{"id":9108,"date":"2024-12-14T03:48:37","date_gmt":"2024-12-13T19:48:37","guid":{"rendered":"https:\/\/www.andrewsun.net\/panta_rhei\/?p=9108"},"modified":"2024-12-14T03:48:37","modified_gmt":"2024-12-13T19:48:37","slug":"leon-b-lucy","status":"publish","type":"post","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/9108","title":{"rendered":"Leon B. Lucy"},"content":{"rendered":"
\n\n

[latexpage]<\/p>\n\n\n\n

\u80cc\u666f<\/h3>\n\n\n\n

\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<\/a>\uff0c\u5b83\u88ab\u540e\u4e16\u96c6\u4e2d\u7528\u4e8e\u56fe\u50cf\u53bb\u566a\u3002\u4f8b\u5982\uff0cMATLAB\u7684Image Processing Toolbox\u6709\u4e00\u4e2adeconvlucy<\/code>\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<\/p>\n\n\n\n

\u4f46\u662f\uff0cLucy\u7684\u539f\u6587[1]<\/a><\/span>\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<\/p>\n\n\n\n

\\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}<\/p>\n\n\n\n

\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>\n\n\n\n

\\[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]\\]<\/p>\n\n\n\n

\u5728\u56fe\u50cf\u53bb\u566a\u7684\u4e0a\u4e0b\u6587\u4e2d\uff0c\u4ee5deconvlucy<\/code>\u4e3a\u4f8b\uff0c\u6838\u51fd\u6570$P\\left(x\\right)$\u662f\u4e00\u4e2a\u56fa\u5b9a\u53c2\u6570$\\xi$\u7684\u51fd\u6570\uff0c\u4e14$\\xi$\u7684\u53d6\u503c\u8303\u56f4\uff08\u5373$P\\left(x\\right)$\u7684\u201c\u5bbd\u5ea6\u201d\uff09\u8fdc\u7a84\u4e8e$x$\u7684\u8303\u56f4\uff08\u5728\u56fe\u50cf\u8bed\u5883\u4e2d\u662f\u56fe\u50cf\u7684\u5927\u5c0f\uff09\u3002\u4f46\u5728Lucy\u539f\u6587\u7684\u8bed\u5883\u4e2d\uff0c$\\phi\\left(x\\right)$\u3001$P\\left(x\\middle|\\xi\\right)$\u548c$\\psi\\left(\\xi\\right)$\u90fd\u662f\u652f\u6491\u4e3a\u6574\u4e2a\u5b9e\u6570\u7684\u5206\u5e03\u5bc6\u5ea6\u51fd\u6570\u3002\u53ef\u4ee5\u8bf4\uff0c\u56fe\u50cf\u53bb\u566a\u5e94\u7528\uff0c\u53ea\u662fLucy\u539f\u6587\u7b97\u6cd5\u7684\u5176\u4e2d\u4e00\u4e2a\u5f88\u7279\u6b8a\u7684\u4f8b\u5b50\u3002<\/p>\n\n\n\n

\u6570\u503c\u8ba1\u7b97\u7684\u8bbe\u8ba1\u827a\u672f<\/h3>\n\n\n\n

\u5728\u6211\u7684\u7814\u7a76\u4e2d\uff0c\u6070\u597d\u9700\u8981\u89e3\u51b3Lucy\u539f\u6587\u610f\u4e49\u7684\u95ee\u9898\uff0c\u56e0\u6b64\u6211\u662f\u8ba4\u771f\u9605\u8bfb\u4e86Lucy\u7684\u539f\u6587\u591a\u6b21\u7684\uff0c\u6709\u4e9b\u5176\u4ed6\u65b9\u9762\u7684\u611f\u53d7\u3002<\/p>\n\n\n\n

\u539f\u6587\u7684\u6587\u5b57\u6781\u5176\u6e05\u6670\u548c\u6d41\u7545\uff0c\u903b\u8f91\u5341\u5206\u4e25\u5bc6\uff0c\u8bb0\u53f7\u4ed4\u7ec6\uff08\u65e2\u4e0d\u6ee5\u7528\u53c8\u4e0d\u6df7\u7528\uff09\u3002\u6211\u89c9\u5f97\u8fd9\u662f\u6709\u6210\u5c31\u7684\u4f5c\u8005\u7684\u5171\u6027\u3002\u8bfb\u5230\u8fd9\u6837\u7684\u6587\u5b57\u5c31\u80fd\u8bf4\u660e\u4f5c\u8005\u662f\u4e8b\u5b9e\u4e0a\u7684\u5927\u5e08\uff08\u5c3d\u7ba1\u4e16\u4fd7\u540d\u8a89\u4e0a\u672a\u5fc5\uff09\u3002<\/p>\n\n\n\n

\u5728\u8bba\u6587\u53d1\u8868\u76841970\u5e74\u4ee3\uff0c\u7535\u5b50\u8ba1\u7b97\u673a\u5728\u79d1\u5b66\u8ba1\u7b97\u4e2d\u7684\u5e94\u7528\u5df2\u7ecf\u6bd4\u8f83\u666e\u53ca\u3002\u539f\u6587\u6ca1\u6709\u63d0\u53ca\u6240\u62a5\u9053\u7684\u9a8c\u8bc1\u5b9e\u9a8c\u662f\u5728\u4ec0\u4e48\u8ba1\u7b97\u673a\u4e0a\u8fdb\u884c\u7684\uff0c\u53ea\u5728\u81f4\u8c22\u4e2d\u8bf4\u5230\u4e86NASA\u7684Goddard Institute for Space Studies (GISS)\u63d0\u4f9b\u4e86\u673a\u65f6\u3002\u6211\u76f8\u4fe1\uff0c\u8fd9\u5e94\u8be5\u662f\u4e00\u79cd\u9700\u8981\u7533\u8bf7\u8282\u70b9\u7684\u5927\u578b\u8ba1\u7b97\u673a\uff0c\u673a\u65f6\u8d44\u6e90\u5e94\u8be5\u662f\u6bd4\u8f83\u6602\u8d35\u7684\u3002<\/p>\n\n\n\n

\u5728\u4eca\u5929\uff0c\u50cf\u6211\u8fd9\u79cd\u6570\u503c\u8ba1\u7b97\u7684\u5916\u884c\uff0c\u53ef\u4ee5\u5728MATLAB\u5f00\u53d1\u73af\u5883\u4e2d\u91cd\u590d\u8fd0\u884c\u591a\u6b21\u6765\u5b66\u4e60\u4e00\u4e2a\u6ca1\u6709\u4ece\u539f\u7406\u4e0a\u5403\u900f\u7684\u7b97\u6cd5\u7684\u884c\u4e3a\uff0c\u56e0\u4e3a\u5f88\u591a\u8ba1\u7b97\u5728\u4eca\u5929\u7684\u666e\u901a\u7b14\u8bb0\u672c\u7535\u8111\u4e0a\u8fd0\u884c\u90fd\u6beb\u65e0\u538b\u529b\u3002\u4f46\u662f\u5728\u5f53\u5e74\uff0c\u8fd9\u79cd\u201c\u4f5c\u5f0a\u201d\u7684\u505a\u6cd5\u662f\u4e0d\u63d0\u5021\u7684\u3002\u7ed9\u5b9a\u4e00\u4e2a\u7b97\u6cd5\uff0c\u4f60\u7406\u5e94\u52aa\u529b\u5730\u5728\u8349\u7a3f\u7eb8\u4e0a\u5206\u6790\u5b83\u7684\u597d\u5904\u548c\u574f\u5904\u3002\u8fd9\u79cd\u529f\u592b\u6211\u6ca1\u6709\uff0c\u4f46\u4ece\u8fd9\u7bc7\u8bba\u6587\u4e2d\u8fd8\u662f\u9886\u6559\u4e86\u5b83\u7684\u4f18\u96c5\u3002<\/p>\n\n\n\n

\u4ece\u7b97\u6cd5\u7684\u539f\u7406\uff0c\u5c31\u80fd\u770b\u51fa\u5b83\u7684\u7ed3\u679c\u53ea\u5bf9\u957f\u6ce2\u957f\u566a\u97f3\u654f\u611f\uff0c\u800c\u5bf9\u77ed\u6ce2\uff08\u9ad8\u9891\uff09\u566a\u97f3\u8fdf\u949d\uff0c\u5b83\u5728\u5934\u51e0\u4e2a\u8fed\u4ee3\u5c31\u80fd\u5feb\u901f\u6536\u655b\uff0c\u5b83\u5728\u6837\u672c\u6570$N$\u592a\u5c11\u65f6\u4f1a\u6709\u4ec0\u4e48\u5f02\u5e38\u2026\u2026\u7b49\u7b49\uff0c\u90fd\u901a\u8fc7\u5206\u6790\uff0c\u5728\u4e0d\u653e\u5230\u8ba1\u7b97\u673a\u4e2d\u778e\u8bd5\u4e4b\u524d\uff0c\u5c31\u90fd\u6e05\u695a\u4e86\u3002\u800c\u6211\u7684\u505a\u6cd5\uff0c\u5e38\u5e38\u662f\u8fb9\u8bd5\u8fb9\u6539\u7b97\u6cd5\u3002\u6781\u5ea6\u6d6a\u8d39\u8ba1\u7b97\u8d44\u6e90\u6765\u8fc1\u5c31\u6211\u5728\u5206\u6790\u4e0a\u7684\u61d2\u60f0\u3002\u8fd9\u5728\u4eca\u5929\u4e5f\u8bb8\u4e0d\u662f\u4ec0\u4e48\u7f6a\u8fc7\uff0c\u751a\u81f3\u4f5c\u4e3a\u4e00\u4e2a\u6570\u503c\u8ba1\u7b97\u7684\u5916\u884c\uff0c\u8fd9\u53ef\u4ee5\u8bf4\u662f\u5728\u806a\u660e\u5730\u8282\u7701\u65f6\u95f4\uff0c\u4f46\u6211\u4ecd\u7136\u5bf9\u539f\u6587\u8fd9\u79cd\u201c\u6570\u503c\u8ba1\u7b97\u7684\u8bbe\u8ba1\u827a\u672f\u201d\u611f\u5230\u656c\u4f69\u3002<\/p>\n\n\n\n

\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<\/p>\n\n\n\n

\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[2]<\/a><\/span>\u548c\u7eaa\u5ff5\u6587\u7ae0[3]<\/a><\/span>\u3002<\/p>\n

References<\/h2>\n
    \n
  1. <\/a>\nL.B. Lucy, \"An iterative technique for the rectification of observed distributions\", The Astronomical Journal<\/i>, vol. 79, pp. 745, 1974. http:\/\/dx.doi.org\/10.1086\/111605<\/a>\n\n\n<\/li>\n
  2. <\/a>\nD. Baade, J. Danziger, R. Hook, and J. Walsh, \"Leon B. Lucy (1938\u20132018)\", Bulletin of the AAS<\/i>, vol. 54, 2022. http:\/\/dx.doi.org\/10.3847\/25c2cfeb.88cfeeba<\/a>\n\n\n<\/li>\n<\/ol>\n\n<\/div> ","protected":false},"excerpt":{"rendered":"

    [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]\\] \u5728\u56fe\u50cf\u53bb\u566a\u7684\u4e0a\u4e0b\u6587\u4e2d\uff0c\u4ee5deconvlucy\u4e3a\u4f8b\uff0c\u6838\u51fd\u6570$P\\left(x\\right)$\u662f\u4e00\u4e2a\u56fa\u5b9a\u53c2\u6570$\\xi$\u7684\u51fd\u6570\uff0c\u4e14$\\xi$\u7684\u53d6\u503c\u8303\u56f4\uff08\u5373$P\\left(x\\right)$\u7684\u201c\u5bbd\u5ea6\u201d\uff09\u8fdc\u7a84\u4e8e$x$\u7684\u8303\u56f4\uff08\u5728\u56fe\u50cf\u8bed\u5883\u4e2d\u662f\u56fe\u50cf\u7684\u5927\u5c0f\uff09\u3002\u4f46\u5728Lucy\u539f\u6587\u7684\u8bed\u5883\u4e2d\uff0c$\\phi\\left(x\\right)$\u3001$P\\left(x\\middle|\\xi\\right)$\u548c$\\psi\\left(\\xi\\right)$\u90fd\u662f\u652f\u6491\u4e3a\u6574\u4e2a\u5b9e\u6570\u7684\u5206\u5e03\u5bc6\u5ea6\u51fd\u6570\u3002\u53ef\u4ee5\u8bf4\uff0c\u56fe\u50cf\u53bb\u566a\u5e94\u7528\uff0c\u53ea\u662fLucy\u539f\u6587\u7b97\u6cd5\u7684\u5176\u4e2d\u4e00\u4e2a\u5f88\u7279\u6b8a\u7684\u4f8b\u5b50\u3002 \u6570\u503c\u8ba1\u7b97\u7684\u8bbe\u8ba1\u827a\u672f 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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":6981,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/6981","url_meta":{"origin":9108,"position":1},"title":"\u975e\u9ad8\u65af\u7cfb\u6570\u7b26\u53f7\u4e3a\u4ec0\u4e48\u6709\u4e2a\u4e0b\u68072\uff1f","author":"Andrew","date":"2016\u5e743\u670831\u65e5","format":"aside","excerpt":"[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 3600\u8d85\u7ea7\u8ba1\u7b97\u673a\u4e0a\u8ba1\u7b97\u4e86864\u4e2aLennard-Jones\u4f5c\u7528\u52bf\u7684\u539f\u5b50\uff0c\u6a21\u62df\u6db2\u6c29\u3002","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":7025,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/7025","url_meta":{"origin":9108,"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":4792,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/4792","url_meta":{"origin":9108,"position":3},"title":"\u65f6\u6e29\u53e0\u52a0\u5f97\u5230\u7684\u677e\u5f1b\u65f6\u95f4\u662f\u4ec0\u4e48\u677e\u5f1b\u65f6\u95f4\uff1f","author":"Andrew","date":"2012\u5e747\u670814\u65e5","format":false,"excerpt":"\u8fd9\u4e2a\u95ee\u9898\u641e\u4e86\u6211\u4e00\u4e2a\u901a\u5bb5\uff0c\u8bb0\u5f55\u4e00\u4e0b\u3002 \u65f6\u6e29\u53e0\u52a0\u6027\u8d28\u662f\u6307\u4f53\u7cfb\u5728\u4e0d\u540c\u6e29\u5ea6\u6d4b\u91cf\u5f97\u5230\u7684\u76f8\u5173\u51fd\u6570\u6216\u54cd\u5e94\u51fd\u6570\u53ef\u5728\u65f6\u95f4\u8f74\u5e73\u79fb\u53e0\u52a0\u6210\u4e00\u6761\u4e3b\u66f2\u7ebf\u3002\u5e73\u79fb\u56e0\u5b50aT\u5e38\u5e38\u8ba4\u4e3a\u662f\u677e\u5f1b\u65f6\u95f4\u7684\u6bd4\u503c\uff1a \u4f46\u662f\uff0c\u5b9e\u9645\u6837\u54c1\u5f80\u5f80\u5177\u6709\u4e00\u4e2a\u8f83\u5bbd\u7684\u677e\u5f1b\u65f6\u95f4\u8c31\u3002\u90a3\u4e48\u4e0a\u5f0f\u4e2d\u7684\u677e\u5f1b\u65f6\u95f4\u5230\u5e95\u662f\u54ea\u4e2a\u503c\uff0c\u662f\u5e73\u5747\u503c\u8fd8\u662f\u6700\u5927\u503c\u8fd8\u662f\u4ec0\u4e48\uff1f \u73bb\u7483\u6001\u4f53\u7cfb\u7684\u76f8\u5173\u51fd\u6570\u6216\u54cd\u5e94\u51fd\u6570\u5e38\u5e38\u7b26\u5408Kohlrausch-Williams-Watts\uff08KWW\uff09\u5173\u7cfb\uff1a \u5982\u679c\u4f53\u7cfb\u53ea\u6709\u4e00\u4e2a\u677e\u5f1b\u65f6\u95f4\u503c\uff0c\u5219\u03b2 = 1\uff0c\u03c40\u5c31\u662f\u677e\u5f1b\u65f6\u95f4\u3002\u5982\u679c0 < \u03b2 < 1\uff0c\u5219\u53cd\u6620\u4f53\u7cfb\u6709\u4e00\u4e2a\u5bbd\u5206\u5e03\u7684\u677e\u5f1b\u65f6\u95f4\u8c31\u3002\u03c40\u53ea\u80fd\u662f\u4e00\u4e2a\u7528\u6765\u63cf\u8ff0\u76f8\u5173\u51fd\u6570\u6216\u54cd\u5e94\u51fd\u6570\u7684\u7279\u5f81\u65f6\u95f4\u3002\u65f6\u6e29\u53e0\u52a0\u5f97\u5230\u7684\u5c31\u662f\u03c40\u7684\u6bd4\u503c\u3002\u03c40\u8ddf\u677e\u5f1b\u65f6\u95f4\u8c31\u6709\u4ec0\u4e48\u5173\u7cfb\u5462\uff1f\u5b83\u662f\u4e0d\u662f\u5c31\u662f\u5e73\u5747\u677e\u5f1b\u65f6\u95f4<\u03c4>\u5462\uff1f \u4ee5\u5e94\u529b\u677e\u5f1b\u4e3a\u4f8b\uff0c\u5047\u8bbe\u6837\u54c1\u53ef\u8868\u793a\u6210\u65e0\u6570\u4e2aMaxwell\u6a21\u578b\u7684\u548c\uff1a \u5176\u4e2dg(\u03c4)\u662f\u677e\u5f1b\u65f6\u95f4\u5206\u5e03\u51fd\u6570\u3002\u5b83\u4e0e\u66f4\u5e38\u7528\u7684\u677e\u5f1b\u65f6\u95f4\u8c31\u7684\u5173\u7cfb\u662fH(\u03c4)=\u03c4g(\u03c4)\u3002\u7531\u4e8eG(t)\u7b26\u5408KWW\u65b9\u7a0b\uff0c\u5219\u6709 \u4ee4u=1\/\u03c4\uff0c\u4e0a\u5f0f\u5c31\u53d8\u6210\u4e00\u4e2aLaplace\u53d8\u6362\uff1a \u4e3a\u4e86\u5f97\u5230g(\u03c4)\uff0c\u987b\u8ba1\u7b97\u4ee5\u4e0b\u53cdLaplace\u53d8\u6362\uff1a g(\u03c4)\u7684n\u9636\u77e9<\u03c4n>\uff1a \u6240\u4ee5 \u5373\u5bf9\u4e8e\u4e00\u4e2a\u6ee1\u8db3KWW\u578b\u54cd\u5e94\u51fd\u6570G(t)\u7684\u4f53\u7cfb\uff0c\u5176\u5e73\u5747\u677e\u5f1b\u65f6\u95f4<\u03c4>\u4e0eKWW\u65b9\u7a0b\u7684\u7279\u5f81\u65f6\u95f4\u03c40\u6210\u6b63\u6bd4\uff0c\u56e0\u6b64\u5e73\u79fb\u56e0\u5b50aT\u786e\u5b9e\u662f\u5e73\u5747\u677e\u5f1b\u65f6\u95f4\u7684\u6bd4\u503c \u3002 \u66f4\u9ebb\u70e6\u7684\u95ee\u9898\u662f\uff0c\u505a\u52a8\u6001\u6d4b\u8bd5\u7684\u65f6\u5019\uff0cG' = G''\u5904\u6240\u5bf9\u5e94\u7684\u9891\u7387\u5012\u6570\u662f\u4ec0\u4e48\u65f6\u95f4\uff1f\u5f88\u591a\u5730\u65b9\u8bf4\u8fd9\u662f\u6700\u5927\u677e\u5f1b\u65f6\u95f4\uff08maximum relaxation time\uff09\u03c4m\uff0c\u5373\u5f53\u89c2\u5bdf\u65f6\u95f4\u65f6\u95f4\u8d85\u8fc7\u03c4m\uff0c\u5c31\u53ef\u4ee5\u770b\u5230\u6837\u54c1\u7684\u6d41\u52a8\u3002\u53ef\u662f\uff0c\u4e00\u4e2a\u54cd\u5e94\u51fd\u6570\u7b26\u5408KWW\u65b9\u7a0b\u7684\u6837\u54c1\uff0c\u662f\u5426\u5177\u6709\u786e\u5b9a\u7684\u6700\u5927\u677e\u5f1b\u65f6\u95f4\uff1f\u4e5f\u5c31\u662f\u8bf4\uff0c\u5176\u677e\u5f1b\u65f6\u95f4\u5206\u5e03\u51fd\u6570g(\u03c4)\u662f\u5426\u6709\u786e\u5b9a\u7684\u5b9a\u4e49\u57df\uff0c\u4f7f\u5f97\u5f53\u03c4>\u03c4m\u65f6g(\u03c4) = 0\uff1f\u8fd9\u4e2a\u90fd\u4e0d\u597d\u8bf4\u3002\u201c\u6700\u5927\u677e\u5f1b\u65f6\u95f4\u201d\u7684\u6982\u5ff5\u8ddf\u677e\u5f1b\u65f6\u95f4\u8c31\u7684\u5173\u7cfb\uff0c\u591a\u6570\u89c1\u4e8e\u4eba\u4e3a\u7ed9\u5b9a\u7684\u677e\u5f1b\u65f6\u95f4\u8c31\uff0c\u4f8b\u5982Wedge-Box distribution\u3001BSW distribution\u7b49\u7b49\u3002\u7ed9\u5b9ag(\u03c4)\uff0c\u5012\u662f\u53ef\u4ee5\u5206\u522b\u8ba1\u7b97G'\u548cG'' \u4ee4G' = G''\uff0c\u628a\u4e0a\u9762\u4e24\u5f0f\u8054\u8d77\u6765\u89e3\u51fa\u03c9\u6765\uff0c\u5c31\u80fd\u6c42\u5f97\u6240\u8c13\u7684\u201c\u6700\u5927\u677e\u5f1b\u65f6\u95f4\u201d\u662f\u5565\uff0c\u4f46\u8fd9\u4e2a\u592a\u96be\u4e86\u2026\u2026\u53c8\u636e\u8bf4\uff0cG' = G''\u65f6\u7684\u03c9\u4e5f\u5c31\u662fG''\u6700\u5927\u503c\u65f6\u7684\u03c9\u3002\u5219\u4e0a\u9762\u7b2c\u4e8c\u4e2a\u5f0f\u5b50\u6c42\u5bfc\uff0c\u5c31\u5269\u4e0b\u88ab\u79ef\u51fd\u6570\u4e86\u3002\u90a3\u5768\u4e1c\u897f\u4e2d\u53f3\u8fb9\u7684\u5206\u5f0f\u662f\u975e\u96f6\u7684\uff0c\u6240\u4ee5\u552f\u6709\u8ba9g(\u03c4)=0\u3002\u8fd9\u5c31\u53c8\u56de\u5230\u4e4b\u524d\u95ee\u7684\u90a3\u4e2a\u95ee\u9898\u4e86\uff1ag(\u03c4)\u662f\u5426\u6709\u786e\u5b9a\u7684\u5b9a\u4e49\u57df\uff0c\u4f7f\u5f97\u5f53\u03c4>\u03c4m\u65f6g(\u03c4) = 0\uff1f \u6709\u4eba\u7ed9\u51fa\uff08\u89c1\u53c2\u8003\u6587\u732e\uff09\u4e86g(\u03c4)\u7684\u8fd1\u4f3c\u8868\u8fbe\u5f0f\uff0c\u662f\u4e2a\u542bGamma\u51fd\u6570\u7684\u7ea7\u6570\u3002\u8981\u6211\u89e3g(\u03c4) = 0\u51e0\u4e4e\u662f\u4e0d\u53ef\u80fd\u7684\u2026\u2026 \u8fd8\u6709\u4e00\u4e2a\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6cd5\uff0c\u5c31\u662f\u91c7\u7528\u677e\u5f1b\u65f6\u95f4\u8c31\u7684\u4e00\u4e9b\u8fd1\u4f3c\u7b97\u6cd5\u3002\u4f8b\u5982\uff1a \u8fd9\u5c31\u53ef\u76f4\u63a5\u4eceG(t)\u53bb\u8ba1\u7b97\u677e\u5f1b\u65f6\u95f4\u8c31H(\u03c4)=\u03c4g(\u03c4)\uff0c\u7136\u540e\u4ee3\u5165\u4e0a\u9762\u7684\u5f0f\u5b50\u7b97\u51faG'\u548cG''\uff0c\u89e3\u51fa\u201c\u6700\u5927\u677e\u5f1b\u65f6\u95f4\u201d\u6765\u3002\u4f46\u603b\u4e4b\u90a3\u4e24\u4e2a\u79ef\u5206\u8fd8\u662f\u5f88\u96be\u641e \u6211\u731c\u6d4b\u8ba1\u6700\u5927\u677e\u5f1b\u65f6\u95f4\u03c4m\u4e5f\u662f\u8ddf\u03c40\u548c\u03b2\u6709\u5173\uff0c\u4f46\u662f\u5426\u6b63\u6bd4\u4e8e\u03c40\u5c31\u4e0d\u77e5\u9053\u4e86\u3002\u771f\u7684\u4e0d\u884c\u7684\u8bdd\uff0c\u6211\u53ea\u80fd\u5728MATLAB\u6570\u503c\u7b97\u4e00\u4e0b\uff0c\u8003\u5bdf\u4e00\u4e0b\u4e0d\u540c\u03b2\u503c\u4e0b\u7684\u6700\u5927\u677e\u5f1b\u65f6\u95f4\u03c4m\u3001\u5e73\u5747\u677e\u5f1b\u65f6\u95f4<\u03c4>\u548cKWW\u7684\u7279\u5f81\u65f6\u95f4\u03c40\u4e4b\u95f4\u7684\u8868\u89c2\u5173\u7cfb\uff0c\u770b\u80fd\u5426\u8fd1\u4f3c\u8ba4\u4e3a\u03c4m\u8ddf\u03c40\u6210\u6b63\u6bd4\u3002\u5982\u679c\u80fd\uff0c\u5219\u65f6\u6e29\u53e0\u52a0\u5f97\u5230\u7684\u5e73\u79fb\u56e0\u5b50\uff0c\u540c\u65f6\u4e5f\u662f\u6700\u5927\u677e\u5f1b\u65f6\u95f4\u4e4b\u6bd4\uff1a \u53c2\u8003\u6587\u732e\uff1aJ. Chem. Phys. 73:3348\uff0cText. Res. J. 21:404","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":4999,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/4999","url_meta":{"origin":9108,"position":4},"title":"\u5bf9\u65f6X\u53e0\u52a0\u539f\u7406\u548c\u975e\u5e73\u8861\u8fc7\u7a0b\u7684\u4e00\u4e9b\u8ba4\u8bc6","author":"Andrew","date":"2012\u5e7411\u670811\u65e5","format":false,"excerpt":"\u677e\u5f1b\u65f6\u95f4\u5206\u5e03\u7684\u7a33\u5b9a\u6027\u662f\u600e\u4e48\u6765\u7684\uff1f \u9ad8\u5206\u5b50\u677e\u5f1b\u8fc7\u7a0b\u5e38\u89c1\u7684\u65f6\u6e29\u53e0\u52a0\u7684\u73b0\u8c61\u662f\u6837\u54c1\u5728\u4e0d\u540c\u5e73\u8861\u6e29\u5ea6\u4e0b\u7684\u529b\u5b66\u54cd\u5e94\u51fd\u6570\u53ef\u5728\u54cd\u5e94\u65f6\u95f4t\u6216\u54cd\u5e94\u9891\u7387\u03c9\u8f74\u4e0a\u5e73\u79fb\u53e0\u52a0\u6210\u4e3b\u66f2\u7ebf\u3002\u5b9e\u9a8c\u62a5\u9053\u7ecf\u5e38\u4f7f\u7528\u7684\u662f\u9891\u57df\u54cd\u5e94\uff0c\u6240\u4ee5\u9891\u7387\u662f\u201c\u65f6\u6e29\u53e0\u52a0\u201d\u4e2d\u7684\u201c\u65f6\u201d\u3002\u8fd9\u4e00\u73b0\u8c61\u4e5f\u76f8\u5f53\u4e8e\u8bf4\u6e29\u5ea6\u4e0d\u6539\u53d8\u54cd\u5e94\u51fd\u6570\u7684\u5f62\u72b6\u3002 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landscape\u4e2d\u627e\u5230\u5230\u8fbe\u7ec8\u6781\u5e73\u8861\u6001\u7684\u8def\u5f84\u3002\u56e0\u6b64\u5f88\u96be\u60f3\u8c61\uff0c\u5904\u4e8e\u4e0d\u540c\u72b6\u6001\u7684\u677e\u5f1b\u5355\u5143\u5728\u5411\u7740\u5e73\u8861\u6001\u8fdb\u53d1\u7684\u65f6\u5019\u603b\u662f\u201c\u4fdd\u6301\u4e00\u81f4\u7684\u961f\u5f62\u201d\uff0c\u56e0\u4e3a\u4ed6\u4eec\u6240\u5904\u7684\u73af\u5883\u5f80\u5f80\u4e0d\u4e00\u6837\uff0c\u6709\u7684\u80fd\u627e\u5230\u6377\u5f84\u6709\u7684\u8981\u8d70\u5f2f\u8def\u624d\u66f4\u7b26\u5408\u4eba\u7684\u76f4\u89c9\uff0c\u5404\u5355\u5143\u7684\u677e\u5f1b\u5982\u679c\u6709\u5feb\u6709\u6162\u7684\u8bdd\uff0c\u677e\u5f1b\u65f6\u95f4\u7684\u5206\u5e03\u5c31\u4f1a\u968f\u7740\u8001\u5316\u65f6\u95f4\u4e00\u76f4\u53d8\u5316\uff0c\u4f46\u4e8b\u5b9e\u5374\u4e0d\u662f\u8fd9\u6837\u3002time\u2013aging time superposition\u7684\u73b0\u8c61\u7684\u81ea1978\u5e74Struik\u63d0\u51fa\u4ee5\u6765\u5230\u73b0\u5728\uff0c\u5df2\u7ecf\u4e3a\u4eba\u6240\u719f\u77e5\uff0c\u4f46\u662f\u6211\u4f3c\u4e4e\u6ca1\u6709\u542c\u8bf4\u8fc7\u5bf9\u6b64\u73b0\u8c61\u7684\u89e3\u91ca\u3002\u4e5f\u8bb8\u662f\u5728\u4e8e\u8fdc\u79bb\u5e73\u8861\u6001\u7684\u975e\u5e73\u8861\u7269\u7406\u603b\u4f53\u8fd8\u6709\u5f85\u53d1\u5c55\uff1f \u6211\u7684\u7406\u8bba\u57fa\u7840\u6bd4\u8f83\u5dee\uff0c\u4e0d\u662f\u592a\u719f\u6089\u7406\u8bba\u7269\u7406\u65b9\u9762\u7684\u7814\u7a76\u73b0\u72b6\u3002\u76ee\u524d\u5982\u679c\u60f3\u5c31\u6b64\u95ee\u9898\u5728\u5b9e\u9a8c\u4e0a\u66f4\u6df1\u5165\u4e00\u6b65\u7684\u8bdd\uff0c\u53ea\u80fd\u53d8\u7740\u6cd5\u5b50\u53bb\u8bd5\u63a2time\u2013aging time superposition\u7684\u9002\u7528\u8303\u56f4\u9650\u5ea6\u5728\u54ea\u91cc\u3002\u5c31\u597d\u50cf\u5bf9\u70ed\u6d41\u53d8\u590d\u6742\u6027\u7684\u7814\u7a76\u90a3\u6837\uff0c\u5982\u679c\u628atime\u2013aging time 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":5244,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/5244","url_meta":{"origin":9108,"position":5},"title":"Fourier\u53d8\u6362\uff08\u4e00\uff09","author":"Andrew","date":"2013\u5e744\u67088\u65e5","format":false,"excerpt":"[mathjax] \u4e3a\u4e86\u5e26\u5e08\u5f1f\u505a\u5b9e\u9a8c\uff0c\u7b80\u8981\u603b\u7ed3\u4e00\u4e0b\u5728MATLAB\u91cc\u8fdb\u884cFourier\u53d8\u6362\u7684\u4e00\u4e9b\u4ee3\u7801\uff0c\u4ee5\u4f9b\u5927\u5bb6\u53c2\u8003\u3002 Discrete Fourier transform (DFT): The sequence of N complex numbers x0, ..., xN-1 is transformed into an N-periodic sequence of complex numbers according to the DFT formula . \u4e00\u822c\u628a\u53d8\u6362\u524d\u7684\u4fe1\u53f7xn\u5f53\u6210\u65f6\u57df\u4fe1\u53f7\uff0c\u53d8\u6362\u540e\u7684Xk\u5f53\u6210\u9891\u57df\u4fe1\u53f7\u3002\u53ef\u89c1\uff0c\u65f6\u57df\u662fN\u4e2a\u5355\u5143\u7684\u4fe1\u53f7\uff0c\u53d8\u6362\u540e\u9891\u57df\u8fd8\u662fN\u4e2a\u5355\u5143\u7684\u4fe1\u53f7\u3002\u4ec0\u4e48\u5468\u671f\u554a\u9891\u7387\u554a\u8fd9\u4e9b\u4fe1\u606f\u662f\u6682\u65f6\u6ca1\u6709\u7684\u3002\u4f8b\u5982xn\u4e3a\u4ee5\u4e0b\u8fd9\u4e2a\u5e8f\u5217\uff08Fig. 1\uff09\uff0c\u6a2a\u5750\u6807\u5c31\u662f\u5e8f\u53f7n\u3002\u8fd9\u6bb5\u5e8f\u5217\uff0c\u53ef\u4ee5\u662f\u4e00\u6bb510\u79d2\u949f\u7684\u7684\u4fe1\u53f7\uff0c\u4e5f\u53ef\u4ee5\u662f\u4e00\u6bb5100\u79d2\u949f\u7684\u4fe1\u53f7\u3002\u5f53\u7136\uff0c\u5b9e\u9645\u60c5\u51b5\u4e0b\u4e00\u6bb5\u4fe1\u53f7\u662f\u591a\u5c11\u79d2\u662f\u4e8b\u5148\u77e5\u9053\u7684\uff0c\u51fa\u6765\u7684\u4fe1\u53f7\u4e00\u5171\u6709\u591a\u5c11\u4e2a\u5355\u5143\u503c\uff08N\uff09\u5219\u53d6\u51b3\u4e8e\u6211\u4eec\u9694\u591a\u957f\u65f6\u95f4\u53d6\u4e00\u6b21\u6837\u3002\u5982\u679c\u4e00\u6bb510\u79d2\u949f\u7684\u4fe1\u53f7\uff0c\u6211\u4eec\u6bcf\u79d2\u53d6\u4e00\u6b21\u6837\uff0c\u8fd9\u4e2a\u5e8f\u5217\u5c31\u670910\u4e2a\u5355\u5143\uff08N = 10\uff09\u3002\u53cd\u4e4b\uff0cFig. 1\u7684\u8fd9\u4e2a\u5e8f\u5217N = 93\uff0c\u5047\u5982\u8fd9\u4e2a\u4fe1\u53f7\u662f\u6bcf\u96940.1\u79d2\u53d6\u4e00\u6b21\u6837\u5f97\u5230\u7684\uff0c\u8fd9\u4e2a\u4fe1\u53f7\u7684\u603b\u65f6\u957f\u5c31\u662f9.2\u79d2\uff1b\u800c\u5047\u5982\u8fd9\u4e2a\u4fe1\u53f7\u662f\u6bcf\u969410\u79d2\u53d6\u4e00\u6b21\u6837\u5f97\u5230\u7684\uff0c\u8fd9\u4e2a\u4fe1\u53f7\u603b\u65f6\u957f\u5c31\u662f920\u79d2\u4e86\u3002\u6bcf\u9694\u591a\u957f\u65f6\u95f4\u53d6\u4e00\u6b21\u6837\uff0c\u53ef\u4ee5\u7528\u53d6\u6837\u9891\u7387fs\u6765\u8868\u793a\u3002fs = 10 Hz\u5c31\u662f\u6bcf\u96940.1 s\u53d6\u4e00\u6b21\u6837\u3002\u6240\u4ee5\uff0c\u4e3a\u4e86\u7ed9\u4e00\u5bf9\u79bb\u6563Fourier\u53d8\u6362xn\u548cXk\u8d4b\u4ee5\u5468\u671f\u6216\u9891\u7387\u7684\u4fe1\u606f\uff0c\u5fc5\u987b\u7ed9\u65f6\u57df\u4fe1\u53f7\u89c4\u5b9a\u4e00\u4e2afs\u3002\u4e00\u65e6fs\u89c4\u5b9a\u4e86\uff0c\u5468\u671f\u548c\u9891\u7387\u4e5f\u5c31\u89c4\u5b9a\u4e86\u3002\u8fd8\u62ffFig. 1\u7684\u5e8f\u5217\u4e3a\u4f8b\uff0c\u8bbe\u8be5\u5e8f\u5217\u7684\u53d6\u6837\u9891\u7387fs = 10 Hz\uff0c\u5219\u8be5\u5e8f\u5217\u6bcf\u4e2a\u5355\u5143\u9694\u4e860.1 s\u3002\u6211\u4eec\u53ef\u4ee5\u6570\u4e00\u4e0b\u8fd9\u4e2a\u5e8f\u5217\u4e00\u4e2a\u5468\u671f\u6709\u591a\u5c11\u4e2a\u5355\u5143\u2014\u201420\u4e2a\uff0c\u90a3\u4e48\u8be5\u5e8f\u5217\u7684\u5468\u671fT = 2 s\uff0c\u9891\u7387f =\u2026","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":"Figure 1","src":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/f1.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/f1.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/f1.png?resize=525%2C300 1.5x, https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/f1.png?resize=700%2C400 2x"},"classes":[]}],"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/posts\/9108","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/comments?post=9108"}],"version-history":[{"count":11,"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/posts\/9108\/revisions"}],"predecessor-version":[{"id":9167,"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/posts\/9108\/revisions\/9167"}],"wp:attachment":[{"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/media?parent=9108"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/categories?post=9108"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/tags?post=9108"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}