{"id":6981,"date":"2016-03-31T02:49:34","date_gmt":"2016-03-30T18:49:34","guid":{"rendered":"http:\/\/www.andrewsun.net\/panta_rhei\/?p=6981"},"modified":"2020-02-04T23:31:02","modified_gmt":"2020-02-04T15:31:02","slug":"%e9%9d%9e%e9%ab%98%e6%96%af%e7%b3%bb%e6%95%b0%e7%ac%a6%e5%8f%b7%e4%b8%ba%e4%bb%80%e4%b9%88%e6%9c%89%e4%b8%aa%e4%b8%8b%e6%a0%872%ef%bc%9f","status":"publish","type":"post","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/6981","title":{"rendered":"\u975e\u9ad8\u65af\u7cfb\u6570\u7b26\u53f7\u4e3a\u4ec0\u4e48\u6709\u4e2a\u4e0b\u68072\uff1f"},"content":{"rendered":"
\n

[latexpage]<\/p>\n

\u5728\u7528\u7c92\u5b50\u793a\u8e2a\uff08particle tracking)\u65b9\u6cd5\u7814\u7a76\u8f6f\u7269\u8d28\u7684\u8bba\u6587\u4e2d\u7ecf\u5e38\u7528van Hove\u51fd\u6570\u7684\u975e\u9ad8\u65af\u6027\u6765\u8868\u5f81\u4f53\u7cfb\u7684\u52a8\u6001\u4e0d\u5747\u5300\u6027\u3002\u5f80\u5f80\u8fd8\u6709\u4e00\u4e2a\u201c\u975e\u9ad8\u65af\u53c2\u6570\u201d\u6765\u8868\u5f81\u5b9e\u9a8c\u7ed3\u679c\u504f\u79bb\u9ad8\u65af\u5206\u5e03\u7684\u7a0b\u5ea6\uff0c\u5e38\u5e38\u7528$\\alpha_2$\u6765\u8868\u793a\u3002\u5b83\u5176\u5b9e\u662f\u7531\u5cf0\u5ea6\uff08kurtosis\uff09\u5b9a\u4e49\u7684\uff0c\u628a\u5b9e\u9a8c\u5206\u5e03\u7684\u5cf0\u5ea6\u62ff\u9ad8\u65af\u5206\u5e03\u7684\u5cf0\u5ea6\u4e00\u76f8\u9664\uff0c\u518d\u51cf1\uff0c\u4f7f\u5f97\u4e00\u4e2a\u9ad8\u65af\u5206\u5e03\u7684$\\alpha_2$\u7684\u503c\u4e3a0\uff0c\u975e\u96f6\u7684$\\alpha_2$\u5c31\u6807\u5fd7\u7740\u975e\u9ad8\u65af\u7684\u5206\u5e03\u3002<\/p>\n

\\begin{equation}\\label{eq:1}
\n\\alpha_2=\\frac{3\\langle\\Delta x^4\\rangle}{5\\langle\\Delta x^2\\rangle^2}-1
\n\\end{equation}<\/p>\n

\u65e2\u7136\u662f\u8fd9\u6837\u7684\u8bdd\uff0c\u4e3a\u4ec0\u4e48\u4e0d\u76f4\u63a5\u7528\u4e00\u4e2a\u7b80\u5355\u7684\u5b57\u6bcd\u6765\u8868\u793a\uff0c\u800c\u662f\u8981\u7528\u4e00\u4e2a\u5e26\u6709\u6570\u5b57\u4e0b\u68072\u7684\u7b26\u53f7\u5462\uff1f\u96be\u9053\u8fd8\u6709$\\alpha_1$\u3001$\\alpha_3$\uff1f\u6211\u4e00\u76f4\u6709\u8fd9\u4e2a\u7591\u95ee\uff0c\u4eca\u5929\u6211\u8003\u7a76\u4e86\u4e00\u4e0b\u3002\u5176\u5b9e\uff0c\u4f7f\u7528$\\alpha_2$\u6765\u8868\u793a\u8fd9\u4e2a\u91cf\uff0c\u6700\u65e9\u662f\u51fa\u4e8eA. Rahman\u7684\u5de5\u4f5c[1]<\/a><\/span>[2]<\/a><\/span>\u3002\u4f5c\u8005\u63a8\u5bfc\u4e86\u6db2\u4f53\u7684\u65f6\u95f4\u4f9d\u8d56\u76f8\u5173\u51fd\u6570\u7684\u5c55\u5f00\u5f0f\u3002\u7b2c\u4e00\u9879\u5f53\u7136\u662f\u7eaf\u5e03\u6717\u8fd0\u52a8\u7684\u9ad8\u65af\u9879\uff0c\u7b2c\u4e8c\u9879\u5f00\u59cb\u7684\u9ad8\u9636\u9879\u90fd\u662f\u975e\u9ad8\u65af\u9879\uff0c\u5176\u4e2d\u5404\u7684\u7cfb\u6570\u53ef\u4ee5\u7528$\\alpha_n$\u6765\u8868\u793a\uff0c\u5176\u901a\u5f0f\u662f<\/p>\n

\\begin{equation}\\label{eq:2}
\n\\alpha_n=\\frac{\\langle r^{2n}\\rangle}{C_n\\langle r^2\\rangle^n}-1
\n\\end{equation}<\/p>\n

\u5176\u4e2d$C_n=1\\cdot3\\cdot5\\cdots\\left(2n+1\\right)\/3^n$\u3002\u56e0\u6b64$\\alpha_2$\u6070\u597d\u5c31\u662f\u5f0f(\\ref{eq:1})\u3002\u6216\u8005\u8bf4\uff0c\u901a\u8fc7\u5c55\u5f00\u5f97\u5230\u7684\u7b2c\u4e8c\u9879\u7cfb\u6570\uff0c\u6070\u597d\u5c31\u662f\u4e00\u4e2a\u8ddf\u5cf0\u5ea6\u6709\u5173\u7684\u91cf\u3002<\/p>\n

\u4e5f\u5c31\u662f\u8bf4\uff0c$\\alpha_2$\u5176\u5b9e\u662f\u65f6\u95f4\u76f8\u5173\u51fd\u6570\u5c55\u5f00\u5f0f\u7684\u7b2c2\u9879\uff0c\u6240\u4ee5\u4e0b\u6807\u6709\u4e2a2\u3002<\/p>\n

\u81ea\u6587\u732e[2]<\/a><\/span>\u540e\uff0c$\\alpha_2$\u5f88\u5feb\u88ab\u91c7\u7528\u4e3a\u201c\u975e\u9ad8\u65af\u7cfb\u6570\u201d\u7528\u6765\u63cf\u8ff0\u6db2\u4f53\u6027\u8d28\uff0c\u6210\u4e3a\u6db2\u4f53\u7406\u8bba\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u53c2\u6570[3]<\/a><\/span>\u3002\u540e\u6765\u80f6\u4f53\u7269\u7406\u7684\u5174\u8d77\uff0c\u7531\u4e8e\u5176\u7406\u8bba\u57fa\u7840\u57fa\u672c\u4e0a\u662f\u6db2\u4f53\u7269\u7406\uff0c\u6240\u4ee5\u5f88\u81ea\u7136\u5c31\u7528\u4e86\u76f8\u4f3c\u7684\u8ba8\u8bba\u8303\u5f0f\uff0c\u4f7f\u7528$\\alpha_2$\u8fd9\u4e2a\u7b26\u53f7\u3002<\/p>\n

\u518d\u4e00\u6b21\u8bf4\u660e\uff0c\u8981\u60f3\u505a\u597d\u80f6\u4f53\u7269\u7406\u7684\u7814\u7a76\uff0c\u5f88\u5e94\u8be5\u5e94\u8be5\u4ece\u6db2\u4f53\u7269\u7406\u5f00\u59cb\u8865\u4e60\u3002<\/p>\n

\u4e8b\u5b9e\u4e0a\uff0c\u6587\u732e[1]<\/a><\/span>[2]<\/a><\/span>\u662f\u6700\u65e9\u7528\u8ba1\u7b97\u673a\u6765\u8ba1\u7b97\u6db2\u4f53\u7684\u76f8\u5173\u51fd\u6570\u7684\u5de5\u4f5c\u3002\u4f5c\u8005Aneesur Rahman<\/a>\u662f\u201c\u5206\u5b50\u52a8\u529b\u5b66\u4e4b\u7236\u201d\u3002\u5728\u6709\u8ba1\u7b97\u673a\u4e4b\u524d\uff0c\u76f8\u5173\u51fd\u6570\u53ea\u6709\u7406\u60f3\u6c14\u4f53\u548c\u683c\u5b50\u6676\u4f53\u624d\u80fd\u8ba1\u7b97\uff0c\u6db2\u4f53\u7684\u76f8\u5173\u51fd\u6570\u7531\u4e8e\u591a\u4f53\u6548\u5e94\u800c\u96be\u4ee5\u8ba1\u7b97\u3002Aneesur Rahman\u5728\u5f53\u65f6\u7684CDC 3600\u8d85\u7ea7\u8ba1\u7b97\u673a\u4e0a\u8ba1\u7b97\u4e86864\u4e2aLennard-Jones\u4f5c\u7528\u52bf\u7684\u539f\u5b50\uff0c\u6a21\u62df\u6db2\u6c29\u3002<\/p>\n

References<\/h2>\n
    \n
  1. <\/a>\nA. Rahman, \"Correlations in the Motion of Atoms in Liquid Argon\", Physical Review<\/i>, vol. 136, pp. A405-A411, 1964. http:\/\/dx.doi.org\/10.1103\/PhysRev.136.A405<\/a>\n\n\n<\/li>\n
  2. <\/a>\nB. Nijboer, and A. Rahman, \"Time expansion of correlation functions and the theory of slow neutron scattering\", Physica<\/i>, vol. 32, pp. 415-432, 1966. http:\/\/dx.doi.org\/10.1016\/0031-8914(66)90068-1<\/a>\n\n\n<\/li>\n
  3. <\/a>\nT. Tsang, \"Non-Gaussian corrections to incoherent-scattering functions in liquids\", Physical Review A<\/i>, vol. 17, pp. 393-398, 1978. http:\/\/dx.doi.org\/10.1103\/PhysRevA.17.393<\/a>\n\n\n<\/li>\n<\/ol>\n\n<\/div> ","protected":false},"excerpt":{"rendered":"

    [latexpage] \u5728\u7528\u7c92\u5b50\u793a\u8e2a\uff08particle tracking)\u65b9\u6cd5\u7814\u7a76\u8f6f\u7269\u8d28\u7684\u8bba\u6587\u4e2d\u7ecf\u5e38\u7528van Hove\u51fd\u6570\u7684\u975e\u9ad8\u65af\u6027\u6765\u8868\u5f81\u4f53\u7cfb\u7684\u52a8\u6001\u4e0d\u5747\u5300\u6027\u3002\u5f80\u5f80\u8fd8\u6709\u4e00\u4e2a\u201c\u975e\u9ad8\u65af\u53c2\u6570\u201d\u6765\u8868\u5f81\u5b9e\u9a8c\u7ed3\u679c\u504f\u79bb\u9ad8\u65af\u5206\u5e03\u7684\u7a0b\u5ea6\uff0c\u5e38\u5e38\u7528$\\alpha_2$\u6765\u8868\u793a\u3002\u5b83\u5176\u5b9e\u662f\u7531\u5cf0\u5ea6\uff08kurtosis\uff09\u5b9a\u4e49\u7684\uff0c\u628a\u5b9e\u9a8c\u5206\u5e03\u7684\u5cf0\u5ea6\u62ff\u9ad8\u65af\u5206\u5e03\u7684\u5cf0\u5ea6\u4e00\u76f8\u9664\uff0c\u518d\u51cf1\uff0c\u4f7f\u5f97\u4e00\u4e2a\u9ad8\u65af\u5206\u5e03\u7684$\\alpha_2$\u7684\u503c\u4e3a0\uff0c\u975e\u96f6\u7684$\\alpha_2$\u5c31\u6807\u5fd7\u7740\u975e\u9ad8\u65af\u7684\u5206\u5e03\u3002 \\begin{equation}\\label{eq:1} \\alpha_2=\\frac{3\\langle\\Delta x^4\\rangle}{5\\langle\\Delta x^2\\rangle^2}-1 \\end{equation} \u65e2\u7136\u662f\u8fd9\u6837\u7684\u8bdd\uff0c\u4e3a\u4ec0\u4e48\u4e0d\u76f4\u63a5\u7528\u4e00\u4e2a\u7b80\u5355\u7684\u5b57\u6bcd\u6765\u8868\u793a\uff0c\u800c\u662f\u8981\u7528\u4e00\u4e2a\u5e26\u6709\u6570\u5b57\u4e0b\u68072\u7684\u7b26\u53f7\u5462\uff1f\u96be\u9053\u8fd8\u6709$\\alpha_1$\u3001$\\alpha_3$\uff1f\u6211\u4e00\u76f4\u6709\u8fd9\u4e2a\u7591\u95ee\uff0c\u4eca\u5929\u6211\u8003\u7a76\u4e86\u4e00\u4e0b\u3002\u5176\u5b9e\uff0c\u4f7f\u7528$\\alpha_2$\u6765\u8868\u793a\u8fd9\u4e2a\u91cf\uff0c\u6700\u65e9\u662f\u51fa\u4e8eA. Rahman\u7684\u5de5\u4f5c\u3002\u4f5c\u8005\u63a8\u5bfc\u4e86\u6db2\u4f53\u7684\u65f6\u95f4\u4f9d\u8d56\u76f8\u5173\u51fd\u6570\u7684\u5c55\u5f00\u5f0f\u3002\u7b2c\u4e00\u9879\u5f53\u7136\u662f\u7eaf\u5e03\u6717\u8fd0\u52a8\u7684\u9ad8\u65af\u9879\uff0c\u7b2c\u4e8c\u9879\u5f00\u59cb\u7684\u9ad8\u9636\u9879\u90fd\u662f\u975e\u9ad8\u65af\u9879\uff0c\u5176\u4e2d\u5404\u7684\u7cfb\u6570\u53ef\u4ee5\u7528$\\alpha_n$\u6765\u8868\u793a\uff0c\u5176\u901a\u5f0f\u662f \\begin{equation}\\label{eq:2} \\alpha_n=\\frac{\\langle r^{2n}\\rangle}{C_n\\langle r^2\\rangle^n}-1 \\end{equation} \u5176\u4e2d$C_n=1\\cdot3\\cdot5\\cdots\\left(2n+1\\right)\/3^n$\u3002\u56e0\u6b64$\\alpha_2$\u6070\u597d\u5c31\u662f\u5f0f(\\ref{eq:1})\u3002\u6216\u8005\u8bf4\uff0c\u901a\u8fc7\u5c55\u5f00\u5f97\u5230\u7684\u7b2c\u4e8c\u9879\u7cfb\u6570\uff0c\u6070\u597d\u5c31\u662f\u4e00\u4e2a\u8ddf\u5cf0\u5ea6\u6709\u5173\u7684\u91cf\u3002 \u4e5f\u5c31\u662f\u8bf4\uff0c$\\alpha_2$\u5176\u5b9e\u662f\u65f6\u95f4\u76f8\u5173\u51fd\u6570\u5c55\u5f00\u5f0f\u7684\u7b2c2\u9879\uff0c\u6240\u4ee5\u4e0b\u6807\u6709\u4e2a2\u3002 \u81ea\u6587\u732e\u540e\uff0c$\\alpha_2$\u5f88\u5feb\u88ab\u91c7\u7528\u4e3a\u201c\u975e\u9ad8\u65af\u7cfb\u6570\u201d\u7528\u6765\u63cf\u8ff0\u6db2\u4f53\u6027\u8d28\uff0c\u6210\u4e3a\u6db2\u4f53\u7406\u8bba\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u53c2\u6570\u3002\u540e\u6765\u80f6\u4f53\u7269\u7406\u7684\u5174\u8d77\uff0c\u7531\u4e8e\u5176\u7406\u8bba\u57fa\u7840\u57fa\u672c\u4e0a\u662f\u6db2\u4f53\u7269\u7406\uff0c\u6240\u4ee5\u5f88\u81ea\u7136\u5c31\u7528\u4e86\u76f8\u4f3c\u7684\u8ba8\u8bba\u8303\u5f0f\uff0c\u4f7f\u7528$\\alpha_2$\u8fd9\u4e2a\u7b26\u53f7\u3002 \u518d\u4e00\u6b21\u8bf4\u660e\uff0c\u8981\u60f3\u505a\u597d\u80f6\u4f53\u7269\u7406\u7684\u7814\u7a76\uff0c\u5f88\u5e94\u8be5\u5e94\u8be5\u4ece\u6db2\u4f53\u7269\u7406\u5f00\u59cb\u8865\u4e60\u3002 \u4e8b\u5b9e\u4e0a\uff0c\u6587\u732e\u662f\u6700\u65e9\u7528\u8ba1\u7b97\u673a\u6765\u8ba1\u7b97\u6db2\u4f53\u7684\u76f8\u5173\u51fd\u6570\u7684\u5de5\u4f5c\u3002\u4f5c\u8005Aneesur Rahman\u662f\u201c\u5206\u5b50\u52a8\u529b\u5b66\u4e4b\u7236\u201d\u3002\u5728\u6709\u8ba1\u7b97\u673a\u4e4b\u524d\uff0c\u76f8\u5173\u51fd\u6570\u53ea\u6709\u7406\u60f3\u6c14\u4f53\u548c\u683c\u5b50\u6676\u4f53\u624d\u80fd\u8ba1\u7b97\uff0c\u6db2\u4f53\u7684\u76f8\u5173\u51fd\u6570\u7531\u4e8e\u591a\u4f53\u6548\u5e94\u800c\u96be\u4ee5\u8ba1\u7b97\u3002Aneesur Rahman\u5728\u5f53\u65f6\u7684CDC 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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":9108,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/9108","url_meta":{"origin":6981,"position":1},"title":"Leon B. Lucy","author":"Andrew","date":"2024\u5e7412\u670814\u65e5","format":false,"excerpt":"[latexpage] \u80cc\u666f \u5728\u6570\u503c\u8ba1\u7b97\u9886\u57df\u6709\u4e00\u4e2a\u6bd4\u8f83\u77e5\u540d\u7684\u53bb\u5377\u79ef\u8fed\u4ee3\u7b97\u6cd5\u2014\u2014Lucy--Richardson\u7b97\u6cd5\uff0c\u5b83\u88ab\u540e\u4e16\u96c6\u4e2d\u7528\u4e8e\u56fe\u50cf\u53bb\u566a\u3002\u4f8b\u5982\uff0cMATLAB\u7684Image Processing Toolbox\u6709\u4e00\u4e2adeconvlucy\u547d\u4ee4\uff0c\u58f0\u79f0\u5c31\u662f\u7528Lucy--Richardson\u7b97\u6cd5\u5bf9\u7ed9\u5b9a\u56fe\u7247\uff08\u50cf\u7d20\u77e9\u9635\uff09\u4f5c\u7ed9\u5b9a\u70b9\u6269\u6563\u51fd\u6570\u7684\u53bb\u566a\u3002 \u4f46\u662f\uff0cLucy\u7684\u539f\u6587\u6240\u9488\u5bf9\u7684\u95ee\u9898\uff0c\u6bd4\u73b0\u5728\u4e00\u822c\u5e94\u7528\u66f4\u5e7f\u4e49\u3002\u5047\u5b9a$X$\u662f\u4e00\u4e2a\u8fde\u7eed\u53d6\u503c\u968f\u673a\u53d8\u91cf\u3002\u5b83\u7406\u5e94\u6309\u7167\u5206\u5e03\u5bc6\u5ea6\u51fd\u6570$\\phi\\left(x\\right)$\u3002\u6211\u4eec\u60f3\u628a$\\phi\\left(x\\right)$\u89c6\u4e3a\u67d0\u79cd\u7b80\u5355\u5206\u5e03$P\\left(x\\middle|\\xi\\right)$\u6309\u6743\u91cd\u8c31$\\psi\\left(\\xi\\right)$\u7684\u53e0\u52a0\u7ed3\u679c\uff1a \\begin{equation}\\label{eq:original_phi_expression}\\phi\\left(x\\right)=\\int P\\left(x\\middle|\\xi\\right)\\psi\\left(\\xi\\right)\\mathrm{d}\\xi\\end{equation} \u800c\u6211\u4eec\u60f3\u5f97\u77e5\u7ed9\u5b9a\u5f62\u5f0f\u7684\u6838$P\\left(x\\middle|\\xi\\right)$\u6240\u5bf9\u5e94\u7684\u6743\u91cd\u8c31$\\psi\\left(\\xi\\right)$\u3002\u5728\u8fd9\u91cc\uff0c$\\xi$\u662f\u6838\u51fd\u6570$P$\u7684\u53c2\u6570\u3002\u6bd4\u5982\uff0c\u6211\u4eec\u5173\u5fc3\u9ad8\u65af\u6838\u51fd\u6570\u7684\u60c5\u51b5\uff0c\u90a3\u4e48$P\\left(x\\middle|\\xi\\right)$\u53ef\u80fd\u662f\u4ee5$\\xi$\u4e3a\u6807\u51c6\u5dee\u7684\u9ad8\u65af\u51fd\u6570 \\[P\\left(x\\middle|\\xi\\right)=\\frac{1}{\\sqrt{2\\pi\\xi^2}}\\exp\\left[-\\frac{\\left(x-\\mu\\right)^2}{2\\xi^2}\\right]\\] 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\u6709\u90a3\u4e48\u4e00\u4e9b\u8bba\u6587\uff0c\u6211\u662f\u79f0\u4e4b\u4e3a\u201c\u6559\u5b66\u8bba\u6587\u201d\u7684\uff0c\u5c31\u662f\u5b83\u597d\u5230\u53ef\u4ee5\u62ff\u6765\u4f5c\u4e3a\u5178\u8303\uff0c\u7ed9\u7814\u7a76\u751f\u4f5c\u4e3a\u8303\u6587\uff0c\u53bb\u5b66\u4e60\u5f88\u591a\u8d85\u51fa\u8bba\u6587\u5177\u4f53\u5185\u5bb9\u7684\u4e1c\u897f\uff0c\u6bd4\u5982\u8bba\u6587\u7ed3\u6784\u3001\u5b66\u672f\u82f1\u8bed\u5199\u4f5c\u3001\u6279\u5224\u6027\u903b\u8f91\u3001\u4e43\u81f3\u79d1\u5b66\u7cbe\u795e\u3002Lucy\u7684\u8fd9\u7bc7\u8bba\u6587\u5c31\u53ef\u4ee5\u79f0\u4e4b\u4e3a\u4e00\u7bc7\u201c\u6559\u5b66\u8bba\u6587\u201d\u3002 \u8be5\u4f5c\u8005\u7684\u5168\u540d\u662fLeon Brian Lucy\u3002\u5173\u4e8e\u4ed6\u7684\u8be6\u7ec6\u4fe1\u606f\uff0c\u53ef\u89c1\u5176\u4e00\u7bc7\u8ba3\u544a\u548c\u7eaa\u5ff5\u6587\u7ae0\u3002","rel":"","context":"In "\u4e2a\u4eba\u6587\u96c6"","block_context":{"text":"\u4e2a\u4eba\u6587\u96c6","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/featured"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":7025,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/7025","url_meta":{"origin":6981,"position":2},"title":"\u6781\u5750\u6807\u4e0b\u7684\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70","author":"Andrew","date":"2016\u5e746\u670811\u65e5","format":false,"excerpt":"[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}\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":"Rice distributions of varying expetation","src":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2016\/06\/untitled-480x360.png?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":8923,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/8923","url_meta":{"origin":6981,"position":3},"title":"\u975e\u5e73\u8861\u7edf\u8ba1\u529b\u5b66\u7684\u53d1\u5c55\u6982\u51b5\u4e0e\u57fa\u672c\u5185\u5bb9","author":"Andrew","date":"2023\u5e748\u67088\u65e5","format":false,"excerpt":"[latexpage] \u6ce8\uff1a\u672c\u6587\u4f5c\u8005\u4e0d\u662f\u6211\uff0c\u800c\u662f\u4e00\u672c\u4f5a\u540d\u8bb2\u4e49\u7684\u7b2c\u4e00\u8282\u3002\u6211\u672c\u4eba\u5bf9\u8fd9\u6bb5\u6587\u5b57\u7684comment\u662f\uff0c\u867d\u5e74\u4ee3\u6709\u70b9\u8fc7\u65f6\uff0c\u4f46\u662f\u975e\u5e38accurate\uff0c\u5728\u90a3\u4e2a\u5e74\u4ee3\u7684\u4e2d\u6587\u8d44\u6599\u4e2d\u662f\u7f55\u89c1\u7684\u3002\u6401\u73b0\u5728\u4e5f\u662f\u975e\u5e38\u9ad8\u8d28\u91cf\u7684historical review + perspectives\u3002\u5e0c\u671b\u6709\u4eba\u80fd\u5e2e\u6211\u627e\u51fa\uff0c\u5b83\u7684\u4f5c\u8005\u662f\u8c01\u3002 \u975e\u5e73\u8861\u7edf\u8ba1\u529b\u5b66\u7684\u53d1\u5c55\u59cb\u4e8e1872\u5e74\u3002\u5728\u8fd9\u4e00\u5e74Boltzmann\u63d0\u51fa\u4e86\u73b0\u5728\u4ee5\u4ed6\u7684\u540d\u5b57\u547d\u540d\u7684\u65b9\u7a0b\u2014\u2014Boltzmann\u65b9\u7a0b\u3002Boltzmann\u65b9\u7a0b\u662f\u63cf\u8ff0\u7a00\u8584\u6c14\u4f53\u975e\u5e73\u8861\u73b0\u8c61\u7684\u91cd\u8981\u65b9\u7a0b\u3002\u7531\u5b83\u5bfc\u51fa\u7684H\u5b9a\u7406\u7ed9\u70ed\u529b\u5b66\u7b2c\u4e8c\u5b9a\u5f8b\u4ee5\u7edf\u8ba1\u89e3\u91ca\u3002Boltzmann\u65b9\u7a0b\u81ea\u5176\u63d0\u51fa\u4e4b\u65e5\u8d77\u5c31\u53d7\u5230\u4e86\u7269\u7406\u5b66\u5bb6\u548c\u6570\u5b66\u5bb6\u7684\u653b\u51fb\uff0c\u8bf4\u5b83\u5728\u529b\u5b66\u7cfb\u7edf\u4e2d\u5f15\u5165\u4e86\u6982\u7387\u7684\u6982\u5ff5\uff0c\u56e0\u800c\u4e0d\u80fd\u88ab\u8ba4\u4e3a\u662f\u771f\u6b63\u7684\u7269\u7406\u7406\u8bba\u3002\u7531\u4e8e\u5f15\u5165\u4e86\u6982\u7387\u7684\u6982\u5ff5\uff0cBoltzmann\u65b9\u7a0b\u7684\u4e00\u4e9b\u57fa\u672c\u7ed3\u8bba\u5c31\u4e0e\u529b\u5b66\u786e\u5b9a\u8bba\u7684\u63cf\u8ff0\u7edd\u7136\u76f8\u53cd\u3002\u5728\u8fd9\u4e9b\u653b\u51fb\u4e2d\u7684\u4e24\u4e2a\u4e3b\u8981\u7684\u5f02\u8bae\u662fLoschmidt\u53cd\u8bba\uff081876\u5e74\uff09\u4e0eZermelo\u53cd\u8bba\uff081896\u5e74\uff09\u3002\u8fd9\u4e24\u4e2a\u53cd\u8bba\u6307\u51fa\uff1aHamilton\u7cfb\u7edf\u662f\u65f6\u95f4\u53cd\u6f14\u4e0d\u53d8\u7684\uff0c\u4e14\u5177\u6709Poincare\u5faa\u73af\uff0c\u800cBoltzmann\u65b9\u7a0b\u7834\u574f\u4e86\u65f6\u95f4\u53cd\u6f14\u4e0d\u53d8\u6027\uff0c\u5176\u89e3\u662f\u5355\u8c03\u53d8\u5316\u4e14\u8d8b\u5411\u5e73\u8861\u7684\u3002\u5c3d\u7ba1Boltzmann\u65b9\u7a0b\u53d7\u5230\u4e86\u4f17\u591a\u7684\u653b\u51fb\uff0c\u5b83\u4ecd\u5728\u89e3\u91ca\u7a00\u8584\u6c14\u4f53\u975e\u5e73\u8861\u6027\u8d28\u65b9\u9762\u53d6\u5f97\u4e86\u6781\u5927\u7684\u6210\u529f\u3002\u540c\u5b83\u5bfc\u51fa\u7684\u6709\u5173\u8f93\u8fd0\u8fc7\u7a0b\u7684\u7ed3\u8bba\uff0c\u4e0d\u4f46\u5728\u5b9a\u6027\u4e0a\uff0c\u800c\u4e14\u5728\u5b9a\u91cf\u4e0a\u90fd\u4e0e\u5b9e\u9a8c\u4e00\u81f4\u3002\u56e0\u6b64\uff0c\u975e\u5e73\u8861\u6001\u7edf\u8ba1\u529b\u5b66\u57fa\u672c\u4e0a\u6cbf\u7740\u4e24\u4e2a\u65b9\u5411\u53d1\u5c55\uff0c\u4e00\u65b9\u9762\u662fBoltzmann\u65b9\u7a0b\u53ca\u5176\u4ed6\u52a8\u529b\u5b66\u65b9\u7a0b\u7684\u6c42\u89e3\u4e0e\u5e94\u7528\uff1b\u53e6\u4e00\u65b9\u9762\u662fBoltzmann\u65b9\u7a0b\u53ca\u975e\u5e73\u8861\u6001\u7edf\u8ba1\u529b\u5b66\u57fa\u7840\u7684\u7814\u7a76\u3002 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\u5728\u7b26\u5408fluctuation-dissipation theory\uff08FDT\uff09\u7684\u6761\u4ef6\u4e0b\uff0c\u54cd\u5e94\u51fd\u6570\u548c\u76f8\u5173\u51fd\u6570\u4e4b\u95f4\u6709\u7b80\u5355\u7684\u5173\u7cfb\u3002\u5047\u5982\u54cd\u5e94\u51fd\u6570\u6709\u65f6\u6e29\u53e0\u52a0\u6027\uff0c\u76f8\u5173\u51fd\u6570\u4e5f\u4f1a\u6709\u65f6\u6e29\u53e0\u52a0\u6027\u3002\u4f46\u662f\u5728\u8fdc\u79bb\u5e73\u8861\u6001\uff08off-equilibrium\uff09\u7684\u6761\u4ef6\u4e0b\u5f80\u5f80FDT\u6761\u4ef6\u4e0d\u518d\u6ee1\u8db3\uff0c\u4f46\u662f\u6211\u4eec\u4ecd\u7136\u80fd\u591f\u770b\u5230\uff0c\u76f8\u5173\u51fd\u6570\u548c\u54cd\u5e94\u51fd\u6570\u90fd\u4f1a\u7b26\u5408time\u2013aging time superposition\u3002\u76ee\u524d\u6709\u4e00\u4e2a\u201c\u5e7f\u4e49FDT\u201d\u7684\u552f\u8c61\u6846\u67b6\uff0c\u4fdd\u6301\u4e86FDT\u7684\u57fa\u672c\u5f62\u5f0f\uff0c\u53ef\u4ee5\u89e3\u91ca\u7269\u7406\u8001\u5316\u8fd9\u4e00\u8fdc\u79bb\u5e73\u8861\u6001\u8fc7\u7a0b\u4e0b\u76f8\u5173\u51fd\u6570\u548c\u54cd\u5e94\u51fd\u6570\u4e4b\u95f4\u7684\u5173\u8054\u7cfb\uff0c\u5f15\u5165\u4e86\u7b49\u6548\u6e29\u5ea6Teff\u7684\u6982\u5ff5\u3002\u4f46\u662f\u5177\u4f53\u662f\u4ec0\u4e48\u7ed3\u6784\u5728\u63a7\u5236\u7740Teff\uff0c\u8fd8\u7f3a\u4e4f\u80fd\u56de\u7b54\u6b64\u95ee\u9898\u7684\u7ed3\u6784\u6a21\u578b\u3002\u73b0\u5728\u4e00\u4e9b\u80fd\u9884\u6d4b\u7269\u7406\u8001\u5316\u7684\u6a21\u578b\uff0c\u6700\u5e38\u7528\u7684\u8981\u6570\u57fa\u4e8etrap model\u7684soft glassy 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superposition\u79f0\u4e3a\u201c\u7b80\u5355\u8001\u5316\u201d\uff0c\u90a3\u4e48\u662f\u5426\u80fd\u591f\u63a2\u7d22\u5177\u6709\u201c\u590d\u6742\u8001\u5316\u201d\u7684\u4f53\u7cfb\uff1f\u6839\u636e\u4e0a\u6587\u7684\u4e00\u4e9b\u731c\u6d4b\uff0c\u5b9e\u9a8c\u8bbe\u8ba1\u7684\u65b9\u5411\u53ef\u80fd\u5728\u4e8e\u529b\u6c42\u7834\u574fMaxwell-Boltzmann\u5206\u5e03\u7684\u9002\u7528\u6027\u3002\u5982\u679c\u80fd\u591f\u4ece\u5b9e\u9a8c\u4e0a\u5c55\u793a\u51fa\u4ec0\u4e48\u65f6\u540e\u6709\u7b80\u5355\u8001\u5316\uff0c\u4ec0\u4e48\u65f6\u5019\u6709\u590d\u6742\u8001\u5316\uff0c\u4e5f\u8bb8\u5bf9\u7269\u7406\u8001\u5316\u7406\u8bba\u6a21\u578b\u7684\u7814\u7a76\u5177\u6709\u53c2\u8003\u610f\u4e49\u3002","rel":"","context":"In "\u5de5\u4f5c\u7b14\u8bb0"","block_context":{"text":"\u5de5\u4f5c\u7b14\u8bb0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/personal\/%e5%b7%a5%e4%bd%9c%e7%ac%94%e8%ae%b0"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":7088,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/7088","url_meta":{"origin":6981,"position":5},"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 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