Relaxation Rates and Collision Integrals for Bose-Einstein Condensates

Abstract

Near equilibrium, the rate of relaxation to equilibrium and the transport properties of excitations (bogolons) in a dilute Bose-Einstein condensate (BEC) are determined by three collision integrals, , , and . All three collision integrals conserve momentum and energy during bogolon collisions, but only conserves bogolon number. Previous works have considered the contribution of only two collision integrals, and . In this work, we show that the third collision integral makes a significant contribution to the bogolon number relaxation rate and needs to be retained when computing relaxation properties of the BEC. We provide values of relaxation rates in a form that can be applied to a variety of dilute Bose-Einstein condensates.

Appendix: The Form of the Kernels Used in Computation

To calculate the values of the kernel K _l (c ₁,c ₂) in Eq. (35), we split the calculation into six parts, one for each of the individual kernels in Eqs. (25)–(28). In this appendix, we show how to obtain an expression that is well-suited to numerical quadrature for the kernel $Q_{A}^{l}(c_{1}, c_{2})$. Similar procedures can be used with the other five kernels as well. We begin with the definition

(49)

First let us perform the integration over c ₄,

(50)

The resulting integrand only depends on the magnitude of c ₄, where c ₄=|c ₁+c ₂−c ₃|. We now perform the integration over c ₃ in spherical coordinates, with the z-axis oriented parallel to c ₁+c ₂,

(51)

Now notice that our choice of spherical coordinates for the c ₃ integration allows us to write c ₄ as $c_{4}=\sqrt{|\mathbf{c}_{1} + \mathbf{c}_{2}|^{2} + c_{3}^{2} - 2 c_{3} |\mathbf{c}_{1} + \mathbf{c}_{2}| z_{3}}$. We can change variables to write the z ₃ integration as an integration over c ₄ with $d c_{4} = -\frac{2 c_{3} |\mathbf{c}_{1} + \mathbf{c}_{2}|}{2 c_{4}} d z_{3}$,

(52)

Since the integrand now only depends on c ₁, c ₂ and $\hat{\mathbf{c}}_{1} \cdot \hat{\mathbf{c}}_{2}$, let us define $c_{A} = \sqrt{c_{1}^{2} + c_{2}^{2} + 2 c_{1} c_{2} (\hat{\mathbf{c}}_{1} \cdot \hat{\mathbf{c}}_{2})}$ and use c _A as a change of variables for the $\hat{\mathbf{c}}_{1} \cdot \hat{\mathbf{c}}_{2}$ integration. This results in

(53)

To handle the integration over c _A , we write the integration limits in terms of Heaviside theta functions,

(54)

and notice that

$$ \theta\bigl(|c_A - c_3| \leq c_4 \leq c_A + c_3\bigr) = \theta\bigl(|c_3 - c_4| \leq c_A \leq c_3 + c_4\bigr). $$

(55)

This allows us to move the c _A integration through all of the others and write

(56)

where

$$ w^l_{1,2,3,4} = \int_{\max[|c_1 - c_2|, |c_3 - c_4|]}^{\min[c_1 + c_2, c_3 + c_4]} d c_A P_l\biggl(\frac{c_A^2 - c_1^2 - c_2^2}{2 c_1 c_2}\biggr). $$

(57)

The final delta function of energy can now be handled in several ways, but each of them will lead to a well-behaved integrand. Still, we can make a few observations that will help the quadrature go faster.

First, notice that the range of integration on c ₃ must satisfy . This corresponds to . In fact, the symmetry between c ₃ and c ₄ shows that the whole integral is equal to twice the integral from 0≤c ₃≤c _h where . Also, since it is symmetric in c ₁ and c ₂, we can assume that c ₁>c ₂ in Eq. (57) and swap c ₁ and c ₂ if c ₂>c ₁.

Analysis of the function $w^{0}_{1,2,3,4}$ shows that under the constraint that , $w^{0}_{1,2,3,4} = 2 \min[c_{1}, c_{2}, c_{3}, c_{4}]$. This is only generally true for $Q_{A}^{l}$ and not the other kernels. Furthermore, $w^{l}_{1,2,3,4}$ for l≥1 can always be written as the product of $w^{0}_{1,2,3,4}$ and another finite function. The integrand will therefore always have a discontinuity of its derivative when c ₃=c ₂, and the integration region should be split at this point.

With all of these considerations, we finally write the best form as

(58)

where c ₄ takes the value that makes . Three of the other kernels (Q _B , Q _C , R _A ) can be similarly reduced to a single integration with a well-behaved integrand, while the other two (T _A , T _B ) can be reduced to explicit functions of c ₁ and c ₂.

A final point concerning the kernels involves the function $R_{A}^{0}(c_{1}, c_{2})$. Though this function is undefined when c ₁=c ₂, integrals over the entire kernel K _l (c ₁,c ₂) still converge. In the numerical method of Sect. (6), we use the fact that acting alone conserves bogolon number to determine the values of $R_{A}^{0}(c_{1}, c_{1})$.