{"id":39252,"date":"2022-12-17T19:53:37","date_gmt":"2022-12-17T11:53:37","guid":{"rendered":"https:\/\/swarma.org\/?p=39252"},"modified":"2022-12-17T19:53:37","modified_gmt":"2022-12-17T11:53:37","slug":"prl%e5%89%8d%e6%b2%bf%ef%bc%9a%e6%8e%a2%e7%a9%b6%e9%9d%9e%e5%b9%b3%e8%a1%a1%e6%80%81%e8%be%93%e8%bf%90%e4%b8%bb%e5%af%bc%e8%b7%af%e5%be%84%e6%96%b0%e6%96%b9%e6%b3%95-%e5%9c%88","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=39252","title":{"rendered":"PRL\u524d\u6cbf\uff1a\u63a2\u7a76\u975e\u5e73\u8861\u6001\u8f93\u8fd0\u4e3b\u5bfc\u8def\u5f84\u65b0\u65b9\u6cd5\u2014\u2014\u5708\u6d41\u6392\u5e8f"},"content":{"rendered":"
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Cycle Flux Ranking of Network Analysis in Quantum Thermal Devices<\/span><\/p>\n

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2. \u56fe\u8bba\u8bed\u8a00\u63cf\u8ff0\u590d\u6742\u7f51\u7edc\u7ed3\u6784<\/span><\/strong><\/strong><\/p>\n<\/section>\n

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2.1 \u8282\u70b9\u4e0e\u8fb9<\/span><\/strong><\/span><\/section>\n

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2.2 \u56fe\u7684\u751f\u6210\u6811 (Spanning Tree)<\/span><\/strong><\/span><\/section>\n

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\u56fe2. \u6839\u5728\u8282\u70b91\u4e0a\u7684\u751f\u6210\u6811<\/span><\/section>\n

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2.3 \u56fe\u7684\u62c9\u666e\u62c9\u65af\u77e9\u9635<\/span><\/strong><\/span><\/section>\n

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\u63a5\u4e0b\u6765\u6211\u4eec\u5c06\u4ecb\u7ecd\u63cf\u8ff0\u56fe\u7684\u4e00\u4e2a\u91cd\u8981\u77e9\u9635\uff1a\u62c9\u666e\u62c9\u65af\u77e9\u9635\u3002\u7136\u800c\u5728\u4ecb\u7ecd\u62c9\u666e\u62c9\u65af\u77e9\u9635\u4e4b\u524d\uff0c\u6211\u4eec\u9700\u8981\u77e5\u9053\u4e24\u4e2a\u63cf\u8ff0\u56fe\u7279\u5f81\u7684\u77e9\u9635\uff1a\u5ea6\u77e9\u9635\u4e0e\u90bb\u63a5\u77e9\u9635\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u56feG\u67094\u4e2a\u8282\u70b9\uff0c\u5176\u5ea6\u77e9\u9635\u4e3a4*4\u7ef4\u77e9\u9635\uff0c\u4e3b\u5bf9\u89d2\u5143\u4f9d\u6b21\u4e3a\u5404\u4e2a\u8282\u70b9\u5ea6\u6570\uff0c\u5176\u4ed6\u5143\u7d20\u4e3a0\uff0c\u56feG\u7684\u5ea6\u77e9\u9635D\u5982\u4e0b\u6240\u793a\uff1a<\/span><\/section>\n

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\u90bb\u63a5\u77e9\u9635\u662f\u8868\u793a\u9876\u70b9\u4e4b\u95f4\u76f8\u90bb\u5173\u7cfb\u7684\u77e9\u9635\u3002\u5bf9\u4e8e\u65e0\u5411\u56fe\uff0c\u82e5\u8282\u70b9(i,j)\u4e4b\u95f4\u6709\u8fb9\u8fde\u63a5\uff0c\u5219\u5728\u77e9\u9635i\u884cj\u5217\u548cj\u884ci\u5217\u5904\u5747\u53d6\u503c\u4e3a1\uff0c\u82e5\u65e0\u8fde\u63a5\u5219\u4e3a0\uff0c\u5bf9\u65e0\u5411\u56fe\u800c\u8a00\uff0c\u90bb\u63a5\u77e9\u9635\u4e00\u5b9a\u662f\u5bf9\u79f0\u7684\uff0c\u800c\u4e14\u4e3b\u5bf9\u89d2\u7ebf\u4e00\u5b9a\u4e3a\u96f6<\/span>\uff08\u5728\u6b64\u4ec5\u8ba8\u8bba\u65e0\u5411\u7b80\u5355\u56fe\uff09<\/span>\uff0c\u6709\u5411\u56fe\u5219\u4e0d\u4e00\u5b9a\u5982\u6b64\u3002\u56feG\u7684\u90bb\u63a5\u77e9\u9635A\u8868\u793a\u5982\u4e0b\uff1a<\/span><\/section>\n

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\u56feG\u7684\u62c9\u666e\u62c9\u65af\u9635\u88ab\u5b9a\u4e49\u4e3a\uff1aL=D-A\u3002<\/span><\/section>\n

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\u756a\u5916\u5c0f\u77e5\u8bc6\uff1a\u62c9\u666e\u62c9\u65af\u77e9\u9635\u4e0e\u62c9\u666e\u62c9\u65af\u7b97\u5b50\u7684\u8054\u7cfb<\/span><\/strong><\/span><\/section>\n

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\u4ee5\u4e8c\u5143\u51fd\u6570f(x,y)\u4e3a\u4f8b:<\/span><\/section>\n
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\u82e5\u53d6\u95f4\u9694\u0394x=\u0394y=1\uff0c\u5219<\/span><\/section>\n


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\u56fe3.\u51fd\u6570f\uff08x,y\uff09\u79bb\u6563\u5316\u793a\u610f\u56fe<\/span><\/span><\/section>\n

<\/section>\n
\u5176\u4e2d\u524d\u56db\u9879\u6620\u5c04\u5230\u56fe\u7684\u90bb\u63a5\u77e9\u9635\uff0c\u6700\u540e\u4e00\u9879\u6620\u5c04\u5230\u5ea6\u77e9\u9635\u3002<\/span><\/section>\n

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2.3 \u62c9\u666e\u62c9\u65af\u77e9\u9635\u4e0e\u751f\u6210\u6811\u8ba1\u6570\uff08\u77e9\u9635-\u6811\u5b9a\u7406\uff09<\/span><\/strong><\/span><\/section>\n

<\/span><\/section>\n
\u5728\u4e86\u89e3\u4e86\u56fe\u7684\u751f\u6210\u6811\u548c\u62c9\u666e\u62c9\u65af\u77e9\u9635\u4e4b\u540e\uff0c\u672c\u5c0f\u7ed3\u5c06\u4ecb\u7ecd\u4e8c\u8005\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u5982\u4f55\u901a\u8fc7\u62c9\u666e\u62c9\u65af\u77e9\u9635\u5feb\u901f\u8ba1\u7b97\u751f\u6210\u6811\u4e2a\u6570\u3002\u4e0e\u5bf9\u4e8e\u4e00\u5e45\u65e0\u5411\u56fe\uff0c\u6211\u4eec\u53ef\u4ee5\u6570\u5f97\u5bf9\u5e94\u7684\u6240\u6709\u751f\u6210\u6811\u4e2a\u6570\uff0c\u4f8b\u5982\u4e0b\u56fe\uff0c\u5171\u67098\u4e2a\u751f\u6210\u6811\uff1a<\/span><\/section>\n

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<\/p>\n

\u56fe4.\u56feG\u5bf9\u5e94\u7684\u6240\u6709\u751f\u6210\u6811<\/span><\/section>\n

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\u7136\u800c\u6211\u4eec\u4e5f\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635-\u6811\u5b9a\u7406<\/strong>\u5feb\u901f\u8ba1\u7b97\u8be5\u56fe\u751f\u6210\u6811\u7684\u6570\u76ee\u3002<\/span><\/section>\n

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\u77e9\u9635-\u6811\u5b9a\u7406\uff08matrix-tree theorem\uff09\uff1a<\/span><\/strong>\u8be5\u5b9a\u7406\u662f\u4e00\u4e2a\u8ba1\u6570\u5b9a\u7406\u3002\u82e5\u65e0\u5411\u56feG\u7684\u62c9\u666e\u62c9\u65af\u77e9\u9635\u4e3aL\uff0c\u5219L\u7684\u6bcf\u4e2a\u4ee3\u6570\u4f59\u5b50\u5f0f\u76f8\u7b49\uff0c\u4e14\u7b49\u4e8e\u56feG\u7684\u751f\u6210\u6811\u7684\u6570\u76ee\u3002\u8fd9\u5c31\u662f\u77e9\u9635\u4e00\u6811\u5b9a\u7406\uff0c\u6211\u4eec\u4eec\u6709\u65f6\u4e5f\u79f0\u62c9\u666e\u62c9\u65af\u77e9\u9635L\u4e3a\u57fa\u5c14\u970d\u592b\u77e9\u9635\u3002\u5bf9\u4e8e\u4e0a\u56feG\uff0c\u901a\u8fc7\u77e9\u9635-\u6811\u5b9a\u7406\u8ba1\u7b97\u5f97\u77e5\u5176\u751f\u6210\u6811\u6570\u76ee\u4e3a8 \u3002<\/span><\/section>\n

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\u5bf9\u4e8e\u6709\u5411\u56fe\u4e5f\u6709\u7c7b\u4f3c\u7684\u7ed3\u8bba\uff0c\u4f8b\u5982\u4e0b\u56fe\uff0c\u6709\u5411\u56feG\u4e2d\uff0c\u6211\u4eec\u6570\u5f97\u6839\u5728\u8282\u70b92\u4e0a\u7684\u751f\u6210\u6811\u5171\u6709\u4e09\u4e2a\u3002\u5176\u5b9e\u8fd9\u4e09\u4e2a\u751f\u6210\u6811\u5728\u8f93\u8fd0\u8fc7\u7a0b\u4e2d\u4ee3\u8868\u4e86\u7c92\u5b50\u901a\u8fc7\u4e0d\u540c\u8def\u5f84\uff0c\u5386\u7ecf\u5176\u5b83\u8282\u70b9\uff0c\u6700\u7ec8\u6d41\u5165\u8282\u70b92\u7684\u4e09\u79cd\u60c5\u51b5\u3002<\/span><\/section>\n

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\u56fe5. \u56feG\u2019\u548c\u6240\u6709\u6839\u5728\u8282\u70b92\u4e0a\u7684\u751f\u6210<\/span>\u6811<\/span><\/section>\n

<\/span><\/section>\n
\uff0c\u5220\u9664\u7b2c2\u884c\u548c\u7b2c2\u5217\u6c42\u884c\u5217\u5f0f\uff0c\u5f97\u5230<\/span>\uff0c\u4e0e\u6211\u4eec\u6240\u6570\u7ed3\u679c\u4e00\u81f4\u3002<\/span><\/section>\n

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\n
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3. \u5708\u6d41\u4e0e\u975e\u5e73\u8861\u6001\u8f93\u8fd0<\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n
 <\/span><\/section>\n
\u975e\u5e73\u8861\u8f93\u8fd0\u53ef\u4ee5\u6620\u5c04\u5230\u56fe\u548c\u590d\u6742\u7f51\u7edc\u7ed3\u6784\u4e0a\uff0c\u5982\u56fe6(a)\u6240\u793a\uff0c\u4ece\u800c\u7528\u5708\u7684\u5206\u89e3\u8fd9\u4e00\u65b0\u89c6\u89d2\u6765\u7814\u7a76\u8f93\u8fd0\u6027\u8d28 [1]\u3002\u5728\u8fd9\u91cc\u6211\u4eec\u5c06\u4ecb\u7ecd\u4e00\u4e2a\u65b0\u7684\u6982\u5ff5\uff1a\u5708\u6d41\u3002\u5708\u6d41\u662f\u7528\u6765\u8861\u91cf\u7cfb\u7edf\u5728\u6001\u7a7a\u95f4\u4e2d\u7684\u95ed\u5408\u8def\u5f84\uff08\u5708\uff09\u4e0a\u72b6\u6001\u6539\u53d8\u7684\u5feb\u6162\u7a0b\u5ea6\u7684\u7269\u7406\u6982\u5ff5\uff0c\u4ece\u800c\u53cd\u80fd\u5e94\u51fa\u8fd9\u6761\u8def\u5f84\u7684\u6d3b\u8dc3\u7a0b\u5ea6\uff0c\u95f4\u63a5\u4f53\u73b0\u5bf9\u4e8e\u6574\u4e2a\u8f93\u8fd0\u8fc7\u7a0b\u7684\u8d21\u732e\u91cd\u8981\u7a0b\u5ea6<\/strong>\u3002\u901a\u8fc7\u5708\u6d41\u5927\u5c0f\u7684\u6392\u5e8f\uff0c\u4f53\u73b0\u4e86\u4e0d\u540c\u8def\u5f84\u7684\u91cd\u8981\u7a0b\u5ea6\uff0c\u4ece\u800c\u6293\u4f4f\u4e3b\u8981\u673a\u5236\uff0c\u7b80\u5316\u5206\u6790\u8fc7\u7a0b\u3002<\/span><\/section>\n

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\u56fe6.(a) \u975e\u5e73\u8861\u8f93\u8fd0\u7cfb\u7edf\u6620\u5c04\u5230\u6001\u7a7a\u95f4\u7684\u590d\u6742\u7f51\u7edc\u3002(b) \u590d\u6742\u7f51\u7edc\u5bf9\u5e94\u7684\u6240\u6709\u95ed\u5408\u5708\u8f68\u8ff9\u3002\uff08\u56fe\u7247\u6765\u81ea\u6587\u732e[6]\uff09<\/span><\/p>\n


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3.1 \u5229\u7528\u5708\u5206\u89e3\u975e\u5e73\u8861\u6001\u8f93\u8fd0<\/span><\/strong><\/span><\/section>\n

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\u975e\u5e73\u8861\u9a71\u52a8\u4e0b\u7684\u8f93\u8fd0\u7cfb\u7edf\u53ef\u4ee5\u6620\u5c04\u5230\u4e00\u5e45\u56fe\u4e0a\uff0c\u4f8b\u5982\u4e0b\u56feG\u201d\u4e2d\uff0c\u7cfb\u7edf\u72b6\u6001\u6620\u5c04\u5230\u8282\u70b9\uff0c\u72b6\u6001\u6539\u53d8\u5bf9\u5e94\u4e8e\u8fb9\uff0c\u72b6\u6001\u6539\u53d8\u901f\u7387\u5bf9\u5e94\u4e8e\u8fb9\u7684\u6743\u91cd\uff1a<\/span><\/section>\n

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\u56fe7. \u56feG\u201d<\/span><\/section>\n

<\/span><\/section>\n
\u5bf9\u4e8e\u6211\u4eec\u6240\u611f\u5174\u8da3\u7684\u529f\u80fd\u5668\u4ef6\u6216\u8005\u5176\u4ed6\u52a8\u529b\u5b66\u7cfb\u7edf\uff0c\u5728\u975e\u5e73\u8861\u9a71\u52a8\u4e0b\u80fd\u4fdd\u6301\u5728\u7a33\u6001\u7a33\u5b9a\u8fd0\u884c\uff0c\u8fd9\u5c31\u6697\u542b\u5176\u5728\u5b9e\u7a7a\u95f4<\/span>\uff08\u6216\u8005\u76f8\u7a7a\u95f4\u3001\u72b6\u6001\u7a7a\u95f4\u7b49\uff09<\/span>\u7684\u8fd0\u884c\u8f68\u8ff9\u5fc5\u987b\u662f\u4e2a\u95ed\u5408\u56de\u8def\uff0c\u5373\u5b8c\u6210\u4e00\u4e2a\u5faa\u73af\uff0c\u5426\u5219\u5c31\u4f1a\u5728\u67d0\u4e9b\u8282\u70b9\u6301\u7eed\u79ef\u7d2f\u5230\u7206\u70b8\u7834\u574f\u7a33\u6001\u3002\u6240\u4ee5Schnakenberg\u5728\u6587\u732e[1]\u4e2d\u63d0\u4f9b\u4e86\u4e00\u79cd\u57fa\u4e8e\u7f51\u7edc\u7406\u8bba\u4e2d\u7684\u5708\u5206\u6790<\/span>\uff08\u5faa\u73af\uff09<\/span>\u89c6\u89d2\u7814\u7a76\u975e\u5e73\u8861\u6001\u8f93\u8fd0\u7684\u65b0\u601d\u8def\u3002\u5bf9\u4e8e\u56feG\u4e2d\u4efb\u4f55\u4e00\u4e2a\u5708c\u53ef\u4ee5\u901a\u8fc7\u57fa\u672c\u5708<\/span>{c1<\/span>, c<\/span>2<\/span>, \u2026, c<\/span>\u03bd<\/span><\/span>}\u7684\u7ebf\u6027\u7ec4\u5408\u5f97\u5230\uff0c\u7528\u4e0b\u9762\u8fd9\u4e2a\u4f8b\u5b50\u505a\u8be6\u7ec6\u8bf4\u660e\u3002<\/span><\/section>\n

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<\/section>\n
\u56fe8. \u56feG\u4e2d\u5229\u7528\u57fa\u672c\u5708\u7ebf\u6027\u7ec4\u5408\u5f97\u5230\u65b0\u5708C<\/span><\/section>\n

<\/section>\n
\u5728\u56feG\u4e2d\uff0c\u6211\u4eec\u9009\u5b9a{c<\/span>1<\/span>, <\/span>c<\/span>2<\/span>}<\/span>\u4f5c\u4e3a\u57fa\u672c\u5708\uff0c\u6765\u5f97\u5230\u5708c\uff0c\u9075\u5faa\u8fd0\u7b97\u89c4\u5219\uff1ac=<\/sub>\uff0c\u5176\u4e2d<\/sub>\uff0c\u5e76\u4e14\u5f53c<\/span>\u03bd<\/span><\/span>\u4e2d\u6ca1\u6709\u4efb\u610f\u4e00\u6761\u5c5e\u4e8ec\u65f6\uff0c<\/span>\u03c1<\/span>\u03bd<\/span><\/span><\/span>=0\uff0c\u5426\u5219\u03c1<\/span>\u03bd<\/span><\/span>=1\u3002<\/span><\/section>\n

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3.2 \u5708\u6d41\u53ca\u5176\u56fe\u8bba\u8868\u8ff0<\/span><\/strong><\/span><\/section>\n

<\/section>\n
\u5728\u5708\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u8fdb\u4e00\u6b65\u4ecb\u7ecd\u4e00\u4e2a\u65b0\u7684\u91cf\uff1a\u5708\u6d41\u3002\u5708\u6d41\u7684\u6982\u5ff5\u662f\u7531T. L. Hill\u57281966\u5e74\u7684\u8bba\u6587\u4e2d[2]\u9996\u6b21\u63d0\u51fa\uff0c\u5e76\u4f7f\u7528\u5728\u751f\u5316\u52a8\u529b\u5b66\u56fe\u7684\u7814\u7a76\u5f53\u4e2d[3]\uff0c\u968f\u540eKohler\u5728\u5176\u8bba\u6587[4]\u4e2d\u8fdb\u884c\u4e86\u8fdb\u4e00\u6b65\u4ecb\u7ecd\uff0c\u6b64\u65b9\u6cd5\u5df2\u7ecf\u5927\u91cf\u8fd0\u7528\u4e8e\u751f\u5316\u7cfb\u7edf\u52a8\u529b\u5b66\u7684\u7814\u7a76\u4e4b\u4e2d\u3002\u8fd9\u4e9b\u52a8\u529b\u5b66\u7cfb\u7edf\u5e38\u5e38\u53ef\u4ee5\u901a\u8fc7\u4e3b\u65b9\u7a0b\u63cf\u8ff0\uff1a<\/span><\/section>\n

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\u5f53\u7cfb\u7edf\u5230\u8fbe\u7a33\u6001\u65f6\uff0c<\/sub>\uff0c\u5e76\u4e14\u6ee1\u8db3\u5f52\u4e00\u5316\u6761\u4ef6\uff1a<\/sub>\u3002\u6839\u636e\u514b\u62c9\u9ed8\u6cd5\u5219\uff0c\u6211\u4eec\u53ef\u4ee5\u8ba1\u7b97\u51fa\u6bcf\u4e2a\u72b6\u6001\u7684\u6982\u7387\uff1a\u3002Hill\u63d0\u51fa\u4e86\u4e00\u79cd\u56fe\u8bba\u8bed\u8a00\u7684\u65b9\u6cd5\uff1a<\/sub>\uff0c\u5176\u4e2d<\/sub>\u5728\u539f\u6587\u4e2d\u88ab\u63cf\u8ff0\u4e3asum of directional diagrams of state i\uff0c\u2211<\/span>\u5728\u539f\u6587\u4e2d\u88ab\u63cf\u8ff0\u4e3asum of directional diagrams of all states\u3002\u5728\u56fe\u8bba\u4e2d<\/sub>\u5bf9\u5e94\u4e8e\u6240\u6709\u6839\u5728\u8282\u70b9i\u7684\u751f\u6210\u6811\uff0c\u5148\u5c06\u5176\u5404\u81ea\u6743\u91cd\u8fde\u4e58\u518d\u6c42\u548c\u7684\u503c\uff0c\u5176\u7269\u7406\u56fe\u50cf\u662f\u7cfb\u7edf\u6700\u7ec8\u6f14\u5316\u5230\u72b6\u6001i\u7684\u6240\u6709\u53ef\u80fd\u8fc7\u7a0b\uff1b<\/span>\u2211<\/span>\u662f\u6839\u5728\u6bcf\u4e00\u4e2a\u8282\u70b9\u4e0a\u7684\u6240\u6709\u751f\u6210\u6811\uff0c\u5148\u5c06\u5176\u5404\u81ea\u6743\u91cd\u8fde\u4e58\u5728\u6c42\u548c\u5f97\u5230\u7684\u503c\u3002\u6839\u636e\u4ee5\u4e0a\u63cf\u8ff0\uff0c\u8fdb\u4e00\u6b65\u62d3\u5c55\u53ef\u4ee5\u5f97\u5230\u5708\u6d41\u7684\u56fe\u8bba\u8868\u8fbe\u5f62\u5f0f\u5982\u4e0b\uff1a<\/span><\/section>\n

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<\/sub>\u662f\u6cbf\u5708C\u987a\u65f6\u9488\uff08+\uff09\u65b9\u5411\u6216\u8005\u9006\u65f6\u9488\u65b9\u5411\uff08-\uff09\u7684\u5708\u6d41\u503c\uff0c\u5176<\/span>\u4e2d<\/span><\/sub>\u662f\u6cbf\u5708C\u987a\u65f6\u9488\uff08+\uff09\u65b9\u5411\u6216\u8005\u9006\u65f6\u9488\u65b9\u5411\uff08-\uff09\u7684\u6743\u91cd\u5927\u5c0f\u3002<\/span><\/sub>\u63cf\u8ff0\u4e86\u7cfb\u7edf\u6700\u7ec8\u6f14\u5316\u5230\u5708C\u4e0a\u4efb\u4f55\u4e00\u4e2a\u8282\u70b9\u7684\u53ef\u80fd\u6027\uff0c<\/span><\/sub>\u5219\u4f53\u73b0\u4e3a\u7cfb\u7edf\u6cbf\u5708<\/sub>\u72b6\u6001\u8dc3\u8fc1\u7684\u901f\u7387\u3002<\/span>\u56e0\u6b64\uff0c\u5708\u6d41<\/sub><\/span>\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u89c6\u4e3a\u7cfb\u7edf\u72b6\u6001\u5728\u5355\u4f4d\u65f6\u95f4\u5185\u6cbf\u5708<\/span><\/sub>\u8dd1\u7684\u5708\u6570\uff0c\u5373\u9891\u7387\u3002<\/span><\/section>\n

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3.3 \u5229\u7528\u5708\u6d41\u5206\u89e3\u8fb9\u6d41<\/span><\/strong><\/span><\/section>\n

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\u72b6\u6001i\u4e0ej\u4e4b\u95f4\u7684\u8dc3\u8fc1\u5bf9\u5e94\u4e8e\u56fe\u4e2d\u662f\u4e00\u6761\u8fde\u63a5\u8282\u70b9i\u4e0ej\u7684\u53cc\u5411\u8fb9\u3002\u4ecei\u5230j\u7684\u51c0\u8fb9\u6d41\u53ef\u4ee5\u8868\u8fbe\u4e3a<\/sub>\uff0cki, j<\/sub>\u662f\u4ecei\u5230j\u7684\u8dc3\u8fc1\u7387\u3002\u901a\u8fc7\u5708\u6d41\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u5176\u8fdb\u884c\u5206\u89e3\u5bf9\u5e94\u5708\u6d41\u7684\u8868\u8fbe<\/sub>\uff0c\u5176\u4e2d<\/sub>\u4e3a\u6240\u6709\u5305\u542bi\u5230j\u7684\u5708\u6d41\u603b\u548c\uff0c<\/sub>\u662f\u6240\u6709\u5305\u542bj\u5230i\u7684\u5708\u6d41\u603b\u548c\u3002\u4f8b\u5982\u4e0b\u56fe\u4e2d\u4ece3\u52302\u7684\u51c0\u8fb9\u6d41\u53ef\u4ee5\u5206\u89e3\u4e3a\u82e5\u5e72\u4e2a\u5708\u6d41\u7684\u52a0\u51cf\uff1a<\/span><\/section>\n

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\u56fe9.\u8fb9\u6d41\u5206\u89e3\u4e0e\u5708\u6d41\u5bf9\u5e94\u5173\u7cfb<\/span><\/section>\n

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\u5728\u4f20\u7edf\u7814\u7a76\u4e2d\uff0c\u6211\u4eec\u53ea\u80fd\u5c40\u9650\u4e8e\u770b\u5230\u70b9\u5230\u70b9\u7684\u8fb9\u6d41\uff0c\u901a\u8fc7\u5708\u6d41\u5206\u89e3\u65b9\u6cd5\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u77e5\u9053\u54ea\u4e2a\u5708\u8f68\u8ff9<\/span>\uff08\u7269\u7406\u8fc7\u7a0b\uff09<\/span>\u5bf9\u4e8e\u72b6\u6001i-j\u7684\u8dc3\u8fc1\uff0c\u751a\u81f3\u4e8e\u6574\u4e2a\u52a8\u529b\u5b66\u8fc7\u7a0b\u8d77\u4e3b\u5bfc\u6027\u4f5c\u7528\uff0c\u6613\u4e8e\u5206\u6790\u8f93\u8fd0\u7cfb\u7edf\u7684\u4e3b\u8981\u673a\u5236\u3002<\/span><\/section>\n

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3.4 \u5708\u6d41\u7684\u4ee3\u6570\u56fe\u8bba\u8868\u8ff0\u53ca\u56fe\u8bba\u5bf9\u5e94<\/span><\/strong><\/span><\/section>\n

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\u901a\u8fc7\u4ee5\u4e0a\u7684\u4ecb\u7ecd\u7684\u56fe\u8bba\u65b9\u6cd5\uff0c\u6211\u4eec\u7684\u786e\u53ef\u4ee5\u7b97\u51fa\u6bcf\u4e2a\u5708\u7684\u5708\u6d41\uff0c\u7136\u800c\u9762\u4e34\u4e00\u4e2a\u95ee\u9898\uff1a\u5f53\u56fe\u53d8\u5f97\u590d\u6742\u65f6\uff0c\u5708\u7684\u6570\u91cf\u4f1a\u6025\u5267\u589e\u52a0\uff0c\u627e\u51fa\u6839\u5728\u5404\u4e2a\u8282\u70b9\u7684\u751f\u6210\u6811\u5219\u53d8\u5f97\u4e0d\u518d\u5177\u6709\u53ef\u64cd\u4f5c\u6027\u3002\u8fd9\u65f6\uff0c\u6211\u4eec\u5219\u9700\u8981\u4e00\u4e2a\u65b0\u7684\u65b9\u6cd5\u6765\u8ba1\u7b97\u5708\u6d41\u3002\u5728\u6587\u732e[5]\u4e2d\u7ed9\u51fa\u4e86\u4e00\u79cd\u65b0\u7684\u4ee3\u6570\u56fe\u8bba\u63cf\u8ff0\uff0c\u53ef\u4ee5\u5feb\u901f\u8ba1\u7b97\u6240\u6709\u5708\u6d41\u7684\u5927\u5c0f\u3002<\/span><\/section>\n

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\u8fd9\u91ccL\u662f\u5c06\u8f93\u8fd0\u7cfb\u7edf\u6620\u5c04\u5230\u56fe\u540e\u5bf9\u5e94\u7684\u62c9\u666e\u62c9\u65af\u77e9\u9635\uff0c\u6ee1\u8db3<\/sub>\uff0c\u5176\u4e2d<\/span><\/sub>\uff0c<\/sub><\/span>\u662f\u6cbf<\/sub>\u4e0a\u6240\u6709\u8dc3\u8fc1\u7387\u8fde\u4e58\uff0cL[I,j]\u662f\u6307\u5728L\u77e9\u9635\u4e2d\u5212\u53bb\u7b2ci\u884c\u548c\u7b2cj\u5217\u3002\u2211<\/span>c<\/sub><\/span>\u4e0edet(L[C; <\/span>C])<\/span>\u662f\u7b49\u4ef7\u7684\uff1b<\/span>\u2211<\/span>\u4e0e\u2211<\/span>i<\/sub>det(L[i; i]) \u662f\u7b49\u4ef7\u7684\u3002<\/span>\u5173\u4e8e\u5177\u4f53\u63a8\u5bfc\uff0c\u611f\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u9605\u8bfb\u94b1\u5b66\u6d3e\u7684\u8bba\u6587[7]\u3002<\/span>\u4ee5\u4e0b\u7684\u4f8b\u5b50\u5c06\u5e2e\u52a9\u6211\u4eec\u66f4\u76f4\u89c2\u7684\u611f\u53d7\u3002<\/span>\u5982\u4e0b\u56fe\u6240\u793a\u7f51\u7edc\u7ed3\u6784\uff0c\u9ed1\u8272\u52a0\u6df1\u8fb9\u56f4\u6210\u7684\u8def\u5f84\u662f\u6211\u4eec\u5f85\u8ba1\u7b97\u7684\u5708\uff1a<\/span><\/section>\n
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\u56fe10. \u56feG\uff0c\u9ed1\u8272\u52a0\u6df1\u8fb9\u56f4\u6210\u7684\u8def\u5f84\u4e3a\u6211\u4eec\u5f85\u8ba1\u7b97\u7684\u5708\u3002<\/span><\/section>\n

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\u8be5\u56fe\u5bf9\u5e94\u7684\u62c9\u666e\u62c9\u65af\u77e9\u9635\u4e3a\uff1a<\/span><\/section>\n

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\u5f53\u5212\u53bb\u5708C\u6240\u5305\u542b\u7684{1,2,3,4,5}\u5bf9\u5e94\u7684\u884c\u548c\u5217\u540e\uff0cdet(L[C;C])=\u2211<\/span>j\u22606<\/sub>k6, j<\/sub>=k<\/span>6, 5<\/sub>+k<\/span>6, 2<\/sub>+k<\/span>6, 4<\/sub>\uff0c\u521a\u597d\u5bf9\u5e94\u4e8e\u6bcf\u4e00\u4e2a\u6839\u5728\u5708C\u4e0a\u7684\u751f\u6210\u6811\u7684\u6743\u91cd\u5148\u8fde\u4e58\u518d\u6c42\u548c\uff0c\u5373\u2211<\/span>c<\/sub>\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u4e5f\u7b49\u4ef7\u4e8e\u5c06\u5708C\u634f\u6210\u4e00\u4e2a\u70b9d<\/span>\uff08\u56fe7\u5927\u7eff\u8272\u5706\u5f62\uff09<\/span>\uff0c\u5f62\u6210\u4e00\u4e2a\u65b0\u56fe\uff0c\u65b0\u62c9\u666e\u62c9\u65af\u77e9\u9635\u4e3a\uff1a<\/span><\/section>\n

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\u5220\u9664\u70b9d\u540e\uff0cdet(L\u2019[d;d])=\u2211<\/span>j\u22606<\/sub>k<\/span>6, j<\/sub>=<\/span>k<\/span>6, 5<\/sub>+<\/span>k<\/span>6, 2<\/sub>+<\/span>k<\/span>6, 4<\/sub>\uff0c\u5177\u4f53\u751f\u6210\u6811\u5982\u4e0b\u56fe\u6240\u793a\uff1a<\/span><\/section>\n

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\u56fe11<\/span>.<\/span> \u2211c<\/sub>\u4e0e<\/span>det(L[C;C])\u7684\u5bf9\u5e94\u5173\u7cfb\uff0c\u4ee5\u53ca\u634f\u70b9\u4e4b\u540e\u5f97\u5230\u65b0\u56fe\u4e0e\u539f\u56fe\u7684\u7b49\u4ef7\u5173\u7cfb\u3002<\/span><\/section>\n

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4. \u5708\u6d41\u89c6\u89d2\u4e0b\u7684\u91cf\u5b50\u70ed\u8f93\u8fd0\u65b0\u53d1\u73b0<\/span><\/strong><\/strong><\/p>\n<\/section>\n

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\u540c\u6d4e\u5927\u5b66\u58f0\u5b50\u4e2d\u5fc3\u7684\u4efb\u6377\u6559\u6388Xonics\u8bfe\u9898\u7ec4\u805a\u7126\u4e8e\u591a\u7269\u7406\u573a\u8026\u5408\u4f53\u7cfb\uff0c\u8fd1\u671f\u53d1\u8868\u7684\u5de5\u4f5cCycle Flux Ranking of Network Analysis in Quantum Thermal Devices<\/em>\uff0cPhys.<\/em> Rev. Lett. 128, 067701 (2022)<\/em>\uff0c\u901a\u8fc7\u5708\u6d41\u7684\u6392\u5e8f\u7684\u65b9\u6cd5\u5206\u6790\u4e3b\u5bfc\u5708\u7684\u884c\u4e3a\u7279\u5f81\uff0c\u53ef\u4ee5\u5256\u6790\u591a\u7269\u7406\u573a\u8026\u5408\u4f53\u7cfb\u4e2d\u91cf\u5b50\u70ed\u8f93\u8fd0\u884c\u4e3a\u7684\u672c\u8d28\uff0c\u901a\u8fc7\u5206\u6790\u5176\u7279\u5f81\u6765\u8fdb\u884c\u4f18\u5316\u3002\u5728\u6b64\u5de5\u4f5c\u4e2d\uff0c\u901a\u8fc7\u5206\u6790\u4e0e\u4f18\u5316\u4e24\u4e2a\u91cf\u5b50\u6a21\u578b\uff0c\u6765\u9a8c\u8bc1\u4e86\u6b64\u65b9\u6cd5\u7684\u9ad8\u6548\u6027\u4e0e\u5f3a\u5927\u3002<\/span><\/section>\n

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\u5728\u7b2c\u4e00\u4e2a\u201c\u70ed\u62d6\u62fd\u5bfc\u81f4\u7684\u81ea\u65cb\u8d5b\u8d1d\u514b\u6cf5\u6d66\u6548\u5e94\u201d\u7684\u6a21\u578b\u4e2d\uff0c\u901a\u8fc7\u6b64\u65b9\u6cd5\u6210\u679c\u627e\u5230\u4e86\u4e3b\u5bfc\u5708\u7684\u8f68\u8ff9\uff0c\u5e76\u4e14\u8bba\u8bc1\u5904\u4e8e\u5e73\u8861\u6001\u5b50\u7cfb\u7edf\u4e2d\u7684\u81ea\u65cb\u6d41\u662f\u7531\u4e8e\u70ed\u9a71\u52a8\u4ea7\u751f\u7684\uff0c\u800c\u4e0d\u662f\u5e38\u89c1\u7684\u5e93\u4f26\u62d6\u62fd\u6548\u5e94\u5bfc\u81f4\u7684\u3002<\/span><\/section>\n

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\u56fe12.\uff08a\uff09\u201c\u70ed\u62d6\u62fd\u5bfc\u81f4\u7684\u81ea\u65cb\u8d5b\u8d1d\u514b\u6cf5\u6d66\u201d\u6a21\u578b\u793a\u610f\u56fe\u3002\uff08b\uff09\u6001\u7a7a\u95f4\u91cf\u5b50\u8dc3\u8fc1\u4e09\u68f1\u67f1\u3002\uff08c\uff09\uff08d\uff09\u7535\u6d41\u4e0e\u81ea\u65cb\u6d41\u968f\u6e29\u5dee\u4e0e\u5e93\u4f26\u76f8\u4e92\u4f5c\u7528\u5f3a\u5ea6\u7684\u53d8\u5316\u3002<\/span><\/p>\n


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\u56fe12\uff08a\uff09\u7684\u7cfb\u7edf\u662f\u7531\u4e0a\u4e0b\u4e24\u4e2a\u5b50\u7cfb\u7edf\u901a\u8fc7\u5e93\u4f26\u4f5c\u7528\u8026\u5408\u800c\u6210\uff0c\u5176\u54c8\u5bc6\u987f\u91cf\u4e3a\uff1a<\/span><\/section>\n

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\u4e0a\u9762\u7684\u91cf\u5b50\u70b9\u5de6\u53f3\u8026\u5408\u4e24\u4e2a\u65e0\u81ea\u65cb\u7684\u7535\u5b50\u5e93\uff0c\u5e95\u4e0b\u7684\u91cf\u5b50\u70b9\u5de6\u4fa7\u8026\u5408\u4e00\u4e2a\u6709\u81ea\u65cb\u7684\u7535\u5b50\u5e93\uff0c\u53f3\u4fa7\u8026\u5408\u4e00\u4e2a\u78c1\u6fc0\u5b50\u5e93\u3002\u540c\u65f6\u5bf9\u6bd4\u56fe12\uff08c\uff09\uff08d\uff09\uff0c\u6211\u4eec\u53ef\u4ee5\u53d1<\/span>\u73b0\u5728\u03b5u<\/sub>\u22480<\/span>\u9644\u8fd1\uff0c\u4e0a\u9762\u5b50\u7cfb\u7edf\u4e2d\u7684\u7535\u6d41\u51e0\u4e4e\u4e3a0\uff0c\u7136\u800c\u5728\u4e0b\u9762\u8fd9\u4e2a\u6ca1\u6709\u4efb\u4f55\u975e\u5e73\u8861\u529b\u9a71\u52a8\u7684\u5b50\u7cfb\u7edf\u4e2d\u51fa\u73b0\u4e86\u660e\u663e\u7684\u81ea\u65cb\u6d41\uff0c\u5e76\u4e14\u8fd8\u968f\u7740T1<\/sub>\u7684\u6e29\u5ea6\u6539\u53d8\u800c\u53cd\u5411\u3002\u8fd9\u8bc1\u660e\u4e86\u5904\u4e8e\u5e73\u8861\u6001\u5b50\u7cfb\u7edf\u4e2d\u7684\u81ea\u65cb\u6d41\u662f\u7531\u4e8e\u70ed\u9a71\u52a8\u4ea7\u751f\u7684\uff0c\u800c\u4e0d\u662f\u5e38\u89c1\u7684\u5e93\u4f26\u62d6\u62fd\u6548\u5e94\u5bfc\u81f4\u7684\u3002<\/span><\/section>\n

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\u53c2\u8003\u6587\u732e\uff1a<\/span><\/strong><\/span><\/section>\n
[1]  J. Schnakenberg, Rev. Mod. Phys. 48, 571 (1976)<\/span><\/section>\n
[2] T. L. Hill, J. Theor. Biol. 10(3), 442 (1966).<\/span><\/section>\n
[3] T. L. Hill and Y.-D. Chen, Proc. Natl. Acad. Sci. U.S.A. 72, 1291 (1975).<\/span><\/section>\n
[4] H.-H. Kohler and E. Vollmerhaus, J. Math. Biol. 9, 275 (1980).<\/span><\/section>\n
[5] J. Ren, Front. Phys. 12, 1 (2017).<\/span><\/section>\n
[6] L. Wang, Z. Wang, C. Wang and J. Ren, Phys. Rev. Lett. 128, 067701 (2022)<\/span><\/section>\n
[7] D.-Q. Jiang, M. Qian, and M.-P. Qian, Mathematical Theory of Nonequilibrium Steady States: On the Frontier of Probability and Dynamical Systems (Springer Science &Business Media, New York, 2004).<\/span><\/section>\n

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