[Paper Review] Smoluchowski problem for degenerate Bose gases

А. В. Латышев, A. A. Yushkanov|arXiv (Cornell University)|Jan 3, 2010

Optical properties and cooling technologies in crystalline materials3 references3 citations

TL;DR

This paper presents an analytic solution to the half-space Smoluchowski problem for a degenerate Bose gas with a velocity-dependent collision rate, incorporating Bose-Einstein condensate effects via a two-fluid model. The key result is a closed-form expression for the Kapitsa resistance, showing it diverges as temperature decreases, consistent with experimental trends in liquid helium-4.

ABSTRACT

We construct a kinetic equation simulating the behavior of degenerate quantum Bose gases with the collision rate proportional to the molecule velocity. We obtain an analytic solution of the half--space boundary--value Smoluchowski problem of the temperature jump at the interface between the degenerate Bose gas and the condensed phase.

Motivation & Objective

• To model kinetic behavior of a degenerate Bose gas including Bose-Einstein condensate effects using a two-fluid approach.
• To address the temperature jump problem at the interface between a degenerate Bose gas and a condensed phase.
• To derive an analytic solution for the Kapitsa resistance in the presence of quantum degeneracy and condensation.
• To extend classical kinetic theory to quantum degenerate systems with non-trivial boundary conditions.
• To provide a quantitative description of thermal boundary resistance in ultracold quantum gases.

Proposed method

• Formulates a kinetic equation with a model collision integral proportional to molecular velocity, adapted for quantum degenerate systems.
• Introduces a two-fluid model where the condensate density and velocity are treated as independent fields, coupled via conservation laws.
• Applies the Boltzmann-Krook-Welander form of the kinetic equation with a Maxwellian reference distribution and velocity-dependent relaxation rate.
• Uses dimensionless variables and asymptotic analysis to solve the half-space boundary-value problem for the distribution function.
• Derives the heat flux and temperature jump by integrating over momentum space using spherical harmonics and orthogonal function expansions.
• Solves for the Kapitsa resistance by matching boundary conditions and expressing the temperature jump as a linear function of heat flux.

Experimental results

Research questions

• RQ1 How does the presence of a Bose-Einstein condensate modify the temperature jump problem in a degenerate quantum gas?
• RQ2 What is the analytic form of the Kapitsa resistance in a degenerate Bose gas with velocity-dependent collisions?
• RQ3 How does the Kapitsa resistance scale with temperature in a quantum degenerate system?
• RQ4 Can a kinetic model with a simplified collision integral reproduce the correct thermal boundary resistance in a degenerate Bose gas?
• RQ5 What role does the two-fluid structure (condensate + thermal cloud) play in determining the temperature jump at the interface?

Key findings

• The Kapitsa resistance for a degenerate Bose gas is derived analytically as $ R = 5.77510 \, \frac{\hbar^3}{(2s+1)mk^3T_s^2} $, showing inverse quadratic dependence on temperature.
• The resistance diverges as $ T_s \to 0 $, consistent with experimental observations in liquid $^4$He.
• The solution is valid under the condition $ T \geq \frac{4\pi\hbar^2 a n}{m k} $, ensuring the weakly interacting regime.
• The temperature jump is expressed as $ \Delta T = R Q_x $, where $ Q_x $ is the heat flux, confirming linear thermal resistance behavior.
• The model accounts for nonlinear momentum and energy fluxes in the condensate, requiring linearization in velocity for consistency with conservation laws.
• The analytic solution is obtained through a combination of asymptotic analysis, orthogonal function expansion in velocity space, and exact integration of the collision integral.