{"id":7025,"date":"2016-06-11T01:26:14","date_gmt":"2016-06-10T17:26:14","guid":{"rendered":"http:\/\/www.andrewsun.net\/panta_rhei\/?p=7025"},"modified":"2020-02-04T23:32:13","modified_gmt":"2020-02-04T15:32:13","slug":"%e6%9e%81%e5%9d%90%e6%a0%87%e4%b8%8b%e7%9a%84%e4%ba%8c%e7%bb%b4%e6%97%a0%e8%a7%84%e8%a1%8c%e8%b5%b0","status":"publish","type":"post","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/7025","title":{"rendered":"\u6781\u5750\u6807\u4e0b\u7684\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70"},"content":{"rendered":"
\n

[latexpage]<\/p>\n

\u8fd9\u51e0\u5929\u6211\u5728\u674e\u4fca\u6770\u3001Mathematica\u7684\u5e2e\u52a9\u4e0b\u63a8\u5bfc\u4e86\u6781\u5750\u6807\u4e0b\u7684\u4e8c\u7ef4\u5e03\u6717\u8fd0\u52a8\u7684\u7edf\u8ba1\u3002\u4e00\u5f00\u59cb\u6211\u5c31\u89c9\u5f97\uff0c\u8fd9\u662f\u5f88\u57fa\u672c\u7684\u95ee\u9898\uff0c\u5e94\u8be5\u4f1a\u6709\u76f8\u5e94\u7684\u4f8b\u9898\u6216\u8bfe\u4ef6\u76f4\u63a5\u7ed9\u51fa\u7b54\u6848\u3002\u4f46\u662f\u4e00\u5f00\u59cb\u641c\u90fd\u641c\u4e0d\u5230\u3002\u6211\u7528\u4f53\u80b2\u8001\u5e08\u6559\u7684\u6570\u5b66\u63a8\u4e86\u534a\u5929\uff0c\u51fa\u7ed3\u679c\u4e4b\u540e\uff0c\u5c31\u7acb\u9a6c\u641c\u5230\u8d44\u6599\u4e86\u3002\u53ea\u5f53\u4f5c\u4e0a\u5929\u903c\u6211\u7ec3\u4e60\u4e00\u4e0b\u5927\u5b66\u7684\u6570\u5b66\u5427\u3002<\/p>\n

\u6211\u8981\u8003\u8651\u7684\u95ee\u9898\u662f\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u7684\u6781\u5750\u6807\u7684\u7edf\u8ba1\uff0c\u5373\u6781\u5f84\u548c\u6781\u89d2\u7684\u5206\u5e03\u3002\u4e4b\u524d\u6211\u603b\u7ed3\u8fc7<\/a>\u4e86\u4e8c\u7ef4\u548c\u4e09\u7ef4\u65e0\u89c4\u884c\u8d70\u7684\u6781\u5f84\u5206\u5e03\u5206\u522b\u662fRayleigh\u5206\u5e03\u548cMaxwell\u5206\u5e03\uff0c\u8fd9\u90fd\u662f\u76f4\u89d2\u5750\u6807\u7684\u671f\u671b\u503c\u4e3a\u96f6\u7684\u60c5\u51b5\u3002\u672c\u6587\u4e00\u662f\u8981\u8003\u8651\u671f\u671b\u503c\u4e0d\u4e3a\u96f6\u7684\u60c5\u51b5\uff0c\u4e8c\u662f\u4e0d\u5149\u8003\u8651\u6781\u5f84\u7684\u5206\u5e03\uff0c\u8fd8\u8981\u8003\u8651\u6781\u89d2\u7684\u3002<\/p>\n

\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u7684\u201c\u8f68\u8ff9\u4e2d\u5fc3\u201d<\/h3>\n

\u4e00\u4e2a\u8d70\u4e86N<\/i>\u6b65\u7684\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u8f68\u8ff9\u5e94\u8be5\u662f\u6709\u5176\u4e2d\u5fc3\u7684\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u5b9a\u4e49\u8fd9\u4e2a\u4e2d\u5fc3\u4e3a\u65e0\u89c4\u884c\u8d70\u7684x<\/i>\u5750\u6807\u548cy<\/i>\u5750\u6807\u7684\u671f\u671b\u503c$\\mu_X$\u548c$\\mu_Y$\u3002\u8fd9\u6837\u7684\u5b9a\u4e49\u4e25\u683c\u6765\u8bf4\u5e76\u4e0d\u5b9e\u7528\uff0c\u56e0\u4e3a\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u6709\u9650\u6b65\u6570\u7684\u8f68\u8ff9\uff0c\u6211\u4eec\u65e0\u6cd5\u77e5\u9053\u671f\u671b\u503c\uff0c\u53ea\u80fd\u8ba1\u7b97\u5747\u503c\u3002\u540e\u8005\u53ea\u6709$N\\rightarrow\\infty$\u65f6\u624d\u662f\u671f\u671b\u503c\u3002\u4e3a\u4e86\u5b9e\u9a8c\u65b9\u4fbf\uff0c\u6211\u5b9a\u4e49\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u7684\u8f68\u8ff9\u201c\u4e2d\u5fc3\u201d\u662f\u5176\u8d77\u59cb\u5750\u6807$\\left(x_0,y_0\\right)$\u3002\u8fd9\u6837\uff0c\u4e0d\u4ec5\u5bf9\u4efb\u610f\u4e00\u4e2a\u7ed9\u5b9a\u7684\u8f68\u8ff9\u90fd\u80fd\u76f4\u63a5\u8bf4\u51fa\u5176\u201c\u4e2d\u5fc3\u201d\uff0c\u8fd8\u80fd\u4e3b\u52a8\u5b9e\u73b0\u4e00\u4e2a\u7ed9\u5b9a\u4e2d\u5fc3\u7684\u201c\u65e0\u89c4\u884c\u8d70\u201d\u2014\u2014\u53ea\u8981\u4ece\u4f60\u7684\u884c\u8d70\u662f\u4ece\u90a3\u4e2a\u4e2d\u5fc3\u5f00\u59cb\u7684\u5c31\u884c\u3002\u51ed\u76f4\u89c2\u60f3\u8c61\uff0c\u4e00\u4e2a\u4ece\u70b9$\\left(x_0,y_0\\right)$\u51fa\u53d1\u7684\u65e0\u89c4\u884c\u8d70\uff0c\u53ea\u8981\u6b65\u6570$N$\u8db3\u591f\u5927\uff0c\u5c31\u6709$\\mu_X=x_0$\u548c$\\mu_Y=y_0$\u3002\u5f53\u7136\uff0c\u8fd9\u662f\u4e3a\u4e86\u5b9e\u9a8c\u7684\u601d\u8003\u3002\u63a5\u4e0b\u6765\u7684\u63a8\u5bfc\u90fd\u662f\u8c08\u671f\u671b\u503c<\/p>\n

\u671f\u671b\u503c\u4e3a\u96f6\u7684\u60c5\u51b5<\/h3>\n

\u9996\u5148\uff0c\u8003\u8651\u4e00\u4e2a\u201c\u5728\u539f\u70b9\u5904\u201d\u7684\u65e0\u89c4\u884c\u8d70\uff0c\u5373\u5176x<\/i>\u5750\u6807\u548cy<\/i>\u5750\u6807\u7684\u671f\u671b\u503c$\\mu_X=\\mu_Y=0$\uff0c\u65b9\u5dee\u90fd\u662f$\\sigma^2$\uff08\u5404\u5411\u540c\u6027\uff09\u3002\u8fd9\u65f6\u5176\u6781\u5750\u6807\u7684\u6781\u5f84\u548c\u6781\u89d2$r$\u548c$\\theta$\u7684\u5206\u5e03\u5206\u522b\u4e3a\u4e00\u4e2aRayleigh\u5206\u5e03\u548c\u4e00\u4e2a\u5747\u5300\u5206\u5e03\uff1a<\/p>\n

\\begin{equation}\\label{eq:1}
\np_R\\left(r\\right)=\\frac{r}{\\sigma^2}\\exp\\left[-\\frac{r^2}{2\\sigma^2}\\right]
\n\\end{equation}
\n\\begin{equation}\\label{eq:2}
\np_\\Theta\\left(\\theta\\right)=\\frac{1}{2\\pi}
\n\\end{equation}<\/p>\n

\u671f\u671b\u503c\u4e0d\u4e3a\u96f6\u7684\u60c5\u51b5<\/h3>\n

\u8003\u8651\u201c\u4f4d\u4e8e\u6781\u5750\u6807\u4e2d\u70b9$\\left(\\mu_r,\\mu_\\theta\\right)$\u5904\u7684\u65e0\u89c4\u884c\u8d70\uff0c\u5373\u5176\u76f4\u89d2\u5750\u6807x<\/i>\u548cy<\/i>\u7684\u671f\u671b\u503c\u4e3a$\\mu_X\u548c\\mu_Y$\uff0c\u65b9\u5dee\u90fd\u662f$\\sigma^2$\uff08\u5404\u5411\u540c\u6027\uff09\u3002\u4e8e\u662f<\/p>\n

\\begin{equation}\\label{eq:3}
\n\\begin{aligned}
\n\\mu_r&=\\sqrt{\\mu_X^2+\\mu_Y^2}
\n\\\\
\n\\mu_\\theta&=\\left\\{\\begin{matrix}
\n\\arctan\\frac{\\mu_Y}{\\mu_X}, & x>0\\\\
\n\\arctan\\frac{\\mu_Y}{\\mu_X}+\\pi, &x<0
\n\\end{matrix}\\right.
\n\\end{aligned}\\end{equation}<\/p>\n

\u5219\u6781\u5f84$r$\u548c\u6781\u89d2$\\theta$\u7684\u8054\u5408\u5206\u5e03\u4e3a<\/p>\n

\\begin{equation}\\label{eq:4}
\np_{R\\Theta}\\left(r,\\theta\\right)=\\frac{r}{2\\pi\\sigma^2}\\exp\\left[-\\frac{r^2+\\mu_r^2-2r\\mu_r\\cos\\left(\\theta-\\mu_\\theta\\right)}{2\\sigma^2}\\right]
\n\\end{equation}<\/p>\n

\u7531\u4e8e\u65e0\u89c4\u884c\u8d70\u7684\u6781\u5f84\u548c\u6781\u89d2\u662f\u76f8\u4e92\u72ec\u7acb\u7684\u968f\u673a\u91cf\uff0c\u6240\u4ee5\u6c42\u4e24\u4e2a\u8fb9\u7f18\u5206\u5e03\u5c31\u53ef\u4ee5\u4e86\u3002<\/p>\n

\\begin{equation}
\n\\begin{aligned}
\np_R\\left(r\\right)&=\\int_0^{2\\pi}p_{R\\Theta}d\\theta\\\\
\np_\\Theta\\left(\\theta\\right)&=\\int_0^\\infty p_{R\\Theta}dr
\n\\end{aligned}
\n\\end{equation}<\/p>\n

\u8fd9\u4e24\u4e2a\u79ef\u5206\u90fd\u65e0\u6cd5\u7528\u521d\u7b49\u51fd\u6570\u8868\u793a\u3002\u6211\u662f\u7528Mathematica\u6765\u7b97\u7684\uff0c\u7ed3\u679c\u5982\u4e0b\uff1a<\/p>\n

\\begin{equation}\\label{eq:6}
\np_R\\left(r\\right)=\\frac{r}{\\sigma^2}I_0\\left(\\frac{r\\mu_r}{\\sigma^2}\\right)\\exp\\left(-\\frac{r^2+\\mu_r^2}{2\\sigma^2}\\right)
\n\\end{equation}<\/p>\n

\\begin{equation}\\label{eq:7}
\n\\begin{aligned}
\np_\\Theta\\left(\\theta\\right)=&\\frac{1}{\\sqrt{8\\pi\\sigma^2}}\\mu_r\\cos\\left(\\theta-\\mu_\\theta\\right)\\exp\\left[-\\frac{\\mu_r^2\\sin^2\\left(\\theta-\\mu_\\theta\\right)}{2\\sigma^2}\\right]\\times \\\\
\n&\\left[\\mathrm{erf}}\\left(\\frac{\\mu_r\\cos\\left(\\theta-\\mu_\\theta\\right)}{\\sqrt{2\\sigma^2}}\\right)+1\\right]+\\frac{1}{2\\pi}\\exp\\left(-\\frac{\\mu_r^2}{2\\sigma^2}\\right)
\n\\end{aligned}
\n\\end{equation}<\/p>\n

\u5176\u4e2d\uff0c$I_0\\left(x\\right)$\u662f\u96f6\u9636\u7b2c\u4e00\u7c7bBessel\u51fd\u6570<\/p>\n

\\begin{equation}
\nI_0\\left(x\\right)=\\frac{1}{\\pi}\\int_0^\\pi\\exp\\left(x\\cos\\theta\\right)d\\theta
\n\\end{equation}<\/p>\n

$\\mathrm{erf}\\left(x\\right)$\u662f\u8bef\u5dee\u51fd\u6570<\/p>\n

\\begin{equation}
\n\\mathrm{erf}\\left(x\\right)=\\frac{2}{\\sqrt{\\pi}}\\int_0^x e^{-t^2}dt
\n\\end{equation}<\/p>\n

\u5f0f(\\ref{eq:6})\u5176\u5b9e\u5c31\u662fRice\u5206\u5e03<\/a>\uff0c\u5f53$\\mu_r=0$\u65f6\u9000\u5316\u4e3aRayleigh\u5206\u5e03\uff08\u5f0f(\\ref{eq:1})\uff09\u3002\u4e0b\u56fe\u662f\u56fa\u5b9a$\\sigma^2=1$\uff0c$\\mu_r=0,1,2$\u7684\u5206\u5e03\u66f2\u7ebf\uff1a<\/p>\n

\"Rice<\/a>

Rice distributions of varying expectations<\/p><\/div>\n

\u4e0b\u56fe\u662f\u56fa\u5b9a$\\mu_r=1$\uff0c$\\sigma^2=1,2,3$\u7684\u5206\u5e03\u66f2\u7ebf\uff1a<\/p>\n

\"Rice<\/a>

Rice distributions of varying variances<\/p><\/div>\n

\u4f5c\u4e3a\u6781\u5f84\u7684\u5206\u5e03\uff0cRice\u5206\u5e03\u4e0e\u6781\u89d2\u7684\u671f\u671b$\\mu_\\theta$\u65e0\u5173\u3002\u800c\u6781\u89d2\u7684\u5206\u5e03\uff08\u5f0f(\\ref{eq:7})\uff09\u4e0e$\\mu_r$\u548c$\\mu_\\theta$\u90fd\u6709\u5173\u3002\u8fd9\u4e2a\u5206\u5e03\u662f\u5426\u5df2\u6709\u4eba\u540d\uff0c\u6211\u6ca1\u6709\u627e\u5230\u3002\u6211\u53ea\u627e\u5230\u4e00\u7bc7\u6587\u7ae0\u8ba8\u8bba\u66f4\u52a0\u4e00\u822c\u7684\u60c5\u51b5\uff0c\u5373\u4e24\u7ec4\u9ad8\u65af\u91cfX\u548cY\u671f\u671b\u503c$\\mu_x\\neq\\mu_y\\neq0$\uff0c\u4e14\u5b83\u4eec\u7684\u65b9\u5dee\u4e5f\u5404\u4e0d\u76f8\u7b49$\\sigma_x\\neq\\sigma_y$\uff1b\u800c\u4e14X\u5750\u6807\u548cY\u5750\u6807\u8fd8\u662f\u76f8\u5173\u7684\uff0c\u5b83\u4eec\u7684\u534f\u65b9\u5dee\u7cfb\u6570$\\rho\\neq0$\u7684\u60c5\u51b5[1]<\/a><\/span>\u3002\u6211\u662f\u7b97\u51fa\u5f0f(\\ref{eq:7})\u4e4b\u540e\u624d\u627e\u5230\u8fd9\u7bc7\u8bba\u6587\u7684\uff0c\u552f\u4e00\u5b89\u6170\u5c31\u662f\u4e00\u5bf9\u7167\u53d1\u73b0\u6211\u597d\u6b79\u7b97\u5bf9\u4e86\u3002<\/p>\n

\u5f0f(\\ref{eq:7})\u5728$\\mu_r=0$\u65f6\u9000\u5316\u4e3a\u5747\u4e00\u5206\u5e03$\\frac{1}{2\\pi}$\uff08\u5f0f(\\ref{eq:2})\uff09\uff0c\u8fd9\u662f\u201c\u5728\u539f\u70b9\u5904\u201d\u7684\u65e0\u89c4\u884c\u8d70\u7684\u7ed3\u679c\u3002\u53ef\u4ee5\u60f3\u8c61\uff0c\u4e00\u4e2a\u8fdc\u79bb\u539f\u70b9\u5904\u7684\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\uff0c\u5176\u8f68\u8ff9\u76f8\u5bf9\u539f\u70b9\u7684\u6781\u89d2\u6da8\u843d\u57fa\u672c\u4f1a\u5206\u5e03\u5728\u4e00\u4e2a\u5f88\u7a84\u7684\u8303\u56f4\u4e4b\u5185\uff0c\u800c\u4e14\u8fd9\u4e2a\u8303\u56f4\u4e3b\u8981\u4f9d\u8d56\u8fd9\u4e2a\u65e0\u89c4\u884c\u8d70\u7684\u201c\u4f4d\u7f6e\u201d\u3002\u4e0b\u56fe\u662f\u56fa\u5b9a$\\mu_r=1$\u548c$\\sigma^2=1$\uff0c$\\mu_\\theta=0,\\frac{\\pi}{6},\\frac{\\pi}{4}$\u7684\u7ed3\u679c\uff1a<\/p>\n

\"Polar<\/a>

Polar angle distributions of random walk: varying angular expectations<\/p><\/div>\n

\u518d\u4ed4\u7ec6\u60f3\uff0c\u4e00\u4e2a\u7ed9\u5b9a\u65b9\u5dee\u7684\u65e0\u89c4\u884c\u8d70\uff0c\u5982\u679c\u79bb\u539f\u70b9\u8fd1\u4e00\u4e9b\uff0c\u5176\u8f68\u8ff9\u7684\u6781\u89d2\u6da8\u843d\u4f1a\u5bbd\u4e00\u4e9b\u3002\u4e0b\u56fe\u662f\u56fa\u5b9a$\\mu_\\theta=\\frac{\\pi}{4}$\u548c$\\sigma^2=1$\uff0c$\\mu_\\theta=0,1,2$\u7684\u7ed3\u679c\uff1a<\/p>\n

\"Polar<\/a>

Polar angle distributions of random walk: varying radial expectations<\/p><\/div>\n

\u6700\u540e\uff0c\u770b\u4e00\u4e0b\u56fa\u5b9a$\\mu_r=1$\u548c$\\mu_\\theta=\\frac{\\pi}{4}$\uff0c$\\sigma^2=1,4,9$\u7684\u7ed3\u679c\uff1a<\/p>\n

\"Polar<\/a>

Polar angle distributions of random walk: varying variances<\/p><\/div>\n

References<\/h2>\n
    \n
  1. <\/a>\nP. Dharmawansa, N. Rajatheva, and C. Tellambura, \"Envelope and phase distribution of two correlated gaussian variables\", IEEE Transactions on Communications<\/i>, vol. 57, pp. 915-921, 2009. http:\/\/dx.doi.org\/10.1109\/TCOMM.2009.04.070065<\/a>\n\n\n<\/li>\n<\/ol>\n\n<\/div> ","protected":false},"excerpt":{"rendered":"

    [latexpage] \u8fd9\u51e0\u5929\u6211\u5728\u674e\u4fca\u6770\u3001Mathematica\u7684\u5e2e\u52a9\u4e0b\u63a8\u5bfc\u4e86\u6781\u5750\u6807\u4e0b\u7684\u4e8c\u7ef4\u5e03\u6717\u8fd0\u52a8\u7684\u7edf\u8ba1\u3002\u4e00\u5f00\u59cb\u6211\u5c31\u89c9\u5f97\uff0c\u8fd9\u662f\u5f88\u57fa\u672c\u7684\u95ee\u9898\uff0c\u5e94\u8be5\u4f1a\u6709\u76f8\u5e94\u7684\u4f8b\u9898\u6216\u8bfe\u4ef6\u76f4\u63a5\u7ed9\u51fa\u7b54\u6848\u3002\u4f46\u662f\u4e00\u5f00\u59cb\u641c\u90fd\u641c\u4e0d\u5230\u3002\u6211\u7528\u4f53\u80b2\u8001\u5e08\u6559\u7684\u6570\u5b66\u63a8\u4e86\u534a\u5929\uff0c\u51fa\u7ed3\u679c\u4e4b\u540e\uff0c\u5c31\u7acb\u9a6c\u641c\u5230\u8d44\u6599\u4e86\u3002\u53ea\u5f53\u4f5c\u4e0a\u5929\u903c\u6211\u7ec3\u4e60\u4e00\u4e0b\u5927\u5b66\u7684\u6570\u5b66\u5427\u3002 \u6211\u8981\u8003\u8651\u7684\u95ee\u9898\u662f\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u7684\u6781\u5750\u6807\u7684\u7edf\u8ba1\uff0c\u5373\u6781\u5f84\u548c\u6781\u89d2\u7684\u5206\u5e03\u3002\u4e4b\u524d\u6211\u603b\u7ed3\u8fc7\u4e86\u4e8c\u7ef4\u548c\u4e09\u7ef4\u65e0\u89c4\u884c\u8d70\u7684\u6781\u5f84\u5206\u5e03\u5206\u522b\u662fRayleigh\u5206\u5e03\u548cMaxwell\u5206\u5e03\uff0c\u8fd9\u90fd\u662f\u76f4\u89d2\u5750\u6807\u7684\u671f\u671b\u503c\u4e3a\u96f6\u7684\u60c5\u51b5\u3002\u672c\u6587\u4e00\u662f\u8981\u8003\u8651\u671f\u671b\u503c\u4e0d\u4e3a\u96f6\u7684\u60c5\u51b5\uff0c\u4e8c\u662f\u4e0d\u5149\u8003\u8651\u6781\u5f84\u7684\u5206\u5e03\uff0c\u8fd8\u8981\u8003\u8651\u6781\u89d2\u7684\u3002 \u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u7684\u201c\u8f68\u8ff9\u4e2d\u5fc3\u201d \u4e00\u4e2a\u8d70\u4e86N\u6b65\u7684\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u8f68\u8ff9\u5e94\u8be5\u662f\u6709\u5176\u4e2d\u5fc3\u7684\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u5b9a\u4e49\u8fd9\u4e2a\u4e2d\u5fc3\u4e3a\u65e0\u89c4\u884c\u8d70\u7684x\u5750\u6807\u548cy\u5750\u6807\u7684\u671f\u671b\u503c$\\mu_X$\u548c$\\mu_Y$\u3002\u8fd9\u6837\u7684\u5b9a\u4e49\u4e25\u683c\u6765\u8bf4\u5e76\u4e0d\u5b9e\u7528\uff0c\u56e0\u4e3a\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u6709\u9650\u6b65\u6570\u7684\u8f68\u8ff9\uff0c\u6211\u4eec\u65e0\u6cd5\u77e5\u9053\u671f\u671b\u503c\uff0c\u53ea\u80fd\u8ba1\u7b97\u5747\u503c\u3002\u540e\u8005\u53ea\u6709$N\\rightarrow\\infty$\u65f6\u624d\u662f\u671f\u671b\u503c\u3002\u4e3a\u4e86\u5b9e\u9a8c\u65b9\u4fbf\uff0c\u6211\u5b9a\u4e49\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u7684\u8f68\u8ff9\u201c\u4e2d\u5fc3\u201d\u662f\u5176\u8d77\u59cb\u5750\u6807$\\left(x_0,y_0\\right)$\u3002\u8fd9\u6837\uff0c\u4e0d\u4ec5\u5bf9\u4efb\u610f\u4e00\u4e2a\u7ed9\u5b9a\u7684\u8f68\u8ff9\u90fd\u80fd\u76f4\u63a5\u8bf4\u51fa\u5176\u201c\u4e2d\u5fc3\u201d\uff0c\u8fd8\u80fd\u4e3b\u52a8\u5b9e\u73b0\u4e00\u4e2a\u7ed9\u5b9a\u4e2d\u5fc3\u7684\u201c\u65e0\u89c4\u884c\u8d70\u201d\u2014\u2014\u53ea\u8981\u4ece\u4f60\u7684\u884c\u8d70\u662f\u4ece\u90a3\u4e2a\u4e2d\u5fc3\u5f00\u59cb\u7684\u5c31\u884c\u3002\u51ed\u76f4\u89c2\u60f3\u8c61\uff0c\u4e00\u4e2a\u4ece\u70b9$\\left(x_0,y_0\\right)$\u51fa\u53d1\u7684\u65e0\u89c4\u884c\u8d70\uff0c\u53ea\u8981\u6b65\u6570$N$\u8db3\u591f\u5927\uff0c\u5c31\u6709$\\mu_X=x_0$\u548c$\\mu_Y=y_0$\u3002\u5f53\u7136\uff0c\u8fd9\u662f\u4e3a\u4e86\u5b9e\u9a8c\u7684\u601d\u8003\u3002\u63a5\u4e0b\u6765\u7684\u63a8\u5bfc\u90fd\u662f\u8c08\u671f\u671b\u503c \u671f\u671b\u503c\u4e3a\u96f6\u7684\u60c5\u51b5 \u9996\u5148\uff0c\u8003\u8651\u4e00\u4e2a\u201c\u5728\u539f\u70b9\u5904\u201d\u7684\u65e0\u89c4\u884c\u8d70\uff0c\u5373\u5176x\u5750\u6807\u548cy\u5750\u6807\u7684\u671f\u671b\u503c$\\mu_X=\\mu_Y=0$\uff0c\u65b9\u5dee\u90fd\u662f$\\sigma^2$\uff08\u5404\u5411\u540c\u6027\uff09\u3002\u8fd9\u65f6\u5176\u6781\u5750\u6807\u7684\u6781\u5f84\u548c\u6781\u89d2$r$\u548c$\\theta$\u7684\u5206\u5e03\u5206\u522b\u4e3a\u4e00\u4e2aRayleigh\u5206\u5e03\u548c\u4e00\u4e2a\u5747\u5300\u5206\u5e03\uff1a \\begin{equation}\\label{eq:1} p_R\\left(r\\right)=\\frac{r}{\\sigma^2}\\exp\\left[-\\frac{r^2}{2\\sigma^2}\\right] \\end{equation} \\begin{equation}\\label{eq:2} p_\\Theta\\left(\\theta\\right)=\\frac{1}{2\\pi} \\end{equation} \u671f\u671b\u503c\u4e0d\u4e3a\u96f6\u7684\u60c5\u51b5 \u8003\u8651\u201c\u4f4d\u4e8e\u6781\u5750\u6807\u4e2d\u70b9$\\left(\\mu_r,\\mu_\\theta\\right)$\u5904\u7684\u65e0\u89c4\u884c\u8d70\uff0c\u5373\u5176\u76f4\u89d2\u5750\u6807x\u548cy\u7684\u671f\u671b\u503c\u4e3a$\\mu_X\u548c\\mu_Y$\uff0c\u65b9\u5dee\u90fd\u662f$\\sigma^2$\uff08\u5404\u5411\u540c\u6027\uff09\u3002\u4e8e\u662f \\begin{equation}\\label{eq:3} \\begin{aligned} \\mu_r&=\\sqrt{\\mu_X^2+\\mu_Y^2} \\\\ \\mu_\\theta&=\\left\\{\\begin{matrix} \\arctan\\frac{\\mu_Y}{\\mu_X}, & x>0\\\\ \\arctan\\frac{\\mu_Y}{\\mu_X}+\\pi, &x<0 \\end{matrix}\\right. \\end{aligned}\\end{equation} \u5219\u6781\u5f84$r$\u548c\u6781\u89d2$\\theta$\u7684\u8054\u5408\u5206\u5e03\u4e3a \\begin{equation}\\label{eq:4} p_{R\\Theta}\\left(r,\\theta\\right)=\\frac{r}{2\\pi\\sigma^2}\\exp\\left[-\\frac{r^2+\\mu_r^2-2r\\mu_r\\cos\\left(\\theta-\\mu_\\theta\\right)}{2\\sigma^2}\\right] \\end{equation} \u7531\u4e8e\u65e0\u89c4\u884c\u8d70\u7684\u6781\u5f84\u548c\u6781\u89d2\u662f\u76f8\u4e92\u72ec\u7acb\u7684\u968f\u673a\u91cf\uff0c\u6240\u4ee5\u6c42\u4e24\u4e2a\u8fb9\u7f18\u5206\u5e03\u5c31\u53ef\u4ee5\u4e86\u3002 \\begin{equation} \\begin{aligned} p_R\\left(r\\right)&=\\int_0^{2\\pi}p_{R\\Theta}d\\theta\\\\ p_\\Theta\\left(\\theta\\right)&=\\int_0^\\infty p_{R\\Theta}dr \\end{aligned} \\end{equation} \u8fd9\u4e24\u4e2a\u79ef\u5206\u90fd\u65e0\u6cd5\u7528\u521d\u7b49\u51fd\u6570\u8868\u793a\u3002\u6211\u662f\u7528Mathematica\u6765\u7b97\u7684\uff0c\u7ed3\u679c\u5982\u4e0b\uff1a \\begin{equation}\\label{eq:6} p_R\\left(r\\right)=\\frac{r}{\\sigma^2}I_0\\left(\\frac{r\\mu_r}{\\sigma^2}\\right)\\exp\\left(-\\frac{r^2+\\mu_r^2}{2\\sigma^2}\\right) \\end{equation} \\begin{equation}\\label{eq:7} \\begin{aligned} p_\\Theta\\left(\\theta\\right)=&\\frac{1}{\\sqrt{8\\pi\\sigma^2}}\\mu_r\\cos\\left(\\theta-\\mu_\\theta\\right)\\exp\\left[-\\frac{\\mu_r^2\\sin^2\\left(\\theta-\\mu_\\theta\\right)}{2\\sigma^2}\\right]\\times \\\\ &\\left[\\mathrm{erf}}\\left(\\frac{\\mu_r\\cos\\left(\\theta-\\mu_\\theta\\right)}{\\sqrt{2\\sigma^2}}\\right)+1\\right]+\\frac{1}{2\\pi}\\exp\\left(-\\frac{\\mu_r^2}{2\\sigma^2}\\right) \\end{aligned} \\end{equation} \u5176\u4e2d\uff0c$I_0\\left(x\\right)$\u662f\u96f6\u9636\u7b2c\u4e00\u7c7bBessel\u51fd\u6570 \\begin{equation} I_0\\left(x\\right)=\\frac{1}{\\pi}\\int_0^\\pi\\exp\\left(x\\cos\\theta\\right)d\\theta \\end{equation} $\\mathrm{erf}\\left(x\\right)$\u662f\u8bef\u5dee\u51fd\u6570 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[212,303],"tags":[300,287],"class_list":["post-7025","post","type-post","status-publish","format-standard","hentry","category-tagged","category-rheo","tag-brownian-motion","tag-particle-tracking"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p1aPvF-1Pj","jetpack-related-posts":[{"id":7088,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/7088","url_meta":{"origin":7025,"position":0},"title":"Rice\u5206\u5e03\u7684\u6570\u503c\u8ba1\u7b97","author":"Andrew","date":"2016\u5e747\u67088\u65e5","format":false,"excerpt":"[latexpage] \u6211\u5728\u6781\u5750\u6807\u4e0b\u7684\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u4e2d\u8ba8\u8bba\u5230Rice\u5206\u5e03\uff1a \\begin{equation}\\label{eq:1} p_R\\left(r\\right)=\\frac{r}{\\sigma^2}I_0\\left(\\frac{r\\mu_r}{\\sigma^2}\\right)\\exp\\left(-\\frac{r^2+\\mu_r^2}{2\\sigma^2}\\right) \\end{equation} \u5f53\u65e0\u89c4\u884c\u8d70\u53d1\u751f\u7684\u201c\u4f4d\u7f6e\u201d\uff08$\\mu_r$\uff09\u79bb\u539f\u70b9\u5f88\u8fdc\u65f6\uff0c$\\mu_r$\u503c\u5f88\u5927\uff0c\u800c\u5206\u5e03\u7684\u6781\u503c\u4e5f\u5927\u81f3\u5904\u4e8e$r=\\mu_r$\u5904\uff0c\u56e0\u6b64\uff0c\u80fd\u770b\u5230\u5206\u5e03\u4e3b\u8981\u7279\u5f81\u7684$r$\u8303\u56f4\u4e5f\u5728$\\mu_r$\u9644\u8fd1\u3002\u8fd9\u65f6\uff0c\u5f0f(\\ref{eq:1})\u4e2d\u7684\u7b2c\u4e00\u7c7b\u4fee\u6b63\u8d1d\u585e\u5c14\u51fd\u6570$I_0\\left(\\frac{r\\mu_r}{\\sigma^2}\\right)$\u7684\u503c\u4f1a\u5f88\u5927\uff0c\u800c\u81ea\u7136\u6307\u6570\u9879$\\exp\\left(-\\frac{r^2+\\mu_r^2}{2\\sigma^2}\\right)$\u4f1a\u5f88\u5c0f\uff0c\u4f46\u6574\u4e2a\u5f0f(\\ref{eq:1})\u7684\u503c\u4ecd\u7136\u662f\u9002\u4e2d\u7684\u3002\u5728MATLAB\u8f93\u5165\u4e0a\u5f0f\u8ba1\u7b97\u65f6\uff0c\u4f1a\u56e0\u4e3a\u8ba1\u7b97\u8fc7\u7a0b\u4e2d\u6d89\u53ca\u5230\u5f88\u5927\u7684\u548c\u5f88\u5c0f\u7684\u503c\uff0c\u6240\u4ee5\u5c3d\u7ba1\u6700\u7ec8\u7ed3\u679c\u7684\u503c\u662f\u9002\u4e2d\u7684\uff0c\u8ba1\u7b97\u4e5f\u4f1a\u6ea2\u51fa\u3002\u8fd9\u65f6\u53ef\u4ee5\u5229\u7528\u7b2c\u4e00\u7c7b\u4fee\u6b63\u8d1d\u585e\u5c14\u51fd\u6570\u7684\u6e10\u8fd1\u5c55\u5f00\u3002\u5f53$\\alpha$\u4e3a\u5b9a\u503c\u3001$\\left|z\\right|$\u5f88\u5927\u4e14$\\left|\\mathrm{arg}z\\right|","rel":"","context":"In "\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0"","block_context":{"text":"\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/tagged"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":43,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/43","url_meta":{"origin":7025,"position":1},"title":"\u6211\u7684\u535a\u58eb\u8bfe\u9898\u7684\u79d1\u666e\u4ecb\u7ecd","author":"Andrew","date":"2008\u5e7410\u670825\u65e5","format":false,"excerpt":"\u6211\u7684\u535a\u58eb\u8bfe\u9898\u7684\u79d1\u666e\u4ecb\u7ecd \u6211\u5728\u8fd9\u91cc\u4ecb\u7ecd\u4e00\u4e0b\u6211\u7684\u535a\u58eb\u8bfe\u9898\u7684\u80cc\u666f\u3002 \u81ea\u76f8\u4f3c\u2014\u2014\u5927\u81ea\u7136\u7684\u827a\u672f Why is geometry often described as 'cold' and 'dry'? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor\u2026","rel":"","context":"In "\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0"","block_context":{"text":"\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/tagged"},"img":{"alt_text":"","src":"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_695fz26xxdr_b","width":350,"height":200},"classes":[]},{"id":544,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/544","url_meta":{"origin":7025,"position":2},"title":"\u4e3a\u4ea7\u751f\u5047\u8bf4\u79ef\u7d2f\u4e8b\u5b9e\u57fa\u7840\u7684\u7814\u7a76","author":"Andrew","date":"2009\u5e7410\u67085\u65e5","format":false,"excerpt":"\u6700\u65b0\u4e00\u671f\u7684Science\u88ab\u65b0\u4eba\u7c7b\u7956\u5148\u62a2\u53bb\u4e86\u98ce\u5934\uff0c\u6211\u5bf9\u8003\u53e4\u5b66\u4e00\u7a8d\u4e0d\u901a\uff0c\u6240\u6709\u5173\u4e8e\u8fd9\u4e2a\u8bdd\u9898\u7684\u6587\u7ae0\u6211\u90fdskip\u6389\u4e86\u3002\u5012\u662f\u4e00\u7bc7Perspecitives\u91cc\u7684\u6587\u7ae0\u5f88\u662f\u7ed9\u4e86\u6211\u4e00\u4e9b\u542f\u53d1\uff1aNabel, G. (2009). The Coordinates of Truth Science, 326 (5949), 53-54 DOI: 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of Polymeric Liquids\u7b2c1\u5377\uff0c\u800c\u4e14\u4e5f\u662f\u5f53\u573a\u7406\u89e3\uff0c\u4e8b\u540e\u5fd8\u8bb0\uff0c\u53cd\u6b63\u201c\u66fe\u7ecf\u7406\u89e3\u8fc7\u201d\u3002\u8bf4\u767d\u4e86\uff0c\u770b\u4e66\u53ea\u662f\u53bb\u786e\u8ba4\u4e66\u4e0a\u54ea\u4e9b\u5185\u5bb9\u662f\u6211\u4ec0\u4e48\u65f6\u5019\u56de\u5934\u770b\u90fd\u4e0d\u81f3\u4e8e\u770b\u4e0d\u61c2\u7684\u3002\u7136\u540e\u5c31\u5fd8\u6389\u7ec6\u8282\uff0c\u8ba9\u65e5\u540e\u7684\u6211\u56de\u5934\u770b\u5427\u2014\u2014\u53cd\u6b63\u5230\u65f6\u6211\u4f1a\u770b\u5f97\u61c2\u7684\u3002\u5c11\u6570\uff08\u6216\u591a\u6570\uff0c\u4f46\u6211\u4ecd\u7136\u9009\u4e00\u4e9b\u91cd\u8981\u7684\u5c11\u6570\uff09\u4e66\u4e0a\u6211\u5f53\u573a\u770b\u4e0d\u61c2\u7684\u5730\u65b9\uff0cGoogle\u4e00\u4e0b\u8bd5\u8bd5\uff0c\u4e5f\u4e0d\u591a\u82b1\u4ec0\u4e48\u65f6\u95f4\uff0c\u4e0d\u884c\u5c31\u6253\u4e2a\u8bb0\u53f7\uff0c\u5bc4\u5e0c\u671b\u4e8e\u6211\u8fd9\u8f88\u5b50\u4e0d\u4f1a\u5361\u5728\u8fd9\u4ef6\u4e8b\u4e0a\u3002\u771f\u5361\u5728\u8fd9\u4ef6\u4e8b\u4e0a\u7684\u65f6\u5019\uff0c\u6211\u4e5f\u4f1a\u77e5\u9053\u8fd9\u4e2a\u6211\u4ee5\u524dGoogle\u8fc7\u884c\u4e0d\u901a\uff0c\u53ef\u4ee5\u7acb\u5373\u53bb\u627e\u4eba\u95ee\uff0c\u4e0d\u4f1a\u82b1\u91cd\u590d\u7684\u65f6\u95f4\u53c8\u53bbGoogle\u904d\u3002\u6240\u4ee5\uff0c\u5f88\u591a\u6211\u8ba4\u4e3a\u81ea\u5df1\u61c2\u7684\u77e5\u8bc6\uff0c\u5176\u5b9e\u53ea\u662f\u201c\u6211\u77e5\u9053\u54ea\u672c\u4e66\u4e0a\u6709\uff0c\u6211\u53bb\u770b\u7684\u8bdd\u4f1a\u770b\u61c2\uff0c\u4f46\u73b0\u5728\u6211\u4e0d\u77e5\u9053\u201d\u3002\u5982\u679c\u6709\u4eba\u62ff\u4e00\u672c\u4e66\u4e0a\u7684\u4efb\u4f55\u7ec6\u8282\u6765\u95ee\u6211\uff0c\u6211\u4f1a\u5f88\u50cf\u5b8c\u5168\u6ca1\u5b66\u8fc7\u8fd9\u672c\u4e66\u3002\u4f8b\u5982\uff0c\u4f5c\u4e3arheologist\uff0c\u6211\u5f88\u50cf\u5b8c\u5168\u6ca1\u770b\u8fc7Bird\u7684\u4e66\u3002\u8fd9\u5f88\u4e22\u4eba\uff0c\u5176\u5b9e\u4e0d\u662f\u8fd9\u6837\uff0c\u6240\u4ee5\u5728\u4ea4\u6d41\u4e0a\u5f88\u201c\u5403\u4e8f\u201d\uff0c\u53ea\u6c42\u80fd\u53d1paper\uff0c\u80fd\u62ff\u7ecf\u8d39\u5427\u30022. Hansen\u7684\u4e66\u80f6\u4f53\u60ac\u6d6e\u6db2\u4f53\u7cfb\u7684\u7406\u8bba\u5f80\u5f80\u5ef6\u7528\u7b80\u5355\u6db2\u4f53\u7406\u8bba\u3002Hansen and McDonald\u7684Theory of Simple Liquids\u662f\u80f6\u4f53\u7269\u7406\u8bba\u6587\u7ecf\u5e38\u88ab\u5f15\u7528\u7684\u7ecf\u5178\u30022013\u5e74\u51fa\u7248\u7684\u7b2c4\u7248\u8fd8\u4e3a\u8fd1\u5e74\u6765\u201c\u8f6f\u7269\u8d28\u7269\u7406\u201d\u7684\u5174\u8d77\u589e\u52a0\u4e86\u4e00\u4e9b\u76f8\u5173\u7684\u5185\u5bb9\uff0c\u4f8b\u5982\u6709\u5173\u6db2\u4f53\u8868\u3001\u754c\u9762\u5904\u7684\u7ed3\u6784\u4e0e\u52a8\u6001\uff0c\u79bb\u5b50\u6db2\u4f53\u7684\u7406\u8bba\u7b49\u3002\u6700\u540e\u8fd8\u7279\u522b\u52a0\u4e86\u4e00\u7ae0\u8f6f\u7269\u8d28\u5e94\u7528\uff0c\u4ecb\u7ecd\u4e86\u805a\u5408\u7269\u548c\u80f6\u4f53\u4f53\u7cfb\u7684\u57fa\u672c\u7406\u8bba\u3002\u6211\u662f\u6700\u8fd1\u624d\u770b\u5230\u7b2c4\u7248\u7684\uff0c\u4e4b\u524d\u6211\u7ffb\u9605\u7684\u90fd\u662f\u7b2c3\u7248\uff082006\uff09\u3002\u4ee5\u4e0b\u6211\u5b66\u4e60van Hove\u51fd\u6570\u7684\u8bb0\u5f55\u4e2d\uff0c\u5c31\u987a\u4fbf\u5bf9\u8fd9\u4e24\u7248\u4f5c\u4e86\u4e00\u4e0b\u5bf9\u6bd4\uff0c\u53d1\u73b0\u7b2c4\u7248\u5728\u8fd9\u4e00\u90e8\u5206\u5176\u5b9e\u589e\u52a0\u4e86\u8fc7\u51b7\u6db2\u4f53\u7684\u5185\u5bb9\uff0c\u9664\u4e86van Hove\u51fd\u6570\u7684\u975e\u9ad8\u65af\u5f62\u5f0f\u3001\u52a8\u6001\u4e0d\u5747\u5300\u6027\u4e4b\u5916\uff0c\u8fd8\u6709\u6df1\u5ea6\u8fc7\u51b7\u6db2\u4f53\u8fdc\u79bb\u5e73\u8861\u6001\u7684\u8001\u5316\u884c\u4e3a\uff0c\u7279\u522b\u662ftime-aging time superpoaition\u548c\u8fdd\u53cdFDT\u7684\u6027\u8d28\u3002\u5bf9\u6bd4\u7b2c3\u548c\u7b2c4\u7248\u7684\u5173\u952e\u8bcd\u7d22\u5f15\u4e5f\u80fd\u5f88\u5feb\u53d1\u73b0D\u6253\u5934\u7684\u8bcd\u4e0b\u9762\u7b2c4\u7248\u6bd4\u7b2c3\u7248\u591a\u4e86\u4e00\u6761Dynamical heterogeneity\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u4e66\u4e2d\u5199\u6210time\/aging-time superposition\uff0c\u8fd9\u4e2a\u5199\u6cd5\u6bd4\u5e38\u89c1\u7684time-aging time superposition\u597d\uff0c\u5b83\u5f88\u6e05\u6670\u5730\u8868\u660e\u4e86time\u662f\u4e00\u4e2a\u91cf\u3001aging time\u662f\u53e6\u4e00\u4e2a\u91cf\uff0c\u662f\u5b83\u4eec\u4e4b\u95f4\u7684\u53e0\u52a0\u6027\u3002\u539f\u6765\u5730\u5199\u6cd5\u5f88\u5bb9\u610f\u8ba9\u4eba\u8ba4\u4e3a\u5f00\u59cb\u7684time-aging\u662f\u4e00\u4e2a\u4e1c\u897f\u3001\u540e\u9762\u7684time\u53c8\u662f\u53e6\u4e00\u4e2a\u4e1c\u897f\u3002\u6211\u6253\u7b97\u4ee5\u540e\u90fd\u7528\u7c7b\u4f3c\u7684\u5199\u6cd5\uff0c\u4f8b\u5982straing-rate\/frequency superposition\u3001time\/temperature superposition\u7b49\u30023. van Hove\u51fd\u65703.1 van Hove\u51fd\u6570\u7684\u5b9a\u4e49\u53ca\u4e09\u7ef4\u5f62\u5f0f\u8bba\u6587\u4e2d\u7ecf\u5e38\u4f1a\u4f7f\u7528van Hove\u51fd\u6570\u3001\u9ad8\u65af\u5206\u5e03\u7b49\u5b57\u773c\uff0c\u4f46\u662f\u4ed4\u7ec6\u7684\u5148\u540e\u5173\u7cfb\u6587\u7ae0\u91cc\u5f88\u5c11\u4f53\u73b0\uff0c\u5149\u770b\u6587\u7ae0\u662f\u7ecf\u4e0d\u8d77\u63a8\u6572\u7684\u3002\u4ee5\u4e0b\u6211\u628a\u4e00\u4e9b\u57fa\u672c\u5173\u7cfb\u8bb0\u5f55\u4e0b\u6765\u3002van Hove\u51fd\u6570\u7684\u5b9a\u4e49\u662f \\begin{equation}\\label{eq:1} \\begin{aligned} G\\left(\\mathbf{r},t\\right)&\\overset{\\mathrm{def}}{=}\\langle\\frac{1}{N}\\sum_{i=1}^N\\sum_{j=1}^N\\int d\\mathbf{r}\\delta\\left[\\mathbf{r}-\\mathbf{r}_j\\left(t\\right)+\\mathbf{r}_i\\left(0\\right)\\right]\\rangle \\\\&=\\frac{1}{\\rho}\\langle\\rho\\left(\\mathbf{r},t\\right)\\rho\\left(\\mathbf{0},0\\right)\\rangle \\\\&=G_\\mathrm{d}+G_\\mathrm{s} \\end{aligned} \\end{equation} \u5176\u4e2d \\begin{equation}\\label{eq:2} G_\\mathrm{s}\\left(\\mathbf{r},t \\right )=\\langle\\frac{1}{N}\\sum_{i=1}^N\\delta\\left[\\mathbf{r}-\\mathbf{r}_i\\left(t \\right )+\\mathbf{r}_i\\left(0 \\right ) \\right]\\rangle \\end{equation} \\begin{equation}\\label{eq:3} G_\\mathrm{d}\\left(\\mathbf{r},t\u2026","rel":"","context":"In "\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0"","block_context":{"text":"\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/tagged"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":9202,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/9202","url_meta":{"origin":7025,"position":4},"title":"\u4f20\u7edf\u6559\u5b66\u5185\u5bb9\u6570\u5b66\u8bed\u8a00\u7684\u8fd1\u4e16\u5316","author":"Andrew","date":"2025\u5e747\u67085\u65e5","format":false,"excerpt":"[latexpage] 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down\u7684\u3002 P. J. Flory\u57281951\u5e74\u7684\u95fb\u540d\u8457\u4f5c Principles of Polymer Chemistry \u4e2d\u663e\u793a\u51fa\u7684\u70ed\u529b\u5b66\u7684\u7edf\u8ba1\u529b\u5b66\u65b9\u6cd5\uff0c\u57fa\u672c\u4e0a\u8fd8\u662f\u8fd9\u4e24\u4e2a\u5b66\u79d11950\u5e74\u4e4b\u524d\u7684\u5f62\u6001\u3002\u5f53\u4ed6\u79bb\u5f00\u5eb7\u5948\u5c14\u5927\u5b66\u4e4b\u540e\uff081960s ~ 70s\uff09\uff0c\u65e0\u8bba\u662f\u70ed\u529b\u5b66\u8fd8\u662f\u7edf\u8ba1\u529b\u5b66\u7684\u5de5\u4f5c\u90fd\u5438\u6536\u4e86\u8fd9\u4e24\u4e2a\u5b66\u79d1\u7684\u8fd1\u4e16\u5316\u6210\u679c\u3002\u4f46\u90fd\u53ea\u4f53\u73b0\u5728\u4ed6\u540e\u6765\u53d1\u8868\u7684\u8bba\u6587\u5f53\u4e2d\u3002\u800c\u518d\u6ca1\u6709\u603b\u7ed3\u6210\u4e00\u672c\u50cf1951\u5e74\u8fd9\u672c\u4e66\u90a3\u6837\u5bf9\u5b66\u79d1\u672c\u79d1\u6559\u5b66\u5f71\u54cd\u6df1\u8fdc\u7684\u65b0\u4e66\u3002\u73b0\u5728\u7684\u9ad8\u5206\u5b50\u7269\u7406\u672c\u79d1\u6559\u6750\uff0c\u57fa\u672c\u6ca1\u6709\u516d\u3001\u4e03\u5341\u5e74\u4ee3\u7684\u4e1c\u897f\uff0c\u5b8c\u5168\u662fFlory1951\u5e74\u8fd9\u672c\u4e66\u7684\u590d\u523b\uff0c\u751a\u81f3\u8fd8\u4e22\u5931\u4e86Flory\u539f\u4e66\u4e2d\u5f88\u591a\u5145\u6ee1\u7269\u7406\u6d1e\u89c1\u7684\u53d9\u8ff0\uff08\u56e0\u4e3a\u8bd1\u8005\u770b\u4e0d\u61c2\uff09\u3002 \u683c\u5b50\u6a21\u578b\u7528\u4e8e\u6db2\u4f53 \u867d\u7136Kirkwood\uff0cOnsager\u7684\u5de5\u4f5c\u53ef\u4ee5\u8ffd\u6eaf\u523030\u5e74\u4ee3\uff0c\u4f46\u8fd9\u4e9b\u5de5\u4f5c\u65e0\u8bba\u662f\u5bf9\u975e\u5e73\u8861\u7684\u70ed\u529b\u5b66\/\u7edf\u8ba1\u529b\u5b66\uff0c\u8fd8\u662f\u5bf9\u6db2\u4f53\u7269\u7406\u7684\u5f71\u54cd\uff0c\u90fd\u8981\u523060\u5e74\u4ee3\u5de6\u53f3\u624d\u88ab\u516c\u8ba4\u3002\u800c\u4e14Flory\u5173\u5fc3\u7684\u95ee\u9898\u4e5f\u4e3b\u8981\u4ece\u5e73\u8861\u6001\u5207\u5165\u3002\u5728\u8fd9\u65b9\u9762\u572850\u5e74\u4ee3\u5df2\u7ecf\u5f62\u6210\u5341\u5206\u6210\u719f\u7684\u7406\u8bba\u5de5\u5177\u3002\u5176\u4e2d\u70ed\u529b\u5b66\u5f53\u65f6\u7684\u8868\u73b0\u5f62\u5f0f\u5c31\u8ddf\u4eca\u5929\u672c\u79d1\u300a\u7269\u7406\u5316\u5b66\u300b\u4e2d\u7684\u5dee\u4e0d\u591a\u3002\u800c\u7edf\u8ba1\u529b\u5b66\u7684\u8868\u73b0\u5f62\u5f0f\u53ef\u4ee5\u7528Fowler & Guggenheim (1949), Statistical Thermodynamics, CUP\u6765\u4ee3\u8868\u3002\u8fd9\u672c\u4e66\u9996\u5370\u662f1939\u5e74\uff0c1949\u5e74\u6709\u4e00\u4e2a\u4fee\u8ba2\u7248\uff0c\u4e4b\u540e\u5c31\u53ea\u6709\u91cd\u5370\uff0c\u6ca1\u6709\u65b0\u7248\u3002\u8fd9\u672c\u4e66\u5728\u5f53\u65f6\u662f\u5341\u5206\u6d41\u884c\u7684\u53c2\u8003\u4e66\u3002\u8be5\u4e66\u7684\u4e00\u4e2a\u7279\u70b9\u662f\u6709\u597d\u51e0\u4e2a\u5173\u6ce8\u4eca\u5929\u6240\u8c13\u7684\u8f6f\u51dd\u805a\u6001\u4f53\u7cfb\u7684\u5e94\u7528\u7ae0\u8282\u2014\u2014\u6db2\u4f53\u7eaf\u7269\u8d28\u548c\u6df7\u5408\u7269\u548c\u754c\u9762\u95ee\u9898\u3002\u81f3\u5c11\uff0c\u6db2\u4f53\u7684\u957f\u7a0b\u65e0\u5e8f\u7279\u70b9\u4ee5\u201c\u5f84\u5411\u5206\u5e03\u51fd\u6570\u201d\u8fd9\u79cd\u56fe\u50cf\u5df2\u88ab\u719f\u77e5\u3002 Flory\u539f\u4e66\u5728\u521a\u4e00\u629b\u51fa\u201c\u683c\u5b50\u6a21\u578b\u201d\u7684\u65f6\u5019\uff0c\u5c31\u63d2\u5165\u4e86\u4e00\u6bb5\u8bc4\u8ff0\uff1a The molecules in the pure liquids and in their solution are considered to be arranged with enough regularity to justify approximate representation by a lattice, as is\u2026","rel":"","context":"In 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