{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "Here we describe the the ZNG formalism for extending the GPE to finite temperatures.\n", "\n", "\n", "" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/javascript": [ "// MathJaX customization, custom commands etc.\n", "console.log('Updating MathJax configuration');\n", "MathJax.Hub.Config({\n", " \"HTML-CSS\": {\n", " //availableFonts: [\"Neo-Euler\"], preferredFont: \"Neo-Euler\",\n", " //webFont: \"Neo-Euler\",\n", " //scale: 85, // Euler is a bit big.\n", " mtextFontInherit: true,\n", " matchFontHeight: true,\n", " scale: 90, // STIX is a bit big.\n", "\n", " },\n", " // This is not working for some reason.\n", " \"TeX\": {\n", " Macros: {\n", " d: [\"\\\\mathrm{d}\"],\n", " I: [\"\\\\mathrm{i}\"],\n", " vect: [\"\\\\vec{#1}\", 1],\n", " uvect: [\"\\\\hat{#1}\", 1],\n", " abs: [\"\\\\lvert#1\\\\rvert\", 1],\n", " Abs: [\"\\\\left\\\\lvert#1\\\\right\\\\rvert\", 1],\n", " norm: [\"\\\\lVert#1\\\\rVert\", 1],\n", " Norm: [\"\\\\left\\\\lVert#1\\\\right\\\\rVert\", 1],\n", " ket: [\"|#1\\\\rangle\", 1],\n", " bra: [\"\\\\langle#1|\", 1],\n", " Ket: [\"\\\\left|#1\\\\right\\\\rangle\", 1],\n", " Bra: [\"\\\\left\\\\langle#1\\\\right|\", 1],\n", " braket: [\"\\\\langle#1\\\\rangle\", 1],\n", " op: [\"\\\\mathbf{#1}\", 1],\n", " mat: [\"\\\\mathbf{#1}\", 1],\n", " pdiff: [\"\\\\frac{\\\\partial^{#1} #2}{\\\\partial {#3}^{#1}}\", 3, \"\"],\n", " diff: [\"\\\\frac{\\\\d^{#1} #2}{\\\\d {#3}^{#1}}\", 3, \"\"],\n", " ddiff: [\"\\\\frac{\\\\delta^{#1} #2}{\\\\delta {#3}^{#1}}\", 3, \"\"],\n", " Tr: \"\\\\mathop{\\\\mathrm{Tr}}\\\\nolimits\",\n", " erf: \"\\\\mathop{\\\\mathrm{erf}}\\\\nolimits\",\n", " erfi: \"\\\\mathop{\\\\mathrm{erfi}}\\\\nolimits\",\n", " sech: \"\\\\mathop{\\\\mathrm{sech}}\\\\nolimits\",\n", " order: \"\\\\mathop{\\\\mathcal{O}}\\\\nolimits\",\n", " diag: \"\\\\mathop{\\\\mathrm{diag}}\\\\nolimits\",\n", " floor: [\"\\\\left\\\\lfloor#1\\\\right\\\\rfloor\", 1],\n", " ceil: [\"\\\\left\\\\lceil#1\\\\right\\\\rceil\", 1],\n", "\n", " mylabel: [\"\\\\label{#1}\\\\tag{#1}\", 1],\n", " degree: [\"^{\\\\circ}\"],\n", " },\n", " }\n", "});\n", "\n", "// Jupyter.notebook.config.update({\"load_extensions\":{\"calico-document-tools\":true}});\n", "// Jupyter.notebook.config.update({\"load_extensions\":{\"calico-cell-tools\":true}});\n", "// Jupyter.notebook.config.update({\"load_extensions\":{\"calico-spell-check\":true}});\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\n", "\n", "

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

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

\n", "\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import mmf_setup; mmf_setup.nbinit()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Equations" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", " \\I\\hbar\\pdiff{\\Phi}{t} = \\left(\\op{H} + g(n_c + 2n_0) - \\I R \\right)\\Phi + \\eta(r)\\\\\n", " n_c = \\abs{\\Phi}^2\\\\\n", " R = \\frac{\\hbar\\Gamma_{12}}{2n_c}\\\\\n", " \\Gamma_{12} = 2g^2\\frac{n_c}{(2\\pi)^5}\n", " \\int\\d{\\vect{p}_1}\\d{\\vect{p}_2}\\d{\\vect{p}_3}\n", " \\delta(\\vect{p}_c + \\vect{p}_1-\\vect{p}_2-\\vect{p}_3)\n", " \\delta(\\epsilon_c+\\epsilon_1-\\epsilon_2-\\epsilon_3)\n", " [f_1(1+f_2)(1+f_3) - (1+f_1)f_2f_3]\\\\\n", " f_i = f(\\vect{p}_i, \\vect{r}, t)\\\\\n", " \\vect{p}_c = m\\vect{v}_c\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# References\n", "\n", "* [Griffin:2009]: Textbook describing the approach.\n", "[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) )" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.13" }, "nikola": { "date": "2017-06-08 10:48:55 UTC-07:00", "description": "", "link": "", "slug": "zng", "tags": "", "title": "ZNG", "type": "text" }, "toc": { "colors": { "hover_highlight": "#DAA520", "navigate_num": "#000000", "navigate_text": "#333333", "running_highlight": "#FF0000", "selected_highlight": "#FFD700", "sidebar_border": "#EEEEEE", "wrapper_background": "#FFFFFF" }, "moveMenuLeft": true, "nav_menu": { "height": "48px", "width": "252px" }, "navigate_menu": true, "number_sections": true, "sideBar": true, "threshold": 4, "toc_cell": false, "toc_section_display": "block", "toc_window_display": false, "widenNotebook": false } }, "nbformat": 4, "nbformat_minor": 2 }