{"id":274,"date":"2025-09-22T11:34:10","date_gmt":"2025-09-22T03:34:10","guid":{"rendered":"https:\/\/blog.wysng.cn\/?p=274"},"modified":"2026-01-04T06:25:00","modified_gmt":"2026-01-03T22:25:00","slug":"surfaceinterfacethermo","status":"publish","type":"post","link":"https:\/\/blog.wysng.cn\/index.php\/2025\/09\/22\/surfaceinterfacethermo\/","title":{"rendered":"\u8868\u9762\u4e0e\u754c\u9762\u7684\u7edf\u8ba1\u70ed\u529b\u5b66\u7b14\u8bb0"},"content":{"rendered":"

\u524d\u8a00<\/h2>\n

\u672c\u5b66\u671f\u9009\u4fee\u4e86\u4e2d\u56fd\u79d1\u5b66\u9662\u5927\u5b66\u5f00\u8bbe\u7684\u8868\u9762\u4e0e\u754c\u9762\u7684\u7edf\u8ba1\u70ed\u529b\u5b66\uff0c\u6b63\u597d\u597d\u4e45\u6ca1\u5199\u535a\u5ba2\u4e86\uff0c\u5e72\u8106\u628anote\u653e\u535a\u5ba2\u4e0a\u9762\uff0c\u7b14\u8bb0\u5199\u7684\u5f88\u4e71\uff0c\u53ef\u80fd\u4e0d\u4e25\u8c28\uff0c\u4e00\u5806\u7b14\u8bef\u4e4b\u7c7b\u7684\uff0c\u51d1\u5408\u770b<\/p>\n

\u8868\u754c\u9762\u70ed\u529b\u5b66\u7406\u8bba\u6846\u67b6<\/h2>\n

\u8be5\u8bfe\u7a0b\u7684\u7406\u8bba\u4e3b\u8981\u5305\u542b\u56db\u90e8\u5206\uff0c\u5206\u522b\u662f\u9884\u5907\u77e5\u8bc6\u3001 \u754c\u9762\u5f20\u529b\u53ca\u6da8\u843d\u4e0d\u7a33\u5b9a\u6027\u3001\u754c\u9762\u7684\u6d78\u6da6\u73b0\u8c61\u548c\u754c\u9762\u76f8\u4e92\u4f5c\u7528\u3002\u5176\u4e2d\u9884\u5907\u77e5\u8bc6\u5305\u542b\u57fa\u672c\u7684\u70ed\u529b\u5b66\u548c\u7edf\u8ba1\u7269\u7406\u77e5\u8bc6\uff0c\u5e76\u4f1a\u4ecb\u7ecd\u90e8\u5206\u5fae\u5206\u51e0\u4f55\u77e5\u8bc6\uff08\u53ea\u5305\u542b\u7ecf\u5178\u5fae\u5206\u51e0\u4f55\uff0c\u4e0d\u542b\u9ece\u66fc\u51e0\u4f55\uff09\uff0c\u7b2c\u4e8c\u90e8\u5206\u4e3b\u8981\u5173\u6ce8\u745e\u5229-\u666e\u62c9\u6258\u4e0d\u7a33\u5b9a\u6027\uff1b\u7b2c\u4e09\u90e8\u5206\u5173\u6ce8\u591a\u76f8\u754c\u9762\uff0c\u7b2c\u56db\u90e8\u5206\u4e3b\u8981\u8bb2\u8ff0DLVO theory<\/p>\n

\u7b2c\u4e00\u7ae0 \u9884\u5907\u77e5\u8bc6<\/h2>\n

\u8be5\u90e8\u5206\u4e3b\u8981\u4ecb\u7ecd\u57fa\u672c\u7684\u70ed\u529b\u5b66\u548c\u7edf\u8ba1\u7269\u7406\uff0c\u4ee5\u53ca\u540e\u7eed\u4f1a\u7528\u5230\u7684\u90e8\u5206\u5fae\u5206\u51e0\u4f55\u77e5\u8bc6<\/p>\n

\u70ed\u529b\u5b66\u57fa\u7840\u56de\u987e<\/h3>\n

\u70ed\u529b\u5b66\u5b9a\u5f8b\u662f\u7269\u7406\u5b66\u4e2d\u63cf\u8ff0\u80fd\u91cf\u8f6c\u6362\u548c\u70ed\u529b\u5b66\u8fc7\u7a0b\u7684\u57fa\u672c\u539f\u7406\uff0c\u4f17\u6240\u5468\u77e5\uff0c\u5f53\u6211\u4eec\u63d0\u5230\u70ed\u529b\u5b66\u65f6\uff0c\u901a\u5e38\u4f1a\u8c08\u5230\u70ed\u529b\u5b66\u56db\u5927\u5b9a\u5f8b\uff0c\u5b83\u4eec\u5206\u522b\u662f\uff1a<\/p>\n

\u70ed\u529b\u5b66\u7b2c0\u5b9a\u5f8b\uff1a\u82e5\u4e24\u4e2a\u70ed\u529b\u5b66\u7cfb\u7edf\u5747\u4e0e\u7b2c\u4e09\u4e2a\u7cfb\u7edf\u5904\u4e8e\u70ed\u5e73\u8861\u72b6\u6001\uff0c\u6b64\u4e24\u4e2a\u7cfb\u7edf\u4e5f\u5fc5\u4e92\u76f8\u5904\u4e8e\u70ed\u5e73\u8861\uff1b\u8003\u8651\u4e09\u4e2a\u70ed\u529b\u5b66\u7cfb\u7edf1\u30012\u30013\uff1b\u82e5\u7cfb\u7edf1\u548c3\u5904\u4e8e\u70ed\u5e73\u8861\u72b6\u6001\uff0c\u5373T_1=T_3<\/span>\uff1b\u7cfb\u7edf2\u548c\u7cfb\u7edf3\u4e5f\u5904\u4e8e\u70ed\u5e73\u8861\u72b6\u6001\uff0c\u5373T_2=T_3<\/span>\uff0c\u90a3\u4e48\u7cfb\u7edf1\u548c\u7cfb\u7edf2\u4e5f\u5904\u4e8e\u70ed\u5e73\u8861\u72b6\u6001\uff0c\u5373T_1=T_2<\/span>\uff1b\u70ed\u529b\u5b66\u7b2c0\u5b9a\u5f8b\u63ed\u793a\u4e86\u70ed\u5e73\u8861\u7684\u9012\u79fb\u6027\uff0c\u4ece\u800c\u4e3a\u6e29\u5ea6\u7684\u5b9a\u4e49\u548c\u6d4b\u91cf\u63d0\u4f9b\u4e86\u7406\u8bba\u57fa\u7840<\/p>\n

\u70ed\u529b\u5b66\u7b2c1\u5b9a\u5f8b\uff1a\u8be5\u5b9a\u5f8b\u662f\u80fd\u91cf\u5b88\u6052\u5b9a\u5f8b\u5bf9\u975e\u5b64\u7acb\u7cfb\u7edf\u7684\u6269\u5c55\uff0c\u5176\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n

\\displaystyle dU=dW+dQ<\/span><\/p>\n

\u8fd9\u91cc\u6211\u4eec\u5747\u9009\u53d6\u6307\u5411\u7cfb\u7edf\u7684\u65b9\u5411\u4e3a\u6b63\u65b9\u5411\uff0c\u4fbf\u4e8e\u540e\u7eed\u7edf\u4e00\u5206\u6790\uff08\u5373dW<\/span>\u4ee3\u8868\u5916\u754c\u5bf9\u7cfb\u7edf\u505a\u7684\u529f\uff0cdQ<\/span>\u4ee3\u8868\u5916\u754c\u4f20\u5165\u7cfb\u7edf\u7684\u70ed\u91cf\uff09<\/p>\n

\u70ed\u529b\u5b66\u7b2c2\u5b9a\u5f8b\uff1a\u8be5\u5b9a\u5f8b\u6709\u514b\u52b3\u4fee\u65af\u8868\u8ff0\u548c\u5f00\u5c14\u6587\u8868\u8ff0\uff0c\u4e3a\u5b9a\u91cf\u63cf\u8ff0\u70ed\u529b\u5b66\u7b2c\u4e8c\u5b9a\u5f8b\uff0c\u514b\u52b3\u4fee\u65af\u5f15\u5165\u201c\u71b5\u201d\u8fd9\u4e00\u72b6\u6001\u51fd\u6570\u3002\u5176\u6838\u5fc3\u7279\u5f81\u662f\uff1a\u71b5\u662f\u5e7f\u5ef6\u91cf\uff0c\u4ec5\u7531\u7cfb\u7edf\u7684\u5e73\u8861\u6001\u51b3\u5b9a\uff0c\u4e0e\u8fc7\u7a0b\u65e0\u5173\u3002\u57fa\u4e8e\u71b5\u7684\u5b9a\u4e49\uff0c\u70ed\u529b\u5b66\u7b2c\u4e8c\u5b9a\u5f8b\u53ef\u901a\u8fc7 \u201c\u71b5\u589e\u539f\u7406\u201d \u8f6c\u5316\u4e3a\u6570\u5b66\u5173\u7cfb\uff1a<\/p>\n

\\displaystyle \\Delta S \\geq \\frac{\\Delta Q}{T}<\/span><\/p>\n

\u79fb\u9879\u5f97\\Delta S-\\frac{\\Delta Q}{T}\\geq0<\/span>\uff0c\u5c06\u4e0d\u7b49\u5f0f\u5de6\u8fb9\u5b9a\u4e49\u4e3a\\Delta S^{tot}<\/span>\uff0c\u5176\u4e2d\\Delta S^{tot}=\\Delta S^{sys}+\\Delta S^{res}<\/span>\uff0csys\u4ee3\u8868\u7cfb\u7edf\uff0cres\u4ee3\u8868\u70ed\u6d74\uff0c\u5176\u4e2d\\Delta S^{sys}=\\Delta S<\/span>\uff0c\\Delta S^{res}=-\\frac{\\Delta Q}{T}<\/span>\u3002\u5176\u4e2d\u4e0d\u7b49\u5f0f\u4e2d\u7684\u7b49\u4e8e\u7b26\u53f7\u53d6\u5728\u5e73\u8861\u6001\uff0c\u53ef\u7528\u4e8e\u4f5c\u4e3a\u975e\u5e73\u8861\u7684\u5224\u636e<\/p>\n

\u7ed3\u5408\u7b2c1\u4e0e\u7b2c2\u5b9a\u5f8b\uff0c\u8fdb\u884c\u79fb\u9879\\Delta W=\\Delta U-\\Delta Q<\/span>\uff0c\u4e4b\u540e\u5229\u7528\\Delta W=\\Delta U-T\\Delta S+T\\Delta S-\\Delta Q<\/span>\uff0c\u53ef\u4ee5\u5f97\u5230\u6700\u5927\u529f\u539f\u7406\uff0c\u53ef\u4ee5\u53d1\u73b0\\Delta U-T\\Delta S<\/span>\u5373\u4e3a\\Delta F<\/span>\uff0c\u800c\u540e\u9762\u90e8\u5206\u5176\u5b9e\u5c31\u662fT\\Delta S^{tot}<\/span>\uff0c\u6545\u800c\u6709\\Delta W=\\Delta F+T\\Delta S^{tot}<\/span>\uff0c\u800c\u7b2c\u4e8c\u5b9a\u5f8b\u53c8\u7ed9\u51fa\u4e86\u8fd9\u4e2a\u8868\u8fbe\u5f0f\u7684\u8fb9\u754c\uff0c\u56e0\u6b64\u6709\\Delta W\\geq \\Delta F<\/span>\uff0c\u6539\u5199\u6210-\\Delta W\\leq -\\Delta F<\/span>\uff0c\u4e5f\u5c31\u7ed9\u51fa\u4e86\u6700\u5927\u529f\u539f\u7406\uff08\u4ea5\u59c6\u970d\u5179\u81ea\u7531\u80fd\u6700\u5c0f\u539f\u7406\uff09<\/p>\n

\u4e0a\u8ff0\u70ed\u529b\u5b66\u7b2c2\u5b9a\u5f8b\u8fd8\u53ef\u4ee5\u901a\u8fc7\u6da8\u843d\u5b9a\u7406\u5bfc\u51fa\uff0c\u6216\u8005\u8bf4\u6da8\u843d\u5b9a\u7406\u662f\u70ed\u529b\u5b66\u7b2c2\u5b9a\u5f8b\u7684\u7edf\u8ba1\u8bc1\u660e\uff08\u77e5\u4e4e\u4e0a\u8bf4\u7684\uff09\uff0c\u7531Crooks\u6da8\u843d\u5b9a\u7406\uff1a<\/p>\n

\\displaystyle <e^{-\\Delta s^{tot}}>=1<\/span><\/p>\n

\u6b64\u5904\\Delta s<\/span>\u65e0\u91cf\u7eb2\uff0c\u73bb\u5c14\u5179\u66fc\u5e38\u6570\u53d6k_B=1<\/span>\uff0c\u5229\u7528\u7434\u751f\u4e0d\u7b49\u5f0f\uff08\u5bf9\u4e8e\u51f8\u51fd\u6570\uff0c\u6709<e^{x}>\\geq e^{< x >}<\/span>\uff09\uff0c\u53ef\u77e5<e^{-\\Delta s^{tot}}>\\geq e^{-< \\Delta s^{tot} >}<\/span>\uff0c\u53731 \\geq e^{-< \\Delta s^{tot} >}<\/span>\uff0c\u7531\u4e8e<\\Delta s^{tot}>=\\Delta S^{tot}<\/span>\uff0c\u6545\u5f97\\Delta S^{tot}\\geq0<\/span> (\u6ce8\u610f\u533a\u5206\u5c0fs\u4e0e\u5927S\uff0c\u5c0fs\u4ee3\u8868\u8fdc\u79bb\u5e73\u8861\u4f4d\u7f6e\u7684\u968f\u673a\u71b5\uff0c\u5927S\u5219\u662f\u603b\u7684\u71b5)<\/p>\n

\u540c\u7406\uff0c\u6700\u5927\u529f\u539f\u7406\u53ef\u4ee5\u901a\u8fc7Jarzynski equality\u5bfc\u51fa\uff0cJarzynski equality\u5373:<\/p>\n

\\displaystyle <e^{-w}>=e^{-\\Delta F}<\/span><\/p>\n

\u6839\u636e\u4e0a\u9762\u7c7b\u4f3c\u7684\u64cd\u4f5c\uff0c\u5bb9\u6613\u5f97\u5230e^{-\\Delta F}\\geq e^{-< w >}<\/span>\uff0c\u6700\u540e\u5f97\u5230-\\Delta W\\leq -\\Delta F<\/span><\/p>\n

\u70ed\u529b\u5b66\u7b2c3\u5b9a\u5f8b\uff1a\u7edd\u5bf9\u96f6\u5ea6\u4e0d\u53ef\u8fbe\u5230<\/p>\n

\u4e0a\u9762\u4e3b\u8981\u5173\u6ce8\u70ed\u529b\u5b66\u7b2c2\u5b9a\u5f8b\u53ca\u71b5\u589e\u5b9a\u5f8b\uff0c\u56e0\u4e3a\\Delta S^{tot}<\/span>\u4e5f\u5373\u6307\u660e\u4e86\u71b5\u4ea7\u751f\uff0c\u800c\\Delta S-\\frac{\\Delta Q}{T}\\geq0<\/span>\u5219\u5b9a\u4e49\u4e86\u5e73\u8861\u6001\uff0c\u7528\u4e8e\u4f5c\u4e3a\u5e73\u8861\u6001\u4e0e\u975e\u5e73\u8861\u6001\u7684\u5224\u636e\u3002\u800c\u5bf9\u4e8e\u81ea\u7531\u80fd\uff0c\u76f4\u8bd1\u4fbf\u662f\u201c\u81ea\u7531\u7684\u80fd\u91cf\u201d\uff0c\u4e2d\u8bd1\u4e2d\u5373\u70ed\u529b\u5b66\u81ea\u7531\u80fd\u53ef\u4ee5\u7528\u4e8e\u5bf9\u5916\u505a\u529f\uff0c\u81ea\u7531\u80fd\u662f\u7cfb\u7edf\u7684\u5185\u80fd\u51cf\u53bb\u4e0d\u80fd\u7528\u4e8e\u505a\u529f\u7684\u80fd\u91cf\uff0c\u56e0\u4e3a\u524d\u9762\u63d0\u5230\u4e86\u71b5\u4ea7\u751f\uff0c\u5e76\u4e0d\u662f\u6240\u6709\u5185\u80fd\u90fd\u53ef\u7528\u4e8e\u505a\u529f\uff1a<\/p>\n

\\displaystyle \\begin{cases}
\nF=U-TS\\\\
\nT=\\frac{\\partial U}{\\partial S}
\n\\end{cases}<\/span><\/p>\n

\u8fd9\u4e5f\u548c\u524d\u9762\u63d0\u5230\u7684\u6700\u5927\u529f\u539f\u7406-\\Delta W\\leq -\\Delta F<\/span>\u8054\u7cfb\u8d77\u6765\u4e86<\/p>\n

Legendre\u53d8\u6362<\/h3>\n

\u7ecf\u5178\u70ed\u529b\u5b66\u7684\u76ee\u6807\u662f\u63cf\u8ff0\u5b8f\u89c2\u6027\u8d28\u7684\u5e73\u5747\u884c\u4e3a\uff0c\u800c\u4e0d\u662f\u6bcf\u4e2a\u7c92\u5b50\u6216\u81ea\u7531\u5ea6\u7684\u5fae\u89c2\u7ec6\u8282\u3002\u9700\u8981\u6ce8\u610f\uff0c\u8fd9\u91cc\u7684Legendre\u53d8\u6362\u662f\u4e00\u4e2a\u5e73\u8861\u6001\u91cc\u624d\u6709\u7684\u6982\u5ff5\u3002\u8003\u8651\u4e00\u4e2a\u7cfb\u7edfsys(S,V,N)<\/span>\uff0c\u5176\u4e2dS<\/span>\u662f\u71b5\uff0cV<\/span>\u662f\u4f53\u79ef\uff0cN<\/span>\u662f\u7c92\u5b50\u6570\uff1b\u53ef\u4ee5\u5f97\u5230\u70ed\u529b\u5b66\u7b2c\u4e8c\u5b9a\u5f8b\uff1a<\/p>\n

\\displaystyle
\n\\begin{cases}
\ndU=TdS-PdV+\\mu dN\\\\
\n(\\frac{\\partial U}{\\partial S})_{V,N}\\equiv T, (\\frac{\\partial U}{\\partial V})_{S,N}\\equiv -P,(\\frac{\\partial U}{\\partial N})_{S,V}\\equiv \\mu
\n\\end{cases}<\/span><\/p>\n

\u5f53\u7136\u8fd9\u5e76\u975e\u552f\u4e00\u7684\u9009\u62e9\uff0c\u7531\u4e8e\u71b5\u96be\u4ee5\u6d4b\u91cf\uff0c\u6211\u4eec\u66f4\u5e0c\u671b\u7528T<\/span>\u4f5c\u4e3a\u53d8\u91cf\u800c\u975eS<\/span>\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u505a\u4e0b\u9762\u7684\u64cd\u4f5c\uff1a<\/p>\n

\\displaystyle \\begin{cases}
\nd[U-TS] = -SdT - PdV + \\mu dN = F(T,V,N)\\\\
\nT = \\left( \\frac{\\partial U}{\\partial S} \\right)_{V,N}
\n\\end{cases}<\/span><\/p>\n

\u4ee5\u6b64\u6765\u5f97\u5230F(T,V,N)<\/span>\uff0c\u540c\u7406\u7c7b\u63a8\u4f60\u53ef\u4ee5\u628aV<\/span>\u7528\u7c7b\u4f3c\u7684\u64cd\u4f5c\u66ff\u6362\u6210P<\/span>\uff0c\u628aN<\/span>\u66ff\u6362\u6210\\mu<\/span>\u7b49\uff0c\u4f46\u662f\u8fd9\u4e9b\u70ed\u529b\u5b66\u53d8\u91cf\u5e76\u4e0d\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u8b6c\u5982\u4f60\u5229\u7528d[U-TS+PV-\\mu N]=-SdT+VdP-Nd\\mu<\/span>\uff0c\u4f60\u4f1a\u53d1\u73b0\u4f60\u5f97\u5230\u4e86Gibbs-Duhem equation\uff0c\u4e5f\u5c31\u662f\u8bf4\u4e0a\u9762\u90a3\u4e00\u5806\u662f\u7b49\u4e8e0\u7684\uff0cd[U-TS+PV-\\mu N]=-SdT+VdP-Nd\\mu=0<\/span>\uff0c\u4e00\u79cd\u8bf4\u6cd5\u662f\u5b8f\u89c2\u70ed\u529b\u5b66\u662f\u5177\u6709\u5e7f\u5ef6\u6027\u7684\uff0c\u800cT,P,\\mu<\/span>\u90fd\u662f\u5f3a\u5ea6\u91cf\uff0cS,V,N<\/span>\u624d\u662f\u5e7f\u5ef6\u91cf\u3002\u8bc1\u660e\u8fd9\u4e2a\u9700\u8981\u5229\u7528\u4e00\u4e2a\u70ed\u529b\u5b66\u52bf\u51fd\u6570\u7684\u6027\u8d28\uff0c\u5373\u70ed\u529b\u5b66\u52bf\u51fd\u6570\u90fd\u662f\u6b27\u62c9\u4e00\u9636\u9f50\u6b21\u51fd\u6570<\/p>\n

\u6b27\u62c9n\u9636\u9f50\u6b21\u51fd\u6570\u662f\u6307\uff1a<\/p>\n

\\displaystyle f(\\lambda x_1,\\lambda x_2,\\cdots,\\lambda x_N)=\\lambda^n f(x_1,x_2,\\cdots,x_N)<\/span><\/p>\n

\u5176\u5177\u6709\u8fd9\u6837\u7684\u6027\u8d28\uff1a<\/p>\n

\\displaystyle nf(x_1,x_2,\\cdots,x_N)=\\sum_{i=1}^{N} x_i\\frac{\\partial f}{\\partial x_i}<\/span><\/p>\n

\u8bc1\u660e\u8fc7\u7a0b\u53ea\u9700\u8981\u5bf9\u5176\u5b9a\u4e49\u5f0f\u4e24\u8fb9\u5bf9\\lambda<\/span>\u505a\u504f\u5bfc\u5373\u53ef\uff1a<\/p>\n

\\displaystyle \\sum_{i=1}^{N}\\frac{\\partial f}{\\partial (\\lambda x_i)}\\frac{\\partial (\\lambda x_i)}{\\partial \\lambda}=n\\lambda^{n-1}f(x_1,x_2,\\cdots,x_N)<\/span><\/p>\n

\u6b64\u65f6\u4ee4\\lambda=1<\/span>\u5373\u5f97\u5230nf(x_1,x_2,\\cdots,x_N)=\\sum_{i=1}^{N} x_i\\frac{\\partial f}{\\partial x_i}<\/span><\/p>\n

\u56de\u5230\u6211\u4eec\u5148\u524d\u8ba8\u8bba\u7684Gibbs-Duhem equation\u4e0a\u6765\uff0cU(S,V,N)=\\frac{\\partial U}{\\partial S}S+\\frac{\\partial U}{\\partial V}V+\\frac{\\partial U}{\\partial N}N=TS-PV+\\mu N<\/span>\uff0c\u81ea\u7136\u800c\u7136\u7684\u6709d[U-TS+PV-\\mu N]=0<\/span>\uff0c\u4e5f\u5373\u8bc1\u660e\u4e86Gibbs-Duhem equation<\/p>\n

\u9700\u8981\u6ce8\u610f\u7684\u662f\u8fd9\u662f\u4e00\u4e2a\u5bf9\u4e8e\u5b8f\u89c2\u70ed\u529b\u5b66\u7cfb\u7edf\u624d\u6210\u7acb\u7684\u65b9\u7a0b\uff0c\u5bf9\u4e8e\u7eb3\u7c73\u7cfb\u7edf\u5982\u5206\u5b50\u5c42\u6b21\u7684\u5fae\u5c0f\u7ed3\u6784\uff0c\u6b64\u65f6\u4e0d\u53ef\u4f7f\u7528\u8be5\u5b9a\u5f8b\uff0c\u65b9\u7a0b\u53f3\u8fb9\u53ef\u80fd\u4e0d\u4e3a0\uff0c\u56e0\u4e3a\u6b64\u65f6\u7684\u70ed\u529b\u5b66\u52bf\u51fd\u6570\u53ef\u80fd\u4e0d\u518d\u662f\u6b27\u62c9\u4e00\u9636\u9f50\u6b21\u51fd\u6570<\/p>\n

\u7edf\u8ba1\u529b\u5b66\u57fa\u7840<\/h3>\n

\u7edf\u8ba1\u529b\u5b66\u662f\u7406\u8bba\u7269\u7406\u7684\u4e00\u4e2a\u5927\u5206\u652f\uff0c\u5b83\u5229\u7528\u6982\u7387\u8bba\u548c\u7edf\u8ba1\u5b66\u7684\u65b9\u6cd5\u6765\u7814\u7a76\u7531\u5927\u91cf\u7c92\u5b50\u7ec4\u6210\u7684\u5b8f\u89c2\u7cfb\u7edf\uff1b\u7531\u4e8e\u8be5\u90e8\u5206\u7684\u8bb2\u8ff0\u9488\u5bf9\u5b8c\u5168\u672a\u5b66\u8fc7\u7edf\u8ba1\u529b\u5b66\u7684\u5b66\u751f\uff0c\u6ca1\u4eceBoltzmann\u90a3\u4e00\u5957\u8bb2\u6cd5\u51fa\u53d1\uff0c\u4e8e\u662f\u8fd9\u90e8\u5206\u76f4\u63a5copy\u4e0a\u8bfe\u677f\u4e66<\/p>\n

\u9996\u5148\uff0c\u5728\u7edf\u8ba1\u529b\u5b66\u4e2d\uff0c\u6211\u4eec\u5f15\u5165\u72b6\u6001\u53d8\u91cf\uff08state variable\uff09\uff0c\u72b6\u6001\u53d8\u91cf\u662f\u6307\u5728\u52a8\u6001\u7cfb\u7edf\u4e2d\uff0c\u53ef\u4ee5\u63cf\u8ff0\u7cfb\u7edf\u6570\u5b66\u72b6\u6001\u7684\u4e00\u7ec4\u53d8\u91cf\uff0c\u5b83\u5e94\u80fd\u786e\u5b9a\u7cfb\u7edf\u672a\u6765\u7684\u6f14\u5316\u884c\u4e3a\uff0c\u8fd9\u91cc\u6211\u4eec\u8bb0\u72b6\u6001\u53d8\u91cf\u4e3a\\omega(t)<\/span><\/p>\n

\u6709\u4e86\u72b6\u6001\u53d8\u91cf\u540e\uff0c\u6211\u4eec\u5c31\u6709\u4e86\u6982\u7387\uff0c\u5199\u4e3ap(\\omega)d\\omega<\/span>\uff0c\u5176\u4e2dp(\\omega)<\/span>\u662f\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff0c\u5728\u5168\u7a7a\u95f4\u4e2d\uff0c\u5e94\u5f53\u6709\\int p(\\omega)d\\omega=1<\/span>\uff0c\u4e4b\u540e\u6211\u4eec\u6709\u53ef\u89c2\u5bdf\u91cf\uff08\u67d0\u4e00\u4e2a\u5177\u4f53\u7684\u7269\u7406\u91cf\uff09\uff1aA(\\omega)<\/span>\uff0c\u5219\u5176\u5e73\u5747\u503c< A >=\\int A(\\omega)p(\\omega)d\\omega<\/span>\uff0c\u5176\u5b9e\u8fd9\u5c31\u662f\u7edf\u8ba1\u529b\u5b66\u7684\u6838\u5fc3\uff0c\u5b9a\u4e49\u72b6\u6001\u53d8\u91cf\uff0c\u83b7\u5f97\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff0c\u7136\u540e\u62ff\u53bb\u7b97\u4f60\u9700\u8981\u7b97\u7684\u7269\u7406\u91cf<\/p>\n

\u6211\u4eec\u843d\u5b9e\u5230\u5177\u4f53\u7684physical quantities\u6765\u8bb2\uff0c\u5728\u4e00\u4e2a\u5b64\u7acb\u7684\u7cfb\u7edf\u4e2d\uff0c\u54c8\u5bc6\u987f\u91cf\u901a\u5e38\u4ee3\u8868\u7cfb\u7edf\u7684\u5185\u80fd\uff0c\u5373<\\hat{H}(\\omega)>=U<\/span>\uff0c\u6ce8\u610f\uff0c\u8fd9\u91cc\u7684\\hat{H}<\/span>\u5e76\u975e\u54c8\u5bc6\u987f\u7b97\u5b50\uff0c\u53ea\u662f\u5199\u6210\u8fd9\u6837\u65b9\u4fbf\u533a\u5206\uff0c\u6839\u636e\u73bb\u5c14\u5179\u66fc\u71b5\u516c\u5f0f\u6709S=<-k_B\\log{p(\\omega)}><\/span>\uff0c\u8fd9\u91cc\u7684\\log<\/span>\u5176\u5b9e\u662f\\ln<\/span>\uff0c\u4e3a\u4e86\u548c\u677f\u4e66\u4e00\u81f4\u6211\u5c31\u6ca1\u6709\u66f4\u6539\u3002\u8fd9\u91cc\u7684S<\/span>\u4fbf\u662fGibbs-Shannon entropy\uff0c\u800c\\hat{s}=-k_B\\log{p(\\omega)}<\/span>\u5219\u662fStochastic entropy<\/p>\n

\u63a5\u7740\u6709F=U-TS=<\\hat{H}(\\omega)>+k_BT<\\log{p(\\omega)}>=<\\hat{H}(\\omega)+k_BT\\log{p(\\omega)}><\/span><\/p>\n

\u8fd9\u4e00\u5806\u4e1c\u897f\u8fd8\u53ef\u4ee5\u5199\u6210<\\hat{f}(\\omega)><\/span>\uff0c\u5176\u4e2d\\hat{f}(\\omega)=\\hat{H}(\\omega)+k_BT\\log{p(\\omega)}=\\hat{H}(\\omega)-T\\hat{s}(\\omega)<\/span>\uff0c\u53c8\u79f0Stochastic free energy<\/p>\n

\u63a5\u7740\u8bb2\u4e86Free energy\uff08non-equilibrium\uff09\uff0c\u5176\u53ef\u4ee5\u6839\u636e\u4e0b\u9762\u7684\u516c\u5f0f\u8fdb\u884c\u8ba1\u7b97\uff1a<\/p>\n

\\displaystyle F=\\int d\\omega p(\\omega)[\\beta \\hat{H}(\\omega)+\\log{p(\\omega)}]k_BT\\\\
\n=k_BT\\int d\\omega p(\\omega)\\log\\frac{p(\\omega)}{\\frac{1}{Z}e^{-\\beta\\hat{H}(\\omega)}}-k_BT\\log{Z}\\\\
\n=k_BT\\int d\\omega p(\\omega)\\log{\\frac{p(\\omega)}{p_s(\\omega)}}-k_BT\\log{Z}\\\\
\n=k_BTD_{KL}(p(\\omega)\/\/p_s(\\omega))-k_BT\\log{Z}<\/span><\/p>\n

\u5176\u4e2d\\beta=\\frac{1}{k_BT}<\/span>\uff0c\u53e6\u5916\u6709Z=\\int d\\omega e^{-\\beta\\hat{H}(\\omega)}<\/span>\uff0c\u5e76\u4e14\\frac{1}{Z}e^{-\\beta\\hat{H}(\\omega)}=p_s(\\omega)<\/span>\uff0c\u4e14\\int d\\omega p_s(\\omega)=1<\/span><\/p>\n

\u800cD_{KL}<\/span>\u662fKullback-Leibler Divergence\uff0c\u4e5f\u53ef\u4ee5\u53ebRelative entropy\uff0c\u5176\u5177\u4f53\u5b9a\u4e49\u4e3a\uff0c\u7ed9\u5b9a\u4e24\u4e2a\u6982\u7387\u5bc6\u5ea6\u51fd\u6570PDF\uff0cp_1(\\omega)<\/span>\u548cp_2(\\omega)<\/span>\uff0c\u5176\u76f8\u5bf9\u71b5\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n

\\displaystyle D_{KL}(p_1(\\omega)\/\/p_2(\\omega))=<\\log{\\frac{p_1(\\omega)}{p_2(\\omega)}}>_{p_1(\\omega)}\\\\
\n=\\int d\\omega p_1(\\omega)\\log{\\frac{p_1(\\omega)}{p_2(\\omega)}}<\/span><\/p>\n

\u4e0b\u6807\u5904\u7684p_1(\\omega)<\/span>\u4ee3\u8868\u4ee5p_1<\/span>\u4f5c\u4e3a\u6982\u7387\u6d4b\u5ea6\u3002\u6700\u540e\u81ea\u7531\u80fd\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n

\\displaystyle F-(-k_BT\\log{Z})=D_{KL}(p(\\omega)\/\/p_s(\\omega))<\/span><\/p>\n

\u5176\u4e2dF_{eq}=(-k_BT\\log{Z})<\/span>\u4e3a\u5e73\u8861\u6001\u81ea\u7531\u80fd\uff0c\u800cD_{KL}(p(\\omega)\/\/p_s(\\omega))<\/span>\u4e3a\u51f8\u51fd\u6570\uff0c\u4e14D_{KL}(p(\\omega)\/\/p_s(\\omega))\\geq0<\/span><\/p>\n

\n

\u63a5\u4e0b\u6765\u5148review\u4e00\u4e0b\u4e4b\u524d\u7684\u4e1c\u897f\uff0c\u4e0a\u6b21\u6211\u4eec\u8bf4\u5230\u5bf9\u4e8eNano system\uff0c\u5b8f\u89c2\u7684\u7cfb\u7edf\u5e7f\u5ef6\u6027\u53ef\u80fd\u4f1a\u4e22\u5931\uff0c\u7136\u540e\u6211\u4eec\u63a5\u7740\u5f00\u542f\u4e86\u7edf\u8ba1\u529b\u5b66\u7684\u90e8\u5206<\/p>\n

\u5bf9\u4e8e\u7edf\u8ba1\u529b\u5b66\uff0c\u9996\u5148\u7b2c\u4e00\u6b65\u662f\u786e\u5b9a\u72b6\u6001\u53d8\u91cf\uff0c\u8fd9\u91cc\u4e3e\u4e24\u4e2a\u4f8b\u5b50\uff0c\u7ecf\u5178\uff1a\u76f8\u7a7a\u95f4\u4e2d\u7684\u4efb\u610f\u4e00\u70b9\uff1b\u91cf\u5b50\uff1a\u6ce2\u51fd\u6570\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u72b6\u6001\u7a7a\u95f4\u6709\u8fde\u7eed\u548c\u79bb\u6563\u4e24\u79cd\u533a\u522b\uff0c\u9700\u8981\u5206\u522b\u5904\u7406<\/p>\n

\u5728\u7ed9\u5b9a\u72b6\u6001\u53d8\u91cf\\omega<\/span>\u540e\uff0c\u6211\u4eec\u5c31\u6709\u4e86Probability\uff1a<\/p>\n

\\displaystyle p(\\omega)=\\begin{cases}
\nprobability,discrete \\ space\\\\
\nprobability \\ density \\ function,continuous \\ space
\n\\end{cases}<\/span><\/p>\n

\u6b64\u5904\u53ef\u4ee5\u770b\u51fa\uff0c\u82e5\u662f\u8fde\u7eed\u7a7a\u95f4\uff0c\u90a3\u6b64\u65f6\u67d0\u4e00\u70b9\u7684p(\\omega)=0<\/span>\uff0c\u662f\u65e0\u610f\u4e49\u7684<\/p>\n

\u4e4b\u540e\u7ed9\u5b9a\u4e00\u4e2a\u53ef\u89c2\u6d4b\u91cf\uff1a<\\hat{A}(\\omega)><\/span>\uff0c\u90a3\u4e48\u663e\u7136\u5bf9\u4e8e\u968f\u673a\u53d8\u91cf\\hat{A}(\\omega)<\/span>\u6709\uff1a<\/p>\n

\\displaystyle <\\hat{A}(\\omega)>=\\begin{cases}
\n\\sum_{\\omega}p(\\omega)\\hat{A}(\\omega),discrete\\\\
\n\\int d\\omega p(\\omega)\\hat{A}(\\omega),continuous
\n\\end{cases}<\/span><\/p>\n

\u518d\u6b21\u56de\u5230\u5177\u4f53\u7684physical quantities\u4e0a\u6765\uff0c\u5728\u4e00\u4e2a\u5b64\u7acb\u7684\u7cfb\u7edf\u4e2d\uff0c\u54c8\u5bc6\u987f\u91cf\u901a\u5e38\u4ee3\u8868\u7cfb\u7edf\u7684\u5185\u80fd\uff0c\u5373<\\hat{H}(\\omega)>=U<\/span>\uff0c\u71b5\u5219\u53ef\u4ee5\u5199\u6210S=<\\hat{s}(\\omega)><\/span>\uff0c\u5176\u4e2d\\hat{s}<\/span>\u662fstochastic entropy\uff0c\u6709\\hat{s}=-k_B\\log{p(\\omega)}<\/span><\/p>\n

\u8fd9\u91cc\u5bf9\\hat{s}<\/span>\u505a\u4e00\u70b9\u989d\u5916\u7684\u89e3\u91ca\uff0c\u71b5\u662f\u4fe1\u606f\u7684\u5ea6\u91cf\uff0c\u4e00\u4e2a\u4e8b\u4ef6\u53d1\u751f\u7684\u6982\u7387\u8d8a\u5c0f\u5219\u5176\u6240\u8574\u542b\u7684\u4fe1\u606f\u91cf\u8d8a\u5927\uff0c\u5177\u4f53\u5199\u4e3a\uff1a<\/p>\n

\\displaystyle \\hat{s}=-k_B\\log p(\\omega)=k_B\\log{\\frac{1}{p(\\omega)}}<\/span><\/p>\n

\u663e\u7136\u6709p(\\omega)<\/span>\u8d8a\u5c0f\uff0c\\hat{s}<\/span>\u8d8a\u5927\uff0c\u800c\u5bf9\u4e8e\u73bb\u5c14\u5179\u66fc\u71b5\u53d8\uff0c\u5219\u6709\uff1a<\/p>\n

\\displaystyle S_{Boltzmann}=k_B\\log{W}=k_B\\log{\\frac{1}{\\frac{1}{W}}}<\/span><\/p>\n

\u5176\u4e2d\\frac{1}{W}=p<\/span>\uff0c\u5982\u679c\u4e00\u4e2a\u5fae\u89c2\u72b6\u6001\u7684\u6982\u7387\u5f88\u5c0f\uff0c\u610f\u5473\u7740\u8be5\u72b6\u6001\u8d8a\u4e0d\u5e38\u89c1\uff0c\u7cfb\u7edf\u8d8a\u53ef\u80fd\u5904\u4e8e\u5176\u4ed6\u72b6\u6001\uff0c\u8fd9\u589e\u52a0\u4e86\u7cfb\u7edf\u7684\u6df7\u4e71\u7a0b\u5ea6\uff0c\u56e0\u6b64\u71b5\u5c31\u8d8a\u5927\u3002\u53cd\u4e4b\uff0c\u5982\u679c\u4e00\u4e2a\u5fae\u89c2\u72b6\u6001\u7684\u6982\u7387\u5f88\u5927\uff0c\u610f\u5473\u7740\u8be5\u72b6\u6001\u8d8a\u5e38\u89c1\uff0c\u7cfb\u7edf\u8d8a\u53ef\u80fd\u7a33\u5b9a\u5730\u5904\u4e8e\u8be5\u72b6\u6001\uff0c\u71b5\u5c31\u8d8a\u5c0f\u3002\u6bd4\u5982\u79bb\u6563\u7684\u60c5\u51b5\u4e0b\uff1a<\/p>\n

\\displaystyle <k_B\\log{W}>_p=\\sum_{\\omega}p(\\omega)k_B\\log{W}<\/span><\/p>\n

\u6b64\u5916\uff0c\u5173\u4e8e\u4e3a\u4ec0\u4e48\u71b5\u516c\u5f0f\u4e2d\u542b\u6709\\log<\/span>\u51fd\u6570\uff0c\u6211\u4eec\u8fdb\u884c\u4e86\u5982\u4e0b\u7684\u8ba8\u8bba\uff1a\u7528\u8bed\u8a00\u89e3\u91ca\u5373\u662f\u71b5\u7684\u5e7f\u5ef6\u6027\uff08\u53ef\u52a0\u6027\uff09\uff1b\u5728\u7269\u7406\u4e0a\uff0c\u6211\u4eec\u5148\u4ee4S=k_Bf(\\omega)<\/span>\uff0c\u5176\u4e2df(\\omega)<\/span>\u662f\u4e00\u4e2a\u672a\u77e5\u7684\u51fd\u6570\uff0c\u6b64\u65f6\u8003\u8651\u4e00\u4e2a\u7cfb\u7edfsys<\/span>\uff0c\u5176\u5177\u6709\u5fae\u89c2\u72b6\u6001\u6570W<\/span>\uff0c\u6b64\u65f6\u5c06\u8be5\u7cfb\u7edf\u6cbf\u7740\u4e2d\u95f4\u5207\u6210\u4e24\u534a\uff0c\u5206\u6210\u7cfb\u7edf1\u548c\u7cfb\u7edf2\uff0c\u5e76\u5206\u522b\u5177\u6709\u5fae\u89c2\u72b6\u6001\u6570W_1<\/span>\u548cW_2<\/span>\uff0c\u4e8c\u8005\u53ea\u5728\u754c\u9762\u5904\u6709\u4ea4\u6362\uff0c\u4ee4\u8be5\u70ed\u529b\u5b66\u7cfb\u7edf\u4e3a\u65e0\u7a77\u5927\uff0c\u6b64\u65f6\u4ea4\u6362\u53ef\u4ee5\u5ffd\u7565\uff0c\u90a3\u4e48\u4e8c\u8005\u7684\u5fae\u89c2\u72b6\u6001\u6570\u5e94\u8be5\u6ee1\u8db3\uff1a<\/p>\n

\\displaystyle W_{sys}=W_1\\cdot W_2<\/span><\/p>\n

\u800c\u71b5\u5e94\u8be5\u6ee1\u8db3:<\/p>\n

\\displaystyle S_{sys}=S_1+S_2<\/span><\/p>\n

\u800c\u53ea\u6709\\log<\/span>\u51fd\u6570\u5177\u6709\u8fd9\u6837\u7684\u6027\u8d28\uff0c\u56e0\u4e3a\uff1a<\/p>\n

\\displaystyle S_{sys}=S_{1+2}=k_B\\log{W}=k_B\\log{W_1W_2}\\\\
\n=k_B\\log{W_1}+k_B\\log{W_2}=S_1+S_2<\/span><\/p>\n

\u63a5\u7740\u662f\u81ea\u7531\u80fdF=U-TS<\/span>\uff0cstochastic free energy\u53ef\u4ee5\u5199\u6210\\hat{f}(\\omega)=\\hat{H}(\\omega)-T\\hat{s}(\\omega)<\/span>\uff0c\u7136\u540e\u6709\uff1a<\/p>\n

\\displaystyle F=<\\hat{f}(\\omega)>=<\\hat{H}(\\omega)-T\\hat{s}(\\omega)><\/span><\/p>\n

\u8fd8\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n

\\displaystyle F=k_BTD_{KL}(p(\\omega)\/\/p_s(\\omega))+F_{eq}<\/span><\/p>\n

\u5176\u4e2dF_{eq}=-k_BT\\log{Z}<\/span>\uff0c\u4e3a\u5e73\u8861\u6001\u81ea\u7531\u80fd\uff0cZ<\/span>\u4e3a\u914d\u5206\u51fd\u6570\uff0cZ=\\int d\\omega e^{-\\beta \\hat{H}(\\omega)}<\/span>\uff0c\u5e76\u4e14\u6709\uff1a<\/p>\n

\\displaystyle D_{KL}(p(\\omega)\/\/p_s(\\omega))=\\int d\\omega p(\\omega)\\log{\\frac{p(\\omega)}{p_{s}(\\omega)}}<\/span><\/p>\n

\u800cp_s(\\omega)=\\frac{1}{Z}e^{-\\beta\\hat{H}(\\omega)}<\/span>\uff0c\u4e5f\u5373Gibbs-Boltzmann probability<\/p>\n

KL\u6563\u5ea6<\/h3>\n

\u524d\u9762\u6211\u4eec\u63d0\u5230\u8fc7\uff1a<\/p>\n

\\displaystyle k_BTD_{KL}(p(\\omega)\/\/p_s(\\omega))=F-F_{eq}<\/span><\/p>\n

\u63a5\u7740\u8be6\u7ec6\u8ba8\u8bba\u4e00\u4e0bKL\u6563\u5ea6\uff0cKL\u6563\u5ea6\u8fd8\u5177\u6709\u4e00\u4e9b\u522b\u7684\u6027\u8d28\uff0c\u6bd4\u5982\u975e\u8d1f\u6027\uff0c\u5373\uff1a<\/p>\n

\\displaystyle D_{KL}(p(\\omega)\/\/p_s(\\omega))\\geq 0<\/span><\/p>\n

\u82e5\u6211\u4eec\u628aF<\/span>\u5199\u6210p(\\omega)<\/span>\u7684\u6cdb\u51fd\uff0c\u5219\u6709F[p(\\omega)]-F_{eq}=D_{KL}(p(\\omega)\/\/p_{eq})<\/span>\uff0c\u8ba4\u771f\u89c2\u5bdf\u4e00\u4e0b\u53ef\u4ee5\u53d1\u73b0\u6211\u4eec\u5f97\u5230\u4e86\u70ed\u529b\u5b66\u4e2d\u7684\u6781\u5c0f\u539f\u7406\uff1a\u6700\u5c0f\u81ea\u7531\u80fd\u539f\u7406\u3002\u6211\u4eec\u8fdb\u884c\u7b80\u5355\u7684\u8bc1\u660e\uff0c\u9996\u5148\uff0cD_{KL}<\/span>\u662f\u51f8\u51fd\u6570\uff0c\u6211\u4eec\u5c06\u5176\u6309\u7167\u6570\u5b66\u5b9a\u4e49\u5199\u4e3a\uff1a<\/p>\n

\\displaystyle D_{KL}(p(\\omega)\/\/p_{eq}(\\omega))=\\mathbb{E}^{p(\\omega)}[\\log{\\frac{p(\\omega)}{p_{eq}(\\omega)}}]\\\\
\n= \\mathbb{E}^{p(\\omega)}[-\\log{\\frac{p_{eq}(\\omega)}{p(\\omega)}}]<\/span><\/p>\n

\u5176\u4e2d\\mathbb{E}<\/span>\u4ee3\u8868\u6570\u5b66\u671f\u671b\uff0c\u4e0a\u6807\u4ee3\u8868\u4ee5p(\\omega)<\/span>\u4e3a\u6982\u7387\u6d4b\u5ea6\u3002\u7531\u4e8e-\\log{x}<\/span>\u662f\u51f8\u51fd\u6570\uff0c\u56e0\u6b64\u6211\u4eec\u5229\u7528\u7434\u751f\u4e0d\u7b49\u5f0f\uff0c\u6709\uff1a<\/p>\n

\\displaystyle \\mathbb{E}[-\\log{x}]\\geq -\\log{\\mathbb{E}(x)}<\/span><\/p>\n

\u8fdb\u800c\u53ef\u4ee5\u5f97\u5230\uff1a<\/p>\n

\\displaystyle \\mathbb{E}^{p(\\omega)}[-\\log{\\frac{p_{eq}(\\omega)}{p(\\omega)}}]\\geq -\\log{\\mathbb{E}^{p(\\omega)}[\\frac{p_{eq}(\\omega)}{p(\\omega)}]}<\/span><\/p>\n

\u5176\u4e2d\u7b49\u4e8e\u53f7\u53d6\u5728\uff1a<\/p>\n

\\displaystyle -\\log{\\int d\\omega p(\\omega)\\frac{p_{eq}(\\omega)}{p(\\omega)}}=-\\log{\\int d\\omega p_{eq}(\\omega)}=0<\/span><\/p>\n

\u8fd9\u91cc\u5229\u7528\u4e86\\int d\\omega p_{eq}(\\omega)=1<\/span>\uff0c\u8fd9\u4e5f\u5c31\u8bf4\u660e\u4e86\uff0c\u65e0\u8bba\u7cfb\u7edf\u4ece\u54ea\u4e00\u4e2a\u6001\u51fa\u53d1\uff0c\u6700\u540e\u4f1a\u6f14\u5316\u5230\u552f\u4e00\u7684\u5e73\u8861\u6001\u3002\u8fd9\u91cc\u5176\u5b9e\u8fd8\u5f15\u51fa\u4e86Lyapunov function\uff0c\u4e0d\u8fc7\u6b64\u5904\u4e0d\u518d\u8fdb\u884c\u6df1\u5165\u8ba8\u8bba\u4e86<\/p>\n

\u52a8\u529b\u5b66<\/h3>\n

\u63a5\u4e0b\u6765\u6211\u4eec\u7814\u7a76\u7cfb\u7edf\u7684\u52a8\u529b\u5b66\uff0c\u5373\u7cfb\u7edf\u4ece\u975e\u5e73\u8861\u6001\u5230\u7a33\u6001\uff08Steady state\uff09\u7684\u8fc7\u7a0b\u3002\u8003\u8651\u4e00\u4e2a\u5177\u6709\u904d\u5386\u6027\uff08Ergodicity\uff09\u7684\u7cfb\u7edf\uff0c\u6211\u4eec\u9700\u8981\u6982\u7387p(\\omega,t)<\/span>\u7684\u6f14\u5316\u65b9\u7a0b\uff0c\u5176\u4e2d\u8be5\u7cfb\u7edf\u5168\u6982\u7387\u5b88\u6052\uff0c\u5373\uff1a<\/p>\n

\\displaystyle \\sum_{\\omega}p(\\omega,t)=1<\/span><\/p>\n

\u800c\u6240\u8c13\u904d\u5386\u6027\uff0c\u5728\u6570\u5b66\u4e2d\u6307\u52a8\u529b\u7cfb\u7edf\u7684\u4e00\u79cd\u6027\u8d28\uff0c\u5b83\u8868\u660e\u7cfb\u7edf\u5728\u957f\u65f6\u95f4\u5185\u53ef\u4ee5\u8bbf\u95ee\u5176\u72b6\u6001\u7a7a\u95f4\u7684\u4efb\u610f\u4e00\u90e8\u5206\uff0c\u8868\u73b0\u4e3a\u7cfb\u7edf\u6027\u8d28\u7684\u4e0d\u53d8\u6027\u3002\u63a5\u4e0b\u6765\u6211\u4eec\u5f15\u5165\u4e3b\u65b9\u7a0b\uff1a<\/p>\n

\\displaystyle \\frac{d}{dt}p_{\\omega}(t)=\\sum_{\\omega'}W_{\\omega\\omega'}p_{\\omega'}(t)<\/span><\/p>\n

\u8fd9\u91cc\u5047\u8bbe\u7cfb\u7edf\u7684\u672a\u6765\u53d1\u5c55\u4ec5\u53d6\u51b3\u4e8e\u5176\u5f53\u524d\u72b6\u6001\uff0c\u800c\u4e0d\u662f\u5176\u8fc7\u53bb\u7684\u5386\u53f2\uff0c\u5373\u505a\u4e86\u9a6c\u5c14\u53ef\u592b\u8fd1\u4f3c\uff0c\u53e6\u5916\u4e3a\u4e86\u7b80\u5355\u6211\u4eec\u4ec5\u8003\u8651\u4e86\u79bb\u6563\u7684\u60c5\u5f62\uff0c\u65b9\u7a0b\u4e2dW_{\\omega\\omega'}<\/span>\u662f\u8f6c\u79fb\u901f\u7387\u77e9\u9635\uff0c\u7ed9\u5b9a\u65f6\u95f4t<\/span>\uff0c\u7cfb\u7edf\u7531\\omega'<\/span>\u6001\u6f14\u5316\u5230\\omega<\/span>\u6001\uff0c\u8fd9\u5176\u4e2d\uff0c\u7531\u4e8e\\sum_{\\omega}p(\\omega,t)=1<\/span>\uff0c\u56e0\u6b64\u5bf9\u4e8e\u4efb\u610fp_{\\omega'}(t)<\/span>\uff0c\u53ef\u4ee5\u5f97\u5230\uff1a<\/p>\n

\\displaystyle\\frac{d}{dt}\\sum_{\\omega}p_{\\omega}(t)=\\sum_{\\omega'}(\\sum_{\\omega}W_{\\omega\\omega'})p_{\\omega'}(t)=0<\/span><\/p>\n

\u663e\u7136\u6709\\sum_{\\omega}W_{\\omega\\omega'}=0<\/span>\uff0c\u8fd9\u8bf4\u660e\u4e86\u8be5\u8f6c\u79fb\u901f\u7387\u77e9\u9635\u5728\u4efb\u4e00\u5217\u7684\u52a0\u548c\u4e3a0\uff0c\u7ee7\u7eed\u5199\u5373\uff1a<\/p>\n

\\displaystyle W_{\\omega'\\omega'}+\\sum_{\\omega\\neq\\omega'}W_{\\omega\\omega'}=0<\/span><\/p>\n

\u6700\u540e\u6709W_{\\omega'\\omega'}=-\\sum_{\\omega\\neq\\omega'}W_{\\omega\\omega'}<\/span>\uff0c\u4e5f\u5373\u8be5\u77e9\u9635\u7684\u4efb\u610f\u4e00\u4e2a\u5bf9\u89d2\u5143\u7b49\u4e8e\u8d1f\u7684\u8be5\u5217\u5176\u4ed6\u5143\u7d20\u7684\u548c\u3002\u7269\u7406\u4e0a\uff0c\u8fd9\u8868\u793a\u5bf9\u4e8e\u4efb\u610f\u72b6\u6001\\omega'<\/span>\uff0c\u5176\u6982\u7387\u6d41\u5931\u901f\u7387\uff08\u7531\u5bf9\u89d2\u5143W_{\\omega'\\omega'}<\/span>\u63cf\u8ff0\uff09\u5728\u6570\u503c\u4e0a\u7b49\u4e8e\u4ece\u8be5\u72b6\u6001\u8f6c\u79fb\u5230\u6240\u6709\u5176\u4ed6\u72b6\u6001\u7684\u603b\u901f\u7387\uff08\u7531\u975e\u5bf9\u89d2\u5143\u4e4b\u548c\\sum_{\\omega\\neq\\omega'}W_{\\omega\\omega'}<\/span>\u7ed9\u51fa\uff09<\/p>\n

\u4e3b\u65b9\u7a0b\u7684\u89e3\u7684\u5f62\u5f0f\u4e3a\uff1a<\/p>\n

\\displaystyle p_{\\omega}(t)=\\sum_{\\lambda}C_{\\lambda}e^{-\\lambda t}\\phi_{\\lambda}(\\omega)<\/span><\/p>\n

\u5176\u4e2d\\lambda<\/span>\u4e3a\u7279\u5f81\u503c\u4e14\\lambda >0<\/span>\uff0c\\phi_{\\lambda}(\\omega)<\/span>\u5bf9\u5e94\u7279\u5f81\u503c\\lambda<\/span>\u7684\u7279\u5f81\u51fd\u6570\uff0c\u800cC_{\\lambda}<\/span>\u4e3a\u5c55\u5f00\u7cfb\u6570\uff0c\u7ee7\u7eed\u63a8\u5bfc\u8fd8\u53ef\u4ee5\u5f97\u5230\uff1a<\/p>\n

\\displaystyle \\sum_{\\omega'}W_{\\omega\\omega'}\\phi_{\\lambda}(\\omega')=-\\lambda\\phi_{\\lambda}(\\omega)<\/span><\/p>\n

\u8fd9\u91cc\u5982\u679c\u6211\u4eec\u8003\u8651\u5177\u4f53\u7684\u4f8b\u5b50\uff0c\u6bd4\u5982\u8003\u8651\\omega<\/span>\u4ee3\u8868\u4f4d\u7f6e\uff0c\u5373\\omega\\rightarrow\\vec{x}<\/span>\uff0c\u5219\u5bf9\u4e8e\u8fde\u7eed\u60c5\u51b5\uff0c\u6211\u4eec\u6709\uff1a<\/p>\n

\\displaystyle \\frac{\\partial}{\\partial t}p(\\vec{x},t)=-\\nabla\\cdot\\vec{J}(\\vec{x},t)<\/span><\/p>\n

\u5176\u4e2d\u6d41\u77e2\u91cf\u5199\u6210\uff1a<\/p>\n

\\displaystyle\\vec{J}(\\vec{x},t)=-D\\nabla p(\\vec{x},t)+\\vec{u}(\\vec{x},t)p(\\vec{x},t)<\/span><\/p>\n

\u5982\u679c\u53ea\u770b\u6d41\u77e2\u91cf\u7684\u7b2c\u4e00\u90e8\u5206\uff0c\u5c31\u5f97\u5230\u4e86\u6269\u6563\u65b9\u7a0b\uff0c\u6216\u8005\u8bf4Fick\u5b9a\u5f8b\uff0c\u8fd9\u90e8\u5206\u4e5f\u79f0\u4e3a\u6269\u6563\u6d41\uff0cD<\/span>\u4e3a\u6269\u6563\u7cfb\u6570\u3002\u800c\u6d41\u77e2\u91cf\u7684\u7b2c\u4e8c\u90e8\u5206\u4e3a\u6f02\u79fb\u6d41\uff0c\u4e24\u90e8\u5206\u4e00\u8d77\u8003\u8651\uff0c\u5c31\u5f97\u5230\u4e86Fokker\u2013Planck equation\uff0c\u4e3a\u4e86\u7814\u7a76Local velocity\uff0c\u6211\u4eec\u628a\u6d41\u77e2\u91cf\u5199\u6210\uff1a<\/p>\n

\\displaystyle\\vec{J}(\\vec{x},t)=p(\\vec{x},t)\\vec{\\gamma}(\\vec{x},t)\\\\
\n=p(\\vec{x},t)[\\vec{u}(\\vec{x},t)-D\\frac{1}{p(\\vec{x},t)}\\nabla p(\\vec{x},t)]\\\\
\n=p(\\vec{x},t)[\\vec{u}(\\vec{x},t)-D\\nabla\\log{p(\\vec{x},t)}]<\/span><\/p>\n

\u56e0\u6b64\u6709\uff1a<\/p>\n

\\displaystyle\\vec{\\gamma}(\\vec{x},t)=\\vec{u}(\\vec{x},t)-D\\nabla\\log{p(\\vec{x},t)}<\/span><\/p>\n

\u82e5\\vec{u}(\\vec{x},t)<\/span>\u4e0d\u542b\u65f6\uff0c\u5373\\vec{u}(\\vec{x},t)=\\vec{u}(\\vec{x})<\/span>\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b83\u5199\u6210\\vec{u}(\\vec{x})=\\kappa\\vec{f}(\\vec{x})=-\\kappa\\nabla\\hat{H}(\\vec{x})<\/span>\u7684\u5f62\u5f0f\uff0c\u5176\u4e2d\\vec{f}=-\\nabla\\hat{H}(\\vec{x})<\/span>\uff0c\\kappa<\/span>\u4e3a\u8fc1\u79fb\u7387<\/p>\n

\u7ee7\u7eed\u63a8\u5bfc\u6211\u4eec\u6709\uff1a<\/p>\n

\\displaystyle \\vec{\\gamma}(\\vec{x},t)=-\\kappa\\nabla\\hat{H}(\\vec{x})-D\\nabla\\log{p(\\vec{x},t)}\\\\
\n=-\\kappa\\nabla[\\hat{H}(\\vec{x})+\\frac{D}{\\kappa}\\log{p(\\vec{x},t)}]<\/span><\/p>\n

\u6b64\u65f6\u82e5\u4ee4D=k_BT\\kappa=\\frac{k_BT}{\\xi}<\/span>\uff0c\u5176\u4e2d\\xi=\\kappa^{-1}<\/span>\uff0c\u5219\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u7231\u56e0\u65af\u5766\u5173\u7cfb\uff08Einstein relation\uff09\u5982\u4e0b\uff1a<\/p>\n

\\displaystyle \\vec{\\gamma}(\\vec{x},t)=-\\kappa\\nabla[\\hat{H}(\\vec{x})+k_BT\\log{p(\\vec{x},t)}]\\\\
\n=-\\kappa\\nabla\\hat{f}(\\vec{x},t)<\/span><\/p>\n

\\hat{f}<\/span>\u5373\u524d\u9762\u63d0\u5230\u7684Stochastic free energy\uff0c\u5176\u4e2dD<\/span>\u4e3a\u6269\u6563\u7cfb\u6570\uff0c\u800c\\xi<\/span>\u4e3a\u6469\u64e6\u7cfb\u6570\uff0c\u5f97\u5230\u7231\u56e0\u65af\u5766\u5173\u7cfb\u7684\u524d\u63d0\u8981\u6c42\u662ft\\rightarrow\\infty<\/span>\u5e76\u4e14\\vec{\\gamma}(\\vec{x},t)=0<\/span>\uff0c\u4e5f\u5373-\\hat{H}(\\vec{x})=\\frac{D}{\\kappa}\\log{p(\\vec{x},t)}<\/span>\uff0c\u6216\u8005\u5199\u6210-\\frac{\\kappa}{D}\\hat{H}(\\vec{x})=\\log{p(\\vec{x},t)}<\/span>\uff0c\u6b64\u65f6p(\\vec{x},t)=e^{-\\frac{\\kappa}{D}\\hat{H}(\\vec{x})}<\/span><\/p>\n

\u5176\u5b9e\u4ed4\u7ec6\u56de\u987e\u8fd9\u90e8\u5206\u63a8\u5bfc\u8fd8\u53ef\u4ee5\u770b\u51fa\uff0c\u6ca1\u6709\u5916\u754c\u9a71\u52a8\u7684\u60c5\u51b5\u4e0b\uff0c\u7cfb\u7edf\u4f1a\u6f14\u5316\u5230\u73bb\u5c14\u5179\u66fc\u5206\u5e03\uff0c\u5176\u5b9e\u5c31\u662f\u89e3Fokker\u2013Planck equation\u7684\u5b9a\u6001\u89e3\uff0c\u5f88\u663e\u7136\uff0c\u5e73\u8861\u6001\u65f6\uff0c\u5bc6\u5ea6\u5206\u5e03\u4e0d\u663e\u542b\u65f6\u95f4<\/p>\n

\u8fd9\u91cc\u6211\u81ea\u5df1\u518d\u5570\u55e6\u4e24\u53e5\uff0c\u5728\u7269\u7406\u5b66\u7684\u8bf8\u591a\u9886\u57df\u4e2d\uff0c\u6211\u4eec\u5e38\u4f1a\u89c2\u5bdf\u5230\u4e00\u79cd\u6781\u5177\u5171\u6027\u7684\u6570\u5b66\u7ed3\u6784\uff1a\u8bb8\u591a\u6838\u5fc3\u7684\u7269\u7406\u77e2\u91cf\u573a\uff0c\u90fd\u53ef\u4ee5\u901a\u8fc7\u67d0\u4e2a\u6807\u91cf\u51fd\u6570\u7684\u8d1f\u68af\u5ea6\u6765\u5b9a\u4e49\u3002\u6bd4\u5982F=-\\nabla V<\/span>\uff0c\u4ea6\u6216\u662fE=-\\nabla \\phi<\/span>\uff0c\u4ee5\u53ca\u8fd9\u91cc\u7684\\vec{f}=-\\nabla\\hat{H}(\\vec{x})<\/span>\uff0c\u7b2c\u4e00\u4e2a\u516c\u5f0f\u63ed\u793a\u4e86\u4fdd\u5b88\u529b\u573a\u4e2d\u52bf\u80fd\u4e0e\u4fdd\u5b88\u529b\u7684\u5173\u7cfb\uff0c\u7b2c\u4e8c\u4e2a\u516c\u5f0f\u63ed\u793a\u4e86\u7535\u573a\u4e2d\u7535\u573a\u5f3a\u5ea6\u4e0e\u9759\u7535\u52bf\u7684\u5173\u7cfb\uff0c\u800c\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u78c1\u573a\u4e2d\u5e76\u4e0d\u5b58\u5728\u8fd9\u6837\u7684\u6570\u5b66\u7ed3\u6784\uff0c\u8fd9\u4e00\u672c\u8d28\u5dee\u5f02\uff0c\u6070\u6070\u4ece\u6570\u5b66\u5c42\u9762\u5370\u8bc1\u4e86\u78c1\u573a\u7684\u4e00\u4e2a\u5173\u952e\u7269\u7406\u7279\u6027\uff1a\u5b83\u4e0d\u5b58\u5728\u5e73\u8861\u6001\uff0c\u6216\u8005\u8bf4\u5b83\u65e0\u6cd5\u8fbe\u5230\u771f\u6b63\u7684\u5e73\u8861\u6001<\/p>\n

Rewrite\u4e00\u4e0b\u4e3b\u65b9\u7a0b\uff1a<\/p>\n

\\displaystyle \\frac{d}{dt}p_{\\omega}(t)=\\sum_{\\omega'}W_{\\omega\\omega'}p_{\\omega'}(t)=\\sum_{\\omega\\neq\\omega'}W_{\\omega\\omega'}p_{\\omega'}(t)-\\sum_{\\omega\\neq\\omega'}W_{\\omega'\\omega}p_{\\omega}(t)<\/span><\/p>\n

\u4e0a\u9762\u5f0f\u5b50\u53ef\u4ee5\u5408\u5e76\u4e3a<\/p>\n

\\displaystyle \\frac{d}{dt}p_{\\omega}(t)=\\sum_{\\omega\\neq\\omega'}[W_{\\omega\\omega'}p_{\\omega'}(t)-W_{\\omega'\\omega}p_{\\omega}(t)]<\/span><\/p>\n

\u4e24\u90e8\u5206\u5206\u522b\u8868\u793a\u83b7\u5f97\u548c\u6d41\u5931\u7684\u6982\u7387\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u4ece\u72b6\u6001\\omega<\/span>\u5230\\omega'<\/span>\u7684\u6982\u7387\u6d41\uff1a<\/p>\n

\\displaystyle J_{\\omega\\omega'}=W_{\\omega\\omega'}p_{\\omega'}(t)-W_{\\omega'\\omega}p_{\\omega}(t)=-J_{\\omega'\\omega}<\/span><\/p>\n

\u6b64\u65f6\u4e3b\u65b9\u7a0b\u5199\u4e3a\uff1a<\/p>\n

\\displaystyle \\frac{d}{dt}p_{\\omega}(t)=\\sum_{\\omega\\neq\\omega'}J_{\\omega\\omega'}=-\\sum_{\\omega\\neq\\omega'}J_{\\omega'\\omega}<\/span><\/p>\n

\u5f53\u7cfb\u7edf\u8fbe\u5230\u7a33\u6001\uff08steady state\uff09\u65f6\uff0c\u82e5\u6240\u6709\u72b6\u6001\u7684\u6982\u7387\u4e0d\u518d\u968f\u65f6\u95f4\u53d8\u5316\uff0c\u5373\\frac{d}{dt}p_{\\omega}(t)=0<\/span>\uff0c\u4e5f\u5373\u5bf9\u4e8e\u4efb\u610f\u7684\u72b6\u6001\\omega<\/span>\u548c\\omega'<\/span>\uff0c\u6709J_{\\omega\\omega'}=0<\/span>\uff0c\u6b64\u65f6\u79f0\u7cfb\u7edf\u6ee1\u8db3\u7ec6\u81f4\u5e73\u8861\uff08detailed balance\uff09\u6761\u4ef6\u3002\u6b64\u65f6\uff0c\u6bcf\u6761\u8f6c\u79fb\u8def\u5f84\u4e0a\u7684\u6b63\u5411\u6d41\u4e0e\u53cd\u5411\u6d41\u76f8\u4e92\u62b5\u6d88\uff0c\u7cfb\u7edf\u5904\u4e8e\u5e73\u8861\u6001\uff08equilibrium state\uff09\uff1b\u53cd\u4e4b\uff0c\u82e5\u7a33\u6001\u4e0b\u67d0\u4e9b\u6982\u7387\u6d41\u975e\u96f6\uff0c\u5219\u7cfb\u7edf\u5904\u4e8e\u975e\u5e73\u8861\u7a33\u6001\u3002\u5e73\u8861\u6001\u662f\u7a33\u6001\u7684\u4e00\u79cd\uff0c\u4f46\u662f\u8981\u6c42\u66f4\u4e25\u683c<\/p>\n

\u7531\u4e8e\u65f6\u95f4\u5173\u7cfb\uff0c\u8df3\u8fc7\u6da8\u843d\u5b9a\u7406\u8fd9\u90e8\u5206\uff0c\u540e\u9762\u6709\u65f6\u95f4\u518d\u8865\u5145......<\/p>\n

\u7ecf\u5178\u5fae\u5206\u51e0\u4f55<\/h3>\n

\u754c\u9762\u7684\u7ef4\u5ea6\u548c\u6211\u4eec\u6240\u8003\u8651\u7684\u7ef4\u5ea6\u662fn-1\u7684\u5173\u7cfb\uff0c\u6bd4\u5982\uff0c\u8003\u8651\u4e8c\u7ef4\u7684\u4f53\u7cfb\uff0c\u90a3\u4e48\u53ea\u4f1a\u6709\u4e00\u7ef4\u7684\u754c\u9762\u5b58\u5728\uff0c\u8981\u6e05\u695a\u7684\u8868\u8ff0\u754c\u9762\u5c31\u9700\u8981\u4e00\u4e9b\u5fae\u5206\u51e0\u4f55\uff0c\u9996\u5148\u9700\u8981\u53c2\u6570\u5316\uff0c\u5bf9\u4e8em\u7ef4\u754c\u9762\uff0c\u6211\u4eec\u5f15\u5165m\u4e2a\u5b9e\u53c2\u6570\uff0c\u5c06\u754c\u9762\u8868\u793a\u4e3a\u4ece\u53c2\u6570\u57df\u5230\u7269\u7406\u7a7a\u95f4\u7684\u6620\u5c04\uff0c\u4f8b\u5982\u4e00\u7ef4\u754c\u9762\u53ef\u7528\u5355\u53c2\u6570{r}(u)<\/span>\u8868\u793a\uff0c\u4e8c\u7ef4\u754c\u9762\u53ef\u7528{r}(u,v)<\/span>\u8868\u793a\uff1b\u4ee5\u4e00\u7ef4\u754c\u9762\u4e3a\u4f8b\uff0c\u5f53u\\rightarrow u+du<\/span>\u65f6\uff0c\\vec{r}(u)\\rightarrow \\vec{r}(u+du)<\/span>\uff0cd\\vec{r}=\\vec{r}(u+du)-\\vec{r}(u)=\\vec{r_u}du<\/span>\uff0c\u5176\u4e2d\\vec{r_u}=\\frac{\\partial \\vec{r}}{\\partial u}<\/span>\uff0c\u662f\u5207\u5411\u91cf\uff0c\u5b83\u6307\u5411\u66f2\u7ebf\u5728u<\/span>\u5904\u7684\u5207\u7ebf\u65b9\u5411\uff0c\u800c\u5f27\u957f\u5fae\u5143ds=|\\vec{r_u}|du<\/span>\uff1b\u5b9a\u4e49\u5355\u4f4d\u5207\u5411\u91cf\\hat{t}(u)=\\frac{\\vec{r_u}}{|\\vec{r_u}|}=\\frac{d\\vec{r}}{ds}<\/span>\uff0c\u4e4b\u540e\u5b9a\u4e49\\frac{d\\hat{t}(s)}{ds}=\\kappa_c\\hat{n}(s)<\/span>\uff0c\u5176\u4e2d\\kappa_c<\/span>\u662f\u66f2\u7387\uff0c\u7528\u4e8e\u8861\u91cf\u66f2\u7ebf\u5728\u8be5\u70b9\u504f\u79bb\u76f4\u7ebf\u7684\u7a0b\u5ea6\uff08\u5f2f\u66f2\u7684\u5267\u70c8\u7a0b\u5ea6\uff09\uff0c\\hat{n}(s)<\/span>\u662f\u5355\u4f4d\u4e3b\u6cd5\u5411\u91cf\uff0c\u5b83\u6307\u5411\u66f2\u7ebf\u7684\u51f9\u4fa7\uff08\u5373\u5f2f\u66f2\u7684\u5706\u5fc3\u65b9\u5411\uff09\uff0c\u5e76\u4e14\u4e0e\u5207\u5411\u91cf\u5782\u76f4\uff0c\u5bf9\u4e8e\u4e8c\u7ef4\u4f53\u7cfb\u4e2d\u7684\u4e00\u7ef4\u754c\u9762\uff0c\u5b9a\u4e49\u4e86\u8fd9\u4e9b\u91cf\u4fbf\u8db3\u4ee5\u63cf\u8ff0<\/p>\n

\u5bf9\u4e8e\u4e8c\u7ef4\u754c\u9762\uff0c\u5219\u53ef\u7528{r}(u,v)<\/span>\u8868\u793a\uff0c\u8003\u8651\u4e00\u4e2a\u4e8c\u7ef4\u6da8\u843d\u754c\u9762\uff0c\u5176\u53c2\u6570\u5316\u8868\u793a\\vec{r}(u,v)<\/span>\uff0c\u5f53\u53c2\u6570\u53d1\u751f\u5fae\u5c0f\u53d8\u5316u\\rightarrow u+du,v\\rightarrow v+dv<\/span>\u65f6\uff0c\u4f4d\u7f6e\u77e2\u91cf\u7684\u5fae\u5206\u4e3a\uff1a<\/p>\n

\\displaystyle d\\vec{r}(u,v)=\\frac{\\partial \\vec{r}}{\\partial u}du+\\frac{\\partial \\vec{r}}{\\partial v}dv=\\vec{r_u}du+\\vec{r_v}dv<\/span><\/p>\n

\u5176\u4e2d\\vec{r_u}<\/span>\u548c\\vec{r_v}<\/span>\u662f\u66f2\u9762\u7684\u5207\u5411\u91cf\uff0c\u5b83\u4eec\u5f20\u6210\u8be5\u70b9\u5904\u7684\u5207\u5e73\u9762\uff0c\u5e73\u9762\u7684\u65b9\u5411\u662f\u6cd5\u5411\uff0c\u6cd5\u5411\u91cf\u5b9a\u4e49\u4e3a\uff1a<\/p>\n

\\displaystyle \\hat{n}(u,v)=\\frac{\\vec{r_u}\\times\\vec{r_v}}{|\\vec{r_u}\\times\\vec{r_v}|}<\/span><\/p>\n

\u5b83\u5782\u76f4\u4e8e\u5207\u5e73\u9762\uff0c\u8868\u5f81\u66f2\u9762\u7684\u5c40\u90e8\u53d6\u5411\uff1b\u5207\u5411\u91cf\\vec{r_u}<\/span>\u548c\\vec{r_v}<\/span>\u5f20\u6210\u7684\u5e73\u884c\u56db\u8fb9\u5f62\u9762\u79ef\u5373\u4e3a\u9762\u79ef\u5fae\u5143\uff1a<\/p>\n

\\displaystyle dA=|\\vec{r_u}\\times\\vec{r_v}|dudv<\/span><\/p>\n

\u5f27\u957f\u5fae\u5143ds\u7531\u7b2c\u4e00\u57fa\u672c\u5f62\u5f0f\u7ed9\u51fa\uff1a<\/p>\n

\\displaystyle (ds)^2=d\\vec{r}\\cdot d\\vec{r}=E(du)^2+2Fdudv+G(dv)^2<\/span><\/p>\n

\u5176\u4e2dE=\\vec{r_u}\\cdot\\vec{r_u},F=\\vec{r_u}\\cdot\\vec{r_v},G=\\vec{r_v}\\cdot\\vec{r_v}<\/span>\uff0c\u5f27\u957f\u5fae\u5143\u5199\u6210\u77e9\u9635\u5f62\u5f0f\u4e3a\uff1a<\/p>\n

\\begin{pmatrix} du&dv \\end{pmatrix}
\n\\begin{pmatrix} E&F\\\\F&G \\end{pmatrix}
\n\\begin{pmatrix} du\\\\dv \\end{pmatrix}<\/span><\/p>\n

\u4e2d\u95f4\u7684\u77e9\u9635\u5373I<\/span>\u77e9\u9635\uff0c\u4e5f\u79f0\u4e3a\u5fae\u5206\u51e0\u4f55\u7684\u7b2c\u4e00\u57fa\u672c\u5f62\u5f0f\uff0c\u5176\u884c\u5217\u5f0f\u901a\u5e38\u8bb0\u4e3ag<\/span>\u6216\u8005\u76f4\u63a5\u5199\u4e3adet(I)<\/span>\uff0c\u5176\u4e0e\u9762\u79ef\u5fae\u5143\u7684\u5173\u7cfb\u4e3a\uff1a<\/p>\n

\\displaystyle dA=\\sqrt{g}dudv<\/span><\/p>\n

\u66f2\u9762\u7684\u5f2f\u66f2\u7531\u7b2c\u4e8c\u57fa\u672c\u5f62\u5f0f\u63cf\u8ff0\uff0c\u5373\uff1a<\/p>\n

\\displaystyle L(du)^2+2Mdudv+N(dv)^2<\/span><\/p>\n

\u5176\u4e2dL=\\hat{n}\\cdot\\vec{r_{uu}},M=\\hat{n}\\cdot\\vec{r_{uv}},N=\\hat{n}\\cdot\\vec{r_{vv}}<\/span>\uff0c\u5199\u6210\u77e9\u9635\u5f62\u5f0f\u4e3a\uff1a<\/p>\n

\\begin{pmatrix} du&dv \\end{pmatrix}
\n\\begin{pmatrix} L&M\\\\M&N \\end{pmatrix}
\n\\begin{pmatrix} du\\\\dv \\end{pmatrix}<\/span><\/p>\n

\u4e2d\u95f4\u7684\u77e9\u9635\u5373II<\/span>\u77e9\u9635\uff0c\u4e5f\u79f0\u4e3a\u5fae\u5206\u51e0\u4f55\u7684\u7b2c\u4e8c\u57fa\u672c\u5f62\u5f0f\uff0c\u7ed9\u5b9a\u5207\u65b9\u5411\\vec{r_u}du+\\vec{r_v}dv<\/span>\u540e\uff0c\u6cd5\u66f2\u7387\u4e3a\uff1a<\/p>\n

\\displaystyle \\frac{II}{I}=\\frac{L(du)^2+2Mdudv+N(dv)^2}{E(du)^2+2Fdudv+G(dv)^2}<\/span><\/p>\n

\u6cd5\u66f2\u7387\\kappa(u,v)<\/span>\u4f9d\u8d56\u4e8e\u5207\u65b9\u5411\uff0c\u82e5\\kappa(u,v)>0<\/span>\uff0c\u5219\u4ee3\u8868\u66f2\u9762\u6cbf\u5207\u65b9\u5411\u671d\u6cd5\u5411\u91cf\u7684\u6b63\u4fa7\u5f2f\u66f2\uff08\u51f9\uff09\uff0c\u82e5\\kappa(u,v)<0<\/span>,\u5219\u4ee3\u8868\u66f2\u9762\u6cbf\u5207\u65b9\u5411\u671d\u6cd5\u5411\u91cf\u7684\u8d1f\u4fa7\u5f2f\u66f2\uff08\u51f8\uff09\uff1b\u6240\u6709\u5207\u65b9\u5411\u4e2d\uff0c\u6cd5\u66f2\u7387\u7684\u6781\u503c\u79f0\u4e3a\u4e3b\u66f2\u7387\\kappa_1<\/span>\u548c\\kappa_2<\/span>\uff0c\u5176\u65b9\u5411\u4e92\u76f8\u5782\u76f4\uff0c;\u800c\u9ad8\u65af\u66f2\u7387K=\\kappa_1\\kappa_2<\/span>\u548c\u5e73\u5747\u66f2\u7387H=\\frac{1}{2}(\\kappa_1+\\kappa_2)<\/span>\u5219\u662f\u70b9\u4e0a\u7684\u5185\u8574\u6216\u4e0d\u53d8\u91cf<\/p>\n

\u5728\u66f2\u9762\u67d0\u70b9\uff0c\u5f62\u72b6\u7b97\u5b50\u662f\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\uff0c\u5b83\u5c06\u5207\u5411\u91cf\u6620\u5c04\u5230\u5207\u5411\u91cf\uff0c\u63cf\u8ff0\u4e86\u66f2\u9762\u5728\u8be5\u70b9\u6cbf\u4e0d\u540c\u65b9\u5411\u7684\u5f2f\u66f2\u7a0b\u5ea6\uff1b\u5f62\u72b6\u7b97\u5b50\u7684\u7279\u5f81\u503c\u5c31\u662f\u4e3b\u66f2\u7387\\kappa_1<\/span>\u548c\\kappa_2<\/span>\uff0c\u5bf9\u5e94\u7684\u7279\u5f81\u65b9\u5411\u5c31\u662f\u4e3b\u65b9\u5411\uff1b\u65b9\u7a0bdet(II-\\kappa I)=0<\/span>\u662f\u5f62\u72b6\u7b97\u5b50\u7684\u7279\u5f81\u65b9\u7a0b\u5728\u53c2\u6570\u8868\u793a\u4e0b\u7684\u5f62\u5f0f\u3002\u5b83\u7684\u4e24\u4e2a\u89e3\u7ed9\u51fa\u4e3b\u66f2\u7387\uff0c\u901a\u8fc7\u6c42\u89e3\u4e45\u671f\u884c\u5217\u5f0f\u5f97\u5230\u4e24\u4e2a\u4e3b\u66f2\u7387\u540e\u5c31\u53ef\u4ee5\u5229\u7528\u7b2c\u4e00\u548c\u7b2c\u4e8c\u57fa\u672c\u5f62\u5f0f\u8868\u793a\u9ad8\u65af\u66f2\u7387\u548c\u5e73\u5747\u66f2\u7387\uff1a<\/p>\n

\\displaystyle K=\\kappa_1 \\kappa_2=\\frac{LN-M^2}{EG-F^2}<\/span><\/p>\n

\\displaystyle H=\\frac{1}{2}(\\kappa_1+\\kappa_2)=\\frac{EN+GL-2FM}{2(EG-F^2)}<\/span><\/p>\n

\u7b2c\u4e8c\u7ae0 \u754c\u9762\u95ee\u9898<\/h2>\n

\u5176\u5b9e\u524d\u9762\u8fd8\u6709\u597d\u591a\u6ca1\u5199\u5b8c\uff0c\u6709\u7a7a\u6765\u8865\u5145<\/p>\n

Monge\u53c2\u6570\u5316<\/h3>\n

Monge gauge\uff0c\u53ef\u4ee5\u53ebMonge\u53c2\u6570\u5316\u6216Monge\u89c4\u8303\u5f62\u5f0f\u6216\u8005\u8499\u65e5\u8868\u8c61\uff0c\u5177\u4f53\u662f\u6307\uff0c\u4e00\u4e2a\u66f2\u9762\u88ab\u8868\u793a\u4e3a\u67d0\u4e2a\u5750\u6807\u51fd\u6570\uff08\u901a\u5e38\u662f\u9ad8\u5ea6\uff09\u5173\u4e8e\u53e6\u5916\u4e24\u4e2a\u5750\u6807\u7684\u663e\u5f0f\u51fd\u6570\u3002\u6700\u5e38\u89c1\u7684\u5f62\u5f0f\u662f\uff1a<\/p>\n

\\displaystyle \\vec{r}=\\vec{r}(x,y,h(x,y))<\/span><\/p>\n

\u6216\u8005\u5199\u6210\u5206\u91cf\u5f62\u5f0f\uff1a<\/p>\n

\\displaystyle \\begin{cases}
\nx=u\\\\
\ny=v\\\\
\nz=h(u,v)
\n\\end{cases}<\/span><\/p>\n

\u66f2\u9762\u7684\u4f4d\u7f6e\u5411\u91cf\u7531\u4e24\u4e2a\u6c34\u5e73\u5750\u6807(x,y)<\/span>\u548c\u5b83\u4eec\u6240\u51b3\u5b9a\u7684\u9ad8\u5ea6z=h(x,y)<\/span>\u552f\u4e00\u786e\u5b9a\uff0cMonge\u5f62\u5f0f\u6781\u5927\u5730\u7b80\u5316\u4e86\u66f2\u9762\u7684\u7b2c\u4e00\u548c\u7b2c\u4e8c\u57fa\u672c\u5f62\u5f0f\u7684\u8ba1\u7b97\uff0c\u8ba9\u51e0\u4f55\u91cf\uff08\u5982\u66f2\u7387\uff09\u7684\u8868\u8fbe\u5f0f\u53d8\u5f97\u975e\u5e38\u7b80\u6d01\uff0c\u5207\u5411\u91cf\u975e\u5e38\u7b80\u5355\uff0c\u53ef\u4ee5\u5199\u4e3a\uff1a<\/p>\n

\\displaystyle \\begin{cases}
\nr_x=(1,0,h_x)\\\\
\nr_y=(0,1,h_y)
\n\\end{cases}<\/span><\/p>\n

\u7b2c\u4e00\u57fa\u672c\u5f62\u5f0f\u7cfb\u6570\u4e3a\uff1a<\/p>\n

\\displaystyle \\begin{cases}
\nE=r_x\\cdot r_x=1+h_x^2\\\\
\nF=r_x\\cdot r_y=h_x h_y\\\\
\nG=r_y\\cdot r_y=1+h_y^2
\n\\end{cases}<\/span><\/p>\n

\u7b2c\u4e00\u57fa\u672c\u5f62\u5f0f\u7684\u884c\u5217\u5f0f\u503c\u4e3a\uff1a<\/p>\n

\\displaystyle g=det(I)=EG-F^2=1+h_x^2+h_y^2=1+(\\vec{\\nabla}h)^2<\/span><\/p>\n

\u5355\u4f4d\u6cd5\u5411\u91cf\u4e3a\uff1a<\/p>\n

\\displaystyle \\hat{n}=\\frac{r_x\\times r_y}{\\sqrt{g}}=\\frac{(-h_x,-h_y,1)}{\\sqrt{1+h_x^2+h_y^2}}<\/span><\/p>\n

\u7b2c\u4e8c\u57fa\u672c\u5f62\u5f0f\u7cfb\u6570\u4e3a\uff1a<\/p>\n

\\displaystyle \\begin{cases}
\nL=\\hat{n}\\cdot r_{xx}=\\frac{h_{xx}}{\\sqrt{g}}\\\\
\nM=\\hat{n}\\cdot r_{xy}=\\frac{h_{xy}}{\\sqrt{g}}\\\\
\nN=\\hat{n}\\cdot r_{yy}=\\frac{h_{yy}}{\\sqrt{g}}
\n\\end{cases}<\/span><\/p>\n

\u5c06\u4e0a\u8ff0\u7b80\u5316\u7684\u7cfb\u6570\u4ee3\u5165\u4e00\u822c\u7684\u9ad8\u65af\u66f2\u7387\u548c\u5e73\u5747\u66f2\u7387\u516c\u5f0f\u53ef\u4ee5\u5f97\u5230\uff1a<\/p>\n

\\displaystyle K=\\kappa_1 \\kappa_2=\\frac{LN-M^2}{EG-F^2}=\\frac{\\frac{h_{xx}h_{yy}}{g}-\\frac{h_{xy}^2}{g}}{g}=\\frac{h_{xx}h_{yy}-h_{xy}^2}{g^2}<\/span><\/p>\n

\\displaystyle H=\\frac{EN+GL-2FM}{2(EG-F^2)}=\\frac{(1+h_x^2)h_{yy}+(1+h_y^2)h_{xx}-2h_x h_y h_{xy}}{2g\\sqrt{g}}<\/span><\/p>\n

\u82e5\u6b64\u65f6\u8003\u8651h_x\\ll 1,h_y\\ll 1<\/span>\uff0c\u5373\u66f2\u9762\u975e\u5e38\u5e73\u5766\uff0c\u659c\u7387\u5f88\u5c0f\uff1b\u6211\u4eec\u53ef\u4ee5\u5bf9\u66f2\u7387\u516c\u5f0f\u8fdb\u884c\u7ebf\u6027\u8fd1\u4f3c\uff08\u4e00\u9636\u8fd1\u4f3c\uff09\uff0c\u4e8c\u9636\u5c0f\u91cf\u53ef\u4ee5\u5ffd\u7565\uff0c\u6b64\u65f6\u6709\uff1a<\/p>\n

\\displaystyle K\\approx h_{xx}h_{yy}-h_{xy}^2,H\\approx \\frac{h_{xx}+h_{yy}}{2}<\/span><\/p>\n

\u6bdb\u7ec6\u4e0d\u7a33\u5b9a\u6027<\/h3>\n

\u6bdb\u7ec6\u4e0d\u7a33\u5b9a\u6027\uff08Rayleigh-Plateau\u4e0d\u7a33\u5b9a\u6027\uff09\u662f\u6307\u4e00\u4e2a\u5706\u67f1\u72b6\u7684\u6db2\u4f53\u5c04\u6d41\uff08\u6216\u7ec6\u4e1d\uff09\u5728\u8868\u9762\u5f20\u529b\u4f5c\u7528\u4e0b\uff0c\u4f1a\u81ea\u53d1\u5730\u65ad\u88c2\u6210\u4e00\u4e32\u6db2\u6ef4\u7684\u73b0\u8c61\uff0c\u5176\u80cc\u540e\u7684\u9a71\u52a8\u529b\u6e90\u4e8eLaplace Pressure\uff0c\u6211\u4eec\u901a\u8fc7\u51e0\u79cd\u4e0d\u540c\u65b9\u5f0f\u6765\u5bf9\u5176\u8fdb\u884c\u63a8\u5bfc\u548c\u89e3\u91ca\uff0c\u9996\u5148\u662f\u57fa\u4e8e\u80fd\u91cf\u6700\u5c0f\u5316\u539f\u7406\u7684\u7b80\u5316\u9759\u529b\u5b66\u8bba\u8bc1\uff0c\u8003\u8651\u521d\u59cb\u5706\u67f1\u534a\u5f84R_0<\/span>\uff0c\u957f\u5ea6\u4e3aL<\/span>\uff0c\u4fa7\u8868\u9762\u79ef\uff08\u754c\u9762\u9762\u79ef\uff09\u4e3aA_1=2\\pi R_0 L<\/span>\uff0c\u5047\u8bbe\u7834\u788e\u540e\u6db2\u6ef4\u65ad\u88c2\u6210n<\/span>\u4e2a\u76f8\u540c\u7684\u7403\u72b6\u6db2\u6ef4\uff0c\u6bcf\u4e2a\u534a\u5f84\u4e3ar_0<\/span>\uff0c\u603b\u8868\u9762\u79ef\u4e3aA_2=n4\\pi r_{0}^2<\/span>\uff0c\u5f15\u5165\u4f53\u79ef\u5b88\u6052\uff0c\u5373\uff1a<\/p>\n

\\displaystyle \\pi R_0^2 L=n\\frac{4}{3}\\pi r_{0}^3<\/span><\/p>\n

\u5f97\u5230n=\\frac{3R_0^2L}{4r_0^3}<\/span>\uff0c\u4ee3\u5165\u603b\u8868\u9762\u79ef\uff1a<\/p>\n

\\displaystyle A_2=n4\\pi r_{0}^2=\\frac{3\\pi R_0^2 L}{r_0}<\/span><\/p>\n

\u6bd4\u8f83\u9762\u79ef\uff0c\u82e5A_2<A_1<\/span>\u5219\u7834\u788e\u540e\u8868\u9762\u79ef\u51cf\u5c0f\uff0c\u80fd\u91cf\u964d\u4f4e\uff0c\u5bfc\u51far_0>\\frac{3}{2}R_0<\/span>\uff0c\u5176\u7269\u7406\u542b\u4e49\u4e3a\u8868\u9762\u5f20\u529b\u9a71\u4f7f\u7cfb\u7edf\u8d8b\u5411\u8868\u9762\u79ef\u6700\u5c0f\u7684\u72b6\u6001\u3002\u5f53r_0>\\frac{3}{2}R_0<\/span>\u65f6\uff0c\u6db2\u6ef4\u603b\u8868\u9762\u79ef\u5c0f\u4e8e\u5706\u67f1\u4fa7\u9762\u79ef\uff0c\u7834\u788e\u5728\u70ed\u529b\u5b66\u4e0a\u6709\u5229<\/p>\n

\u63a5\u4e0b\u6765\u662f\u57fa\u4e8e\u7edf\u8ba1\u70ed\u529b\u5b66\u7684\u7ebf\u6027\u7a33\u5b9a\u6027\u5206\u6790\uff0c\u9996\u5148\u8003\u8651\u4e00\u4e2a\u67f1\u72b6\u754c\u9762\uff0c\u5176\u53c2\u6570\u5316\u4e3a\\vec{r}=(x,y,z)=(h\\cos{\\theta},h\\sin{\\theta},z)<\/span>\uff0ch(z)<\/span>\u662f\u67f1\u72b6\u754c\u9762\u5728\u8f74\u5411\u4f4d\u7f6ez<\/span>\u5904\u7684\u534a\u5f84\uff0c\u7b2c\u4e00\u57fa\u672c\u5f62\u5f0f\u7cfb\u6570\uff1aE=r_z\\cdot r_z=1+h_z^2<\/span>\uff0cG=r_{\\theta}\\cdot r_{\\theta}=h^2<\/span>\uff0cF=0<\/span>\uff0c\u9762\u79ef\u5143dA=\\sqrt{g}dzd\\theta=h\\sqrt{1+h_z^2}dzd\\theta<\/span>\uff0c\u603b\u8868\u9762\u79ef\u80fd\uff1a<\/p>\n

\\displaystyle H[h]=\\gamma\\int dA=\\gamma\\int_0^L dz\\int_0^{2\\pi} d\\theta h\\sqrt{1+h_z^2}<\/span><\/p>\n

\u505a\u5c0f\u659c\u7387\u5c55\u5f00\uff0c\u5f53h_z \\ll1<\/span>\u65f6\uff0c\u6709\uff1a<\/p>\n

\\displaystyle H[h]\\approx\\gamma\\int_0^L dz\\int_0^{2\\pi} d\\theta h[1+\\frac{1}{2}h_z^2]<\/span><\/p>\n

\u7ee7\u7eed\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n

\\displaystyle \\gamma[\\int_0^L dz\\int_0^{2\\pi} d\\theta h+\\frac{1}{2}\\int_0^L dz\\int_0^{2\\pi} d\\theta hh_z^2]<\/span><\/p>\n

\u5f15\u5165\u5e73\u5747\u534a\u5f84\uff0c\u7531\u5e73\u5747\u534a\u5f84\u5b9a\u4e49\\bar{h}=\\frac{1}{L}\\int_0^L h(z)dz<\/span>\uff0c\u56e0\u6b64\uff1a<\/p>\n

\\displaystyle H[h]=2\\pi \\gamma L\\bar{h}+\\frac{1}{2}\\int_0^L dz\\int_0^{2\\pi} d\\theta hh_z^2<\/span><\/p>\n

\u7531\u4e8e\u5b8c\u7f8e\u67f1\u72b6\u754c\u9762\u7684\u80fd\u91cf\u4e3aH_0=2\\pi\\gamma Lh_0<\/span>\uff0c\u56e0\u6b64\u80fd\u91cf\u53d8\u5316\u4e3a\uff1a<\/p>\n

\\displaystyle \\Delta H[h]=2\\pi \\gamma L(\\bar{h}-h_0)+\\frac{1}{2}\\int_0^L dz\\int_0^{2\\pi} d\\theta hh_z^2<\/span><\/p>\n

\u5f15\u5165\u4f53\u79ef\u5b88\u6052V=\\int_0^L \\pi h^2dz=\\pi h_0^2 L<\/span>\uff0c\u6709\\int_0^L h^2dz= h_0^2 L<\/span>\uff0c\u7531\u4e8e\u73b0\u5728\u7684\u754c\u9762\u662f\u5df2\u7ecf\u5f00\u59cb\u6709\u6da8\u843d\u7684\u754c\u9762\uff0c\u6211\u4eec\u5c06h(z)<\/span>\u5c55\u5f00\uff0c\u4ee3\u8868\u5df2\u7ecf\u6709\u6da8\u843d\u540e\u7684\u754c\u9762\uff0c\u8bbeh(z)=h_0+\\delta h<\/span>\uff0c\u5176\u4e2d\\delta h<\/span>\u662f\u5fae\u5c0f\u6270\u52a8\uff0c\u4ee3\u5165\u4f53\u79ef\u5b88\u6052\uff1a<\/p>\n

\\displaystyle \\int_0^L[h_0^2+2h_0\\delta h+(\\delta h)^2]dz=h_0^2L<\/span><\/p>\n

\\displaystyle h_0^2L+2h_0\\int_0^L\\delta hdz+\\int_0^L(\\delta h)^2dz=h_0^2L<\/span><\/p>\n

\\displaystyle 2h_{0}\\int_{0}^{L} \\delta hdz + \\int_{0}^{L} (\\delta h)^{2} dz = 0<\/span><\/p>\n

\u53ef\u4ee5\u5bfc\u51fa\uff1a<\/p>\n

\\displaystyle \\int_{0}^{L}\\delta hdz=-\\frac{1}{2h_{0}}\\int_{0}^{L}(\\delta h)^{2}dz<\/span><\/p>\n

\u7531\u4e8e\u5e73\u5747\u534a\u5f84\u53ef\u5199\u4e3a\uff1a<\/p>\n

\\displaystyle \\bar{h}=\\frac{1}{L}\\int_{0}^{L}h(z)dz=h_{0}+\\frac{1}{L}\\int_{0}^{L}\\delta hdz<\/span><\/p>\n

\u6240\u4ee5\uff1a<\/p>\n

\\displaystyle \\bar{h}=h_0-\\frac{1}{2h_0L}\\int_0^L(\\delta h)^2dz<\/span><\/p>\n

\u6700\u540e\u53d1\u73b0\uff1a<\/p>\n

\\displaystyle \\bar{h}-h_{0}=-\\frac{1}{2h_{0}L}\\int_{0}^{L}(\\delta h)^{2}dz<0<\/span><\/p>\n

\u518d\u6b21\u56de\u5230\u8fd9\u4e2a\u5f0f\u5b50\u4e2d\u6765\uff1a<\/p>\n

\\displaystyle \\Delta H[h]=2\\pi \\gamma L(\\bar{h}-h_0)+\\frac{1}{2}\\int_0^L dz\\int_0^{2\\pi} d\\theta hh_z^2<\/span><\/p>\n

\u7531\u4e8e\\bar{h}-h_{0}<0<\/span>\uff0c\u6240\u4ee5\u7b2c\u4e00\u9879\u4e3a\u8d1f\uff0c\u800c\u7b2c\u4e8c\u9879\u88ab\u79ef\u51fd\u6570\u5747\u4e3a\u6b63\uff0c\u6240\u4ee5\u7b2c\u4e8c\u9879\u4e3a\u6b63\uff1b\u56e0\u6b64\u80fd\u91cf\u53d8\u5316\\Delta H<\/span>\u662f\u6b63\u662f\u8d1f\u53d6\u51b3\u4e8e\u54ea\u4e2a\u9879\u5360\u4e3b\u5bfc\uff0c\u6362\u8a00\u4e4b\uff0c\u7b2c\u4e00\u9879\u4e3b\u5bfc\uff0c\u5219\u4f1a\u5bfc\u81f4breaking\uff0c\u82e5\u7b2c\u4e8c\u9879\u4e3b\u5bfc\uff0c\u5219\u662f\u7a33\u5b9a\u7684<\/p>\n

\u5f53\u7136\uff0c\u8fd9\u91cc\u662f\u5728\u8f74\u5bf9\u79f0\u5047\u8bbe\u4e0b\uff0ch<\/span>\u53ea\u662fz<\/span>\u7684\u51fd\u6570\uff0c\u53ef\u4ee5\u5bf9\\theta<\/span>\u79ef\u5206,\u5728\u4e00\u822c\u60c5\u51b5\u4e0b\uff0ch<\/span>\u662fz<\/span>\u548c\\theta<\/span>\u7684\u51fd\u6570\uff0c\u4e0d\u80fd\u76f4\u63a5\u79ef\u5206\u6389\\theta<\/span>\uff0c\u4f46\u7ebf\u6027\u7a33\u5b9a\u6027\u5206\u6790\u663e\u793a\uff0c\u53ea\u6709\u8f74\u5bf9\u79f0\u6a21\u5f0f(m=0)<\/span>\u53ef\u80fd\u5bfc\u81f4\u5931\u7a33\uff0c\u6240\u4ee5\u7814\u7a76\u8f74\u5bf9\u79f0\u6270\u52a8\u662f\u5408\u7406\u7684\u7b80\u5316\uff0c\u5982\u679c\u8003\u8651\u66f4\u4e00\u822c\u7684\u60c5\u51b5\uff0c\u5373h<\/span>\u662fz<\/span>\u548c\\theta<\/span>\u7684\u51fd\u6570\uff0c\u90a3\u4e48\u6b64\u65f6\u7684\u8868\u9762\u79ef\u80fd\u4f1a\u53d8\u5316\u6210\uff1a<\/p>\n

\\displaystyle H[h]=\\gamma\\int dA=\\gamma\\int_0^L dz\\int_0^{2\\pi} d\\theta h\\sqrt{1+h_z^2+\\frac{1}{h^2}h_{\\theta}^2}<\/span><\/p>\n

\u505a\u5c0f\u659c\u7387\u5c55\u5f00\uff0c\u5f53h_z \\ll1,h_{\\theta} \\ll1<\/span>\u65f6\uff0c\u6709\uff1a<\/p>\n

\\displaystyle H[h]=\\gamma\\int dA=\\gamma\\int_0^L dz\\int_0^{2\\pi}d\\theta h[1+\\frac{1}{2}h_z^2+\\frac{1}{2}\\frac{1}{h^2}h_{\\theta}^2]<\/span><\/p>\n

\\displaystyle H[h]=\\gamma 2\\pi L\\bar{h}+\\frac{\\gamma}{2}\\int_0^L dz\\int_0^{2\\pi}d\\theta h[h_z^2+\\frac{1}{h^2}h_{\\theta}^2]<\/span><\/p>\n

\u5f15\u5165\u4f53\u79ef\u5b88\u6052\u6761\u4ef6\\bar{h^2}=h_0^2<\/span>\uff0c\u524d\u9762\u6211\u4eec\u662f\u901a\u8fc7\u5fae\u6270\u53d8\u5206\u6cd5\u5f15\u5165\u7684\uff0c\u5176\u5b9e\u90fd\u662f\u4e00\u6837\u7684\uff0c\u4f53\u79ef\u5b88\u6052\u8981\u6c42\u5e73\u5747\u5e73\u65b9\u534a\u5f84\\bar{h^2}<\/span>\u7b49\u4e8e\u672a\u6270\u52a8\u7684\u5706\u67f1\u534a\u5f84\u7684\u5e73\u65b9h_0^2<\/span>\uff0c\u5176\u4e2d\u5e73\u5747\u88ab\u5b9a\u4e49\u4e3a\uff1a<\/p>\n

\\displaystyle \\bar{(\\cdot)}=\\frac{1}{2\\pi L}\\int_0^L \\int_0^{2\\pi} dzd\\theta (\\cdot)<\/span><\/p>\n

\u5c06\u754c\u9762\u9ad8\u5ea6\u5c55\u5f00\u4e3a\u5085\u91cc\u53f6\u7ea7\u6570\uff1a<\/p>\n

\\displaystyle h(z,\\theta)=\\sum_{m=-\\infty}^{+\\infty}\\sum_qe^{iqz}e^{im\\theta}\\tilde{h}_m(q)<\/span><\/p>\n

\u5176\u4e2dq<\/span>\u662f\u8f74\u5411\u6ce2\u6570\uff0c\u662f\u79bb\u6563\u503c\uff0cm<\/span>\u662f\u6574\u6570\u65b9\u4f4d\u89d2\u6ce2\u6570\uff0c\u7cfb\u6570\u4e3a\uff1a<\/p>\n

\\displaystyle \\tilde{h}_m(q)=\\frac{1}{2\\pi L}\\int_0^L\\int_0^{2\\pi}e^{-iqz-im\\theta}h(z,\\theta)d\\theta dz<\/span><\/p>\n

\u8ba1\u7b97\\overline{h^{2}}<\/span>\uff1a<\/p>\n

\\displaystyle \\overline{h^{2}} = \\frac{1}{2\\pi L} \\int_{0}^{L} \\int_{0}^{2\\pi} h^{2}d\\theta dz<\/span><\/p>\n

\u4ee3\u5165\u5085\u91cc\u53f6\u5c55\u5f00\uff1a<\/p>\n

\\displaystyle h^{2} = \\sum_{q_{1},q_{2}} \\sum_{m_{1},m_{2}} e^{i(q_{1}+q_{2})z} e^{i(m_{1}+m_{2})\\theta} \\tilde{h}_{m_{1}}(q_{1}) \\tilde{h}_{m_{2}}(q_{2})<\/span><\/p>\n

\u5229\u7528\u6b63\u4ea4\u6027\uff1a<\/p>\n

\\displaystyle \\int_{0}^{L} e^{i(q_{1}+q_{2})z}dz = L \\delta_{q_{1},-q_{2}}, \\int_{0}^{2\\pi} e^{i(m_{1}+m_{2})\\theta}d\\theta = 2\\pi \\delta_{m_{1},-m_{2}}<\/span><\/p>\n

\u5f97\u5230\uff1a<\/p>\n

\\displaystyle \\bar{h^2} = \\sum_{q,m} \\tilde{h}_{m}(q) \\tilde{h}_{-m}(-q) = \\sum_{q,m} |\\tilde{h}_{m}(q)|^{2}<\/span><\/p>\n

\u8fd9\u4e2a\u6c42\u548c\u5305\u62ec\u6240\u6709\u6ce2\u6570q<\/span>\u548cm<\/span>\uff0c\u7279\u522b\u5305\u542b\u96f6\u6a21q=0,m=0<\/span>\uff0c\u5176\u7cfb\u6570\\tilde{h_0}(0)=\\bar{h}<\/span>\u662f\u5e73\u5747\u534a\u5f84\uff1b\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u96f6\u6a21\u9879\u5206\u79bb\u51fa\u6765\uff1a<\/p>\n

\\displaystyle \\bar{h^2} = \\bar{h}^{2} + \\sum_{(q,m) \\neq (0,0)} |\\tilde{h}_{m}(q)|^{2}<\/span><\/p>\n

\u7531\u4f53\u79ef\u5b88\u6052\\bar{h^2}=h_0^2<\/span>\uff0c\u6709\uff1a<\/p>\n

\\displaystyle h_{0}^{2} = \\bar{h}^{2} + \\sum_{q,m} |\\tilde{h}_{m}(q)|^{2}<\/span><\/p>\n

\u89e3\u5f97\uff1a<\/p>\n

\\displaystyle h_{0}=\\sqrt{\\overline{h}^{2}+\\sum_{q,m}|\\tilde{h}_{m}(q)|^{2}}=\\overline{h}\\sqrt{1+\\frac{1}{\\overline{h}^{2}}\\sum_{q,m}|\\tilde{h}_{m}(q)|^{2}}<\/span><\/p>\n

\u5047\u8bbe\u6270\u52a8\u5c0f\uff0c\u8fd1\u4f3c\u4e3a\uff1a<\/p>\n

\\displaystyle h_{0}\\approx\\overline{h}\\left(1+\\frac{1}{2\\overline{h}^{2}}\\sum_{q,m}|\\tilde{h}_{m}(q)|^{2}\\right)=\\overline{h}+\\frac{1}{2\\overline{h}}\\sum_{q,m}|\\tilde{h}_{m}(q)|^{2}<\/span><\/p>\n

\u56e0\u6b64:<\/p>\n

\\displaystyle \\overline{h}-h_{0}=-\\frac{1}{2\\overline{h}}\\sum_{q,m}|\\tilde{h}_{m}(q)|^{2}<\/span><\/p>\n

\u7ebf\u6027\u8fd1\u4f3c\uff08\u9ad8\u65af\u8fd1\u4f3c\uff09\u4e0b\u53d6\\bar{h} \\approx h_{0}<\/span>\uff0c\u6709\uff1a<\/p>\n

\\displaystyle \\bar{h} - h_{0} \\approx -\\frac{1}{2h_{0}} \\sum_{q,m} |\\tilde{h}_{m}(q)|^{2}<0<\/span><\/p>\n

\u9762\u79ef\u80fdH=\\gamma\\int dA<\/span>\uff0c\u5f88\u5bb9\u6613\u5199\u51fa\u67f1\u72b6\u754c\u9762\u7684\u9762\u79ef\u5143\u5e76\u5f97\u5230\u603b\u9762\u79ef\uff08\u524d\u9762\u5199\u8fc7\u8fd9\u91cc\u4e0d\u518d\u91cd\u590d\uff09\uff0c\u603b\u9762\u79ef\uff1a<\/p>\n

\\displaystyle A=\\int dA\\approx\\int_{0}^{L}\\int_{0}^{2\\pi}\\left(h+\\frac{h}{2}h_{z}^{2}+\\frac{1}{2h}h_{\\theta}^{2}\\right)d\\theta dz<\/span><\/p>\n

\u6539\u5199\u6210\uff1a<\/p>\n

\\displaystyle A=2\\pi L\\bar{h}+\\int_{0}^{L}\\int_{0}^{2\\pi}\\left(\\frac{h}{2}h_{z}^{2}+\\frac{1}{2h}h_{\\theta}^{2}\\right)d\\theta dz<\/span><\/p>\n

\u6027\u8fd1\u4f3c\uff08\u9ad8\u65af\u8fd1\u4f3c\uff09\u4e0b\uff1a<\/p>\n

\\displaystyle A=2\\pi L\\bar{h}+\\int_{0}^{L}\\int_{0}^{2\\pi}\\left(\\frac{h_0}{2}h_{z}^{2}+\\frac{1}{2h_0}h_{\\theta}^{2}\\right)d\\theta dz<\/span><\/p>\n

\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\uff1a<\/p>\n

\\displaystyle \\int h_z^2d\\theta dz=2\\pi L\\sum_{q,m}q^2|\\tilde{h}_m(q)|^2,\\int h_\\theta^2d\\theta dz=2\\pi L\\sum_{q,m}m^2|\\tilde{h}_m(q)|^2<\/span><\/p>\n

\u603b\u9762\u79ef\uff1a<\/p>\n

\\displaystyle A=2\\pi L\\bar{h}+\\pi Lh_0\\sum_{q,m}\\left(q^2+\\frac{m^2}{h_0^2}\\right)|\\tilde{h}_m(q)|^2<\/span><\/p>\n

\u521d\u59cb\u672a\u6270\u52a8\u9762\u79ef\u4e3aA_0=2\\pi h_0 L<\/span>\uff0c\u6545\u9762\u79ef\u53d8\u5316\uff1a<\/p>\n

\\displaystyle \\Delta A=A-A_0=2\\pi L(\\bar{h}-h_0)+\\pi Lh_0\\sum_{q,m}\\left(q^2+\\frac{m^2}{h_0^2}\\right)|\\tilde{h}_m(q)|^2<\/span><\/p>\n

\u4ee3\u5165\\bar{h} - h_{0} = -\\frac{1}{2h_{0}} \\sum_{q,m} |\\tilde{h}_{m}(q)|^{2}<\/span>\uff0c\u53ef\u4ee5\u5f97\u5230\uff1a<\/p>\n

\\displaystyle \\Delta A=2\\pi L\\left(-\\frac{1}{2h_0}\\sum_{q,m}|\\tilde{h}_m(q)|^2\\right)+\\pi Lh_0\\sum_{q,m}\\left(q^2+\\frac{m^2}{h_0^2}\\right)|\\tilde{h}_m(q)|^2<\/span><\/p>\n

\u5408\u5e76\u6c42\u548c\u9879\uff1a<\/p>\n

\\displaystyle \\Delta A=\\frac{\\pi L}{h_0}\\sum_{q,m}\\left(h_0^2q^2+m^2-1\\right)|\\tilde{h}_m(q)|^2<\/span><\/p>\n

\u80fd\u91cf\u53d8\u5316\\Delta H=\\gamma \\Delta A<\/span>\uff0c\u6545\uff1a<\/p>\n

\\displaystyle \\frac{\\Delta H}{\\gamma}=\\frac{\\pi L}{h_0}\\sum_{q,m}\\left(h_0^2q^2+m^2-1\\right)|\\tilde{h}_m(q)|^2<\/span><\/p>\n

\\displaystyle \\Delta H=\\gamma\\frac{\\pi L}{h_0}\\sum_{q,m}\\left(h_0^2q^2+m^2-1\\right)|\\tilde{h}_m(q)|^2<\/span><\/p>\n

\u5c06\u5f0f\u5b50\u5199\u4e3a:<\/p>\n

\\displaystyle \\Delta H=\\gamma\\pi Lh_0\\sum_{q,m}\\left(q^2+\\frac{1}{h_0^2}m^2-\\frac{1}{h_0^2}\\right)|\\tilde{h}_m(q)|^2<\/span><\/p>\n

\u89c2\u5bdf\u7a33\u5b9a\u6027\u7cfb\u6570\uff0c\u6bcf\u4e2a\u6a21\u5f0f(m,q)<\/span>\u5bf9\u80fd\u91cf\u53d8\u5316\u7684\u8d21\u732e\u7cfb\u6570\u4e3a\uff1a<\/p>\n

\\displaystyle L(m,q)=(q^2+\\frac{1}{h_0^2}m^2-\\frac{1}{h_0^2})<\/span><\/p>\n

\u7a33\u5b9a\u6027\u5224\u636e\uff1a\u82e5L(m,q)>0<\/span>\uff0c\u6a21\u5f0f\u7a33\u5b9a\uff08\u6270\u52a8\u589e\u52a0\u80fd\u91cf\uff09\uff0c\u53cd\u4e4b\u82e5L(m,q)<0<\/span>\uff0c\u6a21\u5f0f\u4e0d\u7a33\u5b9a\uff08\u6270\u52a8\u964d\u4f4e\u80fd\u91cf\uff09\uff0c\u89c2\u5bdf\u53d1\u73b0\u82e5m\\neq0<\/span>\uff0c\u7531\u4e8em^2\\geq1<\/span>\uff0c\u6709L(m,q)\\geq0<\/span>\uff0c\u754c\u9762\u603b\u662f\u7a33\u5b9a\u6216\u4e2d\u6027\uff0c\u5f53m=0<\/span>\uff08\u8f74\u5bf9\u79f0\u6270\u52a8\uff09\u65f6\uff0cL(0,q)=(q^2-\\frac{1}{h_0^2})<\/span>\uff0c\u4e0d\u7a33\u5b9a\u6761\u4ef6\u4e3aL(0,q)<0<\/span>\uff0c\u5373\uff1a<\/p>\n

\\displaystyle q^2<\\frac{1}{h_0^2} \\Rightarrow |q|<\\frac{1}{h_0}<\/span><\/p>\n

\u5bf9\u5e94\u6ce2\u957f\\lambda=2\\pi\/|q|>2\\pi h_0<\/span>\uff0c\u957f\u6ce2\u6270\u52a8\u964d\u4f4e\u7cfb\u7edf\u8868\u9762\u80fd\uff0c\u5bfc\u81f4\u67f1\u72b6\u5c04\u6d41\u65ad\u88c2\u6210\u6db2\u6ef4\uff1b\u66f4\u6df1\u5165\u7684\u8ba8\u8bba\u662f\uff0c\u6ce2\u6570q<\/span>\u51b3\u5b9a\u4e86\u8f74\u5411\u5f2f\u66f2\u7684\u5267\u70c8\u7a0b\u5ea6\uff0c\u5bf9\u5e94\u4e8e\\kappa_1<\/span>\u7684\u53d8\u5316\u5c3a\u5ea6\uff0ch_0^{-1}<\/span>\uff08\u521d\u59cb\u5706\u5468\u66f2\u7387\uff09 \u5219\u662f\u7cfb\u7edf\u7684\u56fa\u6709\u5f2f\u66f2\u5c3a\u5ea6\uff0c\u5bf9\u5e94\u4e8e\\kappa_2<\/span>\u7684\u57fa\u51c6\u503c\uff0c\u7a33\u5b9a\u6027\u6761\u4ef6q^2<\\frac{1}{h_0^2}<\/span>\u610f\u5473\u7740\uff1a\u53ea\u6709\u5f53\u8f74\u5411\u7684\u5f2f\u66f2\u53d8\u5316\u6bd4\u5706\u5468\u7684\u56fa\u6709\u5f2f\u66f2\u66f4\u5e73\u7f13\uff08\u957f\u6ce2\uff09\u65f6\uff0c\u7cfb\u7edf\u624d\u4f1a\u5931\u7a33\u3002\u5982\u679c\u8f74\u5411\u53d8\u5316\u592a\u5267\u70c8\uff08\u77ed\u6ce2\uff09\uff0c\u589e\u52a0\u7684\u66f2\u7387\u80fd\u4f1a\u62b5\u6d88\u8868\u9762\u79ef\u51cf\u5c11\u7684\u597d\u5904<\/p>\n

\u6da6\u6e7f\u73b0\u8c61<\/h3>\n

\u7b2c\u4e09\u7ae0 \u8f93\u8fd0\u4e0e\u754c\u9762\u52a8\u529b\u5b66\u65b9\u7a0b<\/h2>\n

Langevin\u65b9\u7a0b<\/h3>\n

Fokker\u2013Planck\u65b9\u7a0b<\/h3>\n

\u8003\u8651\u6982\u7387\u5bc6\u5ea6p(x,t)<\/span>\u7684\u6f14\u5316\u65b9\u7a0b\uff1a<\/p>\n

\\displaystyle \\frac{\\partial}{\\partial t}p(x,t)=-\\nabla\\cdot J(x,t)<\/span><\/p>\n

\u5176\u4e2d\u6982\u7387\u6d41\u5bc6\u5ea6J(x,t)<\/span>\u4e3a\uff1a<\/p>\n

\\displaystyle J(x,t)=-D\\nabla p+u(x)p=p\\left[u(x)-D\\nabla\\log p\\right]<\/span><\/p>\n

\u5b9a\u4e49\u5c40\u57df\u901f\u5ea6v(x,t)=u(x)-D\\nabla\\log p<\/span>\uff0c\u5219J=pv<\/span>\uff0c\u5047\u8bbe\u6f02\u79fb\u901f\u5ea6u(x)<\/span>\u53ef\u5199\u6210\u68af\u5ea6\u5f62\u5f0f\uff1a<\/p>\n

\\displaystyle u(x)=-\\kappa\\nabla H(x)<\/span><\/p>\n

\u5176\u4e2d\\kappa<\/span>\u4e3a\u8fc1\u79fb\u7387\uff0cH(x)<\/span>\u4e3a\u52bf\u80fd\u51fd\u6570\uff08\u6216\u8005\u4f60\u53ef\u4ee5\u7406\u89e3\u6210\u54c8\u5bc6\u987f\u91cf\uff09\uff0c\u5c06u(x)=-\\kappa\\nabla H(x)<\/span>\u4ee3\u5165J<\/span>\u5f97\u5230\uff1a<\/p>\n

\\displaystyle J=-D\\nabla p-\\kappa\\nabla H(x)p<\/span><\/p>\n

\u5229\u7528\u7231\u56e0\u65af\u5766\u5173\u7cfbD=k_BT\\kappa<\/span>\u4ee3\u5165\u5f97\uff1a<\/p>\n

\\displaystyle J=-\\kappa\\left(k_BT\\nabla p+p\\nabla H\\right)<\/span><\/p>\n

\u63d0\u53d6\u516c\u56e0\u5b50p<\/span>\uff1a<\/p>\n

\\displaystyle J=-\\kappa p\\left(\\frac{k_BT}{p}\\nabla p+\\nabla H\\right)=-\\kappa p\\nabla\\left(H+k_BT\\log p\\right)<\/span><\/p>\n

\u5b9a\u4e49\u968f\u673a\u81ea\u7531\u80fd\uff08\u5176\u5b9e\u662f\u6709\u6548\u81ea\u7531\u80fd\u5bc6\u5ea6\uff09\uff1a<\/p>\n

\\displaystyle \\hat{f}(x,t)=H(x)+k_BT\\log p(x,t)<\/span><\/p>\n

\u5219\u6982\u7387\u6d41\u7b80\u5316\u4e3a\uff1a<\/p>\n

\\displaystyle J(x,t)=-p\\kappa\\nabla\\hat{f}<\/span><\/p>\n

\u4e8e\u662fFokker-Planck\u65b9\u7a0b\u53ef\u91cd\u5199\u4e3a\uff1a<\/p>\n

\\displaystyle \\frac{\\partial p}{\\partial t}=-\\nabla\\cdot J=\\nabla\\cdot\\left(p\\kappa\\nabla\\hat{f}\\right)<\/span><\/p>\n

\u5f15\u5165\u7a33\u6001\u6761\u4ef6\uff0ct\\rightarrow\\infty<\/span>\u65f6\uff0c\u6d41\u7684\u6563\u5ea6\u4e3a0\uff0c\u5373\\nabla\\cdot J=0<\/span>\uff0c\u7531\u4e8e\\kappa<\/span>\u548cp<\/span>\u5747\u5927\u4e8e0\uff0c\u6545J=0<\/span>\u7b49\u4ef7\u4e8e\\nabla \\hat{f}=0<\/span>\uff0c\u56e0\u6b64\\hat{f}=H+k_BT\\log p<\/span>\u4e3a\u5e38\u6570\uff1a<\/p>\n

\\displaystyle H+k_BT\\log p=\\text{const}\\Rightarrow p(x)\\propto e^{-H(x)\/(k_BT)}<\/span><\/p>\n

\\frac{1}{k_BT}<\/span>\u901a\u5e38\u53ef\u4ee5\u5199\u4e3a\\beta<\/span>\uff0c\u56e0\u6b64p(x)\\propto e^{-\\beta H}<\/span>\uff0c\u8fd9\u5c31\u662f\u73bb\u5c14\u5179\u66fc\u5206\u5e03<\/p>\n

Edwards-Wilkinson\u65b9\u7a0b<\/h3>\n

\u7b2c\u56db\u7ae0 \u8865\u5145\u77e5\u8bc6<\/h2>\n

\u968f\u673a\u70ed\u529b\u5b66<\/h3>\n","protected":false},"excerpt":{"rendered":"

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