{"id":55007,"date":"2025-01-09T18:36:04","date_gmt":"2025-01-09T10:36:04","guid":{"rendered":"https:\/\/swarma.org\/?p=55007"},"modified":"2025-01-09T18:36:04","modified_gmt":"2025-01-09T10:36:04","slug":"nature-physics-%e4%bf%a1%e6%81%af%e7%83%ad%e5%8a%9b%e5%ad%a6%e7%bb%8f%e5%85%b8%e7%bb%bc%e8%bf%b0%ef%bc%9a%e9%80%9a%e5%be%80%e4%bf%a1%e6%81%af%e7%9a%84%e7%89%a9%e7%90%86%e6%9c%ac%e8%b4%a8","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=55007","title":{"rendered":"Nature Physics \u4fe1\u606f\u70ed\u529b\u5b66\u7ecf\u5178\u7efc\u8ff0\uff1a\u901a\u5f80\u4fe1\u606f\u7684\u7269\u7406\u672c\u8d28"},"content":{"rendered":"
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Juan M. R. Parrondo, Jordan M. Horowitz & Takahiro Sagawa<\/span> <\/span>| \u4f5c\u8005<\/span><\/p>\n

\u674e\u6c9b\u4e1c<\/span> <\/span>| \u8bd1\u8005<\/span><\/span><\/p>\n

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\u8bba\u6587\u9898\u76ee\uff1aThermodynamics of information<\/span><\/p>\n

\u8bba\u6587\u5730\u5740\uff1ahttps:\/\/www.nature.com\/articles\/nphys3230<\/span><\/section>\n<\/section>\n<\/section>\n<\/blockquote>\n


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\u4ece\u672c\u8d28\u4e0a\u8bb2\uff0c\u70ed\u529b\u5b66\u7b2c\u4e8c\u5b9a\u5f8b\u662f\u6982\u7387\u6027\u7684\uff0c\u56e0\u4e3a\u5176\u8868\u8ff0\u9700\u8981\u5bf9\u7cfb\u7edf\u72b6\u6001\u8fdb\u884c\u6982\u7387\u63cf\u8ff0\u3002\u8fd9\u5f15\u53d1\u4e86\u5173\u4e8e\u7b2c\u4e8c\u5b9a\u5f8b\u5ba2\u89c2\u6027\u7684\u95ee\u9898\uff1a\u4f8b\u5982\uff0c\u5b83\u662f\u5426\u4f9d\u8d56\u4e8e\u6211\u4eec\u5bf9\u7cfb\u7edf\u7684\u4e86\u89e3\uff1f\u4e00\u4e2a\u591a\u4e16\u7eaa\u4ee5\u6765\uff0c\u5927\u91cf\u52aa\u529b\u81f4\u529b\u4e8e\u5c06\u4fe1\u606f\u5f15\u5165\u70ed\u529b\u5b66\uff0c\u5e76\u8bc4\u4f30\u64cd\u7eb5\u4fe1\u606f\u7684\u71b5\u548c\u80fd\u91cf\u6210\u672c\u3002\u6700\u8fd1\uff0c\u8fd9\u4e00\u5177\u6709\u5386\u53f2\u610f\u4e49\u7684\u7406\u8bba\u63a2\u7d22\u5728\u4fe1\u606f\u4e8e\u5c0f\u5c3a\u5ea6\u4e0b\u88ab\u64cd\u7eb5\u7684\u5b9e\u9645\u60c5\u5883\u4e2d\u4f53\u73b0\u51fa\u6765\uff0c\u4f8b\u5982\u5206\u5b50\u4e0e\u7ec6\u80de\u751f\u7269\u5b66\u3001\u4eba\u5de5\u7eb3\u7c73\u88c5\u7f6e\u6216\u91cf\u5b50\u8ba1\u7b97\u3002\u5728\u672c\u6587\u4e2d\uff0c\u6211\u4eec\u4ecb\u7ecd\u4e86\u4e00\u4e2a\u57fa\u4e8e\u968f\u673a\u70ed\u529b\u5b66<\/strong><\/span>\uff08stochastic thermodynamics\uff09<\/span>\u548c\u6da8\u843d\u5b9a\u7406<\/span><\/strong>\uff08fluctuation theorems\uff09<\/span>\u7684\u4fe1\u606f\u70ed\u529b\u5b66\u65b0\u7406\u8bba\u6846\u67b6<\/span><\/strong>\uff0c\u56de\u987e\u4e86\u4e00\u4e9b\u6700\u65b0\u7684\u5b9e\u9a8c\u7ed3\u679c\uff0c\u5e76\u6982\u8ff0\u4e86\u8be5\u9886\u57df\u7684\u524d\u6cbf\u53d1\u5c55\u72b6\u51b5\u3002<\/span><\/section>\n


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\u5728\u70ed\u529b\u5b66\u7b2c\u4e8c\u5b9a\u5f8b\u88ab\u53d1\u73b0\u540e\u4e0d\u4e45\uff0c\u8a79\u59c6\u65af\u00b7\u514b\u62c9\u514b\u00b7\u9ea6\u514b\u65af\u97e6<\/span>\uff08James Clerk Maxwell\uff09<\/span>\u901a\u8fc7\u4e00\u4e2a\u601d\u60f3\u5b9e\u9a8c\u9610\u91ca\u4e86\u8be5\u5b9a\u5f8b\u7684\u6982\u7387\u7279\u6027\uff0c\u8fd9\u4e2a\u5b9e\u9a8c\u540e\u6765\u88ab\u79f0\u4e3a\u9ea6\u514b\u65af\u97e6\u5996<\/span>\uff08Maxwell\u2019s demon\uff09<\/span>[1, 2]\u3002\u4ed6\u63d0\u51fa\uff0c\u5982\u679c\u6709\u4e00\u79cd\u806a\u660e\u7684\u751f\u7269\u2014\u2014\u5373\u5996\u2014\u2014\u638c\u63e1\u4e86\u6c14\u4f53\u4e2d\u7c92\u5b50\u7684\u901f\u5ea6\u548c\u4f4d\u7f6e\u7684\u4fe1\u606f\uff0c\u90a3\u4e48\u8fd9\u4e2a\u5996\u53ef\u4ee5\u5c06\u9ad8\u901f\u3001\u9ad8\u80fd\u7684\u7c92\u5b50\u4ece\u51b7\u5e93<\/span>\uff08cold reservoir\uff09<\/span>\u8f6c\u79fb\u5230\u70ed\u5e93<\/span>\uff08hot reservoir\uff09<\/span>\uff0c\u8fd9\u8868\u9762\u4e0a\u8fdd\u53cd\u4e86\u70ed\u529b\u5b66\u7b2c\u4e8c\u5b9a\u5f8b[1, 2]\u3002<\/span><\/section>\n

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\u9ea6\u514b\u65af\u97e6\u5996\u9996\u6b21\u63ed\u793a\u4e86\u71b5\u4e0e\u4fe1\u606f\u4e4b\u95f4\u7684\u5173\u7cfb\u2014\u2014\u8868\u660e\u901a\u8fc7\u4f7f\u7528\u4fe1\u606f\uff0c\u53ef\u4ee5\u653e\u5bbd\u7b2c\u4e8c\u5b9a\u5f8b\u5bf9\u7cfb\u7edf\u4e0e\u5176\u73af\u5883\u4e4b\u95f4\u80fd\u91cf\u4ea4\u6362\u65bd\u52a0\u7684\u9650\u5236\u3002<\/strong>\u7136\u800c\uff0c\u9c81\u9053\u592b\u00b7\u514b\u52b3\u4fee\u65af<\/span>\uff08Rudolf Clausius\uff09<\/span>\u3001\u5f00\u5c14\u6587\u52cb\u7235<\/span>\uff08Lord Kelvin\uff09<\/span>\u548c\u9a6c\u514b\u65af\u00b7\u666e\u6717\u514b<\/span>\uff08Max Planck\uff09<\/span>[3]\u5bf9\u7b2c\u4e8c\u5b9a\u5f8b\u7684\u8868\u8ff0\u4e2d\u5e76\u6ca1\u6709\u63d0\u53ca\u5230\u4fe1\u606f\u3002\u8c03\u548c\u8fd9\u4e24\u79cd\u56fe\u666f\u9700\u8981\u5b8c\u6210\u4e24\u4e2a\u72ec\u7acb\u7684\u4efb\u52a1\u3002\u9996\u5148\uff0c\u6211\u4eec\u5fc5\u987b\u5b8c\u5584\u7b2c\u4e8c\u5b9a\u5f8b\uff0c\u4f7f\u5176\u660e\u786e\u7eb3\u5165\u4fe1\u606f\u7684\u56e0\u7d20\u3002\u5176\u6b21\uff0c\u6211\u4eec\u5fc5\u987b\u9610\u660e\u4fe1\u606f\u7684\u7269\u7406\u672c\u8d28\uff0c\u4f7f\u5176\u4ee5\u7269\u7406\u5b9e\u4f53\u800c\u975e\u62bd\u8c61\u6982\u5ff5\u7684\u5f62\u5f0f\u8fdb\u5165\u7b2c\u4e8c\u5b9a\u5f8b<\/strong>\u3002\u8fd9\u6837\uff0c\u6d4b\u91cf\u3001\u64e6\u9664\u3001\u590d\u5236\u548c\u53cd\u9988\u7b49\u4fe1\u606f\u64cd\u4f5c\u5c31\u53ef\u4ee5\u88ab\u89c6\u4e3a\u5177\u6709\u70ed\u529b\u5b66\u4ee3\u4ef7\u7684\u7269\u7406\u64cd\u4f5c\u3002<\/span><\/section>\n

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\u7b2c\u4e00\u9879\u4efb\u52a1\u7531\u91cc\u5965\u00b7\u897f\u62c9\u5fb7<\/span>\uff08L\u00e9o Szil\u00e1rd\uff09<\/span>\u90e8\u5206\u5b8c\u6210\uff0c\u4ed6\u8bbe\u8ba1\u4e86\u4e00\u4e2a\u7b80\u5316\u7248\u672c\u7684\u9ea6\u514b\u65af\u97e6\u5996\u3002\u897f\u62c9\u5fb7\u7684\u5996\u5229\u7528\u4e00\u6bd4\u7279\u4fe1\u606f<\/span>\uff08\u5373\u4e00\u4e2a\u65e0\u504f\u7684\u201c\u662f\/\u5426\u201d\u6d4b\u91cf\u7ed3\u679c\uff09<\/span>\u6765\u5b9e\u65bd\u4e00\u4e2a\u5faa\u73af\u8fc7\u7a0b\uff0c\u4ece\u6e29\u5ea6\u4e3a T \u7684\u70ed\u5e93\u4e2d\u4ee5\u529f\u7684\u5f62\u5f0f\u63d0\u53d6\u5927\u5c0f\u4e3a kTln2 \u7684\u80fd\u91cf\uff0c\u5176\u4e2d k \u4e3a\u73bb\u5c14\u5179\u66fc\u5e38\u6570[4]\u3002\u8fd9\u8868\u660e\u5996\u4f7f\u7528\u7684\u4fe1\u606f\u4e0e\u4ece\u5355\u4e00\u70ed\u5e93\u4e2d\u53ef\u63d0\u53d6\u529f\u4e4b\u95f4\u5b58\u5728\u5b9a\u91cf\u5173\u7cfb\u3002<\/span><\/section>\n

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\u4e3a\u89e3\u51b3\u7b2c\u4e8c\u9879\u4efb\u52a1\u6240\u505a\u7684\u52aa\u529b\u591a\u79cd\u591a\u6837[1, 2]\u3002\u83b1\u6602\u00b7\u5e03\u91cc\u6e25\u5b89<\/span>\uff08L\u00e9on Brillouin\uff09<\/span>\u5728\u67d0\u4e9b\u7279\u5b9a\u60c5\u51b5\u4e0b\u91cf\u5316\u4e86\u6d4b\u91cf\u7684\u4ee3\u4ef7\uff1b\u9a6c\u91cc\u5b89\u00b7\u65af\u83ab\u9c81\u970d\u592b\u65af\u57fa<\/span>\uff08Marian Smoluchowski\uff09<\/span>[5]\u548c\u7406\u67e5\u5fb7\u00b7\u8d39\u66fc<\/span>\uff08Richard Feynman\uff09<\/span>[6]\u5219\u8bc1\u660e\uff0c\u6da8\u843d\u73b0\u8c61\u4f1a\u963b\u6b62\u81ea\u4e3b\u5996\u5bf9\u7b2c\u4e8c\u5b9a\u5f8b\u7684\u8868\u9762\u8fdd\u80cc\u3002\u6b64\u5916\uff0c\u7f57\u5c14\u592b\u00b7\u5170\u9053\u5c14<\/span>\uff08Rolf Landauer\uff09<\/span>\u3001\u67e5\u5c14\u65af\u00b7\u8d1d\u5185\u7279<\/span>\uff08Charles Bennett\uff09<\/span>\u548c\u5965\u5229\u5f17\u00b7\u5f6d\u7f57\u65af<\/span>\uff08Oliver Penrose\uff09<\/span>\u8bc1\u660e[7]\uff0c\u6d4b\u91cf\u53ef\u4ee5\u5728\u96f6\u71b5\u589e\u7684\u60c5\u51b5\u4e0b\u5b9e\u73b0\u2014\u2014\u5c3d\u7ba1\u6d4b\u91cf\u64e6\u9664\u9700\u8981\u4ed8\u51fa\u80fd\u91cf\u4ee3\u4ef7\u3002\u8d1d\u5185\u7279\u968f\u540e\u7814\u7a76\u4e86\u4fe1\u606f\u5904\u7406\u7684\u82e5\u5e72\u70ed\u529b\u5b66\u65b9\u9762[8]\uff0c\u5305\u62ec\u538b\u7f29\u3001\u7b97\u6cd5\u590d\u6742\u6027\u3001\u903b\u8f91\u53ef\u9006\u6027\u548c\u6821\u5bf9\u8fc7\u7a0b\u3002<\/span><\/section>\n

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\u5c3d\u7ba1\u4fe1\u606f\u5728\u70ed\u529b\u5b66\u4e2d\u7684\u7814\u7a76\u5386\u53f2\u60a0\u4e45\uff0c\u8fc7\u53bb\u5341\u5e74\u95f4\uff0c\u7531\u4e8e\u6da8\u843d\u5b9a\u7406[9, 10]\u548c\u968f\u673a\u70ed\u529b\u5b66[11, 12]\u7684\u5e94\u7528\uff0c\u8be5\u9886\u57df\u53d6\u5f97\u4e86\u663e\u8457\u8fdb\u5c55\u3002\u8fd9\u4e9b\u7814\u7a76\u4e3a\u4fe1\u606f\u70ed\u529b\u5b66\u63d0\u4f9b\u4e86\u4e00\u4e2a\u901a\u7528\u6846\u67b6\uff0c\u65e2\u6f84\u6e05\u53c8\u63a8\u5e7f\u4e86\u4ee5\u5f80\u7684\u7814\u7a76\u7ed3\u679c\u3002\u4e0e\u8fd9\u4e00\u7406\u8bba\u8fdb\u5c55\u5e76\u884c\u7684\u662f\u6280\u672f\u7684\u8fdb\u6b65\uff0c\u5b83\u5e26\u6765\u4e86\u64cd\u63a7\u5c0f\u578b\u6da8\u843d\u7cfb\u7edf\u7684\u65b0\u5b9e\u9a8c\u6280\u672f\u3002\u56e0\u6b64\uff0c\u8fc7\u53bb\u7684\u601d\u60f3\u5b9e\u9a8c\uff0c\u5982\u9ea6\u514b\u65af\u97e6\u5996\u548c\u897f\u62c9\u5fb7\u5f15\u64ce<\/span>\uff08Szil\u00e1rd\u2019s engine\uff09<\/span>\uff0c\u73b0\u5728\u6b63\u5728\u5b9e\u9a8c\u5ba4\u4e2d\u5f97\u4ee5\u5b9e\u73b0[13, 14]\uff0c\u4f7f\u5f97\u5bf9\u4fe1\u606f\u64cd\u4f5c\u7684\u4e00\u4e9b\u57fa\u672c\u9884\u6d4b\u5f97\u5230\u4e86\u9a8c\u8bc1[15, 16]\u3002<\/span><\/section>\n

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\u672c\u6587\u65e8\u5728\u8865\u5145\u73b0\u6709\u7efc\u8ff0\u6587\u732e [1, 2, 11, 12, 17-19]\uff0c\u5448\u73b0\u6700\u8fd1\u53d1\u5c55\u4e0e\u5b9e\u9a8c\u7684\u6700\u65b0\u6982\u8ff0\uff0c\u540c\u65f6\u4e3a\u5206\u6790\u4fe1\u606f\u7cfb\u7edf\u70ed\u529b\u5b66\u6240\u4f7f\u7528\u7684\u4e3b\u8981\u601d\u60f3\u4e0e\u7406\u8bba\u5de5\u5177\u63d0\u4f9b\u57fa\u672c\u4ecb\u7ecd\u3002\u5728\u56de\u987e\u897f\u62c9\u5fb7\u5f15\u64ce\u540e\uff0c\u6211\u4eec\u5c06\u8ba8\u8bba\u5982\u4f55\u5c06\u4fe1\u606f\u7eb3\u5165\u7b2c\u4e8c\u5b9a\u5f8b\u7684\u7cbe\u786e\u8868\u8ff0\u4e2d\uff0c\u968f\u540e\u7ee7\u7eed\u5206\u6790\u8bb0\u5fc6\u5b58\u50a8\u4e0e\u4fe1\u606f\u5904\u7406\u7684\u7269\u7406\u5b9e\u73b0\u3002<\/span><\/section>\n

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\u897f\u62c9\u5fb7\u5f15\u64ce<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/p>\n

\u4ece\u5386\u53f2\u4e0a\u770b\uff0c\u6709\u5173\u4fe1\u606f\u70ed\u529b\u5b66\u7684\u8bb8\u591a\u7814\u7a76\u90fd\u96c6\u4e2d\u4e8e1929\u5e74\u897f\u62c9\u5fb7\u63d0\u51fa\u7684\u5faa\u73af\u5f15\u64ce\u4e0a[4]\u3002\u5728\u897f\u62c9\u5fb7\u7684\u539f\u59cb\u6784\u60f3\u4e2d\uff0c\u5f15\u64ce\u7684\u5de5\u4f5c\u7269\u8d28\u662f\u4e00\u7bb1\u4f53\u79ef\u4e3a V0<\/sub> \u7684\u5355\u5206\u5b50\u6c14\u4f53<\/span>\uff08\u56fe1a\uff09<\/span>\uff0c\u6d78\u6ca1\u5728\u6e29\u5ea6\u4e3a T \u7684\u70ed\u5e93\u4e2d\uff0c\u5e76\u7531\u5916\u90e8\u5b9e\u4f53<\/span>\uff08external agent\uff09<\/span>\u6216\u201c\u5996\u201d\u8fdb\u884c\u64cd\u63a7\u3002\u5996\u9996\u5148\u5feb\u901f\u5730\u5728\u7bb1\u4f53\u4e2d\u95f4\u63d2\u5165\u4e00\u4e2a\u9694\u677f\uff0c\u5c06\u5176\u5206\u6210\u4e24\u534a\uff0c\u4ece\u800c\u542f\u52a8\u4e00\u4e2a\u5faa\u73af\u3002\u5996\u968f\u540e\u6d4b\u91cf\u5206\u5b50\u88ab\u56f0\u5728\u9694\u677f\u7684\u54ea\u4e00\u4fa7\uff0c\u5e76\u6267\u884c\u4e00\u4e2a\u4ece\u4f53\u79ef V0<\/sub>\/2 \u5230 V0<\/sub> \u7684\u53ef\u9006\u81a8\u80c0\uff0c\u4ece\u800c\u63d0\u53d6\u529f Wext<\/sub> = kTln2\u3002\u901a\u8fc7\u79fb\u9664\u9694\u677f\uff0c\u5faa\u73af\u5b8c\u6210\u3002\u5176\u51c0\u6548\u5e94\u662f\u4ece\u5355\u4e00\u70ed\u5e93\u4e2d\u63d0\u53d6\u80fd\u91cf\u5e76\u5b8c\u6210\u4e00\u4e2a\u5faa\u73af\u8fc7\u7a0b\uff0c\u8fd9\u8868\u9762\u4e0a\u8fdd\u53cd\u4e86\u666e\u6717\u514b\u5bf9\u7b2c\u4e8c\u5b9a\u5f8b\u7684\u8868\u8ff0\u3002<\/span><\/section>\n

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\u897f\u62c9\u5fb7\u5f15\u64ce\u7684\u4e00\u4e9b\u65b9\u9762\u4e4d\u770b\u4e4b\u4e0b\u4f3c\u4e4e\u5f88\u6a21\u7cca\uff1a\u5206\u5b50\u4e0d\u4f1a\u65bd\u52a0\u5e73\u6ed1\u53d8\u5316\u7684\u538b\u529b\uff0c\u800c\u662f\u4ea7\u751f\u968f\u673a\u5f3a\u5ea6\u7684\u649e\u51fb\uff0c\u8fd9\u4f7f\u5f97\u4eba\u4eec\u8d28\u7591\u5728\u8ba1\u7b97\u529f\u65f6\u7406\u60f3\u6c14\u4f53\u5b9a\u5f8b\u7684\u9002\u7528\u6027\u3002\u6d4b\u91cf\u7684\u4f5c\u7528\u4e5f\u4e0d\u660e\u786e\uff0c\u56e0\u4e3a\u538b\u529b\u672c\u8eab\u4f3c\u4e4e\u53ef\u4ee5\u63ed\u793a\u5206\u5b50\u7684\u4f4d\u7f6e\u3002\u4ed4\u7ec6\u7684\u5206\u6790\u5df2\u6f84\u6e05\u4e86\u8fd9\u4e9b\u95ee\u9898[1]\u3002\u6b64\u5916\uff0c\u4efb\u4f55\u7ecf\u5386\u76f8\u7a7a\u95f4\u5206\u88c2\u7684\u7cfb\u7edf\u90fd\u53ef\u4ee5\u7528\u4f5c\u201c\u5de5\u4f5c\u7269\u8d28\u201d<\/span>\uff08working substance\uff09<\/span>[20]\u3002\u4f8b\u5982\uff0c\u897f\u62c9\u5fb7\u5f15\u64ce\u6700\u8fd1\u5df2\u7ecf\u5728\u5b9e\u9a8c\u5ba4\u4e2d\u5b9e\u73b0\uff0c\u6240\u4f7f\u7528\u7684\u7cfb\u7edf\u4e0d\u662f\u5355\u5206\u5b50\u6c14\u4f53\uff0c\u800c\u662f\u80f6\u4f53\u5e03\u6717\u7c92\u5b50\u548c\u5355\u7535\u5b50\u3002\u5f52\u6839\u7ed3\u5e95\uff0c\u4f7f\u897f\u62c9\u5fb7\u5f15\u64ce\u80fd\u591f\u8fd0\u8f6c\u7684\u662f\u4fe1\u606f\u4e0e\u71b5\u4e4b\u95f4\u7684\u4e00\u79cd\u57fa\u672c\u5173\u7cfb\uff0c\u8fd9\u79cd\u5173\u7cfb\u4e13\u95e8\u9002\u7528\u4e8e\u4e00\u7c7b\u975e\u5e73\u8861\u72b6\u6001\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u5c06\u8ba8\u8bba\u8fd9\u4e00\u70b9\u3002<\/span><\/section>\n

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\u56fe 1. \u897f\u62c9\u5fb7\u5f15\u64ce\u53ca\u8fd1\u671f\u7684\u5b9e\u9a8c\u5b9e\u73b0\u3002a. \u5728\u539f\u59cb\u7684\u897f\u62c9\u5fb7\u5f15\u64ce\u4e2d\uff0c\u9694\u677f\u88ab\u63d2\u5165\u5305\u542b\u5355\u4e2a\u5206\u5b50\u7684\u7bb1\u5b50\u4e2d\uff0c\u7bb1\u5b50\u88ab\u6e29\u5ea6\u4e3a T \u7684\u70ed\u5e93\u5305\u56f4\u3002\u7bb1\u5b50\u4e2d\u542b\u6709\u5206\u5b50\u7684\u90a3\u4e00\u534a\u88ab\u6d4b\u91cf\u5230\u540e\uff0c\u9694\u677f\u88ab\u79fb\u52a8\uff0c\u6267\u884c\u7b49\u6e29\u81a8\u80c0\u4ee5\u63d0\u53d6\u529f\u3002b. \u4f7f\u7528\u5355\u7535\u5b50\u76d2\uff08single-electron box\uff09\uff08SEB\uff09\u8fdb\u884c\u7684\u5b9e\u9a8c\u5b9e\u73b0\uff0c\u7531\u95e8\u7535\u538b Vg<\/sub> \u63a7\u5236\u5e76\u901a\u8fc7\u5355\u7535\u5b50\u6676\u4f53\u7ba1\uff08single-electron transistor\uff0cSET\uff09\u76d1\u6d4b[14]\u3002\u4e0a\u56fe\u663e\u793a\u4e86\u5b9e\u9a8c\u88c5\u7f6e\u3002\u4e0b\u56fe\u5c55\u793a\u4e86\u76d2\u5b50\u7684\u80fd\u7ea7\uff0c\u5176\u5927\u5c0f\u53d6\u51b3\u4e8e\u7535\u5b50\u6570 (n=0, 1)\uff0c\u662f\u5f52\u4e00\u5316\u95e8\u7535\u538b ng<\/sub> \u7684\u51fd\u6570\u3002\u5f53\u7535\u5b50\u6570 n \u88ab\u6d4b\u91cf\u5230\u65f6\uff0cng<\/sub> \u88ab\u5feb\u901f\u6539\u53d8\uff0c\u80fd\u91cf\u4e0b\u964d\uff08\u5de6\u56fe\uff09\uff0c\u7136\u540e\u7f13\u6162\u8fd4\u56de\u521d\u59cb\u503c\uff08\u53f3\u56fe\uff09\u3002\u5728\u8fd9\u4e2a\u8fc7\u7a0b\u4e2d\uff0c\u7531\u4e8e n<\/span>g<\/sub> \u4ec5\u5728\u7f13\u6162\u53d8\u5316\u65f6\u5f15\u8d77 n \u7684\u70ed\u6fc0\u53d1\uff0c\u56e0\u6b64\u4ea7\u751f\u4e86\u51c0\u529f\u8f93\u51fa\u3002c. \u4f7f\u7528\u80f6\u4f53\u7c92\u5b50\uff08colloidal particle\uff09\u548c\u4e24\u4e2a\u5149\u5b66\u9631\uff08optical traps\uff09\u7684\u5b9e\u9a8c\u5b9e\u73b0 [16]\u3002\u4e0a\u56fe\u663e\u793a\u4e86\u5b9e\u9a8c\u88c5\u7f6e\uff1a\u4e00\u4e2a\u9631\u56fa\u5b9a\u5728\u4f4d\u7f6e x = 0\uff0c\u53e6\u4e00\u4e2a\u5219\u4ee5\u901f\u5ea6 vtrap<\/sub> \u6c34\u5e73\u79fb\u52a8\u3002\u7531\u4e24\u4e2a\u7535\u6781\u4ea7\u751f\u7684\u53ef\u63a7\u9759\u7535\u573a\u4f7f\u7c92\u5b50\u504f\u5411\u67d0\u4e2a\u9631\u3002\u4e0b\u56fe\u663e\u793a\u4e86\u5f71\u54cd\u7c92\u5b50\u7684\u52bf\u9631\u7684\u7b49\u9ad8\u7ebf\u56fe\uff0c\u8be5\u8fc7\u7a0b\u4e2d\uff0c\u79fb\u52a8\u7684\u9631\u88ab\u79fb\u52a8\u7136\u540e\u56de\u5230\u521d\u59cb\u4f4d\u7f6e\u3002\u7c92\u5b50\u7684\u8f68\u8ff9\u4ee5\u6da8\u843d\u7684\u767d\u7ebf\u5f62\u5f0f\u5448\u73b0\u3002\u897f\u62c9\u5fb7\u5faa\u73af\u901a\u8fc7\u5728\u8fc7\u7a0b\u7684\u4e2d\u95f4\u6d4b\u91cf\u7c92\u5b50\u6240\u5728\u4e8e\u54ea\u4e2a\u9631\uff0c\u5e76\u5728\u8fc7\u7a0b\u7684\u540e\u534a\u6bb5\u504f\u7f6e\u8be5\u9631\u6765\u5b9e\u73b0\u3002d. \u4f7f\u7528\u65cb\u8f6c\u80f6\u4f53\u7c92\u5b50\uff08rotating colloidal particle\uff09\u7684\u5b9e\u9a8c\u5b9e\u73b0[13]\u3002\u4e0a\u56fe\u663e\u793a\u4e86\u5b9e\u9a8c\u88c5\u7f6e\uff0c\u5176\u4e2d\u4e24\u4e2a\u7c92\u5b50\u9644\u7740\u5728\u4e00\u5757\u76d6\u73bb\u7483\u4e0a\u3002\u4e00\u4e2a\u7c92\u5b50\u88ab\u89c6\u4e3a\u65cb\u8f6c\u5e03\u6717\u7c92\u5b50\uff0c\u5e76\u7531\u56db\u4e2a\u7535\u6781\u4ea7\u751f\u7684\u7535\u573a\u63a7\u5236\u3002\u4e0b\u56fe\u663e\u793a\u4e86\u6709\u6548\u52bf\u80fd\u7684\u4e24\u79cd\u5f62\u72b6\uff0c\u5b83\u4eec\u662f\u6b63\u5f26\u52bf\u80fd\u548c\u6052\u5b9a\u626d\u77e9\u7684\u53e0\u52a0\u3002\u5f53\u7c92\u5b50\u7684\u4f4d\u7f6e\u88ab\u6d4b\u91cf\u5230\u5904\u4e8e\u8de8\u8d8a\u52bf\u80fd\u6700\u5c0f\u503c\u540e\u7684\u4e0a\u5761\u65b9\u5411\u65f6\uff0c\u52bf\u80fd\u4ece\u4e00\u79cd\u5f62\u72b6\u5207\u6362\u5230\u53e6\u4e00\u79cd\u5f62\u72b6\u3002<\/span><\/p>\n


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\u5f53\u6211\u4eec\u83b7\u53d6\u5173\u4e8e\u5177\u6709\u5fae\u89c2\u72b6\u6001 x \u7684\u7269\u7406\u7cfb\u7edf\u7684\u65b0\u4fe1\u606f\u65f6\uff0c\u6211\u4eec\u5c06\u7edf\u8ba1\u72b6\u6001\u4ece\u03c1<\/span>(x) \u66f4\u65b0\u4e3a \u03c1<\/span>(x|m)\uff0c\u5176\u4e2d m \u8868\u793a\u65b0\u7684\u4fe1\u606f\uff0c\u4f8b\u5982\u6d4b\u91cf\u7ed3\u679c\u6216\u7528\u4e8e\u51c6\u5907\u72b6\u6001\u7684\u65b9\u6cd5\u3002\u4f8b\u5982\uff0c\u5728\u897f\u62c9\u5fb7\u5f15\u64ce\u4e2d\uff0c\u6d4b\u91cf\u540e\u7edf\u8ba1\u72b6\u6001 \u03c1<\/span>(x|m) \u88ab\u9650\u5236\u5728\u7bb1\u5b50\u7684\u5de6\u534a\u90e8\u5206\u6216\u53f3\u534a\u90e8\u5206\u3002\u4e00\u822c\u6765\u8bf4\uff0c\u5373\u4f7f\u6d4b\u91cf\u524d\u7684\u72b6\u6001 \u03c1<\/span>(x) \u5904\u4e8e\u5e73\u8861\u72b6\u6001\uff0c\u66f4\u65b0\u540e\u7684\u72b6\u6001 \u03c1<\/span>(x|m) \u4ecd\u7136\u662f\u975e\u5e73\u8861\u7684\u3002\u4fe1\u606f\u5c06\u7cfb\u7edf\u63a8\u79bb\u5e73\u8861\uff0c\u800c\u8868\u9762\u4e0a\u65e0\u9700\u4ed8\u51fa\u80fd\u91cf\u4ee3\u4ef7\u3002\u56e0\u6b64\uff0c\u4fe1\u606f\u70ed\u529b\u5b66\u53ef\u4ee5\u88ab\u89c6\u4e3a\u975e\u5e73\u8861\u70ed\u529b\u5b66\u7684\u4e00\u4e2a\u5206\u652f\uff0c\u5904\u7406\u4e00\u7c7b\u7279\u6b8a\u7684\u72b6\u6001\u548c\u8fc7\u7a0b<\/strong>\u3002\u8be5\u7406\u8bba\u7684\u8d77\u70b9\u662f\u5bf9\u71b5\u5b9a\u4e49\u7684\u6269\u5c55\uff0c\u4ee5\u9002\u5e94\u8fd9\u7c7b\u7279\u6b8a\u7684\u975e\u5e73\u8861\u72b6\u6001\u3002<\/span><\/section>\n

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\u9999\u519c\u71b5\uff08Shannon entropy\uff09\u7684\u7269\u7406\u610f\u4e49\u3002<\/strong>\u975e\u5e73\u8861\u72b6\u6001\u7684\u71b5\u5b9a\u4e49\u591a\u5e74\u6765\u4e00\u76f4\u5b58\u5728\u4e89\u8bae\u3002\u5c3d\u7ba1\u5e73\u8861\u6001\u70ed\u529b\u5b66\u71b5\u5728\u67d0\u4e9b\u975e\u5e73\u8861\u60c5\u5883\u4e2d\u5f97\u5230\u4e86\u6210\u529f\u5e94\u7528\uff0c\u4f46\u5bf9\u4e8e\u4e00\u822c\u975e\u5e73\u8861\u72b6\u6001\uff0c\u5982\u4f55\u5b9a\u4e49\u5177\u6709\u7269\u7406\u610f\u4e49\u7684\u71b5\u4ecd\u4e0d\u660e\u786e\u3002\u9999\u519c\u71b5<\/span>\uff08\u6216\u91cf\u5b50\u60c5\u51b5\u4e0b\u7684\u51af\u00b7\u8bfa\u4f9d\u66fc\u71b5\uff09<\/span>\u5df2\u88ab\u5e7f\u6cdb\u4f7f\u7528\uff0c\u4f46\u5728\u8fd9\u4e9b\u60c5\u51b5\u4e0b\uff0c\u5b83\u5f80\u5f80\u88ab\u5bbd\u6cdb\u5730\u89c6\u4e3a\u7b49\u540c\u4e8e\u70ed\u529b\u5b66\u71b5\u3002\u7136\u800c\uff0c\u6700\u8fd1\u968f\u673a\u70ed\u529b\u5b66\u7684\u53d1\u5c55\u4e25\u683c\u8868\u660e\uff0c\u5728\u67d0\u4e9b\u60c5\u5883\u4e0b\uff0c\u9999\u519c\u71b5\u5177\u6709\u660e\u786e\u7684\u7269\u7406\u610f\u4e49\uff1b\u5177\u4f53\u800c\u8a00\uff0c\u5b83\u51b3\u5b9a\u4e86\u4e0e\u4e00\u4e2a\u6216\u591a\u4e2a\u70ed\u5e93\u8026\u5408\u7684\u7cfb\u7edf\u7684\u975e\u5e73\u8861\u8fc7\u7a0b\u7684\u80fd\u91cf\u5b66\u7279\u6027<\/strong>[12]\u3002<\/span><\/section>\n

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\u5728\u4fe1\u606f\u8bba\u4e2d[21]\uff0c\u5177\u6709\u6982\u7387\u5bc6\u5ea6 \u03c1<\/span>(x) \u7684\u968f\u673a\u53d8\u91cf X \u7684\u9999\u519c\u71b5\u5b9a\u4e49\u4e3a\uff1a<\/sub>\u3002\u5f53 X \u8868\u793a\u7269\u7406\u7cfb\u7edf\u7684\u5fae\u89c2\u72b6\u6001\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u9999\u519c\u71b5\u4e58\u4ee5\u73bb\u5c14\u5179\u66fc\u5e38\u6570\uff0c\u4f5c\u4e3a\u975e\u5e73\u8861\u71b5\u7684\u5019\u9009\u8868\u8fbe\u5f0f<\/span><\/section>\n

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\u6211\u4eec\u5c06 S(\u03c1<\/span>) \u548c H(X) \u90fd\u79f0\u4e3a\u9999\u519c\u71b5\uff0c\u56e0\u4e3a\u5b83\u4eec\u672c\u8d28\u4e0a\u662f\u7528\u4e0d\u540c\u5355\u4f4d\u6d4b\u91cf\u7684\u76f8\u540c\u91cf\u3002\u7136\u800c\uff0c\u4fdd\u7559\u8fd9\u4e24\u4e2a\u7b26\u53f7\u53ca\u5176\u5404\u81ea\u7684\u5177\u4f53\u4f9d\u8d56\u5173\u7cfb\u662f\u6709\u76ca\u7684\uff0c\u5b83\u4eec\u5206\u522b\u662f\u70ed\u529b\u5b66\u548c\u4fe1\u606f\u8bba\u4e2d\u7684\u6807\u51c6\u8868\u793a\u3002<\/span><\/section>\n

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\u5bf9\u4e8e\u7ecf\u5178\u5e73\u8861\u6001 <\/sub>\uff0c\u9999\u519c\u71b5\u4e0e\u5e73\u8861\u71b5\u76f8\u7b26\uff0c\u5176\u4e2d <\/sub> \u662f\u54c8\u5bc6\u987f\u91cf\uff0c \u03b2<\/span>=1\/(kT)<\/span>\uff08\u5176\u4e2d T \u662f\u6e29\u5ea6\uff09<\/span>\uff0c\u800c Z<\/span>0<\/sub> \u662f\u914d\u5206\u51fd\u6570\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u6062\u590d\u4e86\u81ea\u7531\u80fd F=E-TS \u7684\u70ed\u529b\u5b66\u5173\u7cfb\uff0c\u5176\u4e2d\u81ea\u7531\u80fd F(\u03c1<\/span>0<\/sub>)=-kTln Z0<\/sub>\uff0c\u5e73\u5747\u5185\u80fd <\/sub>\uff0c\u4ee5\u53ca\u9999\u519c\u71b5 S(\u03c1<\/span>0<\/sub>)\u3002\u6211\u4eec\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5b9a\u4e49\u4e00\u4e2a\u975e\u5e73\u8861\u81ea\u7531\u80fd\uff0c\u7528\u4e8e\u63cf\u8ff0\u4e0e\u70ed\u6d74\u63a5\u89e6\u4e14\u54c8\u5bc6\u987f\u91cf\u4e3a <\/sub> \u7684\u7cfb\u7edf\u7684\u4e00\u822c\u7edf\u8ba1\u72b6\u6001 [22-26] <\/span><\/section>\n

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\u53ef\u4ee5\u8bc1\u660e\uff0c\u5728\u975e\u5e73\u8861\u7b49\u6e29\u8fc7\u7a0b\u4e2d\u7684\u9999\u519c\u71b5\u53ca\u5176\u76f8\u5173\u7684\u975e\u5e73\u8861\u81ea\u7531\u80fd\uff0c\u4e0e\u5176\u5728\u5e73\u8861\u6001\u65f6\u5bf9\u5e94\u7684\u71b5\u548c\u81ea\u7531\u80fd\u662f\u7c7b\u4f3c\u7684\u3002\u5728\u6b64\uff0c\u201c\u7b49\u6e29\u201d\u610f\u5473\u7740\u7cfb\u7edf\u4e0e\u6e29\u5ea6\u4e3a T \u7684\u70ed\u5e93\u63a5\u89e6\uff0c\u5c3d\u7ba1\u7cfb\u7edf\u672c\u8eab\u53ef\u80fd\u6ca1\u6709\u660e\u786e\u7684\u6e29\u5ea6\u3002\u5982\u68461\u6240\u793a\uff0c\u5e73\u5747\u800c\u8a00\uff0c\u4e3a\u4e86\u7b49\u6e29\u5730\u5c06\u7cfb\u7edf\u4ece\u4e00\u4e2a\u4efb\u610f\u72b6\u6001\u9a71\u52a8\u5230\u53e6\u4e00\u4e2a\u72b6\u6001\uff0c\u6240\u9700\u7684\u6700\u5c0f\u529f\u53ea\u662f\u4e24\u4e2a\u72b6\u6001\u4e4b\u95f4\u975e\u5e73\u8861\u81ea\u7531\u80fd\u7684\u5dee <\/sub>\u3002\u76f8\u5bf9\u4e8e\u8fd9\u4e00\u6700\u5c0f\u503c\u7684\u591a\u4f59\u529f\u662f\u8017\u6563\u7684\u6216\u4e0d\u53ef\u9006\u7684\u529f\uff0cWdiss <\/sub>\u3002<\/span><\/section>\n

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\u68461<\/strong>. <\/strong>\u975e\u5e73\u8861\u81ea\u7531\u80fd<\/strong><\/span><\/p>\n


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\u4e00\u4e2a\u7b80\u5355\u7684\u8bba\u8bc1\uff0c\u5982\u56fe\u6240\u793a\uff0c\u6709\u52a9\u4e8e\u9610\u660e\u975e\u5e73\u8861\u81ea\u7531\u80fd\u7684\u7269\u7406\u610f\u4e49[23]\u3002\u8be5\u8bba\u8bc1\u6bd4\u8f83\u4e86\u4e24\u79cd\u7b49\u6e29\u8fc7\u7a0b\u7684\u80fd\u91cf\u7279\u6027\uff0c\u8fd9\u4e24\u79cd\u8fc7\u7a0b\u5c06\u7cfb\u7edf\u4ece\u975e\u5e73\u8861\u72b6\u6001 \u03c1<\/span> \u9a71\u52a8\u5230\u5e73\u8861\u72b6\u6001 <\/sub>\u3002\u7b2c\u4e00\u79cd\u662f\u81ea\u53d1\u7684\u3001\u4e0d\u53ef\u9006\u7684\u5f1b\u8c6b\u8fc7\u7a0b\uff0c\u5176\u4e2d\u70ed\u91cf  <\/sub>\u4ece\u70ed\u5e93\u4f20\u9012\u5230\u7cfb\u7edf\uff0c\u4e14\u4e0d\u6d89\u53ca\u529f\u7684\u4ea7\u751f\u3002\u7b2c\u4e8c\u4e2a\u8fc7\u7a0b\u7531\u5916\u90e8\u5b9e\u4f53\u9a71\u52a8\uff0c\u8be5\u5916\u90e8\u5b9e\u4f53\u77ac\u65f6\u5c06\u54c8\u5bc6\u987f\u91cf\u4ece\u521d\u59cb\u503c <\/sub> \u53d8\u5316\u5230 <\/sub>\uff0c\u7136\u540e\u4ee5\u51c6\u9759\u6001\u548c\u7b49\u6e29\u7684\u65b9\u5f0f\u5c06\u54c8\u5bc6\u987f\u91cf\u6062\u590d\u5230 <\/span><\/sub> <\/span> \u7684\u521d\u59cb\u503c\u3002\u5b8c\u6210\u8fd9\u4e2a\u8fc7\u7a0b\u6240\u505a\u7684\u529f Wdriv<\/sub> \u662f\u6267\u884c\u6dec\u706b<\/span>\uff08quench\uff09<\/span>\u6240\u9700\u7684\u529f <\/sub>\uff0c\u52a0\u4e0a\u5c06\u54c8\u5bc6\u987f\u91cf\u53ef\u9006\u5730\u6062\u590d\u5230\u521d\u59cb\u503c <\/span><\/sub> \u7684\u529f\uff0c\u8fd9\u53ef\u4ee5\u8868\u793a\u4e3a\u5e73\u8861\u81ea\u7531\u80fd\u7684\u5dee\u5f02 F(\u03c1<\/span>0<\/sub>) – F(\u03c1<\/span>) \u3002\u56e0\u6b64\uff0c <\/span><\/p>\n


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<\/p>\n

\u53ef\u4ee5\u8bc1\u660e\uff0c\u8fd9\u4e2a\u529f\u662f\u6240\u6709\u53ef\u80fd\u7684\u7b49\u6e29\u8fc7\u7a0b\u9a71\u52a8\u7cfb\u7edf\u4ece \u03c1<\/span> \u5230 \u03c1<\/span>0<\/sub> \u6240\u9700\u7684\u6700\u5c0f\u529f[23]\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c-Wdriv <\/sub>\u662f\u4ece\u975e\u5e73\u8861\u72b6\u6001 \u03c1<\/span> \u4e2d\u63d0\u53d6\u7684\u6700\u5927\u529f\u3002\u56e0\u6b64\uff0c\u7ed9\u5b9a\u54c8\u5bc6\u987f\u91cf <\/span><\/sub>\uff0c\u975e\u5e73\u8861\u81ea\u7531\u80fd\u5728\u5e73\u8861\u72b6\u6001 \u03c1<\/span>0<\/sub> \u65f6\u8fbe\u5230\u6700\u5c0f\u503c\u3002\u8fd9\u610f\u5473\u7740\uff0c\u5bf9\u4e8e\u4efb\u4f55\u7edf\u8ba1\u72b6\u6001 \u03c1<\/span>\uff0c\u90fd\u6709 <\/sub>\u3002[23] \u8fd9\u4e2a\u8fc7\u7a0b\u5728\u975e\u5e73\u8861\u60c5\u51b5\u4e0b\u8d77\u7740\u4e0e\u5e73\u8861\u70ed\u529b\u5b66\u4e2d\u53ef\u9006\u8fc7\u7a0b\u76f8\u540c\u7684\u4f5c\u7528\u3002\u5b83\u5728\u64cd\u4f5c\u4e0a\u662f\u53ef\u9006\u7684\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u9a71\u52a8\u8fc7\u7a0b\u5728\u65f6\u95f4\u4e0a\u53cd\u8f6c\uff0c\u7cfb\u7edf\u5c06\u6cbf\u7740\u5b8c\u5168\u76f8\u540c\u7684\u7edf\u8ba1\u72b6\u6001\u5e8f\u5217\u53cd\u5411\u884c\u8fdb\uff0c\u4e14\u80fd\u91cf\u7279\u6027\u2014\u2014 W<\/span>driv <\/sub>\u548c Qdriv <\/sub>\u2014\u2014\u662f\u76f8\u7b49\u4e14\u65b9\u5411\u76f8\u53cd\u7684\u3002\u5b83\u8fd8\u4f7f\u63d0\u53d6\u7684\u529f\u6700\u5927\u5316\u3002<\/span><\/p>\n


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

\u5728\u8fd9\u4e2a\u53ef\u9006\u8fc7\u7a0b\u4e2d\uff0c\u603b\u71b5\u589e\u91cf <\/sub> \uff0c\u5b9a\u4e49\u4e3a\u7cfb\u7edf\u4e2d\u9999\u519c\u71b5\u53d8\u5316 \u0394<\/span>Ssys<\/sub> = S(\u03c1<\/span>0<\/sub>) – S(\u03c1<\/span>) \u548c\u70ed\u5e93\u4e2d\u71b5\u53d8\u5316 \u0394<\/span>Sres<\/sub> = -Qdriv<\/sub>\/T \u7684\u603b\u548c\uff0c\u5176\u603b\u548c\u7684\u503c\u4e3a\u96f6\uff0c\u4ece\u70ed\u5e93\u4f20\u9012\u5230\u7cfb\u7edf\u7684\u70ed\u91cf\u4e3a <\/sub><\/span>\u3002<\/span><\/p>\n


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\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u4e0d\u53ef\u9006\u7684\u81ea\u53d1\u5f1b\u8c6b\u8fc7\u7a0b\u7684\u71b5\u589e\u91cf\u4e3a\uff1a<\/span><\/p>\n


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\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u4e0d\u53ef\u9006\u8fc7\u7a0b\u4e2d\uff0c\u71b5\u589e\u91cf\u4e3a\u6b63\uff0c\u8fd9\u610f\u5473\u7740\u81ea\u7531\u80fd\u7684\u51cf\u5c11\u88ab\u6d6a\u8d39\u4e86\uff0c\u4e5f\u5c31\u662f\u8bf4\u6ca1\u6709\u529f\u88ab\u63d0\u53d6\u3002<\/span><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


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\u68462. \u4e92\u4fe1\u606f<\/span><\/strong><\/span><\/p>\n


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\u5728\u4fe1\u606f\u8bba\u4e2d\uff0c\u4e24\u4e2a\u968f\u673a\u53d8\u91cf U \u548c  V \u4e4b\u95f4\u7684\u4e92\u4fe1\u606f\u5b9a\u4e49\u4e3a[21]\uff1a<\/span><\/p>\n


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\u4e92\u4fe1\u606f\u603b\u662f\u975e\u8d1f\u7684\u3001\u5bf9\u79f0\u7684\uff0c\u5e76\u4e14\u4ec5\u5f53 U \u548c V \u7edf\u8ba1\u72ec\u7acb\u65f6\u624d\u4e3a\u96f6\u3002\u5f53\u6211\u4eec\u5c06\u5176\u91cd\u65b0\u8868\u8fbe\u4e3a\u4ee5\u4e0b\u5f62\u5f0f\u65f6\uff0c\u5b83\u7684\u610f\u4e49\u53d8\u5f97\u663e\u800c\u6613\u89c1\uff1a<\/span><\/p>\n


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\u5176\u4e2d\uff1a<\/span><\/p>\n


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<\/p>\n

\u662f\u6761\u4ef6\u71b5\u3002\u56de\u60f3\u4e00\u4e0b\uff0cH(U) \u8861\u91cf\u4e86\u6211\u4eec\u5bf9 U \u7684\u4e0d\u786e\u5b9a\u6027\uff0c\u800c H(U|V) \u8861\u91cf\u4e86\u5728\u5df2\u77e5 V \u7684\u6761\u4ef6\u4e0b\u6211\u4eec\u5bf9 U \u7684\u4e0d\u786e\u5b9a\u6027\u3002\u7531\u6b64\u53ef\u77e5\uff0c\u4e92\u4fe1\u606f\u91cf\u5316\u4e86\u5728\u83b7\u77e5 V \u65f6\uff0c\u6211\u4eec\u5bf9 U \u7684\u4e0d\u786e\u5b9a\u6027\u51cf\u5c11\u91cf\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5bbd\u6cdb\u5730\u8bf4\uff0c\u4e92\u4fe1\u606f\u662f\u5bf9\u76f8\u5173\u6027\u7684\u4e00\u4e2a\u8861\u91cf\u6307\u6807\u3002\u4e00\u4e2a\u91cd\u8981\u7684\u7279\u6b8a\u60c5\u51b5\u662f\uff0c\u5f53 U \u660e\u786e\u51b3\u5b9a V \u65f6<\/span>\uff08\u4f8b\u5982\u65e0\u8bef\u6d4b\u91cf\uff08error-free measurements\uff09\u7684\u60c5\u5f62\uff09<\/span>\uff0c\u6b64\u65f6 H(V|U) = 0 \u4e14 I(U; V) = H(V)\u3002<\/span><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


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\u4e92\u4fe1\u606f\uff08Mutual information\uff09\u4e0e\u7b2c\u4e8c\u5b9a\u5f8b\u3002<\/strong>\u6211\u4eec\u73b0\u5728\u53ef\u4ee5\u89e3\u51b3\u6784\u5efa\u4fe1\u606f\u70ed\u529b\u5b66\u7684\u7b2c\u4e00\u4e2a\u4efb\u52a1\uff1a\u8bc4\u4f30\u6d4b\u91cf\u5bfc\u81f4\u7684\u975e\u5e73\u8861\u81ea\u7531\u80fd\u53d8\u5316\u3002\u5f53\u7cfb\u7edf\u7684\u7edf\u8ba1\u72b6\u6001\u4ece \u03c1<\/span>(x) \u53d8\u4e3a \u03c1<\/span>(x|m) \u65f6\uff0c\u7cfb\u7edf\u9999\u519c\u71b5\u7684\u53d8\u5316 S(\u03c1<\/span>(x|m)) – S(\u03c1<\/span>(x))\uff0c\u5728\u6982\u7387\u4e3a pm<\/sub> \u7684\u6240\u6709\u53ef\u80fd\u7ed3\u679c\u7684\u5e73\u5747\uff0c\u53ef\u4ee5\u5199\u6210[29]\uff1a<\/span><\/section>\n

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\u5176\u4e2d H \u662f\u7b49\u5f0f\uff081\uff09\u4e2d\u7684\u9999\u519c\u71b5\uff0cI \u662f\u7cfb\u7edf\u5fae\u89c2\u72b6\u6001 X \u548c\u6d4b\u91cf\u7ed3\u679c M \u4e4b\u95f4\u7684\u4e92\u4fe1\u606f<\/span>\uff08\u89c1\u68462\uff09<\/span>\u3002<\/span><\/section>\n

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\u4e3a\u4e86\u5c06\u8fd9\u4e00\u89c2\u5bdf\u7ed3\u679c\u8f6c\u5316\u4e3a\u5173\u4e8e\u81ea\u7531\u80fd\u7684\u9648\u8ff0\uff0c\u6211\u4eec\u96c6\u4e2d\u8ba8\u8bba\u90a3\u4e9b\u4e0d\u5f71\u54cd\u54c8\u5bc6\u987f\u91cf\u6216\u7cfb\u7edf\u5fae\u89c2\u72b6\u6001\u7684\u6d4b\u91cf\u3002\u7ed3\u679c\u662f\uff0c\u6d4b\u91cf\u4e0d\u6539\u53d8\u5e73\u5747\u80fd\u91cf\uff0c\u800c\u975e\u5e73\u8861\u81ea\u7531\u80fd\u589e\u52a0\u4e86\uff1a<\/span><\/section>\n

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           (4)<\/span><\/section>\n

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\u7531\u4e8e\u4e92\u4fe1\u606f\u662f\u6b63\u503c\uff0c\u6d4b\u91cf<\/span>\uff08\u6216\u4fe1\u606f\u83b7\u53d6\uff09<\/span>\u603b\u662f\u589e\u52a0\u81ea\u7531\u80fd\uff0c\u4ece\u800c\u589e\u52a0\u53ef\u7b49\u6e29\u63d0\u53d6\u7684\u529f\u3002\u5982\u65b9\u7a0b\uff083\uff09\u6240\u793a\uff0c\u8fd9\u4e00\u589e\u52a0\u5fc5\u987b\u4f34\u968f\u7740\u529f\u7684\u4ea7\u751f\u3002\u9610\u660e\u8fd9\u4e2a\u529f\u7684\u6765\u6e90\u662f\u6211\u4eec\u7684\u7b2c\u4e8c\u4e2a\u4efb\u52a1\uff0c\u5e76\u5c06\u5728\u4e0b\u4e00\u8282\u4e2d\u8ba8\u8bba\u3002\u73b0\u5728\uff0c\u6211\u4eec\u7684\u8ba8\u8bba\u4ec5\u9650\u4e8e\u6d4b\u91cf\u5bf9\u7cfb\u7edf\u548c\u5468\u56f4\u73af\u5883\u4e4b\u95f4\u80fd\u91cf\u4f20\u9012\u7684\u5f71\u54cd\u3002<\/span><\/section>\n

<\/section>\n
\u6211\u4eec\u73b0\u5728\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u770b\u51fa\u5982\u4f55\u4fee\u6b63\u7b2c\u4e8c\u5b9a\u5f8b\u4ee5\u4f7f\u4fe1\u606f\u53d8\u5f97\u6e05\u695a\u660e\u767d\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u8003\u8651\u4e00\u4e2a\u7531\u4e00\u7ec4\u5916\u90e8\u53c2\u6570 \u03bb<\/span> \u8868\u5f81\u7684\u7cfb\u7edf\uff0c\u8fd9\u4e9b\u53c2\u6570\u7531\u201c\u5996\u201d\u8fdb\u884c\u8c03\u8282\u3002\u8d77\u521d\uff0c\u8fd9\u4e9b\u53c2\u6570\u56fa\u5b9a\u4e3a \u03bb<\/span> = A\uff0c\u7cfb\u7edf\u5904\u4e8e\u6e29\u5ea6\u4e3a T \u7684\u6b63\u5219\u5e73\u8861\u6001\u3002\u968f\u540e\uff0c\u201c\u5996\u201d\u5728\u65f6\u95f4\u95f4\u9694 [0, \u03c4<\/span>] \u5185\u6539\u53d8\u5916\u90e8\u53c2\u6570\uff0c\u5e76\u5728\u67d0\u4e2a\u4e2d\u95f4\u65f6\u523b tms<\/sub> \u8fdb\u884c\u6d4b\u91cf\uff0c\u5229\u7528\u6d4b\u91cf\u5f97\u5230\u7684\u4fe1\u606f\u8bbe\u8ba1\u4e00\u4e2a\u5916\u90e8\u53c2\u6570\u534f\u8bae<\/span>\uff08protocol\uff09<\/span>\u03bb<\/span>m<\/sup>(t) \uff0c\u5176\u4ec5\u4f9d\u8d56\u4e8e\u6d4b\u91cf\u7ed3\u679c m \u4e14\u5728 t\u2265<\/span>tms<\/sub> \u65f6\u751f\u6548\u3002\u6ce8\u610f\u5230\u6d4b\u91cf\u4f1a\u5f15\u8d77\u81ea\u7531\u80fd\u7684\u589e\u52a0<\/span>\uff08\u65b9\u7a0b (4)\uff09<\/span>\uff0c\u6211\u4eec\u53ef\u4ee5\u5e94\u7528\u65b9\u7a0b (3) \u6765\u5f97\u5230\u53cd\u9988\u8fc7\u7a0b\u7684\u7b2c\u4e8c\u5b9a\u5f8b[30]\uff1a<\/span><\/section>\n

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\u5176\u4e2d\uff0cW \u662f\u5bf9\u7cfb\u7edf\u6240\u505a\u7684\u5e73\u5747\u529f\uff0c <\/span><\/sub> \u662f\u975e\u5e73\u8861\u81ea\u7531\u80fd\u7684\u5e73\u5747\u53d8\u5316\u3002\u6ce8\u610f\uff0c\u529f\u548c\u81ea\u7531\u80fd\u7684\u53d8\u5316\u90fd\u4f9d\u8d56\u4e8e\u6d4b\u91cf\u7ed3\u679c M\uff0c\u5e76\u4e14\u65b9\u7a0b (5) \u4ec5\u5bf9\u5f97\u5230\u6d4b\u91cf\u7ed3\u679c M \u540e\u7684\u5e73\u5747\u503c\u6709\u6548\u3002\u5728\u4e00\u4e2a\u5faa\u73af\u8fc7\u7a0b\u4e2d\uff0c\u82e5 \u03bb<\/span>m<\/sup>(\u03c4<\/span>)<\/span> = \u03bb<\/span>m<\/sup>(0) = A \u5bf9\u6240\u6709 m \u90fd\u6210\u7acb\uff0c\u5e76\u4e14\u7cfb\u7edf\u88ab\u5141\u8bb8\u5f1b\u8c6b\u56de\u5230\u5e73\u8861\u6001\uff0c\u5219 <\/span><\/sub> <\/span> = 0\uff0c\u5e76\u4e14 W\u2265<\/span>-kT I(X(tms<\/sub>); M)\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u53ef\u4ee5\u63d0\u53d6\u4e00\u90e8\u5206\u529f\uff0c\u5e73\u5747\u540e\uff0c\u5176\u91cf\u4e0e\u6d4b\u91cf\u4e2d\u83b7\u5f97\u7684\u4fe1\u606f\u6210\u6b63\u6bd4\u3002\u5bf9\u4e8e\u65e0\u8bef\u5dee\u7684\u6d4b\u91cf\uff0cM \u7684\u503c\u7531 X \u76f4\u63a5\u51b3\u5b9a\uff0c\u5373\u4e92\u4fe1\u606f\u5c31\u7b49\u4e8e\u6d4b\u91cf\u7684\u9999\u519c\u71b5 I(X(t<\/span>ms<\/sub>); M) = H(M)<\/span>\uff08\u89c1\u68462\uff09<\/span>\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u897f\u62c9\u5fb7\u5f15\u64ce\u53ef\u4ee5\u8fbe\u5230\u8fd9\u4e00\u6781\u9650\uff0c\u56e0\u4e3a M \u4ee3\u8868\u7740\u5de6\u6216\u53f3\uff0c\u6982\u7387\u5747\u4e3a 1\/2\uff0c\u56e0\u6b64 H(M) = ln2 \u3002<\/span><\/section>\n

<\/section>\n
\u6da8\u843d\u5b9a\u7406<\/strong>\uff1a\u5305\u542b\u4fe1\u606f\u7684\u70ed\u529b\u5b66\u7b2c\u4e8c\u5b9a\u5f8b\uff08\u65b9\u7a0b (5)\uff09\u5bf9\u7cfb\u7edf\u7684\u5e73\u5747\u884c\u4e3a\u8bbe\u5b9a\u4e86\u9650\u5236\u3002\u7136\u800c\uff0c\u5982\u679c\u5c06\u6da8\u843d\u8003\u8651\u5728\u5185\uff0c\u8fd9\u4e2a\u4e0d\u7b49\u5f0f\u53ef\u4ee5\u8f6c\u5316\u4e3a\u7b49\u5f0f\u3002\u5047\u8bbe\u91cd\u590d\u4f9d\u8d56\u4e8e\u6d4b\u91cf\u7684\u534f\u8bae \u03bb<\/span>m<\/sup>(t) \u591a\u6b21\uff0c\u6bcf\u6b21\u90fd\u4ece\u5e73\u8861\u6001\u521d\u59cb\u5316\uff0c\u8fdb\u884c\u6d4b\u91cf\u5e76\u6267\u884c\u53cd\u9988\u3002\u70ed\u6da8\u843d\u4f1a\u4f7f\u7cfb\u7edf\u6bcf\u6b21\u5728\u5176\u76f8\u7a7a\u95f4\u4e2d\u63cf\u7ed8\u51fa\u4e0d\u540c\u7684\u5fae\u89c2\u8f68\u8ff9 \u03b3 = {x(t)}\u3002\u6211\u4eec\u5c06\u8f68\u8ff9\u548c\u7ed3\u679c m \u7684\u8054\u5408\u6982\u7387\u8868\u793a\u4e3a <\/sub>\u3002\u8fd9\u79cd\u60c5\u666f\u88ab\u79f0\u4e3a\u6b63\u5411<\/span>\uff08\u53cd\u9988\uff09<\/span>\u8fc7\u7a0b\u3002<\/span><\/section>\n

<\/section>\n
\u6da8\u843d\u5b9a\u7406\u5c06\u6b63\u5411\u8fc7\u7a0b\u4e0e\u5176\u65f6\u95f4\u53cd\u5411\u8fc7\u7a0b\u8fdb\u884c\u6bd4\u8f83\uff0c\u540e\u8005\u662f\u901a\u8fc7\u53cd\u8f6c\u9a71\u52a8\u534f\u8bae<\/span>\uff08driving protocol\uff09<\/span>\u800c\u5f97\u5230\u7684[9, 10, 12]\u3002\u8bc6\u522b\u8fd9\u79cd\u53cd\u5411\u8fc7\u7a0b\u7684\u96be\u70b9\u5728\u4e8e\u5982\u4f55\u9009\u62e9\u53cd\u5411\u534f\u8bae\uff0c\u56e0\u4e3a\u534f\u8bae \u03bb<\/span>m<\/sup>(t)<\/span> \u5728 tms<\/sub> \u4e4b\u540e\u4f9d\u8d56\u4e8e\u6d4b\u91cf\u7ed3\u679c m\u3002\u6587\u732e [31, 32] \u4e2d\u63d0\u51fa\u7684\u89e3\u51b3\u65b9\u6848\u5728\u64cd\u4f5c\u4e0a\u662f\u53ef\u884c\u7684\uff0c\u5e76\u63d0\u4f9b\u4e86\u4e00\u4e2a\u6709\u7528\u7684\u63cf\u8ff0\u529f\u548c\u4fe1\u606f\u6da8\u843d\u7684\u5b9a\u7406\u3002\u5176\u6838\u5fc3\u601d\u60f3\u662f\u4ece\u6b63\u5411\u8fc7\u7a0b\u4e2d\u7ed3\u679c\u7684\u6982\u7387\u5bc6\u5ea6 pm <\/sub>\u4e2d\u968f\u673a\u9009\u62e9 m\uff0c\u7136\u540e\u7b80\u5355\u5730\u8fd0\u884c\u5bf9\u5e94\u534f\u8bae\u7684\u5b8c\u5168\u53cd\u5411\u8fc7\u7a0b <\/sub>\uff0c\u65e0\u9700\u8fdb\u884c\u6d4b\u91cf\u3002<\/span><\/section>\n

<\/section>\n
\u8be6\u7ec6\u7684\u6da8\u843d\u5b9a\u7406\u5c06 <\/span><\/sub> \u4e0e\u53cd\u5411\u8fc7\u7a0b\u4e2d\u89c2\u5bdf\u5230\u53cd\u5411\u8f68\u8ff9  <\/sub>\u7684\u6982\u7387   <\/sub>\u8fdb\u884c\u6bd4\u8f83\uff0c\u5176\u4e2d * \u8868\u793a\u52a8\u91cf\u53cd\u5411[31, 32]\uff1a<\/span><\/section>\n

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         (6)<\/span><\/section>\n

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\u8fd9\u91cc\uff0cw \u662f\u6cbf\u8f68\u8ff9 \u03b3<\/span> \u5bf9\u7cfb\u7edf\u6240\u505a\u7684\u968f\u673a\u529f\uff0c\u540c\u65f6\u5f15\u5165\u4e86\u4e00\u4e2a\u4f9d\u8d56\u4e8e\u8f68\u8ff9\u7684\u4e92\u4fe1\u606f\uff1a<\/span><\/section>\n

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<\/p>\n


<\/span><\/section>\n
\u5176\u5728 \u03b3<\/span> \u548c m \u4e0b\u7684\u5e73\u5747\u540e\u7ed3\u679c\u4e3a\u4e92\u4fe1\u606f\uff0c\u5373 <\/sub><\/span>\uff08\u89c1 \u6846 2\uff09<\/span>\u3002<\/span><\/section>\n

<\/section>\n
\u65b9\u7a0b (6) \u4e2d\u7684\u8be6\u7ec6\u6da8\u843d\u5b9a\u7406\u5c06\u5e95\u5c42\u5fae\u89c2\u52a8\u529b\u5b66\u7684\u57fa\u672c\u65f6\u95f4\u53cd\u6f14\u5bf9\u79f0\u6027\u4e0e\u70ed\u529b\u5b66\u8054\u7cfb\u8d77\u6765\u3002\u57fa\u4e8e\u8fd9\u4e00\u5173\u7cfb\uff0c\u53ef\u4ee5\u5bb9\u6613\u5730\u63a8\u5bfc\u51fa\u5e26\u6709\u53cd\u9988\u7684\u5404\u79cd\u6da8\u843d\u5b9a\u7406[32-34]\u3002\u4f8b\u5982\uff0c\u79ef\u5206\u6da8\u843d\u5b9a\u7406<\/strong>\u4e3a\uff1a<\/span><\/section>\n

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\u8fd9\u4e00\u7ed3\u679c\u5df2\u901a\u8fc7\u7531\u5355\u7535\u5b50\u7ec4\u6210\u7684\u897f\u62c9\u5fb7\u5f15\u64ce\u7684\u5b9e\u9a8c\u9a8c\u8bc1[14]\u3002\u5c06\u8a79\u68ee\u4e0d\u7b49\u5f0f<\/span>\uff08Jensen’s inequality\uff09<\/span>\u5e94\u7528\u4e8e\u65b9\u7a0b (7)\uff0c\u5373\u53ef\u5f97\u5230\u4e0d\u7b49\u5f0f (5)\u3002<\/span><\/section>\n

<\/section>\n
\u65b9\u7a0b (6) \u8868\u660e\uff0c\u5982\u679c\u53cd\u9988\u8fc7\u7a0b\u4e0e\u5176\u53cd\u5411\u8fc7\u7a0b\u4e0d\u53ef\u533a\u5206\u2014\u2014\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u8fc7\u7a0b\u662f\u53ef\u9006\u7684\uff0c\u90a3\u4e48\u4e0d\u7b49\u5f0f (5) \u53ef\u4ee5\u5728\u5355\u4e2a\u8f68\u8ff9\u5c42\u9762\u4e0a\u8fbe\u5230\u6781\u9650\u3002\u8fd9\u4e00\u89c2\u5bdf\u7ed3\u679c\u63d0\u4f9b\u4e86\u4e00\u79cd\u901a\u7528\u65b9\u6cd5\uff0c\u7528\u4e8e\u8bbe\u8ba1\u6700\u4f18\u7684\u53cd\u9988\u534f\u8bae\uff1a\u901a\u8fc7\u627e\u5230\u5728\u53cd\u5411\u8fd0\u884c\u65f6\u770b\u8d77\u6765\u50cf\u53cd\u9988\u8fc7\u7a0b\u7684\u8fc7\u7a0b\u3002\u7279\u522b\u8981\u63d0\u5230\u7684\u662f\uff0c\u53cd\u5411\u8fc7\u7a0b\u5fc5\u987b\u5728\u6d4b\u91cf\u540e\u51c6\u5907\u597d\u6b63\u786e\u7684\u76f8\u5e94\u72b6\u6001\u3002\u5982\u679c\u80fd\u591f\u627e\u5230\u8fd9\u6837\u7684\u53cd\u5411\u8fc7\u7a0b\uff0c\u901a\u8fc7\u53cd\u8f6c\u5b83\uff0c\u5c31\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u53cd\u9988\u534f\u8bae\u3002\u8fd9\u79cd\u5236\u5907\u65b9\u6cd5\u5df2\u88ab\u7528\u4e8e\u8bbe\u8ba1\u591a\u7c92\u5b50\u897f\u62c9\u5fb7\u5f15\u64ce\u7684\u6700\u4f18\u534f\u8bae[35, 36]\u3002\u5176\u4e2d\u4e00\u4e2a\u7248\u672c\u7684\u5236\u5907\u65b9\u6cd5\u7c7b\u4f3c\u4e8e \u6846 1 \u4e2d\u63cf\u8ff0\u7684\u6700\u4f18\u529f\u534f\u8bae\uff0c\u5176\u4ed6\u53ef\u9006\u534f\u8bae\u5219\u88ab\u7528\u4e8e\u53ef\u53cd\u9988\u63a7\u5236\u8c10\u632f\u5b50\u6a21\u578b\u4e2d[37, 38]\u3002<\/span><\/section>\n

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\u56fe2. \u4e00\u4e2a\u5b58\u50a8\u5668\u7684\u73a9\u5177\u6a21\u578b\u3002\u4e00\u4e2a\u5e03\u6717\u7c92\u5b50\u53ef\u4ee5\u88ab\u7a33\u5b9a\u7684\u56da\u7981\u5728\u4f4d\u7f6e\u5750\u6807\u4e3a y \u7684\u53cc\u52bf\u9631\u7684\u5de6\u9631\u6216\u53f3\u9631\u4e2d\uff0c\u5206\u522b\u8868\u793a\u4e00\u4e2a\u6bd4\u7279\u7684\u4ecb\u89c2\u4fe1\u606f\u6001 m=0 \u548c m=1 \u3002\u8fd9\u4e2a\u5b58\u50a8\u5668\u53ef\u4ee5\u662f\u5bf9\u79f0\u7684 (a) \u6216\u4e0d\u5bf9\u79f0\u7684 (b)\u3002<\/span><\/p>\n


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\u4fe1\u606f\u7684\u7269\u7406\u672c\u8d28<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


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\u5b58\u50a8\u4e0e<\/strong>\u5170\u9053\u5c14<\/strong>\u539f\u7406<\/strong>\uff1a\u4e4d\u4e00\u770b\uff0c\u4fe1\u606f\u662f\u4e00\u4e2a\u62bd\u8c61\u7684\u91cf\uff0c\u4e0e\u5176\u7269\u7406\u5b9e\u73b0\u65b9\u5f0f\u65e0\u5173\u3002\u7136\u800c\uff0c\u5f53\u6211\u4eec\u8fdb\u884c\u6d4b\u91cf\u65f6\uff0c\u7ed3\u679c\u5fc5\u987b\u8bb0\u5f55\u5728\u67d0\u79cd\u7269\u7406\u7cfb\u7edf\u4e2d\uff0c\u65e0\u8bba\u662f\u5199\u5728\u7eb8\u4e0a\u8fd8\u662f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u7684\u786c\u76d8\u4e2d\u3002\u8fd9\u4e00\u89c2\u70b9\u6700\u65e9\u7531\u897f\u62c9\u5fb7\u63d0\u51fa[4]\uff0c\u4f46\u5170\u9053\u5c14\u7528\u4ed6\u7684\u540d\u8a00\u5c06\u5176\u8868\u8fbe\u5f97\u6700\u4e3a\u7b80\u6d01\uff1a\u201c\u4fe1\u606f\u662f\u7269\u7406\u7684<\/strong>\u3002\u201d[39]<\/span><\/section>\n

<\/section>\n
\u4e3a\u4e86\u6784\u5efa\u4e00\u4e2a\u5b58\u50a8\u5668\uff0c\u4e00\u4e2a\u7cfb\u7edf\u5fc5\u987b\u5177\u6709\u591a\u4e2a\u53ef\u533a\u5206\u7684\u201c\u4fe1\u606f\u72b6\u6001\u201d\u6765\u5b58\u50a8\u4fe1\u606f\u3002\u4e3a\u4e86\u5b9e\u73b0\u53ef\u9760\u4e14\u4e0d\u8870\u9000\u7684\u4fe1\u606f\u5b58\u50a8\uff0c\u8fd9\u4e9b\u72b6\u6001\u5fc5\u987b\u5177\u6709\u8f83\u957f\u7684\u5bff\u547d\uff0c\u5bf9\u73af\u5883\u566a\u58f0\u5177\u6709\u9c81\u68d2\u6027\uff0c\u5e76\u5728\u76f8\u540c\u7684\u5916\u90e8\u7ea6\u675f\u4e0b\u5f97\u4ee5\u5b9e\u73b0\u3002\u4ece\u7edf\u8ba1\u529b\u5b66\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u8fd9\u4e9b\u8981\u6c42\u610f\u5473\u7740\u4e00\u4e2a\u5177\u6709\u5fae\u89c2\u72b6\u6001 y \u7684\u7cfb\u7edf\u82e5\u8981\u4f5c\u4e3a\u5b58\u50a8\u5668\uff0c\u5219\u5176\u54c8\u5bc6\u987f\u91cf <\/sub> \u4e0b\u5fc5\u987b\u5177\u6709\u591a\u4e2a\u4e0d\u540c\u7684\u4e9a\u7a33\u6001\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5404\u6001\u904d\u5386\u6027<\/span>\uff08ergodicity\uff09<\/span>\u5fc5\u987b\u662f\u7834\u7f3a\u7684\u2014\u2014\u6216\u8005\u5728\u5b58\u50a8\u53ef\u9760\u7684\u65f6\u95f4\u5c3a\u5ea6\u4e0a\u662f\u6709\u6548\u7834\u7f3a\u7684\u3002\u5728\u5b9e\u8df5\u4e2d\uff0c\u8fd9\u610f\u5473\u7740\u5176\u76f8\u7a7a\u95f4 \u0393<\/span> \u88ab\u5206\u5272\u4e3a\u591a\u4e2a\u904d\u5386\u533a\u57df {\u0393<\/span>m<\/sub>}\uff0c\u6bcf\u4e2a\u533a\u57df\u5bf9\u5e94\u4e00\u4e2a\u4fe1\u606f\u72b6\u6001 m [16]\u3002\u8fd9\u79cd\u904d\u5386\u6027\u7684\u4e27\u5931\u53ef\u4ee5\u901a\u8fc7\u96c6\u4f53\u5750\u6807\u4e2d\u7684\u76f8\u53d8\u6765\u5b9e\u73b0\uff0c\u4f8b\u5982\u5728\u6807\u51c6\u78c1\u5b58\u50a8\u4e2d\uff0c\u5c0f\u578b\u94c1\u78c1\u7574\u7684\u78c1\u5316\uff1b\u6216\u8005\u901a\u8fc7\u63d0\u9ad8\u5206\u9694\u5fae\u89c2\u81ea\u7531\u5ea6\u7684\u9ad8\u81ea\u7531\u80fd\u5c4f\u969c\u6765\u5b9e\u73b0\uff0c\u4f8b\u5982\u5728\u5355\u7535\u5b50\u5b58\u50a8\u6216 DNA \u78b1\u57fa\u914d\u5bf9\u4e2d\u3002<\/span><\/section>\n

<\/section>\n
\u5b58\u50a8\u5668\u7684\u7edf\u8ba1\u72b6\u6001\u7531\u5904\u4e8e\u5176\u904d\u5386\u533a\u57df\u4e4b\u4e00\u7684\u6982\u7387 pm<\/sub> \u8868\u5f81\u3002\u8fdb\u4e00\u6b65\u5047\u8bbe\u5b58\u50a8\u5668\u5728\u5c40\u90e8\u5e73\u8861\u6001\u4e0b\u8fd0\u884c\uff0c\u90a3\u4e48\u5728\u7ed9\u5b9a\u4fe1\u606f\u72b6\u6001 m \u7684\u6761\u4ef6\u4e0b\uff0c\u8be5\u6982\u7387\u662f\u76f8\u7a7a\u95f4\u4e2d\u4e0e m \u76f8\u914d\u7684\u533a\u57df \u0393<\/span>m<\/sub> \u4e0a\u7684\u6b63\u5219\u5206\u5e03\u3002\u8be5\u5206\u5e03\u5177\u6709\u5e73\u5747\u80fd\u91cf Em<\/sub> \u548c\u9999\u519c\u71b5 Sm<\/sub>\u3002\u5b58\u50a8 M \u7684\u975e\u5e73\u8861\u81ea\u7531\u80fd\u53ef\u4ee5\u7528\u8fd9\u4e9b\u91cf\u8868\u793a\u4e3a\uff1a<\/span><\/section>\n

<\/span><\/section>\n

          (8)<\/span>    <\/p>\n


<\/p>\n

\u5176\u4e2d\uff0cFm<\/sub> = Em<\/sub> – TSm <\/sub>\u662f\u76f8\u5e94\u72b6\u6001\u7684\u81ea\u7531\u80fd\u3002\u8fd9\u91cc\u5b58\u50a8\u5668\u7684\u71b5 S(\u03c1<\/span>) \u88ab\u5206\u89e3\u4e3a\u5404\u4e2a\u4fe1\u606f\u6001\u4e2d\u7684\u9999\u519c\u71b5\uff0c<\/sub>\uff0c\u548c\u4ed6\u4eec\u5404\u81ea\u5185\u90e8\u71b5 Sm <\/sub>\u7684\u5e73\u5747\u503c\u3002<\/span><\/section>\n

<\/section>\n
\u8fd9\u79cd\u88c5\u7f6e\u7684\u4e00\u4e2a\u76f4\u89c2\u793a\u4f8b\u662f\u56fe 2 \u4e2d\u63cf\u8ff0\u7684\u5355\u6bd4\u7279\u5b58\u50a8\u5668\u73a9\u5177\u6a21\u578b\u3002\u5b58\u50a8\u5668\u7531\u4e00\u4e2a\u5e03\u6717\u7c92\u5b50\u548c\u4e00\u4e2a\u53cc\u52bf\u9631\u7ec4\u6210\u3002\u5f53\u52bf\u5792\u8fdc\u9ad8\u4e8e\u70ed\u8fd0\u52a8\u80fd\u91cf\u65f6\uff0c\u7c92\u5b50\u5c06\u5728\u4efb\u4e00\u52bf\u9631\u4e2d\u505c\u7559\u8f83\u957f\u65f6\u95f4\u3002\u56e0\u6b64\uff0c\u7c92\u5b50\u4f4d\u4e8e\u5de6\u4fa7\u6216\u53f3\u4fa7\u52bf\u9631\u53ef\u4ee5\u5206\u522b\u4f5c\u4e3a\u6bd4\u7279\u7684\u7a33\u5b9a\u4fe1\u606f\u72b6\u6001\u201c0\u201d\u548c\u201c1\u201d\u3002\u56fe 2a \u4e2d\u7684\u5bf9\u79f0\u52bf\u4f7f\u5f97 F0<\/sub> = F1<\/sub>\u3002\u53e6\u4e00\u79cd\u60c5\u51b5\u662f\u56fe 2b \u4e2d\u7684\u4e0d\u5bf9\u79f0\u52bf\uff0c\u6b64\u65f6 F<\/span>0<\/sub> \u2260 F<\/span>1<\/sub>\uff0c\u4e24\u4e2a\u52bf\u9631\u7684\u5e73\u5747\u80fd\u91cf\u548c\u76f8\u7a7a\u95f4\u4f53\u79ef\u4e0d\u540c\u3002<\/span><\/section>\n

<\/section>\n
\u73b0\u5728\u6211\u4eec\u5047\u8bbe\uff0c\u5728\u5bf9\u5b58\u50a8\u5668\u8fdb\u884c\u4efb\u4f55\u64cd\u4f5c\u540e\uff0c\u5176\u8fc7\u7a0b\u59cb\u7ec8\u4ee5\u76f8\u540c\u7684\u521d\u59cb\u54c8\u5bc6\u987f\u91cf <\/span> <\/sub>\u7ed3\u675f\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u552f\u4e00\u76f8\u5173\u7684\u72b6\u6001\u662f\u4fe1\u606f\u72b6\u6001\uff0c\u5e76\u4e14\u64cd\u4f5c\u5b58\u50a8\u5668\u53ca\u5176\u5b58\u50a8\u4fe1\u606f\u7684\u80fd\u91cf\u5b66\u7279\u5f81\u5e94\u8be5\u4ec5\u7528\u672f\u8bed pm<\/sub> \u6765\u8868\u8fbe\u3002\u7279\u522b\u662f\uff0c\u5c06\u5b58\u50a8\u5668\u7684\u7edf\u8ba1\u72b6\u6001\u4ece\u5206\u5e03\u4e3a p<\/span>m<\/sub> \u7684 M \u6539\u53d8\u4e3a\u5206\u5e03\u4e3a p‘<\/span><\/span>m<\/sub> \u7684 M‘<\/span> \u6240\u9700\u7684\u5e73\u5747\u529f\uff0c\u53ef\u4ee5\u6839\u636e\u7b2c\u4e8c\u5b9a\u5f8b (3) \u8868\u793a\u4e3a\uff1a<\/span><\/section>\n

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<\/section>\n

<\/span><\/section>\n
\u5bf9\u4e8e\u5bf9\u79f0\u7684\u5b58\u50a8\u5668\uff0c\u6839\u636e\u65b9\u7a0b (8)\uff0c\u8fd9\u4ec5\u4e0e\u4fe1\u606f\u5185\u5bb9\u7684\u53d8\u5316\u6709\u5173\uff0c\u5373  W\u2265kT (H(M) – H(M’))\u3002<\/span><\/section>\n

<\/section>\n
\u4e00\u4e2a\u8457\u540d\u7684\u7279\u6b8a\u60c5\u51b5\u662f\u5170\u9053\u5c14\u539f\u7406\uff0c\u5b83\u9650\u5236\u4e86\u5728\u5bf9\u79f0\u5b58\u50a8\u5668\u4e2d\u91cd\u7f6e<\/span>\uff08\u6216\u64e6\u9664\uff09<\/span>\u4e00\u4e2a\u968f\u673a\u6bd4\u7279\u6240\u9700\u91ca\u653e\u7684\u70ed\u91cf[40]\u3002\u91cd\u7f6e\u5b58\u50a8\u662f\u6307\uff0c\u65e0\u8bba\u521d\u59cb\u6761\u4ef6\u5982\u4f55\uff0c\u6240\u6709\u4fe1\u606f\u72b6\u6001 m \u90fd\u88ab\u9a71\u52a8\u5230\u4e00\u4e2a\u9884\u9009\u7684\u6807\u51c6\u72b6\u6001\uff0c\u4f8b\u5982 m = 0\uff0c\u5373 p‘<\/span><\/span>0<\/sub> = 1\uff0c\u800c\u6240\u6709\u5176\u4ed6\u7684 p‘<\/span><\/span>m<\/sub> = 0\u3002\u5bf9\u4e8e\u5bf9\u79f0\u7684\u5b58\u50a8\u5668\uff0c\u6839\u636e\u65b9\u7a0b(9)\uff0c\u6267\u884c\u8fd9\u4e00\u201c\u5f52\u96f6\u201d\u64cd\u4f5c\u6240\u9700\u7684\u6700\u5c0f\u529f\u4e3a Wreset <\/sub>\u2265 kTH(M)\uff0c\u56e0\u4e3a H(M’) = 0\u3002\u5bf9\u4e8e\u4e00\u4e2a\u968f\u673a\u6bd4\u7279\uff0cp0<\/sub> = p1<\/sub> = 1\/2\uff0c\u6211\u4eec\u5f97\u5230\u4e86\u5170\u9053\u5c14\u7684\u539f\u59cb\u9650\u5236\uff0c\u8be5\u9650\u5236\u5df2\u901a\u8fc7\u56fe 2 \u4e2d\u73a9\u5177\u6a21\u578b\u5b58\u50a8\u5668\u7684\u5b9e\u9a8c\u5b9e\u73b0\u5f97\u5230\u4e86\u9a8c\u8bc1[15, 41, 42]\u3002\u5bf9\u4e8e\u4e0d\u5bf9\u79f0\u5b58\u50a8\u5668\uff0c<\/span> <\/sub>\u4e0d\u4e00\u5b9a\u7b49\u4e8e -kTH(M)\uff0c\u5e7f\u4e49\u5170\u9053\u5c14\u539f\u7406\u4e3a[43, 44]\uff1a<\/span><\/section>\n

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<\/p>\n


<\/span><\/section>\n
\u5f53\u91cd\u7f6e\u8fc7\u7a0b\u5728\u70ed\u529b\u5b66\u4e0a\u662f\u53ef\u9006\u65f6\uff0c\u7b49\u53f7\u6210\u7acb\u3002\u8fd9\u5e76\u4e0d\u4e0e\u91cd\u7f6e\u7684\u903b\u8f91\u4e0d\u53ef\u9006\u6027\u76f8\u77db\u76fe\uff0c\u540e\u8005\u610f\u5473\u7740\u4fe1\u606f\u72b6\u6001\u7684\u71b5 H(M) \u51cf\u5c11[44]\u3002\u5bf9\u4e8e\u6709\u9650\u65f6\u95f4\u5185\u7684\u64e6\u9664\u64cd\u4f5c\uff0c\u5f53\u65e0\u6cd5\u8fbe\u5230\u65b9\u7a0b (10) \u7684\u6781\u9650\u65f6\uff0c\u4e00\u4e9b\u6a21\u578b\u7814\u7a76\u5728\u9610\u660e\u6548\u7387\u6743\u8861\u65b9\u9762\u53d6\u5f97\u4e86\u6210\u6548[45, 46]\u3002\u540c\u6837\uff0c\u53ef\u4ee5\u4f7f\u7528\u79ef\u5206\u6da8\u843d\u5b9a\u7406\u5728\u8f68\u8ff9\u5c42\u9762\u4e0a\u5206\u6790\u8fd9\u4e00\u9650\u5236[47]\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u4e0e\u201c\u5f52\u96f6\u201d\u76f8\u53cd\u7684\u8fc7\u7a0b\u662f\u901a\u8fc7\u589e\u52a0\u5b58\u50a8\u5668\u7684\u65e0\u5e8f\u6027\u6765\u63d0\u53d6\u529f\u3002\u4f8b\u5982\uff0c\u4e00\u4e2a\u5177\u6709\u4f4e\u9999\u519c\u71b5\u7684 N-\u6bd4\u7279\u5bf9\u79f0\u5b58\u50a8\u5668\u53ef\u4ee5\u88ab\u7f6e\u4e8e\u4e00\u4e2a\u65b0\u7684\u9ad8\u71b5\u72b6\u6001 M’\uff0c\u4ece\u800c\u63d0\u53d6\u529f\u7684\u91cf\u4e3a  kT(H(M’) – H(M))\u3002\u8fd9\u79cd\u5c06\u6709\u5e8f\u5b58\u50a8\u5668\u7528\u4f5c\u201c\u71c3\u6599\u201d\u7684\u65b9\u5f0f\u6700\u521d\u7531 Bennett \u63d0\u51fa[8]\uff0c\u4f46\u6700\u8fd1\u88ab Mandal \u548c Jarzynski \u901a\u8fc7\u4e00\u4e2a\u663e\u5f0f\u6a21\u578b\u5b9e\u73b0[48]\u3002\u8fd9\u4e00\u8fdb\u5c55\u63a8\u52a8\u4e86\u9488\u5bf9\u8fd9\u7c7b\u6709\u5e8f\u5b58\u50a8\u5668\u7684\u65b0\u7406\u8bba\u6846\u67b6\u7684\u5efa\u7acb\uff0c\u8fd9\u4e9b\u5b58\u50a8\u5668\u73b0\u5728\u88ab\u79f0\u4e3a\u4fe1\u606f\u5e93\uff0c\u4e0e\u70ed\u5e93\u6216\u5316\u5b66\u5e93\u7b49\u5176\u4ed6\u70ed\u529b\u5b66\u50a8\u5e93\u4eab\u6709\u5e73\u7b49\u7684\u5730\u4f4d[49-55]\u3002<\/span><\/section>\n

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<\/section>\n

\u56fe 3<\/strong>. <\/strong>\u6d4b\u91cf\u3001\u53cd\u9988\u548c\u91cd\u7f6e\u7684\u793a\u610f\u56fe\u3002<\/strong>\u6d4b\u91cf\u662f\u4e00\u79cd\u5c06\u5b58\u50a8\u5668\u4ece M’\u2192M \u6539\u53d8\uff0c\u83b7\u53d6\u5173\u4e8e\u7cfb\u7edf X \u7684\u4fe1\u606f I \u7684\u70ed\u529b\u5b66\u8fc7\u7a0b\u3002\u5728\u53cd\u9988\u8fc7\u7a0b\u4e2d\uff0c\u7cfb\u7edf\u5728\u5b58\u50a8\u5668\u56fa\u5b9a\u7684\u60c5\u51b5\u4e0b\u505aX\u2192<\/span>X’ \u6f14\u5316\uff0c\u5229\u7528\u5173\u8054\u6765\u63d0\u53d6\u529f -W\u3002\u6700\u540e\uff0c\u5b58\u50a8\u5668\u901a\u8fc7\u4e0e X \u65e0\u5173\u7684\u534f\u8bae\u91cd\u7f6e\u4e3a\u521d\u59cb\u72b6\u6001 M’\u3002<\/span><\/p>\n


<\/section>\n
\u6d4b\u91cf\u7684\u4ee3\u4ef7<\/strong><\/span><\/section>\n

<\/strong><\/span><\/section>\n
\u6211\u4eec\u73b0\u5728\u8ba8\u8bba\u4fe1\u606f\u70ed\u529b\u5b66\u7684\u7b2c\u4e8c\u4e2a\u4efb\u52a1\uff0c\u5373\u7406\u89e3\u7b2c\u4e8c\u5b9a\u5f8b\u4e2d\u4e92\u4fe1\u606f\u7684\u8d77\u6e90<\/span>\uff08\u65b9\u7a0b5\uff09<\/span>\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u5c06\u7814\u7a76\u7684\u7cfb\u7edf X \u548c\u4e00\u4e2a\u5b58\u50a8\u5668 XM \u4f5c\u4e3a\u4e00\u4e2a\u8054\u5408\u8d85\u7cfb\u7edf X\uff0c\u5e76\u5206\u6790\u5176\u5728\u6d4b\u91cf\u548c\u53cd\u9988\u4e24\u4e2a\u70ed\u529b\u5b66\u8fc7\u7a0b\u4e2d\u7684\u80fd\u91cf\u5b66\uff0c\u5982\u56fe 3 \u6240\u793a\u3002\u8fdb\u4e00\u6b65\u7684\u7b80\u5316\u5047\u8bbe\u662f\uff0c\u5b58\u50a8\u5668\u548c\u7cfb\u7edf\u4ec5\u5728\u6267\u884c\u6d4b\u91cf\u548c\u53cd\u9988\u64cd\u4f5c\u65f6\u8fdb\u884c\u76f8\u4e92\u4f5c\u7528\u2014\u2014\u5426\u5219\uff0c\u7cfb\u7edf\u7684\u603b\u54c8\u5bc6\u987f\u91cf\u4e3a <\/sub>\u3002\u5229\u7528\u4e92\u4fe1\u606f\u7684\u5b9a\u4e49<\/span>\uff08\u89c1\u6846 2\uff09<\/span>\uff0c\u5f53\u6ca1\u6709\u4ea4\u4e92\u80fd\u91cf\u65f6\uff0c\u590d\u5408\u7cfb\u7edf\u7684\u975e\u5e73\u8861\u81ea\u7531\u80fd<\/span>\uff08\u65b9\u7a0b8\uff09<\/span>\u53ef\u4ee5\u7b80\u5355\u5730\u8868\u793a\u4e3a\uff1a<\/span><\/section>\n

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<\/p>\n


<\/span><\/section>\n
\u4ece\u8fd9\u4e2a\u65b9\u7a0b\u4e2d\u53ef\u4ee5\u8f7b\u677e\u83b7\u5f97\u6d4b\u91cf\u548c\u53cd\u9988\u7684\u80fd\u91cf\u5b66\u7279\u5f81\u3002<\/span><\/section>\n

<\/section>\n
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<\/span><\/section>\n

<\/p>\n


<\/span><\/section>\n
\u5176\u4e2d\uff0c <\/sub>\u662f\u5b58\u50a8\u5668\u81ea\u7531\u80fd\u7684\u53d8\u5316\u3002\u65b9\u7a0b (3) \u4e2d\u7684\u7b2c\u4e8c\u5b9a\u5f8b\u8868\u660e\uff0c\u65b9\u7a0b (12) \u7ed9\u51fa\u7684\u81ea\u7531\u80fd\u589e\u52a0\u9700\u8981\u505a\u529f[25, 43, 56, 57]\uff1a<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n

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<\/span><\/section>\n
<\/section>\n

<\/span><\/section>\n
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<\/span><\/section>\n
<\/section>\n

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<\/span><\/section>\n

<\/p>\n


<\/span><\/section>\n
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\u52a8\u6001\u4fe1\u606f\u6d41<\/strong><\/span><\/h3>\n


<\/strong><\/span><\/p>\n

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         (15)<\/span><\/section>\n

<\/span><\/section>\n
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\u5c55\u671b\u3001\u6269\u5c55\u4e0e\u524d\u666f<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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\u5173\u4e8e\u91cf\u5b50\u6d4b\u91cf\u548c\u53cd\u9988\u7684\u70ed\u529b\u5b66\u4e5f\u6709\u5927\u91cf\u7814\u7a76\uff0c\u5305\u62ec\u91cf\u5b50\u897f\u62c9\u5fb7\u5f15\u64ce[72, 73]\u3002\u5728\u8fd9\u65b9\u9762\uff0c\u4eba\u4eec\u5df2\u7ecf\u5f97\u5230\u4e86\u7b2c\u4e8c\u5b9a\u5f8b[30, 43, 74]\u548c\u7528\u4e8e\u6d4b\u91cf\u4e0e\u53cd\u9988\u7684\u6da8\u843d\u5b9a\u7406[75-77]\u7684\u91cf\u5b50\u5e7f\u4e49\u5f62\u5f0f\uff0c\u4ece\u800c\u80fd\u591f\u5728\u6b64\u573a\u666f\u4e0b\u5bf9\u7ea0\u7f20\u548c\u91cf\u5b50\u4e0d\u548c\u8c10\u8fdb\u884c\u4e00\u81f4\u7684\u63cf\u8ff0[78-80]\u3002<\/span><\/section>\n

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\u6211\u4eec\u5728\u6b64\u4e3b\u8981\u5173\u6ce8\u9ea6\u514b\u65af\u97e6\u5996\u7684\u4fe1\u606f\u65b9\u9762\u3002\u7136\u800c\uff0c\u9ea6\u514b\u65af\u97e6\u5996\u7684\u539f\u59cb\u8868\u8ff0\u4e5f\u53ef\u4ee5\u88ab\u89c6\u4e3a\u662f\u4e00\u79cd\u65e0\u9700\u660e\u663e\u80fd\u91cf\u6d88\u8017\u5373\u53ef\u8fdb\u884c\u72b6\u6001\u95f4\u8f6c\u53d8\u7684\u7cfb\u7edf\u3002\u8fd9\u79cd\u65b9\u6cd5\u6700\u521d\u7531 Smoluchowski [5]\u548c Feynman [6]\u63a2\u7d22\uff0c\u4ed6\u4eec\u8003\u8651\u4e86\u4e00\u79cd\u80fd\u591f\u901a\u8fc7\u68d8\u8f6e\u673a\u5236\u6821\u6b63\u70ed\u6da8\u843d\u7684\u81ea\u4e3b\u9ea6\u514b\u65af\u97e6\u5996\u7684\u53ef\u80fd\u6027\u3002\u7136\u800c\uff0c\u8fd9\u79cd\u5b9e\u73b0\u65b9\u5f0f\u4ec5\u5728\u7cfb\u7edf\u53ef\u4ee5\u88ab\u4fdd\u6301\u5728\u975e\u5e73\u8861\u72b6\u6001\u65f6\u624d\u6709\u6548\uff0c\u4f8b\u5982\u5b58\u5728\u4e24\u4e2a\u4e0d\u540c\u7684\u70ed\u6d74[6]\u6216\u9ad8\u81ea\u7531\u80fd\u5316\u5b66\u50a8\u5e93\u3002\u5728 Feynman \u7684\u5206\u6790\u4e4b\u540e\u7684\u6570\u5e74\u95f4\uff0c\u8fd9\u4e9b\u6784\u9020\u2014\u2014\u73b0\u5728\u88ab\u79f0\u4e3a\u975e\u5e73\u8861\u5e03\u6717\u68d8\u8f6e\u2014\u2014\u5df2\u4f5c\u4e3a\u4e00\u79cd\u65b0\u578b\u7684\u8fd0\u8f93\u673a\u5236\u88ab\u5e7f\u6cdb\u7814\u7a76[81]\u3002\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u8f6c\u53d8\u4e2d\u7684\u504f\u5411\u6027\u53ef\u4ee5\u7528\u4fe1\u606f\u7684\u672f\u8bed\u6765\u89e3\u91ca\uff0c\u4f8b\u5982\u5728\u5316\u5b66\u5f00\u5173\u4e2d[82]\uff0c\u5176\u8f6c\u53d8\u4f9d\u8d56\u4e8e\u5206\u5b50\u590d\u5408\u7269\u4e2d\u8f6e\u70f7\u73af\u7684\u4f4d\u7f6e\u3002\u6700\u8fd1\uff0c\u5229\u7528\u968f\u673a\u70ed\u529b\u5b66\u5206\u6790\u4e86\u4e00\u4e2a\u6253\u7834\u7ec6\u81f4\u5e73\u8861\u7684\u901a\u7528\u9ea6\u514b\u65af\u97e6\u5996[83]\u3002\u7136\u800c\uff0c\u6709\u8da3\u7684\u662f\uff0c\u8f6c\u53d8\u7684\u7b80\u5355\u504f\u5411\u6027\u4e0d\u8db3\u4ee5\u51b3\u5b9a\u7cfb\u7edf\u7684\u70ed\u529b\u5b66\u3002\u6211\u4eec\u5df2\u7ecf\u8868\u660e\uff0c\u76f8\u540c\u7684\u504f\u5411\u6027\u548c\u76f8\u540c\u7684\u52a8\u529b\u5b66\u53ef\u4ee5\u7531\u5316\u5b66\u71c3\u6599\u6216\u9ea6\u514b\u65af\u97e6\u5996\u5f15\u53d1\uff0c\u4f46\u5176\u70ed\u529b\u5b66\u5374\u622a\u7136\u4e0d\u540c[25]\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\u7684\u5996\u80fd\u591f\u4ee5\u6bd4\u5316\u5b66\u71c3\u6599\u5c11\u5f97\u591a\u7684\u71b5\u589e\u6765\u5b9e\u73b0\u76f8\u540c\u7684\u504f\u5dee\u3002<\/span><\/section>\n

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2. <\/strong>\u751f\u7269\u9ea6\u514b\u65af\u97e6\u5996\uff1a\u751f\u547d\u5982\u4f55\u5229\u7528\u4fe1\u606f\u4ea7\u751f\u79e9\u5e8f<\/a><\/strong><\/span><\/section>\n
3. <\/strong>\u524d\u6cbf\u8fdb\u5c55\uff1a\u5b8f\u89c2\u9ea6\u514b\u65af\u97e6\u5996\u80cc\u540e\u7684\u70ed\u529b\u5b66\u51e0\u4f55\u539f\u7406<\/a><\/strong><\/span><\/section>\n
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