal G}^{31}G31 contains the effects of processes where one bogolon becomes three and vice versa. These processes are allowed because the total number of bogolons is not conserved. Since G31{\cal G}^{31}G31 is of the same order in the interaction strength as G22{\cal G}^{22}G22, we predict that it will significantly influence the dynamics of the bogolon gas, especially the relaxation of the total number of bogolons to its equilibrium value.