自由空间中的自由高斯连系综等价性

参考文献:Soft Matter, 2018, 14, 6857--6866




考虑受拉伸的高斯链。

定长系综

一高斯链,一端位于原点,则配分函数为:

\begin{equation} \mathscr Z_0(\mathbf r)=\mathcal N \int_{\mathbf r(0)=0}^{\mathbf r(L)=\mathbf r}\mathcal D[r(s)]\exp[-\beta\mathscr H_0(\mathbf r(s))] \label{Z0} \end{equation}

式中$\mathscr H_0(\mathbf r(s))$为Edwards 哈密顿量:

\begin{equation} \mathscr H_0=\frac{3}{4l_{\mathrm p}\beta} \int_{0}^{L}\left(\frac{\partial \mathbf r(s)}{\partial s}\right)^2\mathrm ds \label{H0} \end{equation}

其中$l_{\mathrm p}$为驻留长度,周线长度$L=2l_{\mathrm p}N=bN$。

末端距满足高斯分布:

\begin{equation} P_0(\mathbf r)=\left(\frac{3}{4l_{\mathrm p}}\right)^{3/2}\exp\left(-\frac{3\mathbf r^2}{4Ll_{\mathrm p}}\right) \label{P0} \end{equation}

$P_0(\mathbf r)\sim \Omega(\mathbf r)$,$\Omega(\mathbf r)$为构象数。高斯链亥姆霍兹自由能中只有熵的贡献:$F=U-TS=-\ln\Omega(\mathbf r)/\beta$,共轭的熵力为:

\begin{equation} \langle \mathbf f \rangle = \nabla_{\mathbf r}F = \frac{3\mathbf r}{2\beta Ll_{\mathrm p}} \label{fav} \end{equation}

定力系综

高斯链末端施加一个力$\mathbf f $。哈密顿量为:

\begin{equation} \mathscr H=\mathscr H_0-\mathbf f\cdot \mathbf r=\mathscr H_0- \int_{0}^{L}\mathbf f\cdot \left(\frac{\partial \mathbf r(s)}{\partial s}\right)\mathrm ds \label{H} \end{equation}

平均拉伸长度:

\begin{equation} \begin{split} \langle \mathbf r \rangle =& \frac{\int_{\mathbf r(0)=0}\mathcal D[\mathbf r(s)]\mathbf r \exp\left [-\frac{3}{4l_{\mathrm p}} \int_{0}^{L}\left(\frac{\partial \mathbf r(s)}{\partial s}\right)^2\mathrm ds+\beta\int_{0}^{L}\mathbf f\cdot \left(\frac{\partial \mathbf r(s)}{\partial s}\right)\mathrm ds\right]}{\int_{\mathbf r(0)=0} \mathcal D[\mathbf r(s)]\exp\left [-\frac{3}{4l_{\mathrm p}} \int_{0}^{L}\left(\frac{\partial \mathbf r(s)}{\partial s}\right)^2\mathrm ds+\beta\int_{0}^{L}\mathbf f\cdot \left(\frac{\partial \mathbf r(s)}{\partial s}\right)\mathrm ds\right]} \\ =& \frac{\int_{\mathbf r(0)=0}\mathcal D[\mathbf r(s)]\mathbf r \exp\left [-\frac{3}{4l_{\mathrm p}}\int_{0}^{L}\left(\frac{\partial \mathbf r(s)}{\partial s}-\frac{2l_{\mathrm p}\beta}{3}\mathbf f\right)^2\mathrm ds\right]}{\int_{\mathbf r(0)=0} \mathcal D[\mathbf r(s)]\exp\left [-\frac{3}{4l_{\mathrm p}}\int_{0}^{L}\left(\frac{\partial \mathbf r(s)}{\partial s}-\frac{2l_{\mathrm p}\beta}{3}\mathbf f\right)^2\mathrm ds\right]} \\ =& \frac{\int_{\mathbf R(0)=0}\mathcal D[\mathbf R(s)]\left(\mathbf R+\frac{2l_{\mathrm p}\beta}{3}\mathbf fL\right) \exp\left [-\frac{3}{4l_{\mathrm p}}\int_{0}^{L}\left(\frac{\partial \mathbf R(s)}{\partial s}\right)^2\mathrm ds\right]}{\int_{\mathbf R(0)=0} \mathcal D[\mathbf R(s)]\exp\left [-\frac{3}{4l_{\mathrm p}}\int_{0}^{L}\left(\frac{\partial \mathbf R(s)}{\partial s}\right)^2\mathrm ds\right]} \\ =& \frac{2Ll_{\mathrm p}\beta}{3}\mathbf f \end{split} \label{rav} \end{equation}

对比\eqref{fav}和\eqref{rav},定长系综(亥姆霍兹)与定力系综(吉布斯)结果是等价的。

标签: 高斯链, 系综

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