NSR Calculate II

Table of Contents

1. 整体思路

\begin{align} \frac{\Delta F}{NE_n} = \left[ \tilde{\tilde{\Omega}}_{\mathrm{int}} - \tilde{\Omega}_{0}^{\mathrm{M}} + \tilde{\mu} \right]\left( \frac{n_{\varepsilon}}{n} \right)^{2/3} \end{align} \begin{align} \frac{n}{n_{\varepsilon}} =& - \frac{\partial}{\partial\tilde{\mu}}\left[ \tilde{\tilde{\Omega}}_{\mathrm{int}} + \tilde{\Omega}_{0}^{\mathrm{B}} \right] \end{align} \begin{align} \tilde{\tilde{\Omega}}_{\mathrm{int}} = \int \mathrm{d}\tilde{q}\cdot \tilde{q}^2 \int_{-\infty}^{+\infty}\mathrm{d}\tilde{\omega} \cdot \frac{3}{\pi}\cdot \frac{1}{e^{\tilde{\beta}\tilde{\omega}}-1} \delta^p(\vec{q},z) \end{align} \begin{align} \tilde{\Omega}_0^{\mathrm{B}} = \frac{3}{\tilde{\beta}} \int_0^{\infty} \mathrm{d} \tilde{k} \cdot \tilde{k}^2 \ln \left[1 - e^{- \tilde{\beta} \tilde{\xi}_k}\right] \end{align} \begin{align} \delta^p(\vec{q}, z) =& \mathrm{Arg}\left[ \frac{1}{4\pi}\cdot \frac{2R}{k_{\varepsilon}^2 v} + \tilde{z}\cdot\frac{1}{4\pi} + \frac{2 R}{M k_{\varepsilon}^2}\Pi_r(\vec{q},z + \mathrm{i}0^+) \right] \end{align} \begin{align} \frac{2 R}{M k_{\varepsilon}^2}\Pi_r(\vec{q},z) =& \frac{2}{\pi^2} \cdot k_{\varepsilon}R\cdot \int \mathrm{d}\tilde{k} \left\{ \int_{-1}^1 \mathrm{d}x \cdot x^2\left[ 1+n(\xi_{\vec{k}+\vec{q}/2}) + n(\xi_{-\vec{k}+\vec{q}/2}) \right]\frac{3}{2}\frac{\tilde{k}^4} {2\tilde{k}^2 - \tilde{z}} -\frac{1}{2}\tilde{k}^2 - \frac{1}{4}\tilde{z} \right\} \end{align}

带 tilde 的都是以 \(\varepsilon\) 或 \(k_{\varepsilon}\) 为单位的无量纲量.

最终画的是 \((1)\) 式. \((1)\) 式左边以 \(E_n\) 为单位, 右边乘上 \(\left( \frac{n_{\varepsilon}}{n} \right)^{2/3}\) 进行单位转换.

将 \((2)~(5)\) 式依次代入可得结果

还有一些关系: \(\tilde{\xi} = \tilde{k}^2 - \tilde{\mu} ,\quad n(\xi) = \frac{1}{e^{\tilde{\beta} \tilde{\xi}}-1}, \quad \tilde{z} = \tilde{\omega} - \frac{\tilde{q}^2}{2} + 2 \tilde{\mu}\)

2. \(\delta^p(\vec{q}, z)\) 的近似的解析表达式

\(\delta^p(\vec{q}, z)\) 的近似的解析表达式为

\begin{align} -\pi \theta(\omega - a) \end{align}

其中

\begin{align} a = \frac{\tilde{q}^2}{2} - 2 \tilde{\mu} - \frac{2R}{k_{\varepsilon}^2 v} \end{align}

以 \(\tilde{q}\) 和 \(\tilde{\omega}\) 为变量, 数值上画出的 \(\delta^p(\vec{q}, z)\) 的二维图为

2Dfig.png

是一个阶跃函数. 其边界和近似的解析表达式对比

stepLine.png

3. 计算 density I

\begin{align} \frac{n}{n_{\varepsilon}} =& - \frac{\partial}{\partial\tilde{\mu}}\left[ \tilde{\tilde{\Omega}}_{\mathrm{int}} + \tilde{\Omega}_{0}^{\mathrm{B}} \right] \end{align}

density \(\frac{n}{n_{\varepsilon}}\) 与温度 \(\tilde{\beta}\) , \(k_{\varepsilon}R\) , \(\frac{2R}{k_{\varepsilon}^2 v}\) , 化学势 \(\tilde{\mu}\) 有关.

将温度 \(\tilde{\beta}=1\) 和 \(k_{\varepsilon}R=1/30\) 固定, 以 \(\frac{2R}{k_{\varepsilon}^2 v}\) 为横坐标, 画出不同化学势 \(\tilde{\mu}\) 时积分部分的多条曲线.

取 \(\frac{2R}{k_{\varepsilon}^2 v}\in [-2, 2]\) , \(\mu \in [-2, -.0.1]\)

4. 计算 density II

用 \(\delta^p(\vec{q}, z)\) 的近似的解析表达式求 density 的解析表达式

\begin{align} \frac{n}{n_{\varepsilon}} =& - \frac{\partial}{\partial\tilde{\mu}}\left[ \tilde{\tilde{\Omega}}_{\mathrm{int}} + \tilde{\Omega}_{0}^{\mathrm{B}} \right] \\ =& \int_0^{\infty}\mathrm{d}\tilde{q}\cdot 3 \tilde{q}^2\left[ \frac{2}{e^{\tilde{\beta}a}-1} + \frac{1}{e^{\tilde{\beta}(\tilde{q}^2 - \tilde{\mu})}-1} \right] \end{align}

5. 最终结果

将结果与原文中的结果对比

co-result.png

  • 蓝线: 原文结果
  • num 数值结果.
  • ana1 只把 \(\delta^p\) 的边界的边界用数值解代替(analytically calculate \(\delta^p\) ).
  • ana2 直到积分的结果都是解析的, 微分是数值微分(analytically calculate the integral F).
  • ana3 对 \(\mu\) 的微分也用解析结果(analytically calculate \(\mu\) ).

6. 计算 \(T_C\)

6.1. weak coupling

算出 \(\mu = 0\) 时的 density .

对应的纵坐标为

\begin{align} \frac{k_BT_C}{E_n} = \frac{1}{\beta E_n} = \frac{1}{\varepsilon \beta} \frac{\varepsilon}{E_n} = \frac{1}{\tilde{\beta}} \cdot \left(\frac{n_{\varepsilon}}{n}\right)^{2/3} \end{align}

横坐标为

\begin{align} \frac{2 R_p}{k_n^2v_p} = \frac{2 \tilde{R_p}}{k_{\varepsilon}^2 \tilde{v_p}}\cdot \left( \frac{n_{\varepsilon}}{n}\right)^{2/3} \end{align}

Tc.png

TcNum.png

Date: 2019-09-08 Sun 00:00

Created: 2022-04-18 Mon 20:39

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