{"id":39111,"date":"2022-12-13T18:25:41","date_gmt":"2022-12-13T10:25:41","guid":{"rendered":"https:\/\/swarma.org\/?p=39111"},"modified":"2022-12-13T18:25:41","modified_gmt":"2022-12-13T10:25:41","slug":"entropy-%e7%9b%b8%e5%8f%98%e4%b8%8e%e6%b6%8c%e7%8e%b0%e7%8e%b0%e8%b1%a1%e7%89%b9%e5%88%8a%ef%bc%9a%e7%ae%80%e5%8d%95%e4%b8%ad%e5%a6%82%e4%bd%95%e6%b6%8c%e7%8e%b0%e5%a4%8d%e6%9d%82%e5%8f%98%e5%8c%96","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=39111","title":{"rendered":"Entropy \u76f8\u53d8\u4e0e\u6d8c\u73b0\u73b0\u8c61\u7279\u520a\uff1a\u7b80\u5355\u4e2d\u5982\u4f55\u6d8c\u73b0\u590d\u6742\u53d8\u5316\uff1f"},"content":{"rendered":"
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\u590d\u6742\u7cfb\u7edf\u7814\u7a76\u5173\u6ce8\u7b80\u5355\u7684\u90e8\u5206\u5982\u4f55\u76f8\u4e92\u4f5c\u7528\uff0c\u5728\u6574\u4f53\u5c42\u9762\u6d8c\u73b0\u51fa\u590d\u6742\u73b0\u8c61\u3002\u8fd9\u548c\u7edf\u8ba1\u7269\u7406\u901a\u8fc7\u7814\u7a76\u5fae\u89c2\u7c92\u5b50\u7684\u7edf\u8ba1\u884c\u4e3a\u6765\u89e3\u91ca\u5b8f\u89c2\u72b6\u6001\u53d8\u5316\u5177\u6709\u6839\u672c\u4e00\u81f4\u6027\u30022021\u5e74\uff0cEntropy<\/em> \u6742\u5fd7\u63a8\u51fa\u201c\u76f8\u53d8\u4e0e\u6d8c\u73b0\u73b0\u8c61\u7279\u520a\u201d\uff0c\u805a\u7126\u590d\u6742\u7cfb\u7edf\u7684\u7edf\u8ba1\u7269\u7406\u3002\u7279\u520a\u5171\u6536\u5f556\u7bc7\u76f8\u5173\u4e3b\u9898\u8bba\u6587\uff0c\u5305\u62ec2021\u5e74\u8bfa\u8d1d\u5c14\u7269\u7406\u5b66\u5956\u5f97\u4e3b Giorgio Parisi \u5173\u4e8e\u81ea\u65cb\u73bb\u7483\u76f8\u53d8\u7684\u5de5\u4f5c\uff0c\u4ee5\u53ca\u5176\u4ed6\u76f8\u53d8\u4e0e\u4e34\u754c\u73b0\u8c61\u3001\u4f0a\u8f9b\u6a21\u578b\u3001\u590d\u6742\u7f51\u7edc\u7b49\u4ea4\u53c9\u9886\u57df\u7814\u7a76\u3002<\/span><\/strong><\/span><\/strong><\/span><\/strong><\/p>\n<\/section>\n

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\u5173\u952e\u8bcd\uff1a\u76f8\u53d8\u4e0e\u4e34\u754c\u73b0\u8c61\uff0c\u666e\u9002\u6027\uff0c\u6807\u5ea6\u5f8b\uff0c\u4f0a\u8f9b\u6a21\u578b\uff0c\u81ea\u65cb\u73bb\u7483\uff0c\u7edf\u8ba1\u7269\u7406\uff0c\u590d\u6742\u7f51\u7edc<\/span><\/strong><\/span><\/strong><\/span><\/strong><\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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Phase Transitions and Emergent Phenomena: How Change Emerges through Basic Probability Models<\/span><\/p>\n

\u7279\u520a\u94fe\u63a5\uff1a<\/span><\/span><\/p>\n

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1. \u771f\u5b9e\u526f\u672c\u4e4b\u95f4\u7684\u8ddd\u79bb\u53d8\u5316\u5bfc\u81f4\u81ea\u65cb\u73bb\u7483\u76f8\u53d8<\/span><\/strong><\/p>\n

2. \u4f0a\u8f9b\u81ea\u65cb\u73bb\u7483\u7684\u8d85\u6807\u5ea6\u8fdd\u53cd<\/span><\/strong><\/p>\n

3. \uff08\u56db\uff09\u53cc\u5143\u683c\u4e09\u7ef4\u4f0a\u8f9b\u6a21\u578b<\/span><\/strong><\/p>\n

4. \u57fa\u4e8e\u4f0a\u8f9b\u6a21\u578b\u7684\u968f\u673a\u5757\u6a21\u578b\u7684\u7cbe\u786e\u6062\u590d<\/span><\/strong><\/p>\n

5. \u6709\u5411 BA \u7f51\u7edc\u4e0a\u7684\u79bb\u6563\u610f\u89c1\u52a8\u529b\u5b66\u6a21\u578b<\/span><\/strong><\/p>\n

6. \u7f51\u7edc\u751f\u6210\u6811\u7684\u666e\u9002\u6027\u548c\u6709\u9650\u5c3a\u5bf8\u4fee\u6b63<\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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19\u4e16\u7eaa\u672b\uff0c\u73bb\u5c14\u5179\u66fc<\/span>\uff08Ludwig Boltzmann\uff09<\/span>\u548c\u540c\u65f6\u4ee3\u4eba\u5f00\u521b\u4e86\u7edf\u8ba1\u7269\u7406\u5b66\u3002\u8be5\u5b66\u79d1\u7684\u652f\u67f1\u5305\u62ec\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u7684\u76f8\u53d8\u7406\u8bba\u548c\u4e34\u754c\u73b0\u8c61\uff0c\u5b83\u5efa\u7acb\u5728\u5982\u695e\u6b21<\/span>\uff08Wilhelm Lenz\uff09<\/span>\u548c\u4f0a\u8f9b<\/span>\uff08Ernst Ising\uff09<\/span>\u572820\u4e16\u7eaa\u521d\u7684\u60f3\u6cd5\u548c\u5de5\u4f5c\u7684\u57fa\u7840\u4e0a\u3002\u7528\u970d\u91d1\u7684\u8bdd\u6765\u8bf4\uff0c\u6211\u4eec\u6b63\u5904\u4e8e\u201c\u590d\u6742\u6027\u7684\u4e16\u7eaa\u201d\uff0c\u5173\u6ce8\u7684\u95ee\u9898\u4ece\u652f\u914d\u7269\u8d28\u7684\u57fa\u672c\u6cd5\u5219\uff0c\u8f6c\u5411\u4e00\u5207\u4e8b\u7269\u5982\u4f55\u76f8\u4e92\u8054\u7cfb\u3002<\/strong><\/span><\/section>\n
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\u5c3d\u7ba1\u4f0a\u8f9b\u6700\u521d\u7684\u7814\u7a76\u5e76\u6ca1\u6709\u5f97\u5230\u671f\u671b\u7684\u76f8\u53d8\u7ed3\u679c\uff0c\u4f46\u662f\u968f\u673a\u6027\u52a0\u4e0a\u5fae\u89c2\u5c42\u9762\u7684\u603b\u4f53\u7b80\u5316\u53ef\u4ee5\u89e3\u91ca\u5b8f\u89c2\u5c42\u9762\u7684\u72b6\u6001\u53d8\u5316\u8fd9\u4e00\u60f3\u6cd5\u662f\u5f00\u521b\u6027\u7684\u3002\u73b0\u5728\u6211\u4eec\u77e5\u9053\u4e86\u7ef4\u5ea6\u3001\u76f8\u4e92\u4f5c\u7528\u8303\u56f4\u3001\u5bf9\u79f0\u6027\u3001\u6a21\u578b\u4e3a\u7ecf\u5178\u6216\u91cf\u5b50\u3001\u5e73\u8861\u6216\u975e\u5e73\u8861\u7b49\u5728\u7406\u89e3\u7269\u7406\u53d8\u5316\u4e2d\u7684\u91cd\u8981\u6027\u3002<\/span><\/section>\n
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\u5927\u91cf\u7684\u7814\u7a76\u6db5\u76d6\u4e86\u8fd9\u7c7b\u7cfb\u7edf\u7684\u5404\u79cd\u53d8\u91cf\u5982\u4f55\u63cf\u8ff0\u65e5\u76ca\u590d\u6742\u7684\u7cfb\u7edf\uff0c\u4f46\u662f\u5c06\u6982\u7387\u5e94\u7528\u4e8e\u7b80\u5316\u591a\u4f53\u7cfb\u7edf\u7684\u57fa\u672c\u601d\u60f3\u662f\u4ece\u793e\u4f1a\u7cfb\u7edf\u4e2d\u501f\u9274\u6765\u7684\u3002\u8fd1\u5e74\u6765\uff0c\u968f\u7740\u201c\u6d8c\u73b0\u201d\u6982\u5ff5\u7684\u51fa\u73b0\uff0c\u590d\u6742\u7cfb\u7edf\u7684\u7edf\u8ba1\u7269\u7406\u91cd\u65b0\u62e5\u62b1\u4e86\u5b83\u7684\u8de8\u5b66\u79d1\u8bde\u751f\u5904\uff0c\u8865\u5145\u4e14\u62d3\u5c55\u4e86\u7269\u7406\u5b66\uff0c\u5e76\u4e3a\u6211\u4eec\u5bf9\u4e16\u754c\u7684\u7406\u89e3\u505a\u51fa\u4e86\u8d21\u732e\u3002<\/span><\/section>\n
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\u672c\u671f\u7279\u520a\u805a\u7126\u5728\u5fae\u89c2\u5c42\u9762\u4e0a\u7b80\u5355\u4f46\u5728\u5b8f\u89c2\u5c42\u9762\u4e0a\u590d\u6742\u7684\u6a21\u578b<\/strong>\u3002\u7279\u520a\u5bf9\u6b63\u9762\u7ed3\u679c\u548c\u50cf\u4f0a\u8f9b\u8fd9\u6837\u7684\u8d1f\u9762\u7ed3\u679c\u90fd\u611f\u5174\u8da3\uff0c\u53cd\u601d\u7edf\u8ba1\u7269\u7406\u7684\u53d1\u6e90\uff0c\u540c\u65f6\u4e5f\u6b22\u8fce\u8de8\u5b66\u79d1\u7814\u7a76\u548c\u4f20\u7edf\u7269\u7406\u5b66\u3002\u672c\u520a\u5173\u6ce8\u53d8\u5316\u8fd9\u4e00\u6982\u5ff5\u2014\u2014\u65e0\u8bba\u53d8\u5316\u53d1\u751f\u5728\u4f55\u5904\uff0c\u7b80\u5355\u7684\u4e8b\u7269\u5982\u4f55\u901a\u8fc7\u975e\u5e73\u51e1\u7684\u673a\u5236\u6d8c\u73b0\u590d\u6742\u7684\u4e8b\u7269\u3002<\/span><\/section>\n

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1. \u771f\u5b9e\u526f\u672c\u4e4b\u95f4\u7684\u8ddd\u79bb\u53d8\u5316\u5bfc\u81f4\u81ea\u65cb\u73bb\u7483\u76f8\u53d8<\/span><\/strong><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n

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Spin Glasses in a Field Show a Phase Transition Varying the Distance among Real Replicas (And How to Exploit It to Find the Critical Line in a Field)<\/span><\/section>\n
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\u8fd9\u7bc7\u8bba\u6587\u7684\u4f5c\u8005\u4e4b\u4e00\u662f2021\u5e74\u8bfa\u8d1d\u5c14\u7269\u7406\u5b66\u5956\u5f97\u4e3b Giorgio Parisi\u3002\u5728\u7269\u7406\u5b66\u4e2d\uff0cParisi \u7684\u4e00\u9879\u8457\u540d\u5de5\u4f5c\u662f\u81ea\u65cb\u73bb\u7483\u7406\u8bba\u4e2d\u7684\u590d\u672c\u65b9\u6cd5<\/span>\uff08replica method\uff09<\/span>\u3002\u590d\u672c\u65b9\u6cd5\u662f\u5904\u7406\u968f\u673a\u76f8\u4e92\u4f5c\u7528\u7684\u4e00\u4e2a\u6807\u51c6\u6280\u5de7\u3002<\/span><\/section>\n

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\u81ea\u65cb\u73bb\u7483\u7406\u8bba\u6df1\u523b\u5730\u63ed\u793a\u4e86\u65e0\u5e8f\u4f53\u7cfb\u4e2d\u7684\u9690\u85cf\u5bf9\u79f0\u6027\uff0cParisi \u7b49\u4eba\u63d0\u51fa\u4e86\u590d\u672c\u95f4\u7684\u4ea4\u53e0\u5e8f\u53c2\u91cf\u6765\u8868\u5f81\u81ea\u65cb\u73bb\u7483\u7684\u76f8\u53d8\u3002\u8fd9\u7bc7\u8bba\u6587\u8ba8\u8bba\u4e86\u4e00\u79cd\u8fc7\u53bb\u5f88\u5c11\u8003\u8651\u7684\u81ea\u65cb\u73bb\u7483\u6a21\u578b\u4e2d\u7684\u76f8\u53d8\uff0c\u5373\u5f53\u4e24\u4e2a\u771f\u5b9e\u526f\u672c<\/span>\uff08replica\uff09<\/span>\u7684\u8ddd\u79bb\u88ab\u5f3a\u8feb\u6269\u5927\u5230\u6bd4\u6b63\u5e38\u60c5\u51b5\u5927<\/span>\uff08\u91cd\u53e0\u8f83\u5c0f\uff09<\/span>\u65f6\u53ef\u80fd\u53d1\u751f\u7684\u76f8\u53d8\u3002<\/span><\/section>\n

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\u5728\u5de5\u4f5c\u7684\u7b2c\u4e00\u90e8\u5206\uff0c\u4f5c\u8005\u4eec\u901a\u8fc7\u89e3\u6790\u5730\u6c42\u89e3\u63a5\u8fd1\u4e34\u754c\u70b9\u7684\u573a\u4e2d\u7684 Sherrington-Kirkpatrick \u6a21\u578b\uff0c\u53d1\u73b0\u5373\u4f7f\u5728\u987a\u78c1\u76f8\uff0c\u5f3a\u8feb\u4e24\u4e2a\u771f\u5b9e\u526f\u672c\u7684\u91cd\u53e0\u8db3\u591f\u5c0f\u4f1a\u5bfc\u81f4\u6a21\u578b\u53d1\u751f\u76f8\u53d8\uff0c\u526f\u672c\u4e4b\u95f4\u7684\u5bf9\u79f0\u6027\u81ea\u53d1\u7834\u7f3a\u3002\u66f4\u91cd\u8981\u7684\u662f\uff0c\u8fd9\u79cd\u76f8\u53d8\u4e0ede Almeida-Thouless<\/span> (dAT) <\/span>\u4e34\u754c\u7ebf\u6709\u5173\u3002\u5728\u5de5\u4f5c\u7684\u7b2c\u4e8c\u90e8\u5206\uff0c\u4f5c\u8005\u4eec\u5229\u7528\u4e24\u4e2a\u771f\u5b9e\u526f\u672c\u4e4b\u95f4\u91cd\u53e0\u7684\u76f8\u53d8\uff0c\u6765\u8bc6\u522b\u6709\u9650\u7ef4\u81ea\u65cb\u73bb\u7483\u4e2d\u573a\u4e2d\u7684\u4e34\u754c\u7ebf\u3002\u56e0\u4e3a\u9700\u8981\u8fdb\u884c\u5927\u91cf\u7684\u6709\u9650\u5c3a\u5bf8\u4fee\u6b63\uff0c\u8fd9\u4e2a\u8ba1\u7b97\u95ee\u9898\u7684\u56f0\u96be\u4f17\u6240\u5468\u77e5\u3002<\/span><\/section>\n

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\u4f5c\u8005\u4eec\u4e3a\u65e0\u5e8f\u7cfb\u7edf\u5f15\u5165\u4e86\u4e00\u79cd\u5206\u6790\u8499\u7279\u5361\u7f57\u6570\u636e\u7684\u65b0\u65b9\u6cd5\uff0c\u5176\u4e2d\u4e24\u4e2a\u771f\u5b9e\u526f\u672c\u4e4b\u95f4\u7684\u91cd\u53e0\u88ab\u4f5c\u4e3a\u4e00\u4e2a\u6761\u4ef6\u53d8\u91cf\u3002\u7136\u540e\u8fd9\u79cd\u5206\u6790\u88ab\u5e94\u7528\u4e8e d =1 \u81ea\u65cb\u73bb\u7483\u6a21\u578b\u573a\u4e2d\u987a\u78c1\u76f8\u7684\u5e73\u8861\u6d4b\u91cf,  \u8fd9\u91cc\u81ea\u65cb\u73bb\u7483\u6a21\u578b\u5177\u6709\u957f\u7a0b\u76f8\u4e92\u4f5c\u7528\uff0c\u8870\u51cf\u5f97\u8db3\u591f\u5feb\uff0c\u8d85\u51fa\u4e86\u5e73\u5747\u573a\u7406\u8bba\u7684\u6709\u6548\u8303\u56f4\u3002\u8fd9\u9879\u5de5\u4f5c\u5bf9\u573a\u4e2d\u7684\u70ed\u529b\u5b66\u4e34\u754c\u6e29\u5ea6\u63d0\u4f9b\u4e86\u975e\u5e38\u53ef\u9760\u7684\u4f30\u8ba1\u3002<\/span><\/section>\n

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\u56fe1. \u7ea2\u8272\u66f2\u7ebf\u662f dAT \u4e34\u754c\u7ebf\uff0c\u5206\u9694\u4e86\u987a\u78c1\u76f8\u548c\u73bb\u7483\u76f8\u3002\u5728\u987a\u78c1\u76f8\u7684\u84dd\u8272\u66f2\u9762\u4ee5\u4e0b\uff0c\u526f\u672c\u5bf9\u79f0\u6027\u88ab\u6253\u7834\u3002<\/span><\/p>\n


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2. \u4f0a\u8f9b\u81ea\u65cb\u73bb\u7483\u7684\u8d85\u6807\u5ea6\u8fdd\u53cd<\/span><\/strong><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n

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\u8bba\u6587\u9898\u76ee\uff1aHyperscaling Violation in Ising Spin Glasses<\/span><\/section>\n
\u8bba\u6587\u5730\u5740\uff1a<\/span>https:\/\/www.mdpi.com\/1099-4300\/21\/10\/978<\/span><\/section>\n<\/blockquote>\n

<\/section>\n
\u7269\u7406\u7cfb\u7edf\u7684\u5fae\u89c2\u7ed3\u6784\u548c\u5b8f\u89c2\u73b0\u8c61\u53ef\u80fd\u4e0d\u5c3d\u76f8\u540c\uff0c\u4f46\u5728\u4e34\u754c\u70b9\u9644\u8fd1\uff0c\u5374\u53ef\u80fd\u8868\u73b0\u51fa\u7c7b\u4f3c\u7684\u76f8\u53d8\u884c\u4e3a\uff0c\u9075\u5faa\u663e\u8457\u7684\u5e42\u5f8b\u5173\u7cfb\uff0c\u5177\u6709\u76f8\u540c\u7684\u4e34\u754c\u6307\u6570\u3002\u800c\u4e14\u8fd9\u4e9b\u4e34\u754c\u6307\u6570\u5e76\u975e\u5f7c\u6b64\u72ec\u7acb\uff0c\u800c\u662f\u901a\u8fc7\u6807\u5ea6\u5f8b<\/span>\uff08scaling rule\uff09<\/span>\u8054\u7cfb\u8d77\u6765\u3002\u5982\u679c\u518d\u5f15\u5165\u7cfb\u7edf\u7684\u7a7a\u95f4\u7ef4\u6570\uff0c\u5219\u5f97\u5230\u8d85\u6807\u5ea6\u5f8b<\/span>\uff08hyperscaling rule\uff09<\/span>\u3002\u5bf9\u4e8e\u6717\u9053\u76f8\u53d8\u7406\u8bba\uff0c\u8d85\u6807\u5ea6\u5173\u7cfb\u901a\u5e38\u5e76\u4e0d\u6210\u7acb\uff0c\u5f53\u4e14\u4ec5\u5f53\u7a7a\u95f4\u7ef4\u5ea6 d=4 \u65f6\uff0c\u8d85\u6807\u5ea6\u5173\u7cfb\u624d\u6210\u7acb\u3002<\/span><\/section>\n

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Edwards\u2013Anderson \u4f0a\u8f9b\u81ea\u65cb\u73bb\u7483<\/span>\uff08Ising Spin Glass\uff09<\/span>\u7684\u6dec\u706b\u76f8\u4e92\u4f5c\u7528\u88ab\u660e\u786e\u5047\u8bbe\u4e3a\u662f\u968f\u673a\u7684\uff0c\u9075\u5faa\u4e0e\u6807\u51c6\u4f0a\u8f9b\u6a21\u578b\u76f8\u540c\u7684\u57fa\u672c\u6807\u5ea6\u5f8b\u548c\u666e\u9002\u6027\u3002\u81f3\u4e8e\u8d85\u6807\u5ea6\u5173\u7cfb\uff0c\u4f17\u6240\u5468\u77e5\uff0c\u5728\u6807\u51c6\u4f0a\u8f9b\u6a21\u578b\u4e2d\uff0c\u5f53\u7ef4\u5ea6\u8d85\u8fc7\u4e0a\u4e34\u754c\u7ef4\u6570 d=4 \u65f6\uff0c\u8d85\u6807\u5ea6\u5f8b\u574d\u584c \uff1bM. Schwartz \u57281991\u5e74\u6307\u51fa\uff0c\u8d85\u6807\u5ea6\u4e5f\u53ef\u4ee5\u5728\u5177\u6709\u6dec\u706b\u968f\u673a\u76f8\u4e92\u4f5c\u7528\u7684\u4f0a\u8f9b\u7cfb\u7edf\u4e2d\u88ab\u7834\u574f\uff0c\u5c5e\u4e8e\u8fd9\u7c7b\u7cfb\u7edf\u7684\u968f\u673a\u573a\u4f0a\u8f9b\u6a21\u578b\u5df2\u7ecf\u88ab\u6df1\u5165\u7814\u7a76\u3002\u672c\u6587\u4e3b\u8981\u7814\u7a76\u4f0a\u8f9b\u81ea\u65cb\u73bb\u7483\u4e2d\u8d85\u6807\u5ea6\u8fdd\u53cd\u7684\u8bc1\u636e\u3002<\/span><\/section>\n

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\u6587\u7ae0\u7ed9\u51fa\u4e863\u30014\u30015\u30017\u7ef4\u60c5\u51b5\u4e0b\uff0c\u5c06\u5f52\u4e00\u5316 Binder \u7d2f\u79ef\u91cf\u4e0e\u7ea6\u5316\u5173\u8054\u957f\u5ea6\u8054\u7cfb\u8d77\u6765\u7684\u6570\u503c\u4f0a\u8f9b\u81ea\u65cb\u73bb\u7483\u6570\u636e\u3002\u57283\u30014\u7ef4\u4ee5\u53ca\u8d85\u8fc7\u4e0a\u4e34\u754c\u7ef4\u6570\u65f6\uff0c\u8d85\u6807\u5ea6\u660e\u663e\u88ab\u8fdd\u53cd\u3002\u7814\u7a76\u5f97\u5230\u4e86\u5404\u79cd\u6a21\u578b\u4e2d\u8fdd\u53cd\u8d85\u6807\u5ea6\u6307\u6570\u6570\u503c\u7684\u4f30\u8ba1\u3002<\/span><\/section>\n

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3.\uff08\u56db\uff09\u53cc\u5143\u683c\u4e09\u7ef4\u4f0a\u8f9b\u6a21\u578b<\/span><\/strong><\/strong><\/span><\/p>\n<\/section>\n<\/section>\n<\/section>\n

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\u8bba\u6587\u9898\u76ee\uff1a<\/span><\/section>\n
(Four) Dual Plaquette 3D Ising Models<\/span><\/section>\n
\u8bba\u6587\u5730\u5740\uff1a<\/span><\/section>\n
<\/span>https:\/\/www.mdpi.com\/1099-4300\/22\/6\/633<\/span><\/section>\n<\/blockquote>\n

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\u4e09\u7ef4\u5143\u683c<\/span>\uff08Plaquette\uff09<\/span>\u4f0a\u8f9b\u6a21\u578b\u7684\u4e00\u4e2a\u7279\u5f81\u662f\u5176\u5e73\u9762\u5b50\u7cfb\u7edf\u7684\u5bf9\u79f0\u6027\u3002\u5df2\u8bc1\u660e\u8be5\u6a21\u578b\u7684\u91cf\u5b50\u7248\u672c\u901a\u8fc7\u5bf9\u5076\u6027\u4e0e X-Cube \u6a21\u578b\u76f8\u5173\u8054\uff0cX-Cube \u6a21\u578b\u662f\u5feb\u901f\u53d1\u5c55\u7684\u5206\u5f62\u5b50<\/span>\uff08fracton\uff09<\/span>\u9886\u57df\u7684\u4e00\u4e2a\u8303\u4f8b\u3002\u4e09\u7ef4\u5143\u683c\u4f0a\u8f9b\u6a21\u578b\u4e0e X-Cube \u6a21\u578b\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u7c7b\u4f3c\u4e8e\u4e8c\u7ef4\u91cf\u5b50\u6a2a\u5411\u81ea\u65cb\u4f0a\u8f9b\u6a21\u578b\u4e0e Toric \u7f16\u7801\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u8ba1\u7b97\u4e8c\u7ef4\u4f0a\u8f9b\u6a21\u578b\u7684\u5168\u5c40\u5bf9\u79f0\u6027\u5e76\u8003\u8651\u9ad8\u6e29\u76f8\u7684\u89c4\u8303\u4e0d\u53d8\u533a\u4ea7\u751f\u4e86 Toric \u7f16\u7801\uff0c\u800c\u8ba1\u7b97\u4e09\u7ef4\u91cf\u5b50\u6a2a\u5411\u81ea\u65cb\u5143\u683c\u4f0a\u8f9b\u6a21\u578b\u7684\u5b50\u7cfb\u7edf\u5bf9\u79f0\u6027\u5219\u4ea7\u751f\u4e86 X-Cube \u6a21\u578b\u3002<\/span><\/section>\n

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\u5728\u5bf9\u5076\u5316 X-cube \u6a21\u578b\u65e0\u5206\u5f62\u5b50\u90e8\u5206\u7684\u8bed\u5883\u4e0b\uff0c\u4e00\u4e2a\u5229\u7528\u4e09\u79cd\u81ea\u65cb\u7684\u975e\u6807\u51c6\u5bf9\u5076\u5f62\u5f0f\u7684\u4e09\u7ef4\u5143\u683c\u4f0a\u8f9b\u6a21\u578b\u6700\u8fd1\u5f15\u53d1\u8ba8\u8bba\u3002\u672c\u6587\u7814\u7a76\u4e86\u8fd9\u79cd\u975e\u6807\u51c6\u5bf9\u5076\u54c8\u5bc6\u987f\u91cf\u7684\u7ecf\u5178\u81ea\u65cb\u7248\u672c\uff0c\u5e76\u8ba8\u8bba\u4e86\u5b83\u4e0e\u66f4\u4e3a\u719f\u6089\u7684\u7c7b Ashkin-Teller \u5bf9\u5076\u53ca\u5176\u76f8\u5173\u5bf9\u5076\u516c\u5f0f\u7684\u5173\u7cfb\uff0c\u4ee5\u53ca\u66f4\u8fdc\u7684\u5305\u62ec\u8fb9\u548c\u9876\u70b9\u81ea\u65cb\u4e0e\u975e\u4f0a\u8f9b\u81ea\u65cb\u7684\u5bf9\u5076\u5f62\u5f0f\u7684\u5173\u7cfb\u3002<\/span><\/section>\n

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\u56fe2. \u5728\u4e09\u7ef4\u5143\u683c\u4f0a\u8f9b\u6a21\u578b\u7684\u4e00\u4e2a\u7acb\u65b9\u4f53\u7684\u4e00\u4e2a\u9762\u4e0a\uff0c\u7ffb\u8f6c\u81ea\u65cb\u5e76\u4e0d\u4f1a\u6539\u53d8\u80fd\u91cf\u3002\u56fe\u4e2d\u7b2c\u4e00\u4e2a\u7acb\u65b9\u4f53\u662f\u53cd\u94c1\u78c1\u6001\uff0c\u6240\u6709\u81ea\u65cb\u4e3a+\uff1b\u5176\u4ed6\u4e09\u4e2a\u7acb\u65b9\u4f53\u7684\u9634\u5f71\u9762\u7684\u89d2\u4e0a\u7684\u81ea\u65cb\u4e3a\u2212\uff0c\u5176\u4ed6\u81ea\u65cb\u4e3a+\u3002\u6240\u6709\u56db\u79cd\u81ea\u65cb\u6784\u578b\u6709\u76f8\u540c\u80fd\u91cf\u3002<\/span><\/p>\n


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4. \u57fa\u4e8e\u4f0a\u8f9b\u6a21\u578b\u7684\u968f\u673a\u5757\u6a21\u578b\u7684\u7cbe\u786e\u6062\u590d<\/span><\/strong><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n


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\u8bba\u6587\u9898\u76ee\uff1a<\/span><\/section>\n
Exact Recovery of Stochastic Block Model by Ising Model<\/span><\/section>\n
\u8bba\u6587\u5730\u5740\uff1a<\/span><\/section>\n
<\/span>https:\/\/www.mdpi.com\/1099-4300\/23\/1\/65<\/span><\/section>\n<\/blockquote>\n

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\u5728\u7f51\u7edc\u5206\u6790\u4e2d\uff0c\u793e\u56e2\u68c0\u6d4b<\/span>\uff08community detection\uff09<\/span>\u662f\u6307\u63a8\u65ad\u56fe\u4e2d\u8fde\u63a5\u66f4\u5bc6\u96c6\u7684\u9876\u70b9\u96c6\u5408\u3002\u793e\u56e2\u68c0\u6d4b\u5df2\u88ab\u5e94\u7528\u4e8e\u8bb8\u591a\u9886\u57df\uff0c\u5305\u62ec\u63a8\u8350\u7cfb\u7edf\u3001\u5206\u5e03\u5f0f\u8ba1\u7b97\u4e2d\u7684\u4efb\u52a1\u5206\u914d\u3001\u57fa\u56e0\u8868\u8fbe\u7b49\u3002\u968f\u673a\u5757\u6a21\u578b<\/span>\uff08Stochastic Block Model\uff09<\/span>\u662f\u793e\u56e2\u68c0\u6d4b\u95ee\u9898\u4e2d\u6700\u5e38\u7528\u7684\u7edf\u8ba1\u6a21\u578b\u4e4b\u4e00\u3002<\/span><\/section>\n

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\u968f\u673a\u5757\u6a21\u578b\u662f\u968f\u673a\u56fe\u7684\u751f\u6210\u6a21\u578b\uff0c\u5b83\u503e\u5411\u4e8e\u751f\u6210\u5305\u542b\u793e\u56e2\u7684\u56fe\uff0c\u5e38\u4f5c\u4e3a\u6062\u590d\u56fe\u6570\u636e\u4e2d\u793e\u56e2\u7ed3\u6784\u7684\u57fa\u51c6\u3002\u6062\u590d\u793e\u56e2\u7684\u4efb\u52a1\u5206\u4e3a\u7cbe\u786e\u6062\u590d<\/span>\uff08exact recovery\uff09<\/span>\u4e0e\u90e8\u5206\u6062\u590d<\/span>\uff08partial recovery\uff09<\/span>\uff0c\u524d\u8005\u8981\u6c42\u7b97\u6cd5\u4f30\u8ba1\u7684\u793e\u56e2\u4e0e\u968f\u673a\u5757\u6a21\u578b\u7684\u793e\u56e2\u7ed3\u6784\u5b8c\u5168\u76f8\u540c\uff0c\u540e\u8005\u8981\u6c42\u7b97\u6cd5\u5c3d\u53ef\u80fd\u51cf\u5c0f\u9519\u8bef\u5206\u7c7b\u7684\u8282\u70b9\u6bd4\u4f8b\u3002\u901a\u5e38\u4f1a\u5206\u6790\u5f53\u56fe\u7684\u89c4\u6a21\u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u68c0\u6d4b\u8bef\u5dee\u7684\u6e10\u8fd1\u7279\u6027\u3002<\/span><\/section>\n

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\u672c\u6587\u7814\u7a76\u5b9a\u4e49\u5728\u968f\u673a\u5757\u6a21\u578b\u4e0a\u7684\u4f0a\u8f9b\u6a21\u578b\u7684\u76f8\u53d8\u7279\u6027\u3002\u5728\u4f0a\u8f9b\u6a21\u578b<\/span>\uff08\u8282\u70b9\u72b6\u6001\u7684\u6982\u7387\u5206\u5e03\uff09<\/span>\u7684\u57fa\u7840\u4e0a\uff0c\u6587\u7ae0\u63d0\u51fa\u4e86\u4e00\u79cd\u968f\u673a\u4f30\u8ba1\u65b9\u6cd5\u6765\u5b9e\u73b0\u968f\u673a\u5757\u6a21\u578b\u7684\u7cbe\u786e\u6062\u590d\u3002\u968f\u673a\u7b97\u6cd5\u53ef\u4ee5\u8f6c\u5316\u4e3a\u4e00\u4e2a\u4f18\u5316\u95ee\u9898\uff0c\u5176\u4e2d\u5305\u62ec\u6700\u5927\u4f3c\u7136\u548c\u6700\u5927\u6a21\u5757\u5316\u7684\u7279\u6b8a\u60c5\u51b5\u3002\u6b64\u5916\uff0c\u672c\u6587\u7ed9\u51fa\u4e86\u968f\u673a\u5757\u6a21\u578b\u53c2\u6570\u7684\u65e0\u504f\u6536\u655b\u4f30\u8ba1\uff0c\u8be5\u4f30\u8ba1\u53ef\u4ee5\u5728\u6052\u5b9a\u65f6\u95f4\u5185\u8ba1\u7b97\u3002\u6700\u540e\uff0c\u5229\u7528 metropolis \u91c7\u6837\u5b9e\u73b0\u4e86\u968f\u673a\u4f30\u8ba1\uff0c\u5e76\u901a\u8fc7\u5b9e\u9a8c\u9a8c\u8bc1\u4e86\u76f8\u53d8\u73b0\u8c61\u3002<\/span><\/section>\n

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5. \u6709\u5411 BA \u7f51\u7edc\u4e0a\u7684\u79bb\u6563\u610f\u89c1\u52a8\u529b\u5b66\u6a21\u578b<\/span><\/strong><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n


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\u8bba\u6587\u9898\u76ee\uff1a <\/span><\/section>\n
Kinetic Models of Discrete Opinion Dynamics on Directed Barab\u00e1si\u2013Albert Networks<\/span><\/section>\n
\u8bba\u6587\u5730\u5740\uff1a<\/span><\/section>\n
https:\/\/www.mdpi.com\/1099-4300\/21\/10\/942<\/span><\/section>\n<\/blockquote>\n

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\u590d\u6742\u7f51\u7edc\u5e7f\u6cdb\u5e94\u7528\u4e8e\u81ea\u7136\u4e2d\u7684\u7f51\u7edc\u548c\u4eba\u7684\u7f51\u7edc\uff0c\u9664\u6b64\u4e4b\u5916\uff0c\u590d\u6742\u7f51\u7edc\u4e0a\u7684\u81ea\u65cb\u7cfb\u7edf\u4e5f\u53ef\u7528\u4e8e\u7814\u7a76\u76f8\u53d8\u7279\u5f81\uff0c\u4ee5\u53ca\u4e34\u754c\u884c\u4e3a\u5bf9\u5e94\u7684\u666e\u9002\u6027\u7c7b\u3002\u5728\u6709\u5411 Barab\u00e1si-Albert<\/span>\uff08BA\uff09<\/span>\u7f51\u7edc\u4e2d\uff0c\u5df2\u7ecf\u8bc1\u660e\u6700\u8fd1\u90bb\u81ea\u65cb 1\/2 \u4f0a\u8f9b\u6a21\u578b\u6ca1\u6709\u76f8\u53d8\uff0c\u4f46\u975e\u5e73\u8861\u591a\u6570\u6295\u7968\u6a21\u578b<\/span>\uff08majority-vote model\uff09<\/span>\u8868\u73b0\u51fa\u6709\u5e8f-\u65e0\u5e8f\u52a8\u6001\u76f8\u53d8\u30022012\u5e74\uff0cBiswas\u3001Chatterjee\u3001Sen \u5f15\u5165\u4e86\u8fde\u7eed<\/span>\uff08\u6216\u79bb\u6563\uff09<\/span>\u610f\u89c1\u52a8\u529b\u5b66\u6a21\u578b\uff0cBCS \u6a21\u578b\u53ef\u4ee5\u5177\u6709\u6b63\u7684\u6216\u8d1f\u7684\u76f8\u4e92\u4f5c\u7528\uff0c\u8fde\u7eed BCS \u6a21\u578b\u7684\u6570\u503c\u6a21\u62df\u8868\u660e\u5b58\u5728\u666e\u904d\u7684\u8fde\u7eed\u76f8\u53d8\u3002<\/span><\/section>\n

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\u672c\u6587\u5229\u7528 Monte Carlo \u6a21\u62df\u65b9\u6cd5\uff0c\u7814\u7a76\u4e86\u6709\u5411 BA \u7f51\u7edc\u4e0a\u7684\u79bb\u6563\u610f\u89c1\u52a8\u529b\u5b66\u6a21\u578b\uff0c\u5373 BCS \u6a21\u578b\u7684\u79bb\u6563\u7248\u672c\uff0c\u53d1\u73b0\u8be5\u4f53\u7cfb\u5b58\u5728\u8fde\u7eed\u76f8\u53d8\u3002\u7814\u7a76\u5f97\u5230\u4e86\u8fd9\u4e9b\u6709\u5411\u7f51\u7edc\u7684\u51e0\u4e2a\u8fde\u901a\u5ea6\u503c\u7684\u566a\u58f0\u53c2\u6570\u4e34\u754c\u503c\u3002\u6b64\u5916\u8fd8\u8ba1\u7b97\u4e86\u5e8f\u53c2\u6570\u7684\u4e34\u754c\u6307\u6570\u4e4b\u6bd4\u4e0e\u5173\u8054\u957f\u5ea6\u7684\u611f\u5e94\u5f3a\u5ea6\u3002\u4f5c\u8005\u4eec\u6ce8\u610f\u5230\uff0c\u4e34\u754c\u6307\u6570\u4e4b\u6bd4\u968f\u7740\u7f51\u7edc\u8fde\u901a\u6027\u7684\u53d8\u5316\u800c\u53d8\u5316\uff0c\u8fd9\u4e9b\u6709\u5411 BA \u7f51\u7edc\u4e0a\u7684\u52a8\u529b\u5b66\u6a21\u578b\u548c\u591a\u6570\u6295\u7968\u6a21\u578b\u5c5e\u4e8e\u540c\u4e00\u666e\u9002\u6027\u7c7b\u3002<\/span><\/section>\n

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6. \u7f51\u7edc\u751f\u6210\u6811\u7684\u666e\u9002\u6027\u548c\u6709\u9650\u5c3a\u5bf8\u4fee\u6b63<\/span><\/strong><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n


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\u8bba\u6587\u9898\u76ee\uff1a<\/span><\/section>\n
Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks<\/span><\/section>\n
\u8bba\u6587\u5730\u5740\uff1a<\/span><\/section>\n
https:\/\/www.mdpi.com\/1099-4300\/21\/9\/895<\/span><\/section>\n<\/blockquote>\n

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\u666e\u9002\u6027\u7684\u6982\u5ff5\u662f\u4e34\u754c\u73b0\u8c61\u7406\u8bba\u7684\u57fa\u77f3\u3002\u5bf9\u4e8e\u5927\u591a\u6570\u7cfb\u7edf\uff0c\u7279\u522b\u662f\u5728\u70ed\u529b\u5b66\u6781\u9650\u4e0b\uff0c\u666e\u9002\u6027\u90fd\u5f97\u5230\u4e86\u5f88\u597d\u7684\u7406\u89e3\u3002\u7136\u800c\uff0c\u5728\u6709\u9650\u7cfb\u7edf\u7684\u4fee\u6b63\u9879\u4e2d\uff0c\u8fb9\u754c\u7279\u5f81\u662f\u660e\u663e\u7684\u3002\u5bf9\u6709\u9650\u5c3a\u5bf8\u6807\u5ea6\u548c\u6807\u5ea6\u4fee\u6b63\u7684\u7406\u8bba\u7814\u7a76\uff0c\u5305\u62ec\u5916\u63a8\u6709\u9650\u6216\u90e8\u5206\u6709\u9650\u7cfb\u7edf\uff0c\u4ee5\u786e\u5b9a\u5b83\u4eec\u5bf9\u5e94\u7684\u65e0\u9650\u7cfb\u7edf\u7684\u4e34\u754c\u548c\u975e\u4e34\u754c\u6027\u8d28\u3002<\/span><\/section>\n

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\u6709\u9650\u5c3a\u5bf8\u7cfb\u7edf\u5e26\u6765\u4e86\u6311\u6218\u3002\u5373\u4f7f\u5728\u4f4e\u7ef4\uff0c\u8fb9\u548c\u89d2\u5bf9\u81ea\u7531\u80fd\u548c\u54cd\u5e94\u51fd\u6570\u7684\u8d21\u732e\u7684\u666e\u9002\u6027\u4e5f\u5f88\u5c11\u88ab\u7814\u7a76\u548c\u7406\u89e3\u3002\u7279\u522b\u662f\u6709\u4e00\u4e2a\u95ee\u9898\uff0c\u5c5e\u4e8e\u540c\u4e00\u666e\u9002\u6027\u7c7b\u3001\u4f46\u5177\u6709\u975e\u5e38\u4e0d\u540c\u89d2\u51e0\u4f55\u7684\u7cfb\u7edf\uff0c\u5728\u6807\u5ea6\u4fee\u6b63\u4e2d\u5982\u4f55\u4fdd\u6301\u666e\u9002\u6027\u3002<\/span><\/section>\n

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\u4e8c\u7ef4\u51e0\u4f55\u63d0\u4f9b\u4e86\u6700\u7b80\u5355\u7684\u4f8b\u5b50\uff0c\u53ef\u4ee5\u6784\u9020\u6709\u89d2\u548c\u6ca1\u6709\u89d2\u7684\u56fe\u5f62\u3002\u4e3a\u4e86\u7814\u7a76\u89d2\u7684\u5b58\u5728\u548c\u4e0d\u5b58\u5728\u5982\u4f55\u4f53\u73b0\u666e\u9002\u6027\uff0c\u672c\u6587\u5206\u6790\u4e86\u4e24\u79cd\u4e0d\u540c\u7684\u6709\u9650\u7cfb\u7edf\uff0c\u5373\u86db\u7f51<\/span>\uff08cobweb\uff09<\/span>\u548c\u6247\u5f62\u7f51\u7edc<\/span>\uff08fan network\uff09<\/span>\u4e0a\u7684\u751f\u6210\u6811<\/span>\uff08Spanning Tree\uff09<\/span>\u751f\u6210\u51fd\u6570\u3002\u751f\u6210\u6811\u662f\u4e00\u4e2a\u8fde\u901a\u56fe\u7684\u5b50\u56fe\uff0c\u8868\u793a\u5c06\u7ed3\u6784\u4e2d\u6bcf\u4e2a\u8282\u70b9\u94fe\u63a5\u8d77\u6765\u7684\u6700\u6709\u6548\u65b9\u5f0f\u3002\u6709\u9650\u5c3a\u5bf8\u7684\u86db\u7f51\u6ca1\u6709\u89d2\uff0c\u800c\u6247\u5f62\u7f51\u7edc\u6709\u56db\u4e2a\u89d2\u3002\u56e0\u4e3a\u671f\u5f85\u4e86\u89e3\u5b83\u4eec\u7684\u666e\u9002\u7279\u6027\uff0c\u8fd9\u4e9b\u6784\u578b\u7684\u89d2\u81ea\u7531\u80fd\u6fc0\u53d1\u4e86\u4eba\u4eec\u6781\u5927\u7684\u5174\u8da3\u3002<\/span><\/section>\n

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2002\u5e74\uff0cIvashkevich\u3001Izmailian\u3001Hu \u5f00\u53d1\u4e86\u4e00\u79cd\u65b9\u6cd5\uff0c\u53ef\u4ee5\u4e3a\u5404\u79cd\u57fa\u672c\u51fd\u6570\uff0c\u6bd4\u5982\u914d\u5206\u51fd\u6570\u53ca\u5176\u5bfc\u6570\uff0c\u63d0\u4f9b\u7cbe\u786e\u7684\u6709\u9650\u5c3a\u5bf8\u4fee\u6b63\u3002\u8be5\u65b9\u6cd5\u88ab\u5e94\u7528\u4e8e\u591a\u79cd\u7edf\u8ba1\u7269\u7406\u6a21\u578b\u4e2d\uff0c\u5982\u4f0a\u8f9b\u6a21\u578b\u3001\u4e8c\u805a\u4f53\u6a21\u578b\u548c\u9ad8\u65af\u6a21\u578b\uff0c\u7ed3\u679c\u8868\u660e\uff0c\u6bcf\u4e2a\u6a21\u578b\u7684\u914d\u5206\u51fd\u6570\u90fd\u53ef\u4ee5\u5199\u6210\u5177\u6709\u626d\u66f2\u8fb9\u754c\u6761\u4ef6<\/span>\uff08twisted boundary condition\uff09<\/span>\u7684\u4e00\u822c\u914d\u5206\u51fd\u6570\u3002<\/span><\/section>\n

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\u8fd9\u9879\u7814\u7a76\u4f7f\u7528 Ivashkevich-Izmailian-Hu \u65b9\u6cd5\uff0c\u5c06\u4e0d\u540c\u7f51\u7edc\u7684\u751f\u6210\u51fd\u6570\u7edf\u4e00\u4e3a\u5177\u6709\u626d\u66f2\u8fb9\u754c\u6761\u4ef6\u7684\u5355\u4e2a\u914d\u5206\u51fd\u6570\u3002\u8fd9\u79cd\u7edf\u4e00\u65b9\u6cd5\u8868\u660e\uff0c\u6247\u5f62\u7f51\u7edc\u56db\u4e2a\u89d2\u7684\u81ea\u7531\u80fd\u76f8\u4e92\u62b5\u6d88\uff0c\u8d21\u732e\u4e4b\u548c\u4e3a\u96f6\uff0c\u56e0\u6b64\u4e0e\u6ca1\u6709\u89d2\u7684\u86db\u7f51\u7f51\u7edc\u7684\u8d21\u732e\u4e00\u81f4\u3002\u56e0\u6b64\uff0c\u5b83\u4e5f\u7b26\u5408\u5171\u5f62\u7406\u8bba<\/span>\uff08\u5176\u4e2d\u5fc3\u7535\u8377\u4e3a c=\u22122\uff09<\/span>\u548c\u6709\u9650\u5c3a\u5bf8\u6807\u5ea6\u9884\u6d4b\u3002<\/span><\/section>\n

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\u8be5\u7814\u7a76\u9a8c\u8bc1\u4e86 Ivashkevich-Izmailian-Hu \u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002\u5176\u5e7f\u6cdb\u7684\u6709\u7528\u6027\u4f53\u73b0\u5728\u53ef\u4ee5\u5e94\u7528\u5230\u8fc4\u4eca\u4e3a\u6b62\u5c1a\u672a\u89e3\u51b3\u7684\u95ee\u9898\u2014\u2014\u5373\u751f\u6210\u51fd\u6570\u5bf9\u6570\u7684\u7cbe\u786e\u6e10\u8fd1\u5c55\u5f00\uff0c\u4ee5\u53ca\u6247\u5f62\u548c\u86db\u7f51\u51e0\u4f55\u7684\u5171\u5f62\u914d\u5206\u51fd\u6570\u3002\u4f5c\u8005\u4eec\u8fd8\u7814\u7a76\u4e86\u5e26\u72b6\u51e0\u4f55<\/span>\uff08strip geometry\uff09<\/span>\uff0c\u518d\u6b21\u8bc1\u5b9e\u4e86\u5171\u5f62\u573a\u8bba\u7684\u9884\u6d4b\u3002\u56e0\u6b64\uff0c\u666e\u9002\u6027\u96be\u9898\u7684\u89e3\u51b3\u8bc1\u660e\u4e86\u7b97\u6cd5\u7684\u5f3a\u5927\uff0c\u5e76\u5f00\u8f9f\u4e86\u672a\u6765\u7684\u65b0\u5e94\u7528\u3002<\/span><\/section>\n

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\u56fe3. (a) M=3, N=8 \u7684 M\u00d7N \u86db\u7f51\u7f51\u7edc\u3002(b) M=3, N=7 \u7684 M\u00d7N \u6247\u5f62\u7f51\u7edc\u3002<\/span><\/section>\n

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\u96c6\u667a\u4ff1\u4e50\u90e8\u56e0\u679c\u6d8c\u73b0\u793e\u533a<\/strong><\/span><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


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