index.ipynb

Kernel: Python 2

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

import mmf_setup; mmf_setup.nbinit()

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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

• [Griffin:2009]: Textbook describing the approach. [Griffin:2009]: http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521837026 (Allan Griffin, Tetsuro Nikuni, and Eugene Zaremba, "Bose-condensed gases at Finite Temperatures", , (2009) )