{"id":57452,"date":"2025-03-17T19:37:11","date_gmt":"2025-03-17T11:37:11","guid":{"rendered":"https:\/\/swarma.org\/?p=57452"},"modified":"2025-03-17T19:37:11","modified_gmt":"2025-03-17T11:37:11","slug":"%e4%bb%8e%e6%97%a0%e5%ba%8f%e5%88%b0%e6%9c%89%e5%ba%8f%ef%bc%9a2025%e5%b9%b4%e7%8e%bb%e5%b0%94%e5%85%b9%e6%9b%bc%e5%a5%96%e5%be%97%e4%b8%bb%e5%a6%82%e4%bd%95%e6%8f%ad%e7%a4%ba%e8%87%aa%e7%84%b6","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=57452","title":{"rendered":"\u4ece\u65e0\u5e8f\u5230\u6709\u5e8f\uff1a2025\u5e74\u73bb\u5c14\u5179\u66fc\u5956\u5f97\u4e3b\u5982\u4f55\u63ed\u793a\u81ea\u7136\u754c\u7684\u9690\u85cfscaling law"},"content":{"rendered":"
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\u73bb\u5c14<\/strong>\u5179\u66fc\u5956\uff08Boltzmann Medal\uff09<\/strong>\u662f\u7531\u56fd\u9645\u7eaf\u7cb9\u4e0e\u5e94\u7528\u7269\u7406\u5b66\u8054\u5408\u4f1a<\/span>\uff08International Union of Pure and Applied Physics, IUPAP\uff09<\/span>\u4e0b\u5c5e\u7684\u7edf\u8ba1\u7269\u7406\u59d4\u5458\u4f1a<\/span>\uff08C3\uff09<\/span>\u8bc4\u9009\u5e76\u9881\u53d1\u7684\u56fd\u9645\u7edf\u8ba1\u7269\u7406\u5b66\u754c\u6700\u9ad8\u8363\u8a89\u4e4b\u4e00\uff0c\u4e0e\u6602\u8428\u683c\u5956\u9f50\u540d\u3002\u8be5\u5956\u9879\u4ee5\u5965\u5730\u5229\u7269\u7406\u5b66\u5bb6\u5362\u5fb7\u7ef4\u5e0c\u00b7\u73bb\u5c14\u5179\u66fc\uff08Ludwig Boltzmann\uff09<\/strong>\u547d\u540d\uff0c\u65e8\u5728\u8868\u5f70\u5728\u7edf\u8ba1\u7269\u7406\u548c\u76f8\u5e94\u4ea4\u53c9\u9886\u57df\u4f5c\u51fa\u5353\u8d8a\u8d21\u732e\u7684\u79d1\u5b66\u5bb6\u3002\u6839\u636e\u5b98\u65b9\u516c\u544a\uff0c2025\u5e74\u7684\u73bb\u5c14\u5179\u66fc\u5956\u5c06\u4e8e2025\u5e747\u670813\u65e5\u81f318\u65e5\u5728\u610f\u5927\u5229\u4f5b\u7f57\u4f26\u8428\u4e3e\u529e\u7684StatPhys29\u5927\u4f1a\u4e0a\u9686\u91cd\u9881\u53d1\uff0cMehran Kardar<\/span>\uff08\u6885\u8d6b\u5170\u00b7\u5361\u8fbe\u5c14\uff09<\/span>\u548cYoshiki Kuramoto<\/span>\uff08\u85cf\u672c\u7531\u7eaa\uff09<\/span>\u5c06\u5171\u540c\u83b7\u6b64\u6b8a\u8363\uff0c\u4ee5\u8868\u5f70\u4ed6\u4eec\u5728\u7edf\u8ba1\u7269\u7406\u4e0e\u590d\u6742\u7cfb\u7edf\u7814\u7a76\u65b9\u9762\u7684\u6770\u51fa\u8d21\u732e\u3002<\/span><\/section>\n

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\u7edf\u8ba1\u7269\u7406\u9886\u57df\u7684\u4f17\u591a\u9876\u7ea7\u79d1\u5b66\u5bb6<\/span>\uff0c\u5982Rodney J. Baxter<\/span>\uff08Baxter-Wu\u6a21\u578b\u3001\u9876\u70b9\u6a21\u578b\u548cYang-Baxter\u65b9\u7a0b\uff09<\/span>\uff0cMichael E. Fisher<\/span>\uff08Fisher\u6807\u5ea6\u5f8b\u548cFisher\u6307\u6570\uff09<\/span>\uff0cLeo Kadanoff<\/span>\uff08Kadanoff\u5757\u53d8\u6362\uff0c\u91cd\u6574\u5316\u7fa4\u7406\u8bba\u7684\u5960\u57fa\u4eba\uff09<\/span>\uff0cBenjamin Widom<\/span>\uff08Widom\u6807\u5ea6\u5f8b\u4e0e\u6db2\u4f53\u7684\u8868\u9762\u6027\u8d28\uff09<\/span>\uff0cE.G.D. Cohen<\/span> (\u975e\u5e73\u8861\u7a33\u6001\u95ee\u9898)<\/span>\uff0cH. Eugene Stanley<\/span>\uff08O(n)\u6a21\u578b\uff09<\/span>\uff0c<\/span>Kurt Binder<\/span>\uff08\u6709\u9650\u5c3a\u5ea6\u6807\u5ea6\uff09<\/span>\u90fd\u83b7\u5f97\u8fc7\u8fd9\u4e00\u5956\u9879\u3002<\/span><\/section>\n

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Mehran Kardar\uff1a\u754c\u9762\u751f\u957f\u52a8\u529b\u5b66\u7684\u7406\u8bba\u5148\u9a71<\/strong><\/span><\/h2>\n

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Mehran Kardar\uff0c\u6765\u6e90\uff1ahttp:\/\/www.mit.edu\/~kardar\/<\/span><\/section>\n

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Mehran Kardar\u6559\u6388\u751f\u4e8e\u4f0a\u6717\uff0c\u73b0\u4e3a\u9ebb\u7701\u7406\u5de5\u5b66\u9662<\/span>\uff08Massachusetts Institute of Technology, MIT\uff09<\/span>\u7269\u7406\u5b66\u6559\u6388\uff0c\u5728\u7edf\u8ba1\u7269\u7406\u5b66\u3001\u51dd\u805a\u6001\u7269\u7406\u548c\u590d\u6742\u7cfb\u7edf\u9886\u57df\u505a\u51fa\u4e86\u5353\u8d8a\u8d21\u732e<\/strong>\u3002\u4ed6\u4e0eGiorgio Parisi<\/a>\u548c\u5f20\u7ffc\u6210\u6559\u6388<\/span>\uff08https:\/\/pattern.swarma.org\/master\/790\uff09<\/span>\u4e8e1986\u5e74\u5171\u540c\u63d0\u51fa\u7684KPZ\u65b9\u7a0b\u662f\u5176\u6700\u5177\u5f71\u54cd\u529b\u7684\u6210\u679c\u4e4b\u4e00\u3002\u8fd9\u4e00\u65b9\u7a0b\u63cf\u8ff0\u4e86\u754c\u9762\u5728\u751f\u957f\u8fc7\u7a0b\u4e2d\u7684\u52a8\u529b\u5b66\u884c\u4e3a\uff0c\u901a\u8fc7\u968f\u673a\u504f\u5fae\u5206\u65b9\u7a0b\u6355\u6349\u4e86\u754c\u9762\u7c97\u7cd9\u5316\u7684\u666e\u9002\u6027\u8d28\uff0c\u89e3\u91ca\u4e86\u52a8\u529b\u5b66\u6807\u5ea6\u6027\u8d28\uff08Dynamic Scaling Properties\uff09\u7684\u666e\u9002\u6027\u8d77\u6e90<\/strong>\u3002KPZ\u65b9\u7a0b\u4e0d\u4ec5\u9002\u7528\u4e8e\u89e3\u91ca\u6676\u4f53\u751f\u957f\u3001\u71c3\u70e7\u950b\u9762\u4f20\u64ad\u548c\u6e4d\u6d41\u7b49\u7269\u7406\u73b0\u8c61\uff0c\u8fd8\u4e0e\u968f\u673a\u77e9\u9635\u7406\u8bba\u3001\u53ef\u79ef\u7cfb\u7edf\u7b49\u6570\u5b66\u7269\u7406\u9886\u57df\u5efa\u7acb\u4e86\u6df1\u523b\u8054\u7cfb\u3002\u8fd1\u4e09\u5341\u591a\u5e74\u6765\uff0cKPZ\u65b9\u7a0b\u5df2\u53d1\u5c55\u6210\u4e3a\u7814\u7a76\u975e\u5e73\u8861\u7edf\u8ba1\u7269\u7406\u7684\u6838\u5fc3\u7406\u8bba\u6846\u67b6\u4e4b\u4e00<\/strong>\uff0c\u5176\u5f71\u54cd\u529b\u5df2\u8fdc\u8d85\u7269\u7406\u5b66\u8303\u7574\uff0c\u6269\u5c55\u81f3\u6982\u7387\u8bba\u3001\u751f\u7269\u7269\u7406\u548c\u6750\u6599\u79d1\u5b66\u7b49\u591a\u4e2a\u9886\u57df\u3002Kardar\u6559\u6388\u7684\u7814\u7a76\u5de5\u4f5c\u4f53\u73b0\u4e86\u7269\u7406\u5b66\u4e2d\u7b80\u5355\u6570\u5b66\u6a21\u578b\u5982\u4f55\u80fd\u591f\u89e3\u91ca\u590d\u6742\u81ea\u7136\u73b0\u8c61\u7684\u5f3a\u5927\u529b\u91cf\u3002<\/span><\/section>\n

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\u503c\u5f97\u4e00\u63d0\u7684\u662f<\/strong>\uff0c2021\u8bfa\u8d1d\u5c14\u7269\u7406\u5b66\u5956\u83b7\u5f97\u8005Giorgio Parisi\u4e0eKardar\u548c\u8457\u540d\u534e\u4eba\u79d1\u5b66\u5bb6\u3001\u6b27\u6d32\u79d1\u5b66\u9662\u9662\u58eb\u5f20\u7ffc\u6210\u5171\u540c\u63d0\u51fa\u4e86Kardar\u2013Parisi\u2013Zhang (KPZ) equation<\/strong><\/span>\uff08\u5373\u5361\u8fbe\u5c14\u2013\u5e15\u91cc\u897f\u2013\u5f20\u65b9\u7a0b\uff0c\u5b83\u662f\u4e00\u79cd\u63cf\u8ff0\u754c\u9762\u751f\u957f\u7684\u52a8\u529b\u5b66\u6807\u5ea6\u7684\u968f\u673a\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u88ab\u63a8\u6d4b\u4e3a\u4f17\u591a\u6a21\u578b\u7684\u80cc\u540e\u7684\u7edf\u8ba1\u573a\u8bba\u5f62\u5f0f\uff09<\/span><\/section>\n

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\u85cf\u672c\u7531\u7eaa Yoshiki Kuramoto\uff1a\u540c\u6b65\u73b0\u8c61\u7814\u7a76\u7684\u5f00\u62d3\u8005<\/strong><\/span><\/h2>\n

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Yoshiki Kuramoto\uff08\u85cf\u672c\u7531\u7eaa\uff09\u6559\u6388\u662f\u4eac\u90fd\u5927\u5b66\u7269\u7406\u5b66\u5bb6\uff0c\u5728\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u548c\u590d\u6742\u7cfb\u7edf\u9886\u57df\u505a\u51fa\u4e86\u5f00\u521b\u6027\u8d21\u732e<\/strong>\u3002\u4ed6\u4e8e1975\u5e74\u63d0\u51fa\u7684\u85cf\u672c\u6a21\u578b<\/span>\uff08Kuramoto Model\uff09<\/span>\u5f7b\u5e95\u6539\u53d8\u4e86\u79d1\u5b66\u754c\u5bf9\u540c\u6b65\u73b0\u8c61\u7684\u7406\u89e3\u3002\u540c\u6b65<\/span>\uff08synchronization\uff09<\/span>\u662f\u81ea\u7136\u754c\u4e2d\u6781\u4e3a\u666e\u904d\u7684\u73b0\u8c61\uff0c\u4ece\u8424\u706b\u866b\u96c6\u4f53\u95ea\u5149\u3001\u9e1f\u7fa4\u534f\u8c03\u98de\u884c\u5230\u4eba\u7c7b\u5fc3\u810f\u7ec6\u80de\u7684\u8282\u5f8b\u6027\u8df3\u52a8\uff0c\u90fd\u4f53\u73b0\u4e86\u72ec\u7acb\u4e2a\u4f53\u5982\u4f55\u901a\u8fc7\u76f8\u4e92\u4f5c\u7528\u5b9e\u73b0\u96c6\u4f53\u884c\u4e3a\u7684\u534f\u8c03\u4e00\u81f4\u3002Kuramoto\u6559\u6388\u901a\u8fc7\u7b80\u6d01\u4f18\u96c5\u7684\u6570\u5b66\u65b9\u7a0b\uff0c\u6210\u529f\u63cf\u8ff0\u4e86\u5927\u91cf\u8026\u5408\u632f\u5b50\u5982\u4f55\u4ece\u521d\u59cb\u7684\u6742\u4e71\u65e0\u7ae0\u72b6\u6001\uff0c\u968f\u7740\u8026\u5408\u5f3a\u5ea6\u7684\u589e\u52a0\uff0c\u9010\u6e10\u5b9e\u73b0\u76f8\u4f4d\u540c\u6b65\uff0c\u6700\u7ec8\u5f62\u6210\u6709\u5e8f\u96c6\u4f53\u884c\u4e3a\u7684\u8fc7\u7a0b\u3002\u8be5\u6a21\u578b\u4e4b\u6240\u4ee5\u5177\u6709\u9769\u547d\u6027\u610f\u4e49\uff0c\u5728\u4e8e\u5b83\u4e0d\u4ec5\u63d0\u4f9b\u4e86\u7406\u89e3\u540c\u6b65\u73b0\u8c61\u7684\u7406\u8bba\u6846\u67b6\uff0c\u8fd8\u63ed\u793a\u4e86\u8fd9\u4e00\u8fc7\u7a0b\u672c\u8d28\u4e0a\u662f\u4e00\u79cd\u975e\u5e73\u8861\u76f8\u53d8\u73b0\u8c61\uff0c\u5b58\u5728\u7740\u4e34\u754c\u73b0\u8c61\u4e2d\u666e\u904d\u7684scaling law<\/strong>\u30022001\u5e74\uff0c\u4ed6\u8fd8\u53c2\u4e0e\u53d1\u73b0\u4e86\u5d4c\u5408\u4f53\u72b6\u6001<\/span>\uff08chimera state\uff09<\/span>\u2014\u2014\u4e00\u79cd\u5728\u540c\u4e00\u7cfb\u7edf\u4e2d\u540c\u65f6\u5b58\u5728\u540c\u6b65\u548c\u975e\u540c\u6b65\u533a\u57df\u7684\u5947\u7279\u73b0\u8c61\uff0c\u8fdb\u4e00\u6b65\u4e30\u5bcc\u4e86\u6211\u4eec\u5bf9\u590d\u6742\u7cfb\u7edf\u52a8\u529b\u5b66\u7684\u8ba4\u8bc6\u3002<\/span><\/section>\n
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\n

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\u4e0b\u9762\uff0c\u8ba9\u6211\u4eec\u4ece\u4e00\u4e9b\u7b80\u5355\u7684\u6a21\u578b\u548c\u73b0\u8c61\u51fa\u53d1\uff0c\u5206\u522b\u7b80\u8ff0\u4e24\u4f4d\u83b7\u5956\u8005\u4e3b\u8981\u5de5\u4f5c\u7684\u80cc\u666f\u77e5\u8bc6\u3001\u5177\u4f53\u5185\u5bb9\u548c\u79d1\u5b66\u610f\u4e49\u3002<\/span><\/section>\n

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

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

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

\u63a2\u79d8KPZ\u65b9\u7a0b\uff1a<\/strong><\/span><\/strong><\/strong><\/p>\n

\u4ece\u79bb\u6563\u6a21\u578b\u5230\u8fde\u7eed\u4e16\u754c\u7684\u5947\u5999\u4e4b\u65c5<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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

<\/p>\n

\u754c\u9762\u751f\u957f\u6a21\u578b\uff1a\u79bb\u6563\u4e16\u754c\u7684\u751f\u957f\u5965\u79d8<\/span><\/strong><\/span><\/h2>\n

<\/span><\/section>\n
\u4e3a\u4e86\u63cf\u8ff0\u754c\u9762\u7684\u751f\u957f\uff0c\u4eba\u4eec\u8bbe\u8ba1\u4e86\u4e00\u7cfb\u5217\u7edf\u8ba1\u7269\u7406\u7684\u79bb\u6563\u6a21\u578b\u3002<\/span><\/section>\n

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

1. \u968f\u673a\u843d\u4f53\u6a21\u578b<\/span><\/strong><\/h3>\n
<\/section>\n
<\/section>\n

\u968f\u673a\u843d\u4f53\u6a21\u578b\u548c\u9ecf\u6027\u843d\u4f53\u6a21\u578b\u7684\u89c4\u5219\u793a\u610f\u56fe\uff0c\u6765\u6e90\uff1ahttps:\/\/arxiv.org\/abs\/1606.06602<\/span><\/p>\n


<\/span><\/section>\n
\u60f3\u8c61\u4e00\u6392\u65e0\u9650\u5ef6\u4f38\u7684\u5782\u76f4\u67f1\u5b50\uff0c\u6bcf\u4e2a\u67f1\u5b50\u4e0a\u65b9\u7684\u65b9\u5757\u4ee5\u5b8c\u5168\u72ec\u7acb\u7684\u65b9\u5f0f\u4e0b\u843d<\/span>\uff08\u5373A,B\u5b50\u56fe\u6240\u793a\uff09<\/span>\u3002\u65b9\u5757\u4e0b\u843d\u7684\u65f6\u95f4\u95f4\u9694\u670d\u4ece\u53c2\u6570\u4e3a1\u7684\u6307\u6570\u5206\u5e03<\/strong>\uff0c\u4e14\u4e0b\u843d\u8fc7\u7a0b\u77ac\u95f4\u5b8c\u6210\u3002\u7528hr<\/sup>(t, x) \u8868\u793a\u65f6\u95f4t\u65f6\u7b2cx\u5217\u7684\u9ad8\u5ea6\uff0c\u8fd9\u4e00\u8fc7\u7a0b\u6784\u6210\u4e00\u7ec4\u72ec\u7acb\u7684\u6cca\u677e\u8fc7\u7a0b<\/strong>\u3002\u6570\u5b66\u4e0a\u53ef\u4ee5\u8ba1\u7b97\uff0c\u968f\u673a\u843d\u4f53\u6a21\u578b\u7684\u671f\u671b\u9ad8\u5ea6\u4e3aE[h<\/span>r<\/sup>(t, x)] = t\uff0c\u8868\u660e\u9ad8\u5ea6\u968f\u65f6\u95f4\u5300\u901f\u589e\u957f\uff1b\u5176\u6da8\u843d\u7279\u6027\u7531\u65b9\u5deeE[(h<\/span>r<\/sup>(t, x)-t)2<\/sup>] = t \u523b\u753b\uff0c\u6807\u51c6\u5dee\u4e3a\"image.png\"<\/sub>\uff0c\u4e0e\u5e03\u6717\u8fd0\u52a8\u7684\u4f4d\u79fb\u6da8\u843d\u76f8\u4f3c\u3002\u52a8\u6001\u6807\u5ea6\u5f8b\u8fdb\u4e00\u6b65\u63ed\u793a\u5176\u6da8\u843d\u6ee1\u8db32:0:1\u6807\u5ea6<\/strong>\uff1a<\/span><\/section>\n

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

<\/p>\n

<\/section>\n
\u5176\u4e2d\u7a7a\u95f4\u6807\u5ea6\u6307\u6570\u4e3a0\uff0c\u8bf4\u660e\u5404\u5217\u7684\u6da8\u843d\u5b8c\u5168\u72ec\u7acb\uff0c\u4e92\u4e0d\u5f71\u54cd\u3002<\/span><\/section>\n


<\/span><\/h3>\n

2. \u9ecf\u6027\u843d\u4f53\u6a21\u578b<\/span><\/strong><\/h3>\n


<\/span><\/h3>\n

\u5982\u679c\u968f\u673a\u843d\u4f53\u6a21\u578b\u662f\u5b64\u7acb\u7684\u96e8\u6ef4\uff0c\u9ecf\u6027\u843d\u4f53\u6a21\u578b\u5c31\u662f\u6d41\u6dcc\u7684\u8702\u871c\u3002<\/span><\/h3>\n

<\/span><\/section>\n
\u4e24\u8005\u975e\u5e38\u7c7b\u4f3c\uff0c\u6838\u5fc3\u533a\u522b\u53ea\u662f\u5728\u4e8e\u5f15\u5165\u4e86\u65b9\u5757\u7684\u5438\u9644\u89c4\u5219<\/strong>\uff1a\u4e0b\u843d\u7684\u65b9\u5757\u4e0d\u518d\u76f4\u8fbe\u5e95\u90e8\uff0c\u800c\u662f\u9ecf\u9644\u4e8e\u63a5\u89e6\u5230\u7684\u9996\u4e2a\u65b9\u5757\u65c1<\/strong>\u3002<\/span>\uff08\u5373C,D\u5b50\u56fe\u6240\u793a\uff09<\/span>\u8fd9\u79cd\u9ecf\u6027\u5f7b\u5e95\u6539\u53d8\u4e86\u754c\u9762\u52a8\u529b\u5b66\u884c\u4e3a\u3002\u7528hb<\/sup>(t, x)\u8868\u793a\u9ecf\u6027\u843d\u4f53\u6a21\u578b\u7684\u9ad8\u5ea6\u51fd\u6570\uff0c\u5176\u5e73\u5747\u9ad8\u5ea6E[h<\/span>b<\/sup>(t, x)] \u221d t \u4ecd\u968f\u65f6\u95f4\u7ebf\u6027\u589e\u957f\uff0c\u4f46\u6da8\u843d\u7279\u6027\u663e\u8457\u4e0d\u540c<\/span> (\u5373\u4e0b\u56fe\u6240\u793a)<\/span>\u3002\u52a8\u6001\u6807\u5ea6\u5f8b\u8868\u660e\uff0c\u9ad8\u5ea6\u6da8\u843dwb<\/sup>(t, x)\u6ee1\u8db33:2:1\u6807\u5ea6<\/strong>\uff1a<\/span><\/section>\n

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

<\/p>\n

<\/section>\n
\u65f6\u95f4\u6807\u5ea6\u6307\u6570\u03f5<\/span>-3<\/sup>\u4e0e\u7a7a\u95f4\u6807\u5ea6\u6307\u6570\u03f5<\/span>-2<\/sup>\uff0c\u53cd\u6620\u4e86\u9ecf\u6027\u5bfc\u81f4\u7684\u534f\u4f5c\u6548\u5e94\u2014\u2014\u67d0\u4e00\u70b9\u7684\u6da8\u843d\u4f1a\u901a\u8fc7\u5438\u9644\u89c4\u5219\u5411\u90bb\u8fd1\u533a\u57df\u6269\u6563\uff0c\u5f62\u6210\u7a7a\u95f4\u5173\u8054\u3002<\/span><\/section>\n

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

\u968f\u673a\u843d\u4f53\u6a21\u578b\u548c\u9ecf\u6027\u843d\u4f53\u6a21\u578b\u7684\u5178\u578b\u6784\u578b\uff0c\u6765\u6e90\uff1ahttps:\/\/arxiv.org\/abs\/1606.06602<\/span><\/p>\n

<\/section>\n

3. Eden\u6a21\u578b\uff1a\u79cd\u5b50\u751f\u957f\u7684\u968f\u673a\u6269\u5f20<\/strong><\/span><\/h3>\n

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

Eden\u6a21\u578b\u5219\u5c55\u73b0\u4e86\u66f4\u590d\u6742\u7684\u751f\u957f\u56fe\u666f\u3002\u5b83\u6700\u521d\u5b9a\u4e49\u5728\u4e8c\u7ef4\u5e73\u9762\u683c\u70b9\u4e0a\uff0c\u4f46\u5176\u89c4\u5219\u53ef\u63a8\u5e7f\u5230\u4efb\u610f\u7ef4\u5ea6\u3002\u7528hE<\/sup>(t, x)\u8868\u793a\u65f6\u95f4t\u65f6\u7b2cx\u5217\u7684\u9ad8\u5ea6\uff0c\u5176\u6838\u5fc3\u89c4\u5219\u4e3a\uff1a<\/span><\/p>\n

    \n
  1. \n

    \u521d\u59cb\u79cd\u5b50<\/strong>\uff1a\u56fa\u5b9a\u5e95\u8fb9\u7684\u4e00\u6392\u683c\u70b9\uff08hE<\/sup>(0, x) = 0\uff09\u3002<\/span><\/p>\n<\/li>\n

  2. \n

    \u751f\u957f\u89c4\u5219<\/strong>\uff1a\u6bcf\u4e00\u6b65\u4ece\u6240\u6709\u4e0e\u73b0\u6709\u79cd\u5b50\u76f8\u90bb<\/strong>\u7684\u7a7a\u767d\u683c\u70b9\u4e2d\uff0c\u968f\u673a\u9009\u62e9\u4e00\u4e2a<\/strong>\u52a0\u5165\u79cd\u5b50\u96c6\u5408\u3002<\/span><\/p>\n<\/li>\n

  3. \n

    \u65f6\u95f4\u5b9a\u4e49<\/strong>\uff1a\u8bbeN\u4e3a\u5df2\u52a0\u5165\u7684\u683c\u70b9\u6570\uff0c\u65f6\u95f4\u5b9a\u4e49\u4e3at := N\/x\uff0c\u4f7f\u5f97\u5e73\u5747\u9ad8\u5ea6\"image.png\"<\/sub>\u968f\u65f6\u95f4\u7ebf\u6027\u589e\u957f\uff0c\u8fd9\u4e00\u70b9\u548c\u4e4b\u524d\u7684\u6a21\u578b\u662f\u4e00\u81f4\u7684\u3002<\/span><\/p>\n

    <\/p>\n<\/li>\n<\/ol>\n

    <\/section>\n

    \u4e8c\u7ef4\u968f\u673a\u66f2\u7ebf\u4e0a\u7684Eden\u751f\u957f\uff0c\u8272\u5f69\u8868\u793a\u65f6\u95f4\u3002\u6765\u6e90\uff1ahttps:\/\/wap.sciencenet.cn\/blog-863936-1429491.html?mobile=1<\/span><\/p>\n


    <\/span><\/section>\n
    \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9ad8\u5ea6\u6da8\u843d\u7684\u5747\u65b9\u6839\u5b9a\u4e49\u754c\u9762\u7c97\u7cd9\u5ea6<\/strong>\uff1a<\/span><\/section>\n

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

    <\/p>\n

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

    <\/span><\/section>\n
    \u5b9e\u9a8c\u53d1\u73b0\uff0c\u5f53\u7cfb\u7edf\u89c4\u6a21x\u548c\u65f6\u95f4t\u6781\u5927\u65f6\uff0cw(t, x)\u6ee1\u8db3\u666e\u9002\u6807\u5ea6\u5f8b\uff1a<\/span><\/section>\n

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

    <\/p>\n

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

    <\/span><\/section>\n
    \u5176\u4e2d\u6807\u5ea6\u6307\u6570\"image.png\"<\/sub>\uff08\u7c97\u7cd9\u5ea6\u6307\u6570\uff09\uff0cz\u22481.7\uff08\u52a8\u6001\u6307\u6570\uff09\u3002\u4eba\u4eec\u5728\u9ecf\u6027\u843d\u4f53\u6a21\u578b\u4e2d\u53d1\u73b0\u5b58\u5728\u7740\u7c7b\u4f3c\u7684\u6807\u5ea6\u5f62\u5f0f\uff08\u5373scaling law<\/strong>\uff09\u2014\u2014\u8fd9\u4f3c\u4e4e\u6697\u793a\u7740\u867d\u7136\u5fae\u89c2\u89c4\u5219\u7ec6\u8282\u4e0d\u540c<\/strong>\uff0c\u4f46\u662fEden\u6a21\u578b\u548c\u9ecf\u6027\u843d\u4f53\u6a21\u578b\u7684\u957f\u671f\u548c\u5927\u5c3a\u5ea6\u884c\u4e3a\uff0c\u5b58\u5728\u7740\u67d0\u79cd\u666e\u9002\u6027<\/strong><\/span><\/section>\n

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


    <\/strong><\/span><\/h2>\n

    \u4ece\u79bb\u6563\u5230\u8fde\u7eed\uff1aKPZ\u65b9\u7a0b\u7684\u8bde\u751f<\/strong><\/span><\/h2>\n


    <\/span><\/h3>\n

    1. Edwards-Wilkinson\u65b9\u7a0b\uff1a\u8fd1\u5e73\u8861\u6001\u7684\u751f\u957f<\/strong>  <\/span><\/h3>\n

    <\/span><\/section>\n
    \u6211\u4eec\u4e0d\u7981\u8981\u95ee\uff1a\u8868\u9762\u751f\u957f\u6a21\u578b\u7684\u7a7a\u95f4\u8fde\u7eed\u5f62\u5f0f\u662f\u4ec0\u4e48\uff1f\u662f\u5426\u5b58\u5728\u4e00\u4e2a\u7edf\u4e00\u7684\u975e\u5e73\u8861\u7edf\u8ba1\u573a\u8bba\u5f62\u5f0f\u53ef\u4ee5\u666e\u9002\u5730\u63cf\u8ff0\u8bf8\u591a\u79bb\u6563\u6a21\u578b\uff1f<\/strong><\/span><\/section>\n

    <\/strong><\/span><\/section>\n
    \u5728KPZ\u65b9\u7a0b\u63d0\u51fa\u4e4b\u524d\uff0cEdwards\u548cWilkinson\u63d0\u51fa\u4e86\u4e00\u4e2a\u7ebf\u6027\u65b9\u7a0b\uff1a<\/span><\/section>\n

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

    <\/p>\n

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

    <\/span><\/section>\n
    \u5176\u4e2d\u03bd<\/span>\u662f\u6269\u6563\u7cfb\u6570\uff0c\u03b7<\/span>(t, x) \u662f\u65f6\u7a7a\u767d\u566a\u58f0\u3002\u8be5\u65b9\u7a0b\u63cf\u8ff0\u8fd1\u5e73\u8861\u6001\u4e0b\u7684\u754c\u9762\u751f\u957f\uff0c\u5176\u6da8\u843d\u6ee1\u8db3Edwards-Wilkinson\u666e\u9002\u6027\uff0c\u5177\u67094:2:1\u52a8\u6001\u6807\u5ea6\u5f8b<\/strong><\/span><\/section>\n

    <\/strong><\/span><\/section>\n
    Edwards-Wilkinson\u65b9\u7a0b\u53ca\u5176\u53d8\u4f53\uff08\u5982\u6dec\u706bEW\u65b9\u7a0b\uff09\u88ab\u8ba4\u4e3a\u662f\u4e00\u7c7b\u7a7a\u95f4\u79bb\u6563\u7cfb\u7edf\u7684\u573a\u8bba\u5f62\u5f0f\uff1a <\/strong>\u81ea\u7ec4\u7ec7\u4e34\u754c\u7406\u8bba\u4e2d\u7684Oslo\u6a21\u578b\u3001\u81ea\u65cb\u7cfb\u7edf\u4e2d\u7684XY\u6a21\u578b<\/span>\uff08\u5728\u4f4e\u6e29\u81ea\u65cb\u6ce2\u8fd1\u4f3c\u4e0b\uff09<\/span>\u3001\u540c\u6b65\u7cfb\u7edf\u4e2d\u7684Kuramoto\u6a21\u578b<\/span>\uff08\u5728\u5f3a\u8026\u5408\u81ea\u65cb\u6ce2\u8fd1\u4f3c\u4e0b\uff0c\u6211\u4eec\u540e\u6587\u8fd8\u4f1a\u63d0\u5230\u8fd9\u4e2a\u6a21\u578b\uff09<\/span>\u7b49\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u4f46Edwards-Wilkinson\u65b9\u7a0b\u53ea\u80fd\u63cf\u8ff0\u8fd1\u5e73\u8861\u7684\u7ebf\u6027\u6269\u6563\u884c\u4e3a\uff0c\u5982\u4f55\u7edf\u4e00\u5730\u89e3\u91ca\u8fdc\u79bb\u5e73\u8861\u7684\u8868\u9762\u751f\u957f\u95ee\u9898<\/strong>\u5462\uff1f<\/span><\/section>\n

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

    2. KPZ\u65b9\u7a0b\uff1a\u8fdc\u79bb\u5e73\u8861\u8868\u9762\u751f\u957f\u7684\u672c\u8d28<\/strong>  <\/span><\/h3>\n

    <\/span><\/section>\n
    1986\u5e74\u53d1\u8868\u5728\u300a\u7269\u7406\u8bc4\u8bba\u5feb\u62a5\u300b\u4e0a\u7684\u6587\u7ae0\u5f15\u5165\u4e86KPZ\u65b9\u7a0b\uff0c\u5b83\u901a\u8fc7\u5efa\u7acb\u8fde\u7eed\u5316\u573a\u8bba\u6a21\u578b\u63ed\u793a\u4e86\u751f\u957f\u8fc7\u7a0b\u7684\u672c\u8d28\u3002Kardar\u7b49\u4eba\u63d0\u51fa\u7684KPZ\u65b9\u7a0b\u4e3a\uff1a<\/span><\/section>\n

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

    <\/p>\n

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

    <\/span><\/section>\n
    \u5176\u4e2d\uff1a<\/span><\/section>\n
    \u2022 \u6269\u6563\u9879\u03bd<\/span>\u0394h<\/span>\uff1a\u5bf9\u5e94\u7ebf\u6027\u6269\u6563\u884c\u4e3a<\/span><\/section>\n
    \u2022 \u975e\u7ebf\u6027\u9879\"image.png\"<\/sub>\uff1a\u63cf\u8ff0\u5c40\u90e8\u9ad8\u5ea6\u68af\u5ea6\u5bf9\u751f\u957f\u901f\u5ea6<\/span><\/section>\n
    \u7684\u53cd\u9988\uff0c\u5bf9\u5e94Eden\u6a21\u578b\u4e2d\u754c\u9762\u6269\u5f20\u6216\u8005\u9ecf\u6027\u843d\u4f53\u6a21\u578b\u4e2d\u7684\u9ecf\u6027\u6548\u5e94<\/span><\/section>\n
    \u2022 \u566a\u58f0\u9879\uff1a\u03b7<\/span>(t, x)<\/span><\/span><\/section>\n
    <\/section>\n
    <\/section>\n

    KPZ\u65b9\u7a0b\u4e2d\u7684\u975e\u7ebf\u6027\u9879\uff0c\u6765\u6e90\uff1aZhao D. and Liping L., A brief introduction to KPZ equation and KPZ universality, Sci. Sin.-Math. 49, 339 (2019).<\/span><\/p>\n


    <\/span><\/section>\n
    \u5f88\u5bb9\u6613\u53d1\u73b0\uff0c\u5f53KPZ\u65b9\u7a0b\u4e2d\u7684\u975e\u7ebf\u6027\u9879\u7cfb\u6570\u03bb<\/span> = 0\u65f6\uff0c\u9000\u5316\u4e3aEdwards-Wilkinson\u65b9\u7a0b\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    KPZ\u65b9\u7a0b\u5728\u4e00\u7ef4\u60c5\u5f62\u4e0b\u7684\u89e3\u6ee1\u8db33:2:1\u52a8\u6001\u6807\u5ea6\u5f8b<\/strong>\uff0c\u4e0e\u9ecf\u6027\u843d\u4f53\u6a21\u578b\u548cEden\u6a21\u578b\u7684\u5b9e\u9a8c\u7ed3\u679c\u4e00\u81f4\u3002\u8fd9\u4e00\u65b9\u7a0b\u6210\u529f\u7edf\u4e00\u4e86\u79bb\u6563\u6a21\u578b\u7684\u6807\u5ea6\u884c\u4e3a\uff0c\u6210\u4e3a\u975e\u5e73\u8861\u754c\u9762\u751f\u957f\u7406\u8bba\u7684\u57fa\u77f3\u3002<\/span><\/section>\n

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

    <\/span><\/section>\n
    \u4ece\u968f\u673a\u843d\u4f53\u7684\u72ec\u7acb\u6da8\u843d\u3001\u9ecf\u6027\u843d\u4f53\u7684\u534f\u4f5c\u6548\u5e94\uff0c\u5230Eden\u6a21\u578b\u7684\u754c\u9762\u7c97\u7cd9\u5316\uff0c\u8fd9\u4e9b\u79bb\u6563\u6a21\u578b\u63ed\u793a\u4e86\u751f\u957f\u8fc7\u7a0b\u4e2d\u968f\u673a\u6027\u4e0e\u975e\u7ebf\u6027\u7ade\u4e89\u7684\u666e\u9002\u89c4\u5f8b\u3002KPZ\u65b9\u7a0b\u901a\u8fc7\u8fde\u7eed\u5316\u6846\u67b6\uff0c\u5c06\u79bb\u6563\u4e16\u754c\u7684\u5965\u79d8\u8f6c\u5316\u4e3a\u6570\u5b66\u8bed\u8a00\uff0c\u4e3a\u7406\u89e3\u6676\u4f53\u751f\u957f\u3001\u7ec6\u83cc\u7fa4\u843d\u6269\u5f20\u7b49\u81ea\u7136\u73b0\u8c61\u63d0\u4f9b\u4e86\u6838\u5fc3\u5de5\u5177\u3002<\/span><\/section>\n

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


    <\/span><\/h1>\n

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

    Kuramoto Model\uff1a\u540c\u6b65\u73b0\u8c61\u7684\u672c\u8d28\u523b\u753b<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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


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

    \n
    \n
    \n

    \u6211\u4eec\u5728\u96c6\u667a\u767e\u79d1\u4e0a\u521b\u5efa\u4e86Kuramoto Model\u7684\u767e\u79d1\u8bcd\u6761\uff0c\u4e5f\u6b22\u8fce\u5927\u5bb6\u70b9\u51fb\u5b66\u4e60\u66f4\u8be6\u7ec6\u7684\u5173\u4e8e\u8be5\u6982\u5ff5\u7684\u4ecb\u7ecd\uff1a<\/span><\/p>\n

    https:\/\/wiki.swarma.org\/index.php\/Kuromoto%E6%A8%A1%E5%9E%8B<\/span><\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n

    <\/section>\n
    \u590d\u6742\u7cfb\u7edf\u7531\u4f17\u591a\u76f8\u4e92\u4f5c\u7528\u7684\u5355\u5143\u7ec4\u6210\u3002\u5728\u52a8\u6001\u6f14\u5316\u8fdb\u7a0b\u91cc\uff0c\u8fd9\u4e9b\u5355\u5143\u5f7c\u6b64\u534f\u8c03\uff0c\u8fdb\u800c\u6d8c\u73b0<\/strong>\u51fa\u96c6\u4f53\u884c\u4e3a\u6a21\u5f0f\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u540c\u6b65<\/strong>\uff0c\u5373\u4e24\u4e2a\u6216\u66f4\u591a\u4e8b\u4ef6\u540c\u65f6\u53d1\u751f\uff0c\u662f\u81ea\u7136\u754c\u6781\u4e3a\u5e38\u89c1\u7684\u73b0\u8c61\u4e4b\u4e00\u3002\u4ece\u65e0\u610f\u8bc6\u7684\u5b9e\u4f53\u5230\u4eba\u7c7b\uff0c\u540c\u6b65\u73b0\u8c61\u5e7f\u6cdb\u5b58\u5728\uff0c\u751a\u81f3\u4ee5\u82b1\u6837\u6e38\u6cf3\u3001\u8df3\u6c34\u7b49\u5f62\u5f0f\u6210\u4e3a\u5965\u6797\u5339\u514b\u8fd0\u52a8\u9879\u76ee\u3002\u540c\u6b65\u5bf9\u4e8e\u751f\u547d\u800c\u8a00\u81f3\u5173\u91cd\u8981\uff0c\u6bd4\u5982\uff0c\u8d77\u640f\u7ec6\u80de\u5fc5\u987b\u540c\u6b65\u653e\u7535\uff0c\u624d\u80fd\u786e\u4fdd\u4eba\u7c7b\u5fc3\u810f\u6b63\u5e38\u8df3\u52a8\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u540c\u6b65\u5c5e\u4e8e\u975e\u5e73\u8861\u6d8c\u73b0\u5f62\u5f0f\u4e2d\u6700\u4e3a\u7b80\u5355\u7684\u4e00\u79cd<\/strong>\uff0c\u5176\u672c\u8d28\u662f\u901a\u8fc7\u5fae\u89c2\u5c42\u9762\u7684\u8026\u5408<\/strong>\uff0c\u8fbe\u6210\u7cfb\u7edf\u5b8f\u89c2\u72b6\u6001\u7684\u4e00\u81f4\u6027<\/strong>\u3002\u5bf9\u4e8e\u7269\u7406\u5b66\u5bb6\u800c\u8a00\uff0c\u65e0\u751f\u547d\u7cfb\u7edf\u4e2d\u7684\u540c\u6b65\u73b0\u8c61\u6781\u5177\u5438\u5f15\u529b\u3002\u82e5\u5c06\u4e24\u4e2a\u5b8c\u5168\u76f8\u540c\u7684\u8282\u62cd\u5668\u653e\u7f6e\u5728\u67b6\u4e8e\u4e24\u4e2a\u78b3\u9178\u996e\u6599\u7f50\u4e0a\u7684\u6728\u6761\u4e0a\uff0c\u4fbf\u4f1a\u89c2\u5bdf\u5230\uff0c\u8fd9\u4e9b\u539f\u672c\u7528\u4e8e\u97f3\u4e50\u8282\u62cd\u6821\u51c6\u7684\u673a\u68b0\u88c5\u7f6e\uff0c\u5176\u8282\u594f\u80fd\u5728\u77ed\u77ed\u51e0\u5206\u949f\u751a\u81f3\u51e0\u79d2\u949f\u5185\u5b9e\u73b0\u540c\u6b65\u3002\u7136\u800c\uff0c\u8282\u62cd\u5668\u662f\u5982\u4f55\u201c\u6289\u62e9\u201d\u8fbe\u5230\u5171\u540c\u8282\u594f\u7684\uff1f\u66f4\u4e3a\u5173\u952e\u7684\u662f\uff0c\u5b83\u4eec\u4e3a\u4f55\u4f1a\u51fa\u73b0\u8fd9\u79cd\u540c\u6b65\u884c\u4e3a\uff1f<\/span><\/section>\n


    <\/span><\/h2>\n

    \u540c\u6b65\u6982\u5ff5\u7684\u65e9\u671f\u5386\u53f2<\/span><\/strong><\/span><\/h2>\n

    <\/span><\/section>\n
    \u4e3a\u89e3\u7b54\u8fd9\u4e9b\u95ee\u9898\uff0c\u6211\u4eec\u9700\u56de\u6eaf\u81f317\u4e16\u7eaa\uff0c\u8d70\u8fdb\u514b\u91cc\u65af\u8482\u5b89\u00b7\u60e0\u66f4\u65af\u7684\u79d1\u5b66\u4e16\u754c\u3002\u60e0\u66f4\u65af\u582a\u79f0\u8377\u5170\u5386\u53f2\u4e0a\u6700\u4e3a\u6770\u51fa\u7684\u79d1\u5b66\u5bb6\u4e4b\u4e00\uff0c\u4ed6\u4e0d\u4ec5\u5bf9\u5929\u6587\u5b66\u3001\u5149\u5b66\u7814\u7a76\u6ee1\u6000\u70ed\u5ff1\uff0c\u5728\u6570\u5b66\u9886\u57df\u4e5f\u9020\u8be3\u9887\u6df1\u30021673\u5e74\uff0c\u60e0\u66f4\u65af\u51fa\u7248\u4e86\u300a\u6446\u949f\u8bba\u300b\uff0c\u8fd9\u90e8\u8457\u4f5c\u5728\u5f53\u65f6\u7684\u79d1\u5b66\u754c\u5177\u6709\u4e3e\u8db3\u8f7b\u91cd\u7684\u5730\u4f4d\uff0c\u827e\u8428\u514b\u00b7\u725b\u987f\u66f4\u662f\u8d5e\u8a89\u60e0\u66f4\u65af\u4e3a\u201c\u73b0\u4ee3\u6700\u4f18\u96c5\u7684\u4f5c\u5bb6\u201d\u3002\u5b9e\u9645\u4e0a\uff0c\u6446\u949f\u6b63\u662f\u60e0\u66f4\u65af\u7684\u53d1\u660e\u6210\u679c\u3002\u5173\u4e8e\u540c\u6b65\u73b0\u8c61\u7684\u79d1\u5b66\u7814\u7a76\uff0c\u6700\u65e9\u53ef\u8ffd\u6eaf\u81f31673\u5e74\u60e0\u66f4\u65af\u5bf9\u4e24\u4e2a\u8026\u5408\u949f\u6446\u540c\u6b65\u73b0\u8c61\u7684\u53d1\u73b0\u3002\u636e\u4f20\uff0c\u8fd9\u4e00\u53d1\u73b0\u6e90\u4e8e\u4ed6\u5728\u67d0\u6bb5\u5367\u75c5\u5728\u5e8a\u3001\u95f2\u6687\u65e0\u804a\u7684\u65f6\u671f\uff0c\u5229\u7528\u4e24\u628a\u6905\u5b50\u3001\u4e24\u4e2a\u949f\u6446\u548c\u4e00\u4e2a\u6728\u6761\u6240\u5f00\u5c55\u7684\u5b9e\u9a8c\uff1a<\/span><\/section>\n

    <\/span><\/section>\n
    <\/section>\n
    \u60e0\u66f4\u65af\u7684\u5b9e\u9a8c\u793a\u610f\u56fe\uff0cCourtesy: Frederique Swist\/IOP Publishing<\/span><\/section>\n

    <\/span><\/section>\n
    \u5728\u79d1\u5b66\u7814\u7a76\u9886\u57df\uff0c\u73b0\u8c61\u89c2\u5bdf\u4e0e\u5b9e\u9a8c\u7814\u7a76\u5f80\u5f80\u5148\u884c\u4e00\u6b65\u3002\u60e0\u66f4\u65af\u867d\u7136\u89c2\u5bdf\u5230\u4e86\u949f\u6446\u901a\u8fc7\u6728\u6761\u7684\u8026\u5408\u800c\u8d8b\u4e8e\u9891\u7387\u4e00\u81f4\uff0c\u4f46\u5173\u4e8e\u540c\u6b65\u7684\u7406\u8bba\u7814\u7a76\u5374\u6ede\u540e\u3002\u76f4\u523020\u4e16\u7eaa\u521d\uff0c\u8fd9\u4e00\u9886\u57df\u624d\u8fce\u6765\u91cd\u5927\u7a81\u7834\u3002\u8fd9\u662f\u7531\u4e8e\u52a8\u529b\u7cfb\u7edf\u7406\u8bba\u7684\u5efa\u7acb\uff0c\u4eba\u4eec\u5f00\u59cb\u8003\u5bdf\u5fae\u5206\u65b9\u7a0b\u7684\u6e10\u8fdb\u548c\u957f\u65f6\u884c\u4e3a\uff0c\u4ece\u610f\u8bc6\u5230\u4e86\u6781\u9650\u73af<\/strong>\u7684\u6982\u5ff5\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u6781\u9650\u73af\u4f5c\u4e3a\u975e\u7ebf\u6027\u8017\u6563\u52a8\u529b\u7cfb\u7edf\u4e2d\u4e00\u7c7b\u5178\u578b\u7684\u65f6\u95f4\u632f\u8361\u89e3\uff0c\u672c\u8d28\u4e0a\u548c\u5438\u5f15\u5b50\u4e00\u6837\uff0c\u662f\u52a8\u529b\u7cfb\u7edf\u4e0d\u53d8\u96c6\u7684\u4e00\u79cd\u3002\u4ee4\u4eba\u60ca\u5947\u7684\u662f\uff0c\u4f17\u591a\u770b\u4f3c\u6beb\u65e0\u5173\u8054\u7684\u7cfb\u7edf\uff0c\u5176\u65f6\u95f4\u632f\u8361\u7adf\u90fd\u80fd\u7528\u6781\u9650\u73af\u52a0\u4ee5\u63cf\u8ff0\u3002\u7531\u6b64\uff0c\u4eba\u4eec\u9010\u6e10\u610f\u8bc6\u5230\uff0c\u5728\u5f62\u5f62\u8272\u8272\u7684\u540c\u6b65\u884c\u4e3a\u80cc\u540e\uff0c\u6781\u6709\u53ef\u80fd\u6f5c\u85cf\u7740\u5171\u901a\u7684\u7269\u7406\u673a\u7406\u3002\u8bf8\u5982\u8424\u706b\u866b\u540c\u6b65\u95ea\u5149\u3001\u87cb\u87c0\u9f50\u58f0\u9e23\u53eb\uff0c\u4e43\u81f3\u8111\u7535\u6ce2\u7684\u8282\u5f8b\u53d8\u5316\uff0c\u8fd9\u4e9b\u73b0\u8c61\u80cc\u540e\u6216\u8bb8\u90fd\u53d7\u540c\u4e00\u7269\u7406\u89c4\u5f8b\u7684\u652f\u914d \u3002<\/span><\/section>\n

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

    \u53ef\u89e3\u6a21\u578b\u7684\u66d9\u5149<\/span><\/strong><\/span><\/h2>\n


    <\/span><\/h3>\n

    1.Winfree\u6a21\u578b\uff1a\u661f\u8424\u9519\u843d\u540c\u660e\u706d<\/span><\/strong><\/span><\/h3>\n

    <\/strong><\/span><\/section>\n
    \u4e0b\u4e00\u4e2a\u91cd\u5927\u7a81\u7834\u57281967\u5e74\u3002<\/strong>\u8fd9\u4e00\u5e74\uff0c\u7f8e\u56fd\u751f\u7269\u6570\u5b66\u5bb6Winfree\uff0c\u6839\u636e\u8424\u706b\u866b\u7684\u540c\u6b65\u95ea\u70c1\u73b0\u8c61\uff0c\u63d0\u51fa\u4e86Winfree\u6a21\u578b\u2014\u2014\u8fd9\u4e00\u6a21\u578b\u7684\u6838\u5fc3\u6d1e\u89c1\u5728\u4e8e\uff1a\u5728\u540c\u6b65\u95ee\u9898\u4e2d\uff0c\u6781\u9650\u73af\u76f8\u4e92\u4f5c\u7528\u7684\u5173\u952e\u81ea\u7531\u5ea6\u662f\u76f8\u4f4d\uff0c\u56e0\u6b64\u901a\u8fc7\u7814\u7a76\u8026\u5408\u7684\u76f8\u4f4d\u632f\u5b50\u6a21\u578b\uff0c\u5c31\u80fd\u63ed\u793a\u540c\u6b65\u73b0\u8c61\u7684\u52a8\u529b\u5b66\u672c\u8d28\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u8be5\u6a21\u578b\u7684\u52a8\u529b\u5b66\u65b9\u7a0b\u4e3a\uff1a<\/span><\/section>\n

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

    <\/span><\/section>\n
    \u5176\u4e2d\uff0c\u03b8i<\/sub> \u8868\u793a\u7b2c i \u4e2a\u632f\u5b50\u7684\u76f8\u4f4d\uff0c\u03c9 <\/span>\u662f\u632f\u5b50\u7684\u81ea\u7136\u9891\u7387\uff0c\u670d\u4ece\u67d0\u4e2a\u5355\u5cf0\u5bf9\u79f0\u5206\u5e03\uff0c\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3ag(\u03c9\uff09<\/span>\u3002X(\u03b8<\/span>j<\/sub>) \u88ab\u79f0\u4e3a\u76f8\u4e92\u4f5c\u7528\u51fd\u6570<\/strong>\uff0c\u7528\u4e8e\u63cf\u8ff0\u632f\u5b50\u95f4\u76f8\u4e92\u4f5c\u7528\u7684\u673a\u5236\uff1bZ(\u03b8<\/span>i<\/sub>\uff09 \u5219\u662f\u76f8\u4f4d\u54cd\u5e94\u51fd\u6570<\/strong><\/span>\uff08Phase Response Curve\uff0cPRC\uff09<\/span>\u4f53\u73b0\u632f\u5b50\u76f8\u4f4d\u5bf9\u5916\u754c\u4f5c\u7528\u7684\u54cd\u5e94\u7279\u6027\u3002\u5f53\u632f\u5b50\u81ea\u7136\u9891\u7387\u7684\u5206\u5e03\u8f83\u7a84\u65f6\uff0c\u632f\u5b50\u4e4b\u95f4\u4f1a\u51fa\u73b0\u76f8\u4e92\u540c\u6b65\u73b0\u8c61\u3002<\/span><\/section>\n

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

    \u8424\u706b\u866b\u7684\u540c\u6b65\u95ea\u70c1\u73b0\u8c61 \u6765\u6e90\uff1ahttps:\/\/swarma.org\/?p=37648<\/span><\/p>\n


    <\/span><\/section>\n
    \u8fd9\u4e00\u6a21\u578b\u7684\u4ef7\u503c\u5728\u4e8e\uff1a\u4e0a\u8ff0\u4e0d\u540c\u7269\u7406\u80cc\u666f\u3001\u4e0d\u540c\u4f53\u7cfb\u4e2d\u7684\u73b0\u8c61\uff0c\u90fd\u80fd\u501f\u52a9\u8026\u5408\u76f8\u632f\u5b50\u7cfb\u7edf\u7684\u540c\u6b65\u52a8\u529b\u5b66\u5f97\u5230\u89e3\u91ca\u3002\u4f46\u8fd9\u4e2a\u6a21\u578b\u5b9e\u9645\u4e0a\u662f\u4e0d\u53ef\u89e3\u7684\uff0c\u8fd9\u662f\u5176\u79d1\u5b66\u610f\u4e49\u4e0a\u7684\u5c40\u9650\u6027\u3002\u90a3\u4e48\uff0c\u5230\u5e95\u8be5\u5982\u4f55\u4ece\u7406\u8bba\u5c42\u9762\u4e0a\u5f7b\u5e95\u523b\u753b\u548c\u89e3\u6790\u540c\u6b65\u73b0\u8c61\u7684\u6d8c\u73b0\u7684\u8fc7\u7a0b\u5462\uff1f<\/span><\/section>\n

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

    2.Kuramoto\u7684\u7a81\u7834\u6027\u8d21\u732e<\/span><\/strong><\/span><\/h3>\n

    <\/span><\/section>\n
    1975\u5e74\uff0c\u65e5\u672c\u7269\u7406\u5b66\u5bb6\u85cf\u672c\u7531\u7d00<\/span>\uff08Yoshiki Kuramoto\uff09<\/span>\u5728\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u9886\u57df\u5df2\u79ef\u7d2f\u6df1\u539a\u7814\u7a76\u57fa\u7840\u3002\u4ed61970\u5e74\u4e8e\u4eac\u90fd\u5927\u5b66\u53d6\u5f97\u7406\u5b66\u535a\u58eb\u5b66\u4f4d\uff0c\u5e08\u4eceKazuhisa Tomita\u4e0eHajime Mori\uff0c\u65e9\u671f\u94bb\u7814\u76f8\u53d8\u7edf\u8ba1\u529b\u5b66\uff0c\u540e\u56e0\u5bf9Ilya Prigogine\u7b49\u5b66\u8005\u7684\u8017\u6563\u7ed3\u6784\u7814\u7a76\u4ea7\u751f\u601d\u8003\uff0c\u8f6c\u800c\u6df1\u5165\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u9886\u57df\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u6b63\u662f\u5728\u8fd9\u6837\u7684\u5b66\u672f\u63a2\u7d22\u5386\u7a0b\u4e2d\uff0c\u4ed6\u4ee5\u5316\u5b66\u632f\u8361\u73b0\u8c61\u4e3a\u5207\u5165\u70b9\uff0c\u6784\u5efa\u4e86\u610f\u4e49\u91cd\u5927\u7684\u85cf\u672c\u6a21\u578b<\/span>\uff08Kuramoto model\uff09<\/span>\u3002\u8be5\u6a21\u578b\u7a81\u7834\u4f20\u7edf\uff0c\u91c7\u7528\u6700\u7b80\u5f62\u5f0f\u7684\u76f8\u4f4d\u5dee\u6b63\u5f26\u51fd\u6570\u63cf\u8ff0\u632f\u5b50\u8026\u5408\uff0c\u65e2\u7cbe\u51c6\u6355\u6349\u7269\u7406\u6838\u5fc3\uff0c\u53c8\u4e3a\u540e\u7eed\u6570\u5b66\u5206\u6790\u642d\u5efa\u4e86\u6709\u6548\u6846\u67b6\u3002<\/span><\/section>\n

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

    <\/p>\n

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

    <\/span><\/section>\n
    \u548cWinfree\u6a21\u578b\u7c7b\u4f3c\uff0c\u03a6<\/span>j<\/sub>\u548c\u03a6<\/span>i<\/sub>\u5206\u522b\u662f\u4e2a\u4f53j\u548c\u4e2a\u4f53i\u7684\u76f8\u4f4d\uff0c\"image.png\"<\/sub>\u8868\u793a\u4e2a\u4f53i\u76f8\u4f4d\u968f\u65f6\u95f4\u7684\u53d8\u5316\u7387\uff0c\u03c9<\/span>i<\/sub>\u8868\u793a\u4e2a\u4f53i\u7684\u81ea\u7136\u9891\u7387\uff0cK\u662f\u8026\u5408\u5f3a\u5ea6\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u8fd9\u4e2a\u65b9\u7a0b\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5199\u6210\u5e73\u5747\u573a\u5f62\u5f0f\uff1a<\/span><\/section>\n

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

    <\/p>\n

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

    <\/span><\/section>\n
    \u5176\u4e2d<\/span><\/section>\n

    <\/span><\/section>\n
    \"image.png\"<\/sub> \u3002<\/span>\u8fd9\u91ccR<\/span>\u662f\u5e8f\u53c2\u91cf\uff0c\u8868\u5f81\u7cfb\u7edf\u7684\u540c\u6b65\u7a0b\u5ea6\u3002<\/span><\/section>\n
    <\/span><\/section>\n

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

    Kuramoto\u6a21\u578b\u7684\u4e09\u79cd\u72b6\u6001\u7684\u793a\u610f\u56fe\uff08a\uff0cb\uff0cc\uff09\u548c\u5e8f\u53c2\u91cf\u4e0e\u8026\u5408\u5f3a\u5ea6\u7684\u5173\u7cfb\uff0c\u6765\u6e90\uff1aActa Physica Sinica, 69, 080502 (2020)<\/span><\/p>\n


    <\/span><\/section>\n
    \u5728Kuramoto\u6a21\u578b\u4e2d\uff0c\u8026\u5408\u5f3a\u5ea6\u662f\u9a71\u52a8\u632f\u5b50\u7fa4\u4f53\u6f14\u5316\u7684\u6838\u5fc3\u63a7\u5236\u53c2\u91cf\u3002<\/strong>\uff08i\uff09\u5f53\u8026\u5408\u5f3a\u5ea6\u4e3aK=0\u65f6\uff0c\u5927\u91cf\u632f\u5b50\u56e0\u81ea\u7136\u9891\u7387\u4e0d\u540c\uff0c\u76f8\u4f4d\u5747\u5300\u6563\u5e03\u4e8e\u533a\u95f4[0,2\u03c0]\uff0c\u901a\u8fc7\u5e8f\u53c2\u91cfR\u8868\u5f81\uff0c\u5448\u73b0R=0\u5b8c\u5168\u65e0\u5e8f\u72b6\u6001<\/strong>\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    (ii)\u968f\u7740\u8026\u5408\u5f3a\u5ea6K\u9010\u6b65\u589e\u52a0\uff0c\u7cfb\u7edf\u5185\u7684\u540c\u6b65\u673a\u5236\u5f00\u59cb\u663e\u73b0\uff1a\u66f4\u591a\u632f\u5b50\u5f00\u59cb\u5b9e\u73b0\u540c\u6b65\uff0c\u5176\u5e73\u5747\u9891\u7387\u8d8b\u5411\u7edf\u4e00\uff0c\u76f8\u4f4d\u76f8\u4e92\u9760\u8fd1\u5e76\u7ef4\u6301\u56fa\u5b9a (\u79f0\u4e3a\u9501\u76f8\u7fa4\u4f53<\/strong>)\uff0c\u5206\u5e03\u5747\u5300\u6027\u88ab\u6253\u7834\uff0c\u5e8f\u53c2\u91cfR\u8131\u79bb\u96f6\u503c\u3002\u8fd9\u4e00\u8fc7\u7a0b\u7684\u5173\u952e\u9608\u503c\u662f\u4e34\u754c\u8026\u5408\u5f3a\u5ea6Kc<\/sub>\uff1a\u5f53K\u2264K<\/span>c<\/sub>\uff0c\u7cfb\u7edf\u7ef4\u6301\u65e0\u5e8f\uff0cR\u59cb\u7ec8\u51e0\u4e4e\u4e3a0\uff1b\u5f53K\u2265<\/span>K<\/span>c<\/sub>\uff0c\u540c\u6b65\u88ab\u89e6\u53d1\uff0cR\u975e\u96f6\uff0c\u632f\u5b50\u95f4\u5f62\u6210\u6709\u5e8f\u7684\u540c\u6b65\u96c6\u56e2\uff0c\u8fdb\u5165\u90e8\u5206\u540c\u6b65\u6001<\/strong>\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    (iii)\u5728\u5f3a\u8026\u5408\u6761\u4ef6\u4e0b\uff0c\u632f\u5b50\u76f8\u4f4d\u9ad8\u5ea6\u805a\u96c6\uff0c\u5f62\u6210\u6574\u4f53\u540c\u6b65\u6001\uff0c\u8fd9\u672c\u8d28\u4e0a\u5c5e\u4e8e\u975e\u5e73\u8861\u76f8\u53d8\u73b0\u8c61\u2014\u2014\u5927\u91cf\u8026\u5408\u632f\u5b50\u901a\u8fc7\u76f8\u4e92\u4f5c\u7528\uff0c\u514b\u670d\u81ea\u7136\u9891\u7387\u5dee\u5f02\u5f15\u53d1\u7684\u65e0\u5e8f\uff0c\u6700\u7ec8\u6d8c\u73b0\u51fa\u5b8c\u5168<\/strong>\u540c\u6b65<\/strong>\u7684<\/strong>\u6709\u5e8f<\/strong>\u6001<\/strong>\u3002<\/span><\/section>\n

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

    <\/span><\/section>\n
    Kuramoto \u501f\u52a9\u7edf\u8ba1\u7269\u7406\u5b66\u65b9\u6cd5\u4e0e\u81ea\u6d3d\u65b9\u7a0b\uff0c\u901a\u8fc7\u5728\u4e34\u754c\u70b9\u5904\u8fdb\u884c\u6cf0\u52d2\u5c55\u5f00\uff0c\u6210\u529f\u6c42\u89e3\u4e34\u754c\u8026\u5408\u5f3a\u5ea6K<\/span>c<\/sub>\uff0c\u5e76\u63ed\u793aK<\/span>c<\/sub>\u9644\u8fd1\u5e8f\u53c2\u91cfR\u7684\u4e34\u754c\u884c\u4e3a\uff08\u5373\u5728\u4e34\u754c\u533a\u57df\"image.png\" <\/sub><\/span><\/span>\u79f0\u4e3a\u540c\u6b65\u76f8\u53d8\u7684\u4e34\u754c\u6307\u6570\u2014\u2014\u8fd9\u5176\u5b9e\u662f\u5927\u81ea\u7136\u4e2d\u666e\u904d\u5b58\u5728\u7684<\/span>scaling law<\/strong>\u7684\u53c8\u4e00\u4f53\u73b0)\uff0c\u9996\u6b21\u4ece\u7406\u8bba\u5c42\u9762\u7cfb\u7edf\u9610\u91ca\u4e86<\/span>\u96c6\u4f53\u540c\u6b65\u73b0\u8c61\u7684\u590d\u6742\u7269\u7406\u673a\u5236<\/strong>\u3002<\/span><\/section>\n


    <\/span><\/h2>\n

    \u6a21\u578b\u7684\u62d3\u5c55\u4e0e\u5e94\u7528<\/span><\/strong><\/span><\/span><\/h2>\n

    <\/span><\/section>\n
    \u4ee5Kuramoto\u6a21\u578b\u4e3a\u4ee3\u8868\u7684\u8026\u5408\u968f\u673a\u81ea\u7136\u9891\u7387\u632f\u5b50\u7cfb\u7edf\uff0c\u4e5f\u5df2\u6210\u4e3a\u540c\u6b65\u7814\u7a76\u9886\u57df\u7684\u8303\u5f0f\u6027\u6a21\u578b\u3002\u5728\u8fd9\u4e00\u6846\u67b6\u4e0b\uff0c\u632f\u5b50\u901a\u8fc7\u76f8\u4e92\u4f5c\u7528\u514b\u670d\u81ea\u7136\u9891\u7387\u5f02\u8d28\u6027\uff0c\u5b9e\u73b0\u4ece\u65e0\u5e8f\u5230\u6709\u5e8f\u7684\u540c\u6b65\u76f8\u53d8\uff0c\u5176\u7269\u7406\u672c\u8d28\u4e0e\u975e\u5e73\u8861\u7edf\u8ba1\u529b\u5b66\u4e2d\u7684\u76f8\u53d8\u673a\u5236\u9ad8\u5ea6\u5951\u5408\u3002\u968f\u7740\u7814\u7a76\u7684\u6df1\u5165\uff0c\u6a21\u578b\u7684\u5e94\u7528\u8fb9\u754c\u4e0d\u65ad\u62d3\u5c55\uff0c\u5df2\u6e17\u900f\u81f3\u7269\u7406\u5b66\u3001\u751f\u7269\u5b66\u3001\u5de5\u7a0b\u5b66\u7b49\u591a\u4e2a\u9886\u57df\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u540e\u7eed\u7684\u7814\u7a76\u8005\u901a\u8fc7\u8c03\u6574\u6a21\u578b\u53c2\u6570\u4e0e\u7ed3\u6784\uff0c\u6210\u529f\u523b\u753b\u4e86\u73b0\u5b9e\u4e16\u754c\u4e2d\u7eb7\u7e41\u590d\u6742\u7684\u540c\u6b65\u73b0\u8c61\u3002\u4f8b\u5982\uff0c\u901a\u8fc7\u5f15\u5165\u4e0d\u540c\u7684\u8026\u5408\u51fd\u6570<\/strong><\/span>\uff08\u5982\u7c7b\u6b63\u5f26\u51fd\u6570\u3001\u4f59\u5f26\u51fd\u6570\u3001\u8109\u51b2\u8026\u5408\u7b49\uff09<\/span>\uff0c\u6a21\u578b\u80fd\u591f\u63cf\u8ff0\u795e\u7ecf\u8109\u51b2\u540c\u6b65\u3001\u8424\u706b\u866b\u53d1\u5149\u8282\u5f8b\u7b49\u751f\u7269\u73b0\u8c61\uff1b\u6539\u53d8\u8026\u5408\u7f51\u7edc\u7684\u62d3\u6251\u7ed3\u6784<\/strong><\/span>\uff08\u5982\u5c0f\u4e16\u754c\u7f51\u7edc\u3001\u65e0\u6807\u5ea6\u7f51\u7edc\u3001\u591a\u5c42\u7f51\u7edc\u548c\u9ad8\u9636\u7f51\u7edc\uff09<\/span>\uff0c\u5219\u53ef\u63ed\u793a\u7535\u529b\u7cfb\u7edf\u632f\u8361\u3001\u793e\u4ea4\u7f51\u7edc\u610f\u89c1\u4f20\u64ad\u7b49\u96c6\u4f53\u884c\u4e3a\u7684\u52a8\u529b\u5b66\u89c4\u5f8b\u3002\u503c\u5f97\u5173\u6ce8\u7684\u662f\uff0c\u6a21\u578b\u5728\u5f3a\u975e\u7ebf\u6027\u6761\u4ef6\u4e0b\u5448\u73b0\u7684\u73bb\u7483\u6001\u540c\u6b65\u3001\u65f6\u6ede\u6548\u5e94\u5f15\u53d1\u7684\u884c\u6ce2\/\u9a7b\u6ce2\u6001\uff0c\u4ee5\u53ca\u60ef\u6027\u3001\u963b\u632b\u3001\u76f8\u79fb\u548c\u5916\u573a\u7b49\u7269\u7406\u6548\u5e94\u7684\u5f15\u5165\uff0c\u5747\u4e3a\u7406\u89e3\u51dd\u805a\u6001\u7269\u7406\u3001\u6fc0\u5149\u9635\u5217\u3001\u96c6\u7fa4\u8fd0\u52a8\u7b49\u5b9e\u9645\u7cfb\u7edf\u63d0\u4f9b\u4e86\u7406\u8bba\u5de5\u5177\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    Kuramoto\u6a21\u578b\u7684\u79d1\u5b66\u4ef7\u503c\u4e0d\u4ec5\u5728\u4e8e\u5176\u5e7f\u6cdb\u7684\u9002\u7528\u6027\uff0c\u66f4\u5728\u4e8e\u5176\u53ca\u4ee5\u540e\u7684\u53d8\u4f53\u6a21\u578b\u5f15\u53d1\u7684\u65b9\u6cd5\u8bba\u5c42\u9762\u7684\u521b\u65b0\u3002\u81ea\u6d3d\u6027\u65b9\u7a0b<\/strong>\u7684\u63d0\u51fa\u9996\u6b21\u5b9e\u73b0\u4e86\u540c\u6b65\u76f8\u53d8\u7684\u7406\u8bba\u89e3\u6790\uff0c\u4e3b\u7a33\u5b9a\u51fd\u6570\uff08MSF\uff09<\/strong>\u4e3a\u590d\u6742\u7f51\u7edc\u540c\u6b65\u80fd\u529b\u8bc4\u4f30\u63d0\u4f9b\u4e86\u666e\u9002\u6846\u67b6\uff0c\u800cOA\u62df\u8bbe\uff08Ott-Antonsen Ansatz\uff09\u3001WS\u53d8\u6362\uff08Watanabe-Strogatz Transformation\uff09\u4e0e\u91cd\u6574\u5316\u7fa4\u65b9\u6cd5<\/strong>\u7684\u5f15\u5165\uff0c\u5219\u4e3a\u5904\u7406\u9ad8\u7ef4\u975e\u7ebf\u6027\u7cfb\u7edf\u5f00\u8f9f\u4e86\u65b0\u8def\u5f84\u3002\u8fd9\u4e9b\u7406\u8bba\u5de5\u5177\u7684\u53d1\u5c55\uff0c\u4f7fKuramoto\u6a21\u578b\u6210\u4e3a\u8fde\u63a5\u5fae\u89c2\u4e2a\u4f53\u884c\u4e3a\u4e0e\u5b8f\u89c2\u96c6\u4f53\u79e9\u5e8f\u7684\u6865\u6881\uff0c\u4e3a\u7406\u89e3\u81ea\u7136\u754c\u4ece\u7ec6\u80de\u4ee3\u8c22\u8282\u5f8b\u5230\u661f\u7cfb\u65cb\u8f6c\u7b49\u8de8\u5c3a\u5ea6\u540c\u6b65\u73b0\u8c61\u63d0\u4f9b\u4e86\u7edf\u4e00\u7684\u52a8\u529b\u5b66\u89c6\u89d2\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u8fd1\u5e74\u6765\uff0c\u8fd9\u4e00\u9886\u57df\u4e0e\u591a\u4e2a\u5b66\u79d1\u4ea4\u53c9\uff0c\u4ea7\u751f\u4e86\u4e00\u4e9b\u7cfb\u5217\u65b0\u6a21\u578b\u3001\u65b0\u6982\u5ff5\u548c\u65b0\u65b9\u6cd5\uff0c\u5982\u4ece\u7edf\u8ba1\u7269\u7406\u89c6\u89d2<\/strong>\uff0c\u5bf9\u540c\u6b65\u76f8\u53d8\u7684\u4e34\u754c\u6027\u8d28\u7684\u8fdb\u4e00\u6b65\u63a2\u7d22\uff0c\u5373\u8003\u5bdf\u4e34\u754c\u7ef4\u6570\u3001\u9664\u5e8f\u53c2\u91cf\u5916\u5176\u4ed6\u4e34\u754c\u6307\u6570\u4e0e\u666e\u9002\u7c7b<\/span>\uff08Daido,1988PRL; Hong\u7b49, 2007PRL; Qiu\u7b49, Unpublished\uff09<\/span>\uff1b\u5f15\u5165\u65b0\u7684\u5185\u7980\u81ea\u7531\u5ea6<\/strong>\u2014\u2014D\u7ef4<\/span>Kuramoto\u6a21\u578b<\/span> (Chandra\u7b49\uff0c2019PRX\uff1bDai\u7b49\uff0c2021PRL\uff1bZou\u7b49\uff0c2023PRL)<\/span>\uff0c\u53d1\u73b0\u4e86\u5176\u884c\u4e3a\u53d1\u73b0\u4e86\u5176\u884c\u4e3a\u5bf9<\/strong>D\u7684\u5947\u5076\u6027\u7684\u65b0\u9896\u4f9d\u8d56\u6027<\/strong>\uff1b\u540c\u6b65\u4e0e\u96c6\u7fa4\u7684\u4ea4\u53c9\u9886\u57df\u2014\u2014\u96c6\u7fa4\u632f\u5b50\u6a21\u578b<\/strong><\/span> <\/strong>(O\u2019Keeffe\u7b49\uff0c2017NC)<\/span>\u7b49\u3002\u5b83\u4eec\u4e0e\u8bf8\u591a\u9886\u57df\u6df1\u5ea6\u878d\u5408\uff0c\u8fdb\u4e00\u6b65\u52a0\u6df1\u4e86\u6211\u4eec\u5bf9\u4e8e\u540c\u6b65\u73b0\u8c61\u548c\u7406\u8bba\u7684\u7406\u89e3\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \n
    \n
    \u8fd9\u90e8\u5206\u5173\u4e8e\u96c6\u7fa4\u632f\u5b50\u548cKuramoto\u6a21\u578b\u7684\u6269\u5c55\u7684\u4e00\u4e9b\u5185\u5bb9\uff0c\u4f1a\u5728\u96c6\u667a\u4ff1\u4e50\u90e8\u6b63\u5728\u7ec4\u7ec7\u7684\u590d\u6742\u7f51\u7edc\u52a8\u529b\u5b66\u8bfb\u4e66\u4f1a<\/a>\u4e0a\u505a\u4e00\u4e9b\u4e13\u9898\u89e3\u8bfb\u548c\u4ecb\u7ecd\uff0c\u6b22\u8fce\u5927\u5bb6\u52a0\u5165\u8bfb\u4e66\u4f1a\u4e00\u8d77\u6765\u4ea4\u6d41\u3002<\/span><\/section>\n<\/section>\n<\/blockquote>\n
    <\/section>\n
    <\/section>\n

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

    \u7ed3\u8bed<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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

    <\/p>\n

    KPZ\u65b9\u7a0b\u63cf\u8ff0\u7684\u754c\u9762\u751f\u957f\u4e0eKuramoto\u6a21\u578b\u523b\u753b\u7684\u632f\u5b50\u540c\u6b65\uff0c\u524d\u8005\u63ed\u793a\u4e86\u5f00\u653e\u7cfb\u7edf\u5728\u566a\u58f0\u9a71\u52a8\u4e0b\u81ea\u53d1\u5f62\u6210\u590d\u6742\u7ed3\u6784\u7684\u666e\u9002\u6027\uff0c\u540e\u8005\u5219\u8bc1\u660e\u65e0\u5e8f\u4e2a\u4f53\u901a\u8fc7\u7b80\u5355\u89c4\u5219\u5b9e\u73b0\u540c\u6b65\u76f8\u53d8\u7684\u6d8c\u73b0\u673a\u5236\u3002\u8fd9\u4e24\u4e2a\u770b\u4f3c\u8fe5\u5f02\u7684\u7269\u7406\u56fe\u666f\uff0c\u5374\u90fd\u5b58\u5728\u7740\u67d0\u79cd\u5f62\u5f0f\u7684\u6807\u5ea6\u5f8b\uff08scaling law\uff09<\/strong>\u3002\u4e8b\u5b9e\u4e0a\uff0c\u4ece\u7eb3\u7c73\u5c3a\u5ea6\u7684\u91cf\u5b50\u6da8\u843d\u5230\u661f\u7cfb\u5c3a\u5ea6\u7684\u7ed3\u6784\u5f62\u6210\uff0c\u4ece\u795e\u7ecf\u7f51\u7edc\u8bad\u7ec3\u4e2d\u7684\u53c2\u6570\u6d8c\u73b0\u5230\u591a\u667a\u80fd\u4f53\u7cfb\u7edf\u7684\u96c6\u4f53\u51b3\u7b56\uff0c\u4eba\u4eec\u90fd\u53d1\u73b0\u82e5\u5e72\u7b80\u5355\u7684\u6807\u5ea6\u6307\u6570\u53ef\u4ee5\u6982\u62ec\u63cf\u8ff0\u4e0d\u540c\u89c4\u6a21\u7684\u7cfb\u7edf\u884c\u4e3a<\/strong>\u3002\u8fd9\u5c55\u793a\u4e86\u7edf\u8ba1\u7269\u7406\u5b66\u7406\u8bba\u5728\u89e3\u91ca\u590d\u6742\u7cfb\u7edf\u884c\u4e3a\u65b9\u9762\u7684\u5f3a\u5927\u80fd\u529b\uff0c\u6211\u4eec\u6216\u8bb8\u56e0\u6b64\u89e6\u78b0\u5230\u4e86\u81ea\u7136\u754c\u6700\u6df1\u9083\u7684\u5965\u79d8<\/strong>\u2014\u2014\u590d\u6742\u7cfb\u7edf\u7684\u52a8\u529b\u5b66\u591a\u6837\u6027\u80cc\u540e\uff0c\u5b58\u5728\u7740\u8d85\u8d8a\u5177\u4f53\u7269\u8d28\u8f7d\u4f53\u7684\u666e\u9002\u6027\u6cd5\u5219\u3002\u8fd9\u79cd\u62bd\u8c61\u6cd5\u5219\u4e0d\u4ec5\u662f\u6d8c\u73b0\u73b0\u8c61\u80cc\u540e\u6df1\u523b\u7684\u6570\u5b66\u7ed3\u6784\u7684\u66f2\u6298\u5448\u73b0\uff0c\u66f4\u662f\u4f5c\u4e3a\u4ece\u6570\u5b66\u5b9e\u5728\u5230\u771f\u5b9e\u7269\u7406\u4e16\u754c\u7684\u6620\u5c04\u65b9\u6848\u7684\u7269\u7406\u89c4\u5f8b\uff0c\u5728\u8de8\u8d8a\u4e0d\u540c\u5c42\u6b21\u65f6\u7f16\u7801\u7684\u57fa\u672c\u8bed\u6cd5\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u4f5c\u4e3a\u4eba\u7c7b\u7684\u4e00\u5206\u5b50\uff0c\u79d1\u5b66\u5bb6\u4eec\u5c45\u7136\u80fd\u591f\u8ba4\u77e5\u5230\u4e16\u754c\u5982\u6b64\u5e95\u5c42\u7684\u89c4\u5f8b\u3002\u4e8e\u662f\uff0c\u6211\u4eec\u4e0d\u5f97\u4e0d\u611f\u53f9\uff1a<\/span><\/section>\n
    <\/span><\/section>\n
    <\/span><\/section>\n
    \u4eba\uff0c\u662f\u5b87\u5b99\u7684\u610f\u4e49\u5206\u6ccc\u5668\u5b98\u3002<\/strong><\/span><\/section>\n

    <\/p>\n

    \n
    \u672c\u6587\u4e3a\u79d1\u666e\u4e2d\u56fd-\u521b\u4f5c\u57f9\u80b2\u8ba1\u5212\u6276\u6301\u4f5c\u54c1<\/section>\n
    \u4f5c\u8005 | \u90b1\u4ef2\u666e<\/section>\n
    \u5ba1\u6838 | \u6a0a\u4eac\u82b3\uff08\u5317\u4eac\u5e08\u8303\u5927\u5b66\u7cfb\u7edf\u79d1\u5b66\u5b66\u9662\u6559\u6388\uff09<\/span><\/section>\n
    \u51fa\u54c1 | \u4e2d\u56fd\u79d1\u534f\u79d1\u666e\u90e8<\/section>\n
    \u76d1\u5236 | \u4e2d\u56fd\u79d1\u5b66\u6280\u672f\u51fa\u7248\u793e\u6709\u9650\u516c\u53f8\u3001\u5317\u4eac\u4e2d\u79d1\u661f\u6cb3\u6587\u5316\u4f20\u5a92\u6709\u9650\u516c\u53f8<\/section>\n<\/blockquote>\n
    <\/section>\n
    \"\u56fe\u7247\"<\/section>\n
    <\/section>\n
    <\/section>\n
    \n
    \n
    \n
    \n

    \u590d\u6742\u7f51\u7edc\u52a8\u529b\u5b66\u8bfb\u4e66\u4f1a<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

    <\/p>\n

    \u96c6\u667a\u4ff1\u4e50\u90e8\u8054\u5408\u5408\u80a5\u5de5\u4e1a\u5927\u5b66\u7269\u7406\u7cfb\u6559\u6388\u674e\u660e\u3001\u540c\u6d4e\u5927\u5b66\u526f\u6559\u6388\u5f20\u6bc5\u8d85\u3001\u5317\u4eac\u5e08\u8303\u5927\u5b66\u7279\u8058\u526f\u7814\u7a76\u5458\u53f2\u8d35\u5143\u4e0e\u5728\u8bfb\u535a\u58eb\u751f\u90b1\u4ef2\u666e\u3001\u5f20\u7ae0\u5171\u540c\u53d1\u8d77<\/span>\u300c\u590d\u6742\u7f51\u7edc\u52a8\u529b\u5b66\u300d\u8bfb\u4e66\u4f1a<\/a>\u3002\u672c\u6b21\u8bfb\u4e66\u4f1a\u5c06\u63a2\u8ba8\uff1a\u540c\u6b65\u76f8\u53d8\u7684\u4e34\u754c\u6027\u3001\u5982\u4f55\u666e\u9002\u5730\u523b\u753b\u591a\u7a33\u6001\u4e0e\u4e34\u754c\u70b9\u3001\u5982\u4f55\u8bc6\u522b\u5e76\u9884\u6d4b\u4e34\u754c\u8f6c\u53d8\u3001\u5982\u4f55\u901a\u8fc7\u5c40\u90e8\u5e72\u9884\u6765\u8c03\u63a7\u7cfb\u7edf\u4fdd\u6301\u6216\u56de\u5230\u671f\u671b\u7a33\u6001\u3001\u7206\u70b8\u903e\u6e17\u4e34\u754c\u884c\u4e3a\u7684\u5173\u952e\u7279\u5f81\u3001\u4e0d\u540c\u7c7b\u578b\u7684\u7ea7\u8054\u8fc7\u7a0b\u5bf9\u903e\u6e17\u76f8\u53d8\u7684\u5f71\u54cd\u6709\u4f55\u5f02\u540c\u3001\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u5f71\u54cd\u80fd\u5426\u7b49\u6548\u4e3a\u82e5\u5e72\u7b80\u5355\u673a\u5236\u7684\u53e0\u52a0\u3001\u5982\u4f55\u6709\u6548\u5730\u4fc3\u8fdb\u4eba\u7c7b\u4e2a\u4f53\u95f4\u7684\u5408\u4f5c\u7b49\u95ee\u9898\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u8bfb\u4e66\u4f1a\u8ba1\u5212\u4ece3\u67087\u65e5\u5f00\u59cb\uff0c\u6bcf\u5468\u4e94\u665a19:30-21:30\u8fdb\u884c\uff0c\u6301\u7eed8-10\u5468\u3002\u8bda\u631a\u9080\u8bf7\u9886\u57df\u5185\u7814\u7a76\u8005\u3001\u5bfb\u6c42\u8de8\u9886\u57df\u878d\u5408\u7684\u7814\u7a76\u8005\u52a0\u5165\uff0c\u5171\u540c\u63a2\u8ba8\u3002<\/span><\/section>\n


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

    \"\u56fe\u7247\"<\/span><\/a><\/section>\n
    \n


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

    \u8be6\u60c5\u8bf7\u89c1\uff1a\u590d\u6742\u7f51\u7edc\u4e0a\u7684\u81ea\u7ec4\u7ec7\u4e0e\u96c6\u4f53\u884c\u4e3a\uff1a\u4ece\u6269\u6563\u3001\u76f8\u53d8\u5230\u535a\u5f08 | \u8bfb\u4e66\u4f1a\u542f\u52a8<\/a><\/span><\/p>\n<\/section>\n

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

    \u975e\u5e73\u8861\u7edf\u8ba1\u7269\u7406\u8bfb\u4e66\u4f1a\u542f\u52a8\uff01<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

    <\/section>\n

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

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    <\/section>\n
    \"\u56fe\u7247\"<\/span><\/a><\/section>\n

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

    \u8be6\u60c5\u8bf7\u89c1\uff1a<\/span>\u4ece\u70ed\u529b\u5b66\u3001\u751f\u547d\u5230\u4eba\u5de5\u667a\u80fd\u7684\u7edf\u8ba1\u7269\u7406\u4e4b\u8def\uff1a\u975e\u5e73\u8861\u7edf\u8ba1\u7269\u7406\u8bfb\u4e66\u4f1a\u542f\u52a8\uff01<\/a><\/span><\/p>\n

    <\/p>\n

    \n
    <\/section>\n<\/section>\n
    \u63a8\u8350\u9605\u8bfb<\/span><\/strong><\/strong><\/strong><\/section>\n
    <\/strong><\/section>\n
    1. <\/strong>\u65e0\u5e8f\u4e2d\u7684\u5bf9\u79f0\u6027\u2014\u2014\u8bfa\u5956\u5f97\u4e3b\u5e15\u91cc\u897f\u5de5\u4f5c\u89e3\u8bfb<\/a><\/strong><\/span><\/strong><\/section>\n
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    <\/p>\n


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

    \u70b9\u51fb\u201c\u9605\u8bfb\u539f\u6587\u201d\uff0c\u62a5\u540d\u8bfb\u4e66\u4f1a<\/span><\/section>\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

    \u5bfc\u8bed \u81ea\u7136\u754c\u4e2d\uff0c\u4e3a\u4f55\u8424\u706b\u866b\u4f1a\u540c\u6b65\u95ea\u70c1\uff1f\u4e3a\u4f55\u4e0d\u540c\u6750\u6599\u7684\u8868\u9762\u751f\u957f\u9075\u5faa\u76f8\u4f3c\u89c4\u5f8b\uff1f2025\u5e74\u73bb\u5c14\u5179\u66fc\u5956\u6388\u4e88Mehran Kardar\u548cYoshiki Kuramoto\uff0c\u8868\u5f70\u4ed6\u4eec\u5206\u522b\u901a\u8fc7KPZ\u65b9\u7a0b\u548cKuramoto\u6a21\u578b\u89e3\u7b54\u4e86\u8fd9\u4e9b\u96be\u9898\u3002\u8fd9\u4e24\u4e2a\u770b\u4f3c\u7b80\u5355\u7684\u6a21\u578b\u63ed\u793a\u4e86\u975e\u5e73\u8861\u7cfb\u7edf\u4ece\u65e0\u5e8f\u8d70\u5411\u6709\u5e8f\u7684\u666e\u9002\u89c4\u5f8b\uff0c\u4e0d\u4ec5\u7edf\u4e00\u89e3\u91ca\u4e86…<\/p>\n","protected":false},"author":1,"featured_media":57420,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"special":[],"_links":{"self":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts\/57452"}],"collection":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=57452"}],"version-history":[{"count":0,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts\/57452\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/media\/57420"}],"wp:attachment":[{"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=57452"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=57452"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=57452"},{"taxonomy":"special","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fspecial&post=57452"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}