Department of Physics and Astronomy

The Forbes Group

ZNG

$\newcommand{\vect}[1]{\mathbf{#1}} \newcommand{\uvect}[1]{\hat{#1}} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\norm}[1]{\lVert#1\rVert} \newcommand{\I}{\mathrm{i}} \newcommand{\ket}[1]{\left|#1\right\rangle} \newcommand{\bra}[1]{\left\langle#1\right|} \newcommand{\braket}[1]{\langle#1\rangle} \newcommand{\op}[1]{\mathbf{#1}} \newcommand{\mat}[1]{\mathbf{#1}} \newcommand{\d}{\mathrm{d}} \newcommand{\pdiff}[3][]{\frac{\partial^{#1} #2}{\partial {#3}^{#1}}} \newcommand{\diff}[3][]{\frac{\d^{#1} #2}{\d {#3}^{#1}}} \newcommand{\ddiff}[3][]{\frac{\delta^{#1} #2}{\delta {#3}^{#1}}} \DeclareMathOperator{\erf}{erf} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\order}{O} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\sech}{sech} $

Here we describe the the ZNG formalism for extending the GPE to finite temperatures.

In [1]:
import mmf_setup; mmf_setup.nbinit()

This cell contains some definitions for equations and some CSS for styling the notebook. If things look a bit strange, please try the following:

  • Choose "Trust Notebook" from the "File" menu.
  • Re-execute this cell.
  • Reload the notebook.

Equations

$$ \I\hbar\pdiff{\Phi}{t} = \left(\op{H} + g(n_c + 2n_0) - \I R \right)\Phi + \eta(r)\\ n_c = \abs{\Phi}^2\\ R = \frac{\hbar\Gamma_{12}}{2n_c}\\ \Gamma_{12} = 2g^2\frac{n_c}{(2\pi)^5} \int\d{\vect{p}_1}\d{\vect{p}_2}\d{\vect{p}_3} \delta(\vect{p}_c + \vect{p}_1-\vect{p}_2-\vect{p}_3) \delta(\epsilon_c+\epsilon_1-\epsilon_2-\epsilon_3) [f_1(1+f_2)(1+f_3) - (1+f_1)f_2f_3]\\ f_i = f(\vect{p}_i, \vect{r}, t)\\ \vect{p}_c = m\vect{v}_c $$

References

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