{"id":62459,"date":"2025-07-23T20:02:35","date_gmt":"2025-07-23T12:02:35","guid":{"rendered":"https:\/\/swarma.org\/?p=62459"},"modified":"2025-07-23T20:02:35","modified_gmt":"2025-07-23T12:02:35","slug":"%e9%87%8d%e6%95%b4%e5%8c%96%e7%be%a4%e9%81%87%e8%a7%81%e6%9c%ba%e5%99%a8%e5%ad%a6%e4%b9%a0%ef%bc%9a%e5%a4%9a%e5%b0%ba%e5%ba%a6%e8%a7%86%e8%a7%92%e6%8e%a2%e7%b4%a2%e5%a4%8d%e6%9d%82%e7%b3%bb%e7%bb%9f","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=62459","title":{"rendered":"\u91cd\u6574\u5316\u7fa4\u9047\u89c1\u673a\u5668\u5b66\u4e60\uff1a\u591a\u5c3a\u5ea6\u89c6\u89d2\u63a2\u7d22\u590d\u6742\u7cfb\u7edf\u5185\u5728\u7684\u7edf\u4e00\u6027"},"content":{"rendered":"
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dx\/dt = f(x, <\/span>\u03b8<\/span><\/span>)        (1)<\/span><\/span><\/section>\n

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dy\/dt = f'(y, <\/span>\u03b8<\/span><\/span>‘)\uff08\u5e73\u8861\u6001\u7cfb\u7edf\u5219\u53ef\u4ee5\u5199\u6210p'(y, <\/span>\u03b8<\/span><\/span>‘) \uff082\uff09<\/span><\/span><\/section>\n

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\u5728\u8fd9\u4e2a\u65b0\u7684\u63cf\u8ff0\u5f62\u5f0f\u4e0b\uff0c\u7cfb\u7edf\u7684\u53d8\u91cf\u53ef\u4ee5\u548c\u539f\u6765\u4e0d\u4e00\u6837\uff0c\u53c2\u6570\u53ef\u4ee5\u4e0d\u4e00\u6837\uff0c\u5c31\u8fde\u65b9\u7a0b\u7684\u5f62\u5f0f\u4e5f\u53ef\u4ee5\u4e0d\u4e00\u6837<\/span><\/span>\uff08\u4ecef\u53d8\u6210\u4e86f’\uff09<\/span><\/span>\u3002\u5176\u5b9e\u5404\u4e2a\u9886\u57df\u7684\u4f20\u7edf\u65b9\u6cd5\uff0c\u5c31\u662f\u8fd9\u4e48\u53bb\u5904\u7406\u4e00\u4e2a\u7cfb\u7edf\u4e0d\u540c\u5c3a\u5ea6\u7684\u3002\u6211\u4eec\u5728\u5904\u7406\u6bcf\u4e00\u4e2a\u5c3a\u5ea6\u4e0a\u7684\u95ee\u9898\u53ef\u4ee5\u8bf4\u662f\u6e38\u5203\u6709\u4f59\uff0c\u5e76\u4e14\u4e5f\u53d1\u660e\u4e86\u5f88\u591a\u975e\u5e38\u6210\u719f\u7684\u5de5\u5177\u7528\u6765\u5206\u6790\u5404\u79cd\u52a8\u529b\u5b66\u4e2d\u7684\u5fae\u5206\u65b9\u7a0b\u548c\u504f\u5fae\u5206\u65b9\u7a0b\u3002\u7ebf\u6027\u52a8\u529b\u5b66\u5c31\u4e0d\u7528\u8bf4\u4e86\uff0c\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u4e5f\u6709\u8bf8\u5982\u6d41<\/span><\/span>\uff08flow\uff09<\/span><\/span>\u3001\u6df7\u6c8c\u3001\u5438\u5f15\u5b50\u3001\u5206\u5f62\u7b49\u4e30\u5bcc\u7684\u5206\u6790\u5de5\u5177<\/span><\/span>\uff08\u63a8\u8350\u4e66\u7c4d\uff1aStenven Strogatz \u7684 Nonlinear dynamics and chaos\uff09<\/span><\/span>\u3002<\/span><\/span><\/section>\n

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\u4e0d\u8fc7\uff0c\u7269\u7406\u5b66\u5bb6\u5c31\u662f\u8fd9\u4e48\u4e00\u7fa4\u5947\u602a\u7684\u7269\u79cd\uff0c\u603b\u662f\u559c\u6b22\u53e6\u8f9f\u8e4a\u5f84\u505a\u4e00\u4e9b\u5947\u602a\u7684\u4e8b\u60c5\u3002\u4ed6\u4eec\u5728\u5bf9\u7cfb\u7edf\u4e0d\u65ad\u505a\u7c97\u7c92\u5316\u7684\u8fc7\u7a0b\uff0c\u53d1\u660e\u4e86\u91cd\u6574\u5316\u7fa4\u7684\u65b9\u6cd5\u3002\u91cd\u6574\u5316\u7fa4\u8fd9\u4e2a\u5de5\u5177\u8981\u56de\u7b54\u7684\u5b9e\u9645\u4e0a\u5c31\u662f\u8fd9\u6837\u4e00\u4e2a\u95ee\u9898\uff1a\u6211\u4eec\u5728\u4e0d\u65ad\u5bf9\u7cfb\u7edf\u505a\u7c97\u7c92\u5316\u7684\u65f6\u5019\uff0c\u7cfb\u7edf\u4f1a\u53d1\u751f\u4ec0\u4e48\u6837\u7684\u53d8\u5316\uff0c\u4e5f\u5c31\u662f\u8bf4\u5728\u8ffd\u95ee\u7684\u662f\uff1a<\/span>\u7cfb\u7edf\u5728\u4e0d\u540c\u5c3a\u5ea6\u4e4b\u95f4\u5b58\u5728\u4ec0\u4e48\u6837\u7684\u5173\u7cfb\uff1f<\/span><\/strong><\/span><\/section>\n

<\/span><\/strong><\/span><\/section>\n
\u5b9e\u9645\u4e0a\uff0c\u5ef6\u7eed\u4e0a\u6587\u7684\u8bed\u8a00\uff0c\u91cd\u6574\u5316\u7fa4\u8981\u5efa\u6a21\u7684\uff0c\u662f\u4e0d\u540c\u5c3a\u5ea6\u7cfb\u7edf\u53c2\u6570\u7684\u52a8\u529b\u5b66\u3002\u5373<\/span><\/span><\/section>\n

<\/span><\/span><\/section>\n
d<\/span>\u03b8<\/span><\/span>\/dl = g(<\/span>\u03b8<\/span><\/span>, <\/span>\u03b3<\/span><\/span>)    \uff083\uff09<\/span><\/span><\/section>\n

<\/span><\/span><\/section>\n
\u4e0a\u9762\u7684\u516c\u5f0f\u6709\u51e0\u4e2a\u5173\u952e\u70b9\uff0c\u9996\u5148\u81ea\u53d8\u91cf\u7531\u539f\u6765\u7684\u65f6\u95f4t\u53d8\u6210\u4e86\u5c3a\u5ea6l\uff0c\u5176\u6b21\u56e0\u53d8\u91cf\u662f\u03b8\uff0c\u4e5f\u5c31\u662f\u539f\u7cfb\u7edf\u7684\u53c2\u6570\u3002<\/span>\u8fd9\u662f\u5728\u8bf4\uff0c\u65b9\u7a0b\u5efa\u6a21\u7684\u662f\u7cfb\u7edf\u7684\u53c2\u6570\u968f\u5c3a\u5ea6\u7684\u53d8\u5316\u3002<\/span><\/strong>\u5f53\u7136\u6211\u4eec\u80fd\u5199\u6210\u4e0a\u9762\u8fd9\u6761\u516c\u5f0f\u7684\u524d\u63d0\uff0c\u662f\u5047\u8bbe\u4e86\u4e0d\u540c\u5c3a\u5ea6\u7684\u52a8\u529b\u5b66\u5f62\u5f0ff\u548cf’\u53ef\u4ee5\u5199\u6210\u57fa\u672c\u4e00\u81f4\u7684\u5f62\u5f0f<\/span><\/span>\uff08\u57fa\u672c\u4e00\u81f4\u8fd9\u4e2a\u8bf4\u6cd5\u5f88\u5fae\u5999\uff0c\u6211\u4eec\u4f1a\u5728\u4e0b\u6587\u7ed9\u51fa\u89e3\u91ca\uff09<\/span><\/span>\u3002\u8fd9\u4e2a\u5047\u8bbe\u867d\u7136\u770b\u7740\u5f88\u5f3a\uff0c\u4f46\u5176\u5b9e\u5728\u5f88\u591a\u7269\u7406\u7cfb\u7edf\u4e2d\u662f\u975e\u5e38\u5408\u7406\u7684\u3002<\/span><\/span><\/section>\n

<\/span><\/span><\/section>\n
\u6211\u4eec\u4ee5\u7ecf\u5178\u7684\u5e73\u8861\u6001 Ising model \u4e3a\u4f8b<\/span><\/span>\uff08Ising model \u7684\u57fa\u672c\u4ecb\u7ecd\u89c1\uff1a\u4f0a\u8f9b\u6a21\u578b | \u96c6\u667a\u767e\u79d1<\/a>\uff09<\/span><\/span>\uff0c\u5757\u7c97\u7c92\u5316<\/span><\/span>\uff08 block coarse-graining\uff09<\/span><\/span>\u7684\u64cd\u4f5c\u5176\u5b9e\u53ef\u4ee5\u8fd1\u4f3c\u7b49\u4ef7\u4e8e\u8c03\u63a7\u7cfb\u7edf\u7684\u8026\u5408\u7cfb\u6570<\/span><\/span>\uff08\u4e5f\u53ef\u4ee5\u7406\u89e3\u6210\u6e29\u5ea6\uff0c\u8026\u5408\u7cfb\u6570\u548c\u6e29\u5ea6\u6709\u76f4\u63a5\u7684\u5173\u7cfb\uff09<\/span><\/span>\u3002\u6240\u4ee5\u6211\u4eec\u4f1a\u6709\u90a3\u5f20\u7ecf\u5178\u7684\u52a8\u6001 Ising \u91cd\u6574\u5316\u7fa4\u7684\u56fe\u50cf<\/span><\/span>\uff08\u5982\u56fe1\u6240\u793a\uff09<\/span><\/span>\u3002\u800c\u53ef\u4ee5\u8fd9\u4e48\u505a\u7684\u524d\u63d0\u662f\uff0c\u4e0d\u540c\u5c3a\u5ea6\u7684 Ising model \u5b83\u4f9d\u7136\u662f\u4e00\u4e2a Ising model\uff0c \u4e0d\u4f1a\u53d8\u6210\u5176\u4ed6\u522b\u7684\u6bd4\u5982Potts\u6a21\u578b<\/span><\/span>\uff08Potts\u6a21\u578b\u7684\u53d8\u91cf\u4e0d\u662f\u4e8c\u503c\u7684\uff09<\/span><\/span>\u3002\u5728\u8fd9\u4e00\u524d\u63d0\u4e0b\uff0c\u516c\u5f0f<\/span><\/span>\uff083\uff09<\/span><\/span>\u5c31\u63cf\u8ff0\u4e86\u4e00\u4e2a\u5168\u65b0\u7684\u52a8\u529b\u7cfb\u7edf\uff0c\u8fd9\u4e2a\u7cfb\u7edf\u6307\u7684\u5c31\u662f\u539f\u6765\u7cfb\u7edf\u7684\u591a\u5c3a\u5ea6\u89c6\u89d2\u3002\u6211\u4eec\u901a\u8fc7\u4f7f\u7528\u52a8\u529b\u7cfb\u7edf\u7684\u65b9\u6cd5\u6765\u5206\u6790\u8fd9\u4e2a\u7cfb\u7edf\uff0c\u4e5f\u53ef\u4ee5\u5f97\u5230\u4e00\u7cfb\u5217\u7684\u7cfb\u7edf\u5173\u4e8e\u5c3a\u5ea6\u7684\u5b9a\u91cf\u89c4\u5f8b\u3002<\/span><\/span><\/section>\n

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

\u56fe1. Ising \u6a21\u578b2\u7ef4\u7a7a\u95f4\u7684\u5b9e\u7a7a\u95f4\u91cd\u6574\u5316\uff0c\u5916\u573a\u4e3a0\u3002\u6bcf\u4e00\u5217\u4ee3\u8868\u7684\u662f\u4e0d\u540c\u7684\u7ea6\u5316\u6e29\u5ea6\u3002\u6bcf\u4e00\u884c\u4ee3\u8868\u7684\u5206\u522b\u662f\u539f\u59cb\u7cfb\u7edf\uff0c1\u6b21\u91cd\u6574\u5316\uff0c2\u6b21\u91cd\u6574\u5316\u540e\u7684\u7cfb\u7edf\u3002<\/span><\/span><\/p>\n


<\/span><\/span><\/section>\n
\u90a3\u4e48\uff0c\u6211\u4eec\u5df2\u7ecf\u5f97\u5230\u4ec0\u4e48\u6837\u7684\u89c4\u5f8b\u5462\uff1f\u65e2\u7136\u4ed6\u662f\u4e00\u4e2a\u52a8\u529b\u7cfb\u7edf\uff0c\u5c31\u53ef\u4ee5\u7528\u4e00\u4e2a\u76f8\u56fe\u6765\u8868\u793a\uff0c\u56fe2\u5206\u522b\u662f Ising model 1d \u548c 2d \u4e3a\u4f8b\u7684\u91cd\u6574\u5316\u65b9\u7a0b\u76f8\u56fe\u30021d\u60c5\u51b5\u4e0b\u6ca1\u6709\u975e\u5e73\u51e1\u7684\u4e0d\u52a8\u70b9<\/span><\/span>\uff08fixed point\uff09<\/span><\/span>\uff0c2d\u65f6\u5219\u5b58\u5728\u4e00\u4e2a\u975e\u5e73\u51e1\u7684\u4e0d\u52a8\u70b9\u3002\u800c\u53c2\u6570\u7a7a\u95f4\u7684<\/span>\u4e0d\u52a8\u70b9<\/span><\/strong>\u610f\u5473\u7740\u5728\u5c3a\u5ea6\u53d8\u5316\u7684\u8fc7\u7a0b\u4e2d\uff0c\u7cfb\u7edf\u7684\u7684\u53c2\u6570\u4e0d\u53d1\u751f\u53d8\u5316\u3002\u8fd9\u5176\u5b9e\u662f\u4e00\u4e2a\u633a\u5947\u602a\u7684\u73b0\u8c61\uff0c\u610f\u5473\u7740\u7c97\u7c92\u5316\u7684\u8fc7\u7a0b\u4e0d\u4f1a\u672c\u8d28\u6539\u53d8\u7cfb\u7edf\u5185\u5143\u7d20\u7684\u76f8\u4e92\u4f5c\u7528\u57fa\u672c\u89c4\u5219\u3002\u8fd9\u53ea\u6709\u4e24\u79cd\u573a\u666f\u624d\u6709\u53ef\u80fd\u53d1\u751f\uff1a<\/span><\/span><\/section>\n