{"id":5432,"date":"2026-02-21T15:22:21","date_gmt":"2026-02-21T06:22:21","guid":{"rendered":"https:\/\/eternalsphere.net\/echoes\/?p=5432"},"modified":"2026-06-06T06:52:34","modified_gmt":"2026-06-05T21:52:34","slug":"gl5jwtnirgoqp10","status":"publish","type":"post","link":"https:\/\/eternalsphere.net\/echoes\/index.php\/2026\/02\/21\/gl5jwtnirgoqp10\/","title":{"rendered":"\u4f0a\u8f9b\u6a21\u578b\uff1a\u4ece\u5fae\u89c2\u81ea\u65cb\u5230\u5b8f\u89c2\u76f8\u53d8\u7684\u6700\u5c0f\u7406\u8bba"},"content":{"rendered":"\n

\u6458\u8981<\/h2>\n\n\n\n

\u4f0a\u8f9b\u6a21\u578b\uff08Ising model\uff09\u662f\u7edf\u8ba1\u7269\u7406\u4e0e\u76f8\u53d8\u7406\u8bba\u4e2d\u6700\u7ecf\u5178\u7684\u683c\u70b9\u81ea\u65cb\u6a21\u578b\u4e4b\u4e00\uff0c\u7528\u6700\u5c11\u7684\u81ea\u7531\u5ea6\uff08\u6bcf\u4e2a\u683c\u70b9\u53ea\u6709\u4e24\u6001\u81ea\u65cb s<\/mi>i<\/mi><\/msub>=<\/mo>\u00b1<\/mo>1<\/mn><\/mrow>s_i=\\pm1<\/annotation><\/semantics><\/math>si\u200b=\u00b11\uff09\u523b\u753b\u4e86\u201c\u5c40\u57df\u76f8\u4e92\u4f5c\u7528 + \u70ed\u6da8\u843d\u201d\u5982\u4f55\u4ea7\u751f\u5b8f\u89c2\u6709\u5e8f\uff08\u78c1\u5316\uff09\u4ee5\u53ca\u4e34\u754c\u76f8\u53d8\u3002\u672c\u6587\u4ee5\u8bba\u6587\u7efc\u8ff0\u7684\u5199\u6cd5\u7cfb\u7edf\u4ecb\u7ecd\u4f0a\u8f9b\u6a21\u578b\u7684\u5b9a\u4e49\u3001\u7edf\u8ba1\u529b\u5b66\u6846\u67b6\u3001\u5178\u578b\u53ef\u89c2\u6d4b\u91cf\u30011D\/2D \u7684\u5173\u952e\u7ed3\u8bba\u3001\u5e38\u7528\u89e3\u6790\u4e0e\u8fd1\u4f3c\u65b9\u6cd5\u3001\u6570\u503c\u6a21\u62df\u8981\u70b9\u3001\u4e34\u754c\u73b0\u8c61\u4e0e\u666e\u9002\u6027\uff0c\u5e76\u8bf4\u660e\u5176\u4e0e\u6982\u7387\u56fe\u6a21\u578b\/\u73bb\u5c14\u5179\u66fc\u5206\u5e03\u7684\u5bf9\u5e94\u5173\u7cfb\u3002<\/p>\n\n\n\n

\u5173\u952e\u8bcd<\/strong>\uff1a\u4f0a\u8f9b\u6a21\u578b\uff1b\u914d\u5206\u51fd\u6570\uff1b\u76f8\u53d8\uff1b\u4e34\u754c\u73b0\u8c61\uff1bMonte Carlo\uff1b\u91cd\u6574\u5316\u7fa4\uff1b\u666e\u9002\u6027<\/p>\n\n\n\n

\n\n\n\n

1. \u5f15\u8a00<\/h2>\n\n\n\n

\u5728\u8bb8\u591a\u7269\u7406\u7cfb\u7edf\u4e2d\uff0c\u5b8f\u89c2\u6027\u8d28\u5e76\u975e\u7b80\u5355\u53e0\u52a0\uff0c\u800c\u662f\u7531\u5927\u91cf\u81ea\u7531\u5ea6\u5728\u76f8\u4e92\u4f5c\u7528\u4e0e\u6da8\u843d\u4e4b\u95f4\u7ade\u4e89\u6240\u6d8c\u73b0\u3002\u94c1\u78c1\u4f53\u7684\u201c\u81ea\u53d1\u78c1\u5316\u201d\u3001\u6db2-\u6c14\u76f8\u53d8\u3001\u4e8c\u5143\u5408\u91d1\u6709\u5e8f\u5316\u7b49\u73b0\u8c61\u90fd\u5177\u6709\u5171\u540c\u7684\u7edf\u8ba1\u7ed3\u6784\uff1a\u5c40\u57df\u8026\u5408\u503e\u5411\u6709\u5e8f\uff0c\u6e29\u5ea6\u5f15\u5165\u65e0\u5e8f<\/strong>\u3002\u4f0a\u8f9b\u6a21\u578b\u4f5c\u4e3a\u8fd9\u7c7b\u73b0\u8c61\u7684\u6700\u5c0f\u8303\u5f0f\uff0c\u5177\u6709\u4ee5\u4e0b\u4ef7\u503c\uff1a<\/p>\n\n\n\n

    \n
  1. \u6700\u5c0f\u590d\u6742\u5ea6<\/strong>\uff1a\u4e24\u6001\u53d8\u91cf + \u6700\u8fd1\u90bb\u8026\u5408\u5373\u53ef\u51fa\u73b0\u4e34\u754c\u76f8\u53d8\uff08\u4e8c\u7ef4\u53ca\u4ee5\u4e0a\uff09\u3002<\/li>\n\n\n\n
  2. \u53ef\u89e3\u6790\u4e0e\u53ef\u6570\u503c<\/strong>\uff1a1D \u53ef\u4e25\u683c\u6c42\u89e3\u4e14\u65e0\u6709\u9650\u6e29\u5ea6\u76f8\u53d8\uff1b2D \u5728\u96f6\u5916\u573a\u4e0b\u6709 Onsager \u7cbe\u786e\u89e3\uff1b3D \u65e0\u89e3\u6790\u89e3\u4f46\u6570\u503c\u7814\u7a76\u6210\u719f\u3002<\/li>\n\n\n\n
  3. \u8de8\u5b66\u79d1\u6865\u6881<\/strong>\uff1a\u5176\u73bb\u5c14\u5179\u66fc\u5206\u5e03\u5f62\u5f0f\u7b49\u540c\u4e8e\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\uff08MRF\uff09\u4e0e\u80fd\u91cf\u6a21\u578b\uff0c\u5728\u673a\u5668\u5b66\u4e60\u3001\u56fe\u50cf\u5206\u5272\u3001\u793e\u4f1a\u52a8\u529b\u5b66\u4e2d\u5e7f\u6cdb\u590d\u7528\u3002<\/li>\n<\/ol>\n\n\n\n
    \n\n\n\n

    2. \u6a21\u578b\u5b9a\u4e49<\/h2>\n\n\n\n

    2.1 \u683c\u70b9\u4e0e\u81ea\u65cb\u53d8\u91cf<\/h3>\n\n\n\n

    \u5728 d<\/mi><\/mrow>d<\/annotation><\/semantics><\/math>d \u7ef4\u6676\u683c\uff08\u5982\u6b63\u65b9\u683c\u3001\u7acb\u65b9\u683c\uff09\u4e0a\uff0c\u6bcf\u4e2a\u683c\u70b9 i<\/mi>\u2208<\/mo>\u039b<\/mi><\/mrow>i\\in\\Lambda<\/annotation><\/semantics><\/math>i\u2208\u039b \u8d4b\u4e88\u4e8c\u503c\u81ea\u65cbs<\/mi>i<\/mi><\/msub>\u2208<\/mo>{<\/mo>+<\/mo>1<\/mn>,<\/mo>\u2212<\/mo>1<\/mn>}<\/mo>.<\/mi><\/mrow>s_i\\in\\{+1,-1\\}.<\/annotation><\/semantics><\/math>si\u200b\u2208{+1,\u22121}.<\/p>\n\n\n\n

    \u7cfb\u7edf\u89c4\u6a21\u4e3a N<\/mi>=<\/mo>\u2223<\/mi>\u039b<\/mi>\u2223<\/mi><\/mrow>N=|\\Lambda|<\/annotation><\/semantics><\/math>N=\u2223\u039b\u2223\u3002<\/p>\n\n\n\n

    2.2 \u54c8\u5bc6\u987f\u91cf\uff08\u80fd\u91cf\u51fd\u6570\uff09<\/h3>\n\n\n\n

    \u6700\u5e38\u7528\u7684\u6700\u8fd1\u90bb\u4f0a\u8f9b\u6a21\u578b\u54c8\u5bc6\u987f\u91cf\u4e3aH<\/mi>(<\/mo>{<\/mo>s<\/mi>}<\/mo>)<\/mo>=<\/mo>\u2212<\/mo>J<\/mi>\u2211<\/mo>\u27e8<\/mo>i<\/mi>,<\/mo>j<\/mi>\u27e9<\/mo><\/mrow><\/munder>s<\/mi>i<\/mi><\/msub>s<\/mi>j<\/mi><\/msub>\u2212<\/mo>h<\/mi>\u2211<\/mo>i<\/mi><\/munder>s<\/mi>i<\/mi><\/msub>,<\/mo><\/mrow>H(\\{s\\})=-J\\sum_{\\langle i,j\\rangle}s_is_j-h\\sum_i s_i,<\/annotation><\/semantics><\/math>H({s})=\u2212J\u27e8i,j\u27e9\u2211\u200bsi\u200bsj\u200b\u2212hi\u2211\u200bsi\u200b,<\/p>\n\n\n\n

    \u5176\u4e2d\uff1a<\/p>\n\n\n\n

      \n
    • \u27e8<\/mo>i<\/mi>,<\/mo>j<\/mi>\u27e9<\/mo><\/mrow>\\langle i,j\\rangle<\/annotation><\/semantics><\/math>\u27e8i,j\u27e9\uff1a\u6700\u8fd1\u90bb\u683c\u70b9\u5bf9\uff1b<\/li>\n\n\n\n
    • J<\/mi><\/mrow>J<\/annotation><\/semantics><\/math>J\uff1a\u4ea4\u6362\u8026\u5408\u5e38\u6570\n
        \n
      • J<\/mi>><\/mo>0<\/mn><\/mrow>J>0<\/annotation><\/semantics><\/math>J>0\uff1a\u94c1\u78c1\u8026\u5408\uff08\u8d8b\u5411\u540c\u5411\u6392\u5217\uff0c\u964d\u4f4e\u80fd\u91cf\uff09<\/li>\n\n\n\n
      • J<\/mi><<\/mo>0<\/mn><\/mrow>J<0<\/annotation><\/semantics><\/math>J<0\uff1a\u53cd\u94c1\u78c1\u8026\u5408\uff08\u8d8b\u5411\u53cd\u5411\u6392\u5217\uff09<\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • h<\/mi><\/mrow>h<\/annotation><\/semantics><\/math>h\uff1a\u5916\u78c1\u573a\uff08\u504f\u7f6e\u9879\uff09<\/li>\n<\/ul>\n\n\n\n

        \u7269\u7406\u89e3\u91ca<\/strong><\/p>\n\n\n\n

          \n
        • \u82e5\u4e24\u76f8\u90bb\u81ea\u65cb\u76f8\u540c\uff1as<\/mi>i<\/mi><\/msub>s<\/mi>j<\/mi><\/msub>=<\/mo>+<\/mo>1<\/mn><\/mrow>s_is_j=+1<\/annotation><\/semantics><\/math>si\u200bsj\u200b=+1\uff0c\u5728 J<\/mi>><\/mo>0<\/mn><\/mrow>J>0<\/annotation><\/semantics><\/math>J>0 \u65f6\u8be5\u5bf9\u8d21\u732e \u2212<\/mo>J<\/mi><\/mrow>-J<\/annotation><\/semantics><\/math>\u2212J\uff08\u66f4\u4f4e\u80fd\uff09<\/li>\n\n\n\n
        • \u82e5\u76f8\u53cd\uff1as<\/mi>i<\/mi><\/msub>s<\/mi>j<\/mi><\/msub>=<\/mo>\u2212<\/mo>1<\/mn><\/mrow>s_is_j=-1<\/annotation><\/semantics><\/math>si\u200bsj\u200b=\u22121\uff0c\u8d21\u732e +<\/mo>J<\/mi><\/mrow>+J<\/annotation><\/semantics><\/math>+J\uff08\u66f4\u9ad8\u80fd\uff09
          \u5916\u573a\u9879 \u2212<\/mo>h<\/mi>\u2211<\/mo>i<\/mi><\/msub>s<\/mi>i<\/mi><\/msub><\/mrow>-h\\sum_i s_i<\/annotation><\/semantics><\/math>\u2212h\u2211i\u200bsi\u200b \u4f7f\u78c1\u5316\u65b9\u5411\u5411 h<\/mi><\/mrow>h<\/annotation><\/semantics><\/math>h \u504f\u7f6e\u3002<\/li>\n<\/ul>\n\n\n\n
          \n\n\n\n

          3. \u7edf\u8ba1\u529b\u5b66\u6846\u67b6<\/h2>\n\n\n\n

          3.1 \u73bb\u5c14\u5179\u66fc\u6743\u91cd\u4e0e\u914d\u5206\u51fd\u6570<\/h3>\n\n\n\n

          \u5728\u6e29\u5ea6 T<\/mi><\/mrow>T<\/annotation><\/semantics><\/math>T \u4e0b\uff0c\u7cfb\u7edf\u5904\u4e8e\u914d\u7f6e {<\/mo>s<\/mi>}<\/mo><\/mrow>\\{s\\}<\/annotation><\/semantics><\/math>{s} \u7684\u6982\u7387\u4e3aP<\/mi>(<\/mo>{<\/mo>s<\/mi>}<\/mo>)<\/mo>=<\/mo>1<\/mn>Z<\/mi><\/mfrac>exp<\/mi>\u2061<\/mo>(<\/mo>\u2212<\/mo>\u03b2<\/mi>H<\/mi>(<\/mo>{<\/mo>s<\/mi>}<\/mo>)<\/mo>)<\/mo>,<\/mo><\/mspace>\u03b2<\/mi>=<\/mo>1<\/mn>k<\/mi>B<\/mi><\/msub>T<\/mi><\/mrow><\/mfrac>,<\/mo><\/mrow>P(\\{s\\})=\\frac{1}{Z}\\exp(-\\beta H(\\{s\\})), \\quad \\beta=\\frac{1}{k_BT},<\/annotation><\/semantics><\/math>P({s})=Z1\u200bexp(\u2212\u03b2H({s})),\u03b2=kB\u200bT1\u200b,<\/p>\n\n\n\n

          \u914d\u5206\u51fd\u6570Z<\/mi>=<\/mo>\u2211<\/mo>{<\/mo>s<\/mi>}<\/mo><\/mrow><\/munder>exp<\/mi>\u2061<\/mo>(<\/mo>\u2212<\/mo>\u03b2<\/mi>H<\/mi>(<\/mo>{<\/mo>s<\/mi>}<\/mo>)<\/mo>)<\/mo>,<\/mo><\/mrow>Z=\\sum_{\\{s\\}}\\exp(-\\beta H(\\{s\\})),<\/annotation><\/semantics><\/math>Z={s}\u2211\u200bexp(\u2212\u03b2H({s})),<\/p>\n\n\n\n

          \u5176\u4e2d\u6c42\u548c\u904d\u5386\u5168\u90e8 2<\/mn>N<\/mi><\/msup><\/mrow>2^N<\/annotation><\/semantics><\/math>2N \u4e2a\u914d\u7f6e\u3002\u5bf9\u5927 N<\/mi><\/mrow>N<\/annotation><\/semantics><\/math>N \u800c\u8a00\uff0cZ<\/mi><\/mrow>Z<\/annotation><\/semantics><\/math>Z \u7684\u7cbe\u786e\u8ba1\u7b97\u901a\u5e38\u4e0d\u53ef\u884c\uff0c\u8fd9\u4e5f\u662f\u76f8\u53d8\u7406\u8bba\u4e0e\u6570\u503c\u6a21\u62df\u7684\u91cd\u8981\u52a8\u673a\u3002<\/p>\n\n\n\n

          3.2 \u81ea\u7531\u80fd\u4e0e\u70ed\u529b\u5b66\u6781\u9650<\/h3>\n\n\n\n

          \u81ea\u7531\u80fdF<\/mi>=<\/mo>\u2212<\/mo>k<\/mi>B<\/mi><\/msub>T<\/mi>ln<\/mi>\u2061<\/mo>Z<\/mi>.<\/mi><\/mrow>F=-k_BT\\ln Z.<\/annotation><\/semantics><\/math>F=\u2212kB\u200bTlnZ.<\/p>\n\n\n\n

          \u70ed\u529b\u5b66\u6781\u9650\u5b9a\u4e49\u4e3a N<\/mi>\u2192<\/mo>\u221e<\/mi><\/mrow>N\\to\\infty<\/annotation><\/semantics><\/math>N\u2192\u221e \u4e14\u5bc6\u5ea6\u4fdd\u6301\u6709\u9650\u3002\u76f8\u53d8<\/strong>\u5e38\u4e0e\u81ea\u7531\u80fd\u7684\u975e\u89e3\u6790\u6027\uff08\u4f8b\u5982\u5bfc\u6570\u4e0d\u8fde\u7eed\u6216\u53d1\u6563\uff09\u76f8\u5173\u3002<\/p>\n\n\n\n
          \n\n\n\n

          4. \u4e3b\u8981\u53ef\u89c2\u6d4b\u91cf\u4e0e\u6da8\u843d\u516c\u5f0f<\/h2>\n\n\n\n

          4.1 \u78c1\u5316\u5f3a\u5ea6\uff08\u5e8f\u53c2\u91cf\uff09<\/h3>\n\n\n\n

          \u603b\u78c1\u5316M<\/mi>=<\/mo>\u2211<\/mo>i<\/mi><\/munder>s<\/mi>i<\/mi><\/msub>,<\/mo><\/mspace>m<\/mi>=<\/mo>1<\/mn>N<\/mi><\/mfrac>\u2211<\/mo>i<\/mi><\/munder>s<\/mi>i<\/mi><\/msub>,<\/mo><\/mrow>M=\\sum_i s_i, \\quad m=\\frac{1}{N}\\sum_i s_i,<\/annotation><\/semantics><\/math>M=i\u2211\u200bsi\u200b,m=N1\u200bi\u2211\u200bsi\u200b,<\/p>\n\n\n\n

          \u70ed\u5e73\u5747\u78c1\u5316\u27e8<\/mo>m<\/mi>\u27e9<\/mo>=<\/mo>\u2211<\/mo>{<\/mo>s<\/mi>}<\/mo><\/mrow><\/munder>m<\/mi>(<\/mo>{<\/mo>s<\/mi>}<\/mo>)<\/mo>P<\/mi>(<\/mo>{<\/mo>s<\/mi>}<\/mo>)<\/mo>.<\/mi><\/mrow>\\langle m\\rangle=\\sum_{\\{s\\}} m(\\{s\\})P(\\{s\\}).<\/annotation><\/semantics><\/math>\u27e8m\u27e9={s}\u2211\u200bm({s})P({s}).<\/p>\n\n\n\n

          \u5728 h<\/mi>=<\/mo>0<\/mn><\/mrow>h=0<\/annotation><\/semantics><\/math>h=0 \u4e14\u4f4e\u6e29\uff08\u4e8c\u7ef4\u53ca\u4ee5\u4e0a\uff09\u53ef\u80fd\u51fa\u73b0 \u81ea\u53d1\u5bf9\u79f0\u6027\u7834\u7f3a<\/strong>\uff1a\u27e8<\/mo>m<\/mi>\u27e9<\/mo>\u2260<\/mo>0<\/mn><\/mrow>\\langle m\\rangle\\neq 0<\/annotation><\/semantics><\/math>\u27e8m\u27e9\ue020=0\u3002<\/p>\n\n\n\n

          4.2 \u5185\u80fd\u4e0e\u6bd4\u70ed<\/h3>\n\n\n\n

          \u5185\u80fdU<\/mi>=<\/mo>\u27e8<\/mo>H<\/mi>\u27e9<\/mo>=<\/mo>\u2212<\/mo>\u2202<\/mi>ln<\/mi>\u2061<\/mo>Z<\/mi><\/mrow>\u2202<\/mi>\u03b2<\/mi><\/mrow><\/mfrac>.<\/mi><\/mrow>U=\\langle H\\rangle=-\\frac{\\partial \\ln Z}{\\partial \\beta}.<\/annotation><\/semantics><\/math>U=\u27e8H\u27e9=\u2212\u2202\u03b2\u2202lnZ\u200b.<\/p>\n\n\n\n

          \u6bd4\u70ed\uff08\u5355\u4f4d\u81ea\u65cb\uff09c<\/mi>=<\/mo>1<\/mn>N<\/mi><\/mfrac>\u2202<\/mi>U<\/mi><\/mrow>\u2202<\/mi>T<\/mi><\/mrow><\/mfrac>=<\/mo>\u03b2<\/mi>2<\/mn><\/msup>N<\/mi><\/mfrac>(<\/mo>\u27e8<\/mo>H<\/mi>2<\/mn><\/msup>\u27e9<\/mo>\u2212<\/mo>\u27e8<\/mo>H<\/mi>\u27e9<\/mo>2<\/mn><\/msup>)<\/mo>,<\/mo><\/mrow>c=\\frac{1}{N}\\frac{\\partial U}{\\partial T} =\\frac{\\beta^2}{N}\\Big(\\langle H^2\\rangle-\\langle H\\rangle^2\\Big),<\/annotation><\/semantics><\/math>c=N1\u200b\u2202T\u2202U\u200b=N\u03b22\u200b(\u27e8H2\u27e9\u2212\u27e8H\u27e92),<\/p>\n\n\n\n

          \u5373\u6bd4\u70ed\u4e0e\u80fd\u91cf\u6da8\u843d\u76f4\u63a5\u76f8\u5173\u3002<\/p>\n\n\n\n

          4.3 \u78c1\u5316\u7387<\/h3>\n\n\n\n

          \u03c7<\/mi>=<\/mo>\u2202<\/mi>\u27e8<\/mo>m<\/mi>\u27e9<\/mo><\/mrow>\u2202<\/mi>h<\/mi><\/mrow><\/mfrac>=<\/mo>\u03b2<\/mi>N<\/mi>(<\/mo>\u27e8<\/mo>m<\/mi>2<\/mn><\/msup>\u27e9<\/mo>\u2212<\/mo>\u27e8<\/mo>m<\/mi>\u27e9<\/mo>2<\/mn><\/msup>)<\/mo><\/mrow>\\chi=\\frac{\\partial \\langle m\\rangle}{\\partial h} =\\beta N\\Big(\\langle m^2\\rangle-\\langle m\\rangle^2\\Big)<\/annotation><\/semantics><\/math>\u03c7=\u2202h\u2202\u27e8m\u27e9\u200b=\u03b2N(\u27e8m2\u27e9\u2212\u27e8m\u27e92)<\/p>\n\n\n\n

          \uff08\u4e0d\u540c\u5b9a\u4e49\u5dee\u4e00\u4e2a N<\/mi><\/mrow>N<\/annotation><\/semantics><\/math>N \u7684\u56e0\u5b50\uff0c\u53d6\u51b3\u4e8e\u662f\u5426\u7528 m<\/mi><\/mrow>m<\/annotation><\/semantics><\/math>m \u6216 M<\/mi><\/mrow>M<\/annotation><\/semantics><\/math>M\uff09\u3002<\/p>\n\n\n\n

          4.4 \u5173\u8054\u51fd\u6570\u4e0e\u5173\u8054\u957f\u5ea6<\/h3>\n\n\n\n

          \u4e24\u70b9\u5173\u8054\u51fd\u6570C<\/mi>(<\/mo>r<\/mi>)<\/mo>=<\/mo>\u27e8<\/mo>s<\/mi>i<\/mi><\/msub>s<\/mi>i<\/mi>+<\/mo>r<\/mi><\/mrow><\/msub>\u27e9<\/mo>\u2212<\/mo>\u27e8<\/mo>s<\/mi>i<\/mi><\/msub>\u27e9<\/mo>\u27e8<\/mo>s<\/mi>i<\/mi>+<\/mo>r<\/mi><\/mrow><\/msub>\u27e9<\/mo>.<\/mi><\/mrow>C(r)=\\langle s_i s_{i+r}\\rangle-\\langle s_i\\rangle\\langle s_{i+r}\\rangle.<\/annotation><\/semantics><\/math>C(r)=\u27e8si\u200bsi+r\u200b\u27e9\u2212\u27e8si\u200b\u27e9\u27e8si+r\u200b\u27e9.<\/p>\n\n\n\n

          \u5728\u975e\u4e34\u754c\u533a\u901a\u5e38\u5448\u6307\u6570\u8870\u51cfC<\/mi>(<\/mo>r<\/mi>)<\/mo>\u223c<\/mo>e<\/mi>\u2212<\/mo>r<\/mi>\/<\/mi>\u03be<\/mi><\/mrow><\/msup>,<\/mo><\/mrow>C(r)\\sim e^{-r\/\\xi},<\/annotation><\/semantics><\/math>C(r)\u223ce\u2212r\/\u03be,<\/p>\n\n\n\n

          \u5176\u4e2d \u03be<\/mi><\/mrow>\\xi<\/annotation><\/semantics><\/math>\u03be \u4e3a\u5173\u8054\u957f\u5ea6\u3002\u4e34\u754c\u70b9\u9644\u8fd1 \u03be<\/mi>\u2192<\/mo>\u221e<\/mi><\/mrow>\\xi\\to\\infty<\/annotation><\/semantics><\/math>\u03be\u2192\u221e \u662f\u4e34\u754c\u73b0\u8c61\u7684\u6838\u5fc3\u7279\u5f81\u4e4b\u4e00\u3002<\/p>\n\n\n\n
          \n\n\n\n

          5. \u57fa\u672c\u7ed3\u8bba\uff1a\u4e0d\u540c\u7ef4\u5ea6\u7684\u76f8\u53d8<\/h2>\n\n\n\n

          5.1 \u4e00\u7ef4\uff081D\uff09\u6700\u8fd1\u90bb\u4f0a\u8f9b\u6a21\u578b\uff1a\u65e0\u6709\u9650\u6e29\u5ea6\u76f8\u53d8<\/h3>\n\n\n\n

          \u5bf9 1D \u6700\u8fd1\u90bb\u3001\u6709\u9650 J<\/mi><\/mrow>J<\/annotation><\/semantics><\/math>J\u3001h<\/mi>=<\/mo>0<\/mn><\/mrow>h=0<\/annotation><\/semantics><\/math>h=0\uff0c\u4e25\u683c\u7ed3\u8bba\u662f\uff1aT<\/mi>c<\/mi><\/msub>=<\/mo>0.<\/mn><\/mrow>T_c=0.<\/annotation><\/semantics><\/math>Tc\u200b=0.<\/p>\n\n\n\n

          \u76f4\u89c2\u539f\u56e0\uff1a1D \u4e2d\u201c\u7574\u58c1\u201d\uff08domain wall\uff09\u80fd\u91cf\u4ee3\u4ef7\u6709\u9650\uff0c\u4f46\u5176\u71b5\u589e\u8db3\u4ee5\u5728\u4efb\u610f T<\/mi>><\/mo>0<\/mn><\/mrow>T>0<\/annotation><\/semantics><\/math>T>0 \u4ea7\u751f\u5927\u91cf\u7574\u58c1\uff0c\u957f\u7a0b\u6709\u5e8f\u88ab\u7834\u574f\u3002<\/p>\n\n\n\n

          5.2 \u4e8c\u7ef4\uff082D\uff09\u65b9\u683c\u4f0a\u8f9b\u6a21\u578b\uff1a\u5b58\u5728\u6709\u9650\u6e29\u5ea6\u76f8\u53d8\uff08Onsager \u7cbe\u786e\u89e3\uff09<\/h3>\n\n\n\n

          \u5bf9 2D \u6b63\u65b9\u6676\u683c\u3001\u6700\u8fd1\u90bb\u3001h<\/mi>=<\/mo>0<\/mn><\/mrow>h=0<\/annotation><\/semantics><\/math>h=0\uff0c\u5b58\u5728\u4e34\u754c\u6e29\u5ea6 T<\/mi>c<\/mi><\/msub><\/mrow>T_c<\/annotation><\/semantics><\/math>Tc\u200b\uff0c\u6ee1\u8db3sinh<\/mi>\u2061<\/mo>\u2009\u2063<\/mtext>(<\/mo>2<\/mn>J<\/mi><\/mrow>k<\/mi>B<\/mi><\/msub>T<\/mi>c<\/mi><\/msub><\/mrow><\/mfrac>)<\/mo><\/mrow>=<\/mo>1.<\/mn><\/mrow>\\sinh\\!\\left(\\frac{2J}{k_BT_c}\\right)=1.<\/annotation><\/semantics><\/math>sinh(kB\u200bTc\u200b2J\u200b)=1.<\/p>\n\n\n\n

          \u8fd9\u662f\u76f8\u53d8\u7406\u8bba\u4e2d\u6700\u8457\u540d\u7684\u7cbe\u786e\u7ed3\u679c\u4e4b\u4e00\u3002\u4f4e\u6e29 T<\/mi><<\/mo>T<\/mi>c<\/mi><\/msub><\/mrow>T<T_c<\/annotation><\/semantics><\/math>T<Tc\u200b \u51fa\u73b0\u81ea\u53d1\u78c1\u5316\uff1b\u9ad8\u6e29 T<\/mi>><\/mo>T<\/mi>c<\/mi><\/msub><\/mrow>T>T_c<\/annotation><\/semantics><\/math>T>Tc\u200b \u78c1\u5316\u4e3a\u96f6\uff08\u5728 h<\/mi>=<\/mo>0<\/mn><\/mrow>h=0<\/annotation><\/semantics><\/math>h=0 \u7684\u5bf9\u79f0\u6761\u4ef6\u4e0b\uff09\u3002<\/p>\n\n\n\n

          5.3 \u4e09\u7ef4\uff083D\uff09\uff1a\u5b58\u5728\u76f8\u53d8\u4f46\u65e0\u89e3\u6790\u89e3<\/h3>\n\n\n\n

          3D \u4f0a\u8f9b\u6a21\u578b\u540c\u6837\u5177\u6709\u6709\u9650\u6e29\u5ea6\u76f8\u53d8\uff0c\u4f46\u7f3a\u4e4f\u7c7b\u4f3c 2D \u7684\u5c01\u95ed\u89e3\u6790\u89e3\u3002\u4e34\u754c\u6307\u6570\u7b49\u9700\u501f\u52a9\u9ad8\u7cbe\u5ea6 Monte Carlo\u3001\u7ea7\u6570\u5c55\u5f00\u4e0e\u91cd\u6574\u5316\u7fa4\u65b9\u6cd5\u786e\u5b9a\u3002<\/p>\n\n\n\n

          \n\n\n\n

          6. \u89e3\u6790\u4e0e\u8fd1\u4f3c\u65b9\u6cd5<\/h2>\n\n\n\n

          6.1 \u8f6c\u79fb\u77e9\u9635\u6cd5\uff08\u4ee5 1D \u4e3a\u4f8b\uff09<\/h3>\n\n\n\n

          1D \u6700\u8fd1\u90bb\u6a21\u578b\u53ef\u5199\u4e3a\u76f8\u90bb\u5bf9\u7684\u6743\u91cd\u4e58\u79ef\uff0c\u4ece\u800c\u5c06\u914d\u5206\u51fd\u6570\u8868\u8fbe\u4e3a\u8f6c\u79fb\u77e9\u9635 T<\/mi><\/mrow>T<\/annotation><\/semantics><\/math>T \u7684\u8ff9\uff1aZ<\/mi>=<\/mo>T<\/mi>r<\/mi><\/mrow>(<\/mo>T<\/mi>N<\/mi><\/msup>)<\/mo>.<\/mi><\/mrow>Z=\\mathrm{Tr}(T^N).<\/annotation><\/semantics><\/math>Z=Tr(TN).<\/p>\n\n\n\n

          \u77e9\u9635\u6700\u5927\u7279\u5f81\u503c \u03bb<\/mi>max<\/mi>\u2061<\/mo><\/msub><\/mrow>\\lambda_{\\max}<\/annotation><\/semantics><\/math>\u03bbmax\u200b \u51b3\u5b9a\u81ea\u7531\u80fd\u5bc6\u5ea6f<\/mi>=<\/mo>\u2212<\/mo>k<\/mi>B<\/mi><\/msub>T<\/mi>ln<\/mi>\u2061<\/mo>\u03bb<\/mi>max<\/mi>\u2061<\/mo><\/msub>.<\/mi><\/mrow>f=-k_BT\\ln \\lambda_{\\max}.<\/annotation><\/semantics><\/math>f=\u2212kB\u200bTln\u03bbmax\u200b.<\/p>\n\n\n\n

          \u8be5\u65b9\u6cd5\u6e05\u6670\u5c55\u793a\uff1a\u81ea\u7531\u80fd\u5728 T<\/mi>><\/mo>0<\/mn><\/mrow>T>0<\/annotation><\/semantics><\/math>T>0 \u89e3\u6790\uff0c\u4ece\u800c\u65e0\u76f8\u53d8\u3002<\/p>\n\n\n\n

          6.2 \u5e73\u5747\u573a\u7406\u8bba\uff08Mean-Field Theory, MFT\uff09<\/h3>\n\n\n\n

          \u5e73\u5747\u573a\u601d\u60f3\uff1a\u5c06\u90bb\u5c45\u81ea\u65cb\u7528\u5e73\u5747\u78c1\u5316 m<\/mi><\/mrow>m<\/annotation><\/semantics><\/math>m \u8fd1\u4f3c\uff0c\u5f97\u5230\u81ea\u6d3d\u65b9\u7a0bm<\/mi>=<\/mo>tanh<\/mi>\u2061<\/mo>(<\/mo>\u03b2<\/mi>(<\/mo>z<\/mi>J<\/mi>m<\/mi>+<\/mo>h<\/mi>)<\/mo>)<\/mo>,<\/mo><\/mrow>m=\\tanh\\big(\\beta(zJm+h)\\big),<\/annotation><\/semantics><\/math>m=tanh(\u03b2(zJm+h)),<\/p>\n\n\n\n

          \u5176\u4e2d z<\/mi><\/mrow>z<\/annotation><\/semantics><\/math>z \u4e3a\u914d\u4f4d\u6570\uff08\u6bcf\u4e2a\u683c\u70b9\u6700\u8fd1\u90bb\u6570\uff0c\u5982 2D \u65b9\u683c z<\/mi>=<\/mo>4<\/mn><\/mrow>z=4<\/annotation><\/semantics><\/math>z=4\uff09\u3002
          \u5728 h<\/mi>=<\/mo>0<\/mn><\/mrow>h=0<\/annotation><\/semantics><\/math>h=0 \u65f6\u7ebf\u6027\u5316 m<\/mi>\u2248<\/mo>\u03b2<\/mi>z<\/mi>J<\/mi>m<\/mi><\/mrow>m\\approx \\beta zJ m<\/annotation><\/semantics><\/math>m\u2248\u03b2zJm \u7ed9\u51fa\u5e73\u5747\u573a\u4e34\u754c\u6e29\u5ea6k<\/mi>B<\/mi><\/msub>T<\/mi>c<\/mi>M<\/mi>F<\/mi><\/mrow><\/msubsup>=<\/mo>z<\/mi>J<\/mi>.<\/mi><\/mrow>k_BT_c^{\\mathrm{MF}}=zJ.<\/annotation><\/semantics><\/math>kB\u200bTcMF\u200b=zJ.<\/p>\n\n\n\n

          \u5e73\u5747\u573a\u80fd\u7ed9\u51fa\u201c\u5b58\u5728\u76f8\u53d8\u201d\u7684\u5b9a\u6027\u56fe\u50cf\uff0c\u4f46\u5bf9\u4f4e\u7ef4\u7cfb\u7edf\u4e0e\u4e34\u754c\u6307\u6570\u5e38\u4e0d\u51c6\u786e\uff08\u5ffd\u7565\u6da8\u843d\uff09\u3002<\/p>\n\n\n\n

          6.3 Landau\uff08\u6717\u9053\uff09\u81ea\u7531\u80fd\u5c55\u5f00\uff08\u5e8f\u53c2\u91cf\u7406\u8bba\uff09<\/h3>\n\n\n\n

          \u5728\u5bf9\u79f0\u6027\u5141\u8bb8\u4e0b\uff0c\u5c06\u81ea\u7531\u80fd\u5bc6\u5ea6\u5c55\u5f00\u4e3af<\/mi>(<\/mo>m<\/mi>)<\/mo>=<\/mo>a<\/mi>(<\/mo>T<\/mi>)<\/mo>m<\/mi>2<\/mn><\/msup>+<\/mo>b<\/mi>m<\/mi>4<\/mn><\/msup>\u2212<\/mo>\u2005\u200a<\/mtext>h<\/mi>m<\/mi>+<\/mo>\u22ef<\/mo>\u2009<\/mtext>,<\/mo><\/mrow>f(m)=a(T)m^2+b m^4-\\;hm+\\cdots,<\/annotation><\/semantics><\/math>f(m)=a(T)m2+bm4\u2212hm+\u22ef,<\/p>\n\n\n\n

          \u5176\u4e2d b<\/mi>><\/mo>0<\/mn><\/mrow>b>0<\/annotation><\/semantics><\/math>b>0 \u4fdd\u8bc1\u7a33\u5b9a\uff0ca<\/mi>(<\/mo>T<\/mi>)<\/mo><\/mrow>a(T)<\/annotation><\/semantics><\/math>a(T) \u5728 T<\/mi>c<\/mi><\/msub><\/mrow>T_c<\/annotation><\/semantics><\/math>Tc\u200b \u9644\u8fd1\u6539\u53f7\u3002
          \u8fd9\u63d0\u4f9b\u4e86\u5bf9\u79f0\u6027\u7834\u7f3a\u4e0e\u4e8c\u9636\u76f8\u53d8\u7684\u901a\u7528\u6846\u67b6\uff0c\u5e76\u80fd\u63a8\u5bfc\u4e00\u7ec4\u201c\u5e73\u5747\u573a\u4e34\u754c\u6307\u6570\u201d\uff0c\u4f46\u4f9d\u7136\u5ffd\u7565\u4e86\u4e34\u754c\u6da8\u843d\u5bf9\u771f\u5b9e\u6307\u6570\u7684\u4fee\u6b63\u3002<\/p>\n\n\n\n

          6.4 \u91cd\u6574\u5316\u7fa4\uff08Renormalization Group, RG\uff09<\/h3>\n\n\n\n

          RG \u5c06\u7cfb\u7edf\u8fdb\u884c\u7c97\u7c92\u5316\u5e76\u7814\u7a76\u8026\u5408\u5e38\u6570\u5728\u5c3a\u5ea6\u53d8\u6362\u4e0b\u7684\u6d41\u52a8\u3002\u5176\u6838\u5fc3\u8d21\u732e\uff1a<\/p>\n\n\n\n

            \n
          • \u89e3\u91ca\u201c\u666e\u9002\u6027\u201d\uff1a\u4e34\u754c\u6307\u6570\u4e3b\u8981\u7531\u7ef4\u5ea6 d<\/mi><\/mrow>d<\/annotation><\/semantics><\/math>d \u4e0e\u5e8f\u53c2\u91cf\u5bf9\u79f0\u6027\uff08\u4f0a\u8f9b\u4e3a Z<\/mi>2<\/mn><\/msub><\/mrow>Z_2<\/annotation><\/semantics><\/math>Z2\u200b\uff09\u51b3\u5b9a\uff1b<\/li>\n\n\n\n
          • \u7ed9\u51fa\u4e34\u754c\u56fa\u5b9a\u70b9\u3001\u76f8\u5173\u7b97\u7b26\u3001\u5c3a\u5ea6\u4e0d\u53d8\u6027\u7b49\u6982\u5ff5\uff1b<\/li>\n\n\n\n
          • \u63d0\u4f9b\u4e34\u754c\u6307\u6570\u7684\u7cfb\u7edf\u8fd1\u4f3c\uff08\u5982 \u03f5<\/mi>=<\/mo>4<\/mn>\u2212<\/mo>d<\/mi><\/mrow>\\epsilon=4-d<\/annotation><\/semantics><\/math>\u03f5=4\u2212d \u5c55\u5f00\u7b49\uff09\u3002<\/li>\n<\/ul>\n\n\n\n
            \n\n\n\n

            7. \u6570\u503c\u6a21\u62df\uff1aMonte Carlo \u7814\u7a76\u4f0a\u8f9b\u6a21\u578b<\/h2>\n\n\n\n

            7.1 \u4e3a\u4ec0\u4e48\u9700\u8981\u6570\u503c<\/h3>\n\n\n\n

            \u5bf9 2D h<\/mi>\u2260<\/mo>0<\/mn><\/mrow>h\\neq 0<\/annotation><\/semantics><\/math>h\ue020=0 \u6216 3D \u7cfb\u7edf\uff0c\u914d\u5206\u51fd\u6570\u4e0e\u70ed\u5e73\u5747\u96be\u4ee5\u89e3\u6790\u6c42\u5f97\uff0cMonte Carlo \u901a\u8fc7\u4ece P<\/mi>(<\/mo>{<\/mo>s<\/mi>}<\/mo>)<\/mo><\/mrow>P(\\{s\\})<\/annotation><\/semantics><\/math>P({s}) \u91c7\u6837\u6765\u8fd1\u4f3c\u70ed\u5e73\u5747\uff1a\u27e8<\/mo>O<\/mi>\u27e9<\/mo>\u2248<\/mo>1<\/mn>N<\/mi>sample<\/mtext><\/msub><\/mfrac>\u2211<\/mo>t<\/mi>=<\/mo>1<\/mn><\/mrow>N<\/mi>sample<\/mtext><\/msub><\/munderover>O<\/mi>(<\/mo>{<\/mo>s<\/mi>}<\/mo>t<\/mi><\/msub>)<\/mo>.<\/mi><\/mrow>\\langle O\\rangle \\approx \\frac{1}{N_{\\text{sample}}}\\sum_{t=1}^{N_{\\text{sample}}} O(\\{s\\}_t).<\/annotation><\/semantics><\/math>\u27e8O\u27e9\u2248Nsample\u200b1\u200bt=1\u2211Nsample\u200b\u200bO({s}t\u200b).<\/p>\n\n\n\n

            7.2 Metropolis \u7b97\u6cd5\uff08\u5355\u81ea\u65cb\u7ffb\u8f6c\uff09<\/h3>\n\n\n\n

            \u5178\u578b\u6b65\u9aa4\uff1a<\/p>\n\n\n\n

              \n
            1. \u968f\u673a\u9009\u53d6\u683c\u70b9 i<\/mi><\/mrow>i<\/annotation><\/semantics><\/math>i\uff0c\u5c1d\u8bd5\u7ffb\u8f6c s<\/mi>i<\/mi><\/msub>\u2192<\/mo>\u2212<\/mo>s<\/mi>i<\/mi><\/msub><\/mrow>s_i\\to -s_i<\/annotation><\/semantics><\/math>si\u200b\u2192\u2212si\u200b\uff1b<\/li>\n\n\n\n
            2. \u8ba1\u7b97\u80fd\u91cf\u5dee \u0394<\/mi>E<\/mi><\/mrow>\\Delta E<\/annotation><\/semantics><\/math>\u0394E\uff08\u53ea\u9700\u770b\u5c40\u57df\u90bb\u5c45\uff09\uff1b<\/li>\n\n\n\n
            3. \u4ee5\u63a5\u53d7\u7387<\/li>\n<\/ol>\n\n\n\n

              p<\/mi>=<\/mo>min<\/mi>\u2061<\/mo>(<\/mo>1<\/mn>,<\/mo>e<\/mi>\u2212<\/mo>\u03b2<\/mi>\u0394<\/mi>E<\/mi><\/mrow><\/msup>)<\/mo><\/mrow><\/mrow>p=\\min\\left(1,e^{-\\beta \\Delta E}\\right)<\/annotation><\/semantics><\/math>p=min(1,e\u2212\u03b2\u0394E)<\/p>\n\n\n\n

              \u63a5\u53d7\u6216\u62d2\u7edd\u3002<\/p>\n\n\n\n

              \u4f18\u70b9\uff1a\u5b9e\u73b0\u7b80\u5355\u3002
              \u7f3a\u70b9\uff1a\u4e34\u754c\u70b9\u9644\u8fd1\u5b58\u5728\u201c\u4e34\u754c\u6162\u5316\u201d\uff08critical slowing down\uff09\uff0c\u81ea\u76f8\u5173\u65f6\u95f4\u53d8\u957f\u3002<\/p>\n\n\n\n

              7.3 \u7fa4\u96c6\u7b97\u6cd5\uff08Wolff \/ Swendsen\u2013Wang\uff09<\/h3>\n\n\n\n

              \u5728\u4e34\u754c\u9644\u8fd1\u901a\u8fc7\u7ffb\u8f6c\u7c07\uff08cluster\uff09\u800c\u975e\u5355\u81ea\u65cb\uff0c\u663e\u8457\u51cf\u5c11\u6162\u5316\u3002
              \u8fd9\u4e9b\u7b97\u6cd5\u5229\u7528\u94c1\u78c1\u8026\u5408\u5bfc\u81f4\u7684\u540c\u5411\u76f8\u5173\uff0c\u6784\u9020\u968f\u673a\u952e\u5e76\u751f\u6210\u7c07\uff0c\u518d\u6574\u4f53\u7ffb\u8f6c\uff0c\u63d0\u9ad8\u91c7\u6837\u6548\u7387\u3002<\/p>\n\n\n\n

              7.4 \u6709\u9650\u5c3a\u5bf8\u6807\u5ea6\uff08Finite-Size Scaling\uff09<\/h3>\n\n\n\n

              \u5b9e\u9645\u6a21\u62df\u53ea\u80fd\u7528\u6709\u9650 L<\/mi>d<\/mi><\/msup><\/mrow>L^d<\/annotation><\/semantics><\/math>Ld \u7cfb\u7edf\u3002\u4e34\u754c\u70b9\u9644\u8fd1\u5e38\u7528\u6709\u9650\u5c3a\u5bf8\u6807\u5ea6\u5206\u6790\uff1a<\/p>\n\n\n\n
                \n
              • \u5173\u8054\u957f\u5ea6 \u03be<\/mi><\/mrow>\\xi<\/annotation><\/semantics><\/math>\u03be \u5728\u6709\u9650\u7cfb\u7edf\u4e2d\u88ab L<\/mi><\/mrow>L<\/annotation><\/semantics><\/math>L \u622a\u65ad\uff1b<\/li>\n\n\n\n
              • \u53ef\u89c2\u6d4b\u91cf\u968f L<\/mi><\/mrow>L<\/annotation><\/semantics><\/math>L \u7684\u6807\u5ea6\u884c\u4e3a\u53ef\u7528\u4e8e\u4f30\u8ba1 T<\/mi>c<\/mi><\/msub><\/mrow>T_c<\/annotation><\/semantics><\/math>Tc\u200b \u4e0e\u4e34\u754c\u6307\u6570\u3002<\/li>\n<\/ul>\n\n\n\n
                \n\n\n\n

                8. \u4e34\u754c\u73b0\u8c61\u4e0e\u666e\u9002\u6027\uff08\u6982\u5ff5\u6027\u603b\u7ed3\uff09<\/h2>\n\n\n\n

                8.1 \u4e34\u754c\u70b9\u7684\u5178\u578b\u884c\u4e3a<\/h3>\n\n\n\n

                \u5728 T<\/mi>\u2192<\/mo>T<\/mi>c<\/mi><\/msub><\/mrow>T\\to T_c<\/annotation><\/semantics><\/math>T\u2192Tc\u200b \u9644\u8fd1\uff0c\u5e38\u89c1\u5e42\u5f8b\uff08\u793a\u610f\uff09\uff1am<\/mi>\u223c<\/mo>(<\/mo>T<\/mi>c<\/mi><\/msub>\u2212<\/mo>T<\/mi>)<\/mo>\u03b2<\/mi>c<\/mi>r<\/mi>i<\/mi>t<\/mi><\/mrow><\/msub><\/msup>,<\/mo><\/mspace>\u03c7<\/mi>\u223c<\/mo>\u2223<\/mi>T<\/mi>\u2212<\/mo>T<\/mi>c<\/mi><\/msub>\u2223<\/mi>\u2212<\/mo>\u03b3<\/mi><\/mrow><\/msup>,<\/mo><\/mspace>\u03be<\/mi>\u223c<\/mo>\u2223<\/mi>T<\/mi>\u2212<\/mo>T<\/mi>c<\/mi><\/msub>\u2223<\/mi>\u2212<\/mo>\u03bd<\/mi><\/mrow><\/msup>.<\/mi><\/mrow>m \\sim (T_c-T)^{\\beta_{\\mathrm{crit}}},\\quad \\chi \\sim |T-T_c|^{-\\gamma},\\quad \\xi \\sim |T-T_c|^{-\\nu}.<\/annotation><\/semantics><\/math>m\u223c(Tc\u200b\u2212T)\u03b2crit\u200b,\u03c7\u223c\u2223T\u2212Tc\u200b\u2223\u2212\u03b3,\u03be\u223c\u2223T\u2212Tc\u200b\u2223\u2212\u03bd.<\/p>\n\n\n\n

                \uff08\u6ce8\u610f\uff1a\u8fd9\u91cc\u7684 \u03b2<\/mi>c<\/mi>r<\/mi>i<\/mi>t<\/mi><\/mrow><\/msub><\/mrow>\\beta_{\\mathrm{crit}}<\/annotation><\/semantics><\/math>\u03b2crit\u200b \u662f\u4e34\u754c\u6307\u6570\uff0c\u548c\u7edf\u8ba1\u529b\u5b66\u4e2d\u7684 \u03b2<\/mi>=<\/mo>1<\/mn>\/<\/mi>k<\/mi>B<\/mi><\/msub>T<\/mi><\/mrow>\\beta=1\/k_BT<\/annotation><\/semantics><\/math>\u03b2=1\/kB\u200bT \u4e0d\u540c\u3002\uff09<\/p>\n\n\n\n

                8.2 \u666e\u9002\u6027\u7c7b<\/h3>\n\n\n\n

                \u4f0a\u8f9b\u6a21\u578b\u5bf9\u5e94 Z2Z_2Z2\u200b<\/strong> \u5bf9\u79f0\u6027\u3001\u6807\u91cf\u5e8f\u53c2\u91cf\u7684\u666e\u9002\u6027\u7c7b\u3002\u8bb8\u591a\u770b\u4f3c\u4e0d\u540c\u7684\u7269\u7406\u7cfb\u7edf\uff0c\u53ea\u8981\u6ee1\u8db3\u76f8\u540c\u5bf9\u79f0\u6027\u4e0e\u7ef4\u5ea6\uff0c\u4e34\u754c\u6307\u6570\u4e0e\u4e34\u754c\u884c\u4e3a\u53ef\u76f8\u540c\u3002<\/p>\n\n\n\n

                \n\n\n\n

                9. \u4e0e\u6982\u7387\u6a21\u578b\/\u673a\u5668\u5b66\u4e60\u7684\u5bf9\u5e94<\/h2>\n\n\n\n

                \u4f0a\u8f9b\u6a21\u578b\u7684\u6982\u7387\u5206\u5e03\u53ef\u91cd\u5199\u4e3a\u80fd\u91cf\u6a21\u578b\u5f62\u5f0f\uff1aP<\/mi>(<\/mo>{<\/mo>s<\/mi>}<\/mo>)<\/mo>=<\/mo>1<\/mn>Z<\/mi><\/mfrac>exp<\/mi>\u2061<\/mo>(<\/mo>\u03b2<\/mi>J<\/mi>\u2211<\/mo>\u27e8<\/mo>i<\/mi>,<\/mo>j<\/mi>\u27e9<\/mo><\/mrow><\/munder>s<\/mi>i<\/mi><\/msub>s<\/mi>j<\/mi><\/msub>+<\/mo>\u03b2<\/mi>h<\/mi>\u2211<\/mo>i<\/mi><\/munder>s<\/mi>i<\/mi><\/msub>)<\/mo><\/mrow>.<\/mi><\/mrow>P(\\{s\\})=\\frac{1}{Z}\\exp\\left(\\beta J\\sum_{\\langle i,j\\rangle}s_is_j+\\beta h\\sum_i s_i\\right).<\/annotation><\/semantics><\/math>P({s})=Z1\u200bexp\u200b\u03b2J\u27e8i,j\u27e9\u2211\u200bsi\u200bsj\u200b+\u03b2hi\u2211\u200bsi\u200b\u200b.<\/p>\n\n\n\n

                \u8fd9\u4e0e\u9a6c\u5c14\u53ef\u592b\u968f\u673a\u573a\uff08MRF\uff09\u3001\u73bb\u5c14\u5179\u66fc\u673a\uff08Boltzmann Machine\uff09\u7ed3\u6784\u4e00\u81f4\uff1a<\/p>\n\n\n\n

                  \n
                • s<\/mi>i<\/mi><\/msub><\/mrow>s_i<\/annotation><\/semantics><\/math>si\u200b \u662f\u4e8c\u5143\u968f\u673a\u53d8\u91cf\uff1b<\/li>\n\n\n\n
                • \u2211<\/mo>s<\/mi>i<\/mi><\/msub>s<\/mi>j<\/mi><\/msub><\/mrow>\\sum s_is_j<\/annotation><\/semantics><\/math>\u2211si\u200bsj\u200b \u662f\u6210\u5bf9\u52bf\u51fd\u6570\uff1b<\/li>\n\n\n\n
                • \u2211<\/mo>s<\/mi>i<\/mi><\/msub><\/mrow>\\sum s_i<\/annotation><\/semantics><\/math>\u2211si\u200b \u662f\u504f\u7f6e\u9879\uff1b<\/li>\n\n\n\n
                • Z<\/mi><\/mrow>Z<\/annotation><\/semantics><\/math>Z \u662f\u5f52\u4e00\u5316\u5e38\u6570\uff08\u673a\u5668\u5b66\u4e60\u4e2d\u5e38\u79f0\u4e3a\u201c\u914d\u5206\u51fd\u6570\u201d\uff0c\u4e5f\u5f80\u5f80\u96be\u7b97\uff09\u3002<\/li>\n<\/ul>\n\n\n\n

                  \u56e0\u6b64\uff0c\u4f0a\u8f9b\u6a21\u578b\u4e0d\u4ec5\u662f\u7269\u7406\u6a21\u578b\uff0c\u4e5f\u662f\u4e00\u7c7b\u901a\u7528\u7684\u4e8c\u503c\u56fe\u6a21\u578b\u3002<\/p>\n\n\n\n

                  \n\n\n\n

                  10. \u8ba8\u8bba\uff1a\u4f0a\u8f9b\u6a21\u578b\u201c\u6700\u5c0f\u4f46\u4e0d\u7b80\u5355\u201d\u7684\u539f\u56e0<\/h2>\n\n\n\n
                    \n
                  1. \u6700\u5c0f\u81ea\u7531\u5ea6\u4ecd\u80fd\u4ea7\u751f\u4e34\u754c\u73b0\u8c61<\/strong>\uff1a\u4e24\u6001\u53d8\u91cf\u5df2\u8db3\u591f\u51fa\u73b0\u81ea\u53d1\u5bf9\u79f0\u6027\u7834\u7f3a\u4e0e\u4e34\u754c\u70b9\u3002<\/li>\n\n\n\n
                  2. \u4f4e\u7ef4\u6da8\u843d\u7684\u51b3\u5b9a\u6027\u4f5c\u7528<\/strong>\uff1a1D \u65e0\u76f8\u53d8\u30012D \u6709\u76f8\u53d8\uff0c\u8bf4\u660e\u7ef4\u5ea6\u4e0e\u6da8\u843d\u5728\u76f8\u53d8\u4e2d\u4e0d\u53ef\u5ffd\u89c6\u3002<\/li>\n\n\n\n
                  3. \u89e3\u6790\u4e0e\u6570\u503c\u4e92\u8865<\/strong>\uff1a2D h<\/mi>=<\/mo>0<\/mn><\/mrow>h=0<\/annotation><\/semantics><\/math>h=0 \u7684\u7cbe\u786e\u89e3\u4e3a\u7406\u8bba\u63d0\u4f9b\u951a\u70b9\uff1b\u66f4\u4e00\u822c\u60c5\u5f62\u4f9d\u8d56\u6570\u503c\u4e0e RG\u3002<\/li>\n\n\n\n
                  4. \u8de8\u5b66\u79d1\u53ef\u8fc1\u79fb<\/strong>\uff1a\u5176\u201c\u80fd\u91cf-\u6982\u7387\u201d\u7ed3\u6784\u4f7f\u4e4b\u80fd\u81ea\u7136\u8fc1\u79fb\u5230\u8ba1\u7b97\u4e0e\u6570\u636e\u5efa\u6a21\u9886\u57df\u3002<\/li>\n<\/ol>\n\n\n\n
                    \n\n\n\n

                    11. \u7ed3\u8bba<\/h2>\n\n\n\n

                    \u4f0a\u8f9b\u6a21\u578b\u7528\u6700\u7cbe\u70bc\u7684\u5f62\u5f0f\u5c55\u793a\u4e86\u7edf\u8ba1\u7269\u7406\u7684\u6838\u5fc3\u903b\u8f91\uff1a\u5c40\u57df\u76f8\u4e92\u4f5c\u7528\u5728\u70ed\u6da8\u843d\u4e0b\u5982\u4f55\u5f62\u6210\uff08\u6216\u7834\u574f\uff09\u5b8f\u89c2\u6709\u5e8f\uff0c\u5e76\u5728\u7279\u5b9a\u7ef4\u5ea6\u4e0e\u6e29\u5ea6\u4e0b\u51fa\u73b0\u4e34\u754c\u76f8\u53d8\u3002\u5b83\u7684\u7406\u8bba\u5de5\u5177\u7bb1\uff08\u914d\u5206\u51fd\u6570\u3001\u8f6c\u79fb\u77e9\u9635\u3001\u5e73\u5747\u573a\u3001Landau \u7406\u8bba\u3001RG\u3001Monte Carlo\u3001\u6709\u9650\u5c3a\u5bf8\u6807\u5ea6\uff09\u6784\u6210\u4e86\u5b66\u4e60\u76f8\u53d8\u4e0e\u4e34\u754c\u73b0\u8c61\u7684\u6807\u51c6\u8def\u5f84\uff0c\u540c\u65f6\u4e5f\u4e3a\u80fd\u91cf\u6a21\u578b\u4e0e\u6982\u7387\u56fe\u6a21\u578b\u63d0\u4f9b\u4e86\u57fa\u7840\u6a21\u677f\u3002<\/p>\n\n\n\n

                    \n\n\n\n

                    \u7b26\u53f7\u8868<\/h2>\n\n\n\n
                      \n
                    • s<\/mi>i<\/mi><\/msub><\/mrow>s_i<\/annotation><\/semantics><\/math>si\u200b\uff1a\u7b2c i<\/mi><\/mrow>i<\/annotation><\/semantics><\/math>i \u4e2a\u683c\u70b9\u81ea\u65cb\uff0c\u53d6 \u00b1<\/mo>1<\/mn><\/mrow>\\pm1<\/annotation><\/semantics><\/math>\u00b11<\/li>\n\n\n\n
                    • \u039b<\/mi><\/mrow>\\Lambda<\/annotation><\/semantics><\/math>\u039b\uff1a\u6676\u683c\u96c6\u5408\uff1bN<\/mi>=<\/mo>\u2223<\/mi>\u039b<\/mi>\u2223<\/mi><\/mrow>N=|\\Lambda|<\/annotation><\/semantics><\/math>N=\u2223\u039b\u2223 \u4e3a\u683c\u70b9\u6570<\/li>\n\n\n\n
                    • \u27e8<\/mo>i<\/mi>,<\/mo>j<\/mi>\u27e9<\/mo><\/mrow>\\langle i,j\\rangle<\/annotation><\/semantics><\/math>\u27e8i,j\u27e9\uff1a\u6700\u8fd1\u90bb\u683c\u70b9\u5bf9\u6c42\u548c<\/li>\n\n\n\n
                    • J<\/mi><\/mrow>J<\/annotation><\/semantics><\/math>J\uff1a\u4ea4\u6362\u8026\u5408\u5e38\u6570<\/li>\n\n\n\n
                    • h<\/mi><\/mrow>h<\/annotation><\/semantics><\/math>h\uff1a\u5916\u78c1\u573a<\/li>\n\n\n\n
                    • H<\/mi>(<\/mo>{<\/mo>s<\/mi>}<\/mo>)<\/mo><\/mrow>H(\\{s\\})<\/annotation><\/semantics><\/math>H({s})\uff1a\u54c8\u5bc6\u987f\u91cf\uff08\u80fd\u91cf\uff09<\/li>\n\n\n\n
                    • T<\/mi><\/mrow>T<\/annotation><\/semantics><\/math>T\uff1a\u6e29\u5ea6\uff1bk<\/mi>B<\/mi><\/msub><\/mrow>k_B<\/annotation><\/semantics><\/math>kB\u200b\uff1a\u73bb\u5c14\u5179\u66fc\u5e38\u6570\uff1b\u03b2<\/mi>=<\/mo>1<\/mn>\/<\/mi>(<\/mo>k<\/mi>B<\/mi><\/msub>T<\/mi>)<\/mo><\/mrow>\\beta=1\/(k_BT)<\/annotation><\/semantics><\/math>\u03b2=1\/(kB\u200bT)<\/li>\n\n\n\n
                    • Z<\/mi><\/mrow>Z<\/annotation><\/semantics><\/math>Z\uff1a\u914d\u5206\u51fd\u6570<\/li>\n\n\n\n
                    • F<\/mi><\/mrow>F<\/annotation><\/semantics><\/math>F\uff1a\u81ea\u7531\u80fd<\/li>\n\n\n\n
                    • M<\/mi>=<\/mo>\u2211<\/mo>i<\/mi><\/msub>s<\/mi>i<\/mi><\/msub><\/mrow>M=\\sum_i s_i<\/annotation><\/semantics><\/math>M=\u2211i\u200bsi\u200b\uff1a\u603b\u78c1\u5316\uff1bm<\/mi>=<\/mo>M<\/mi>\/<\/mi>N<\/mi><\/mrow>m=M\/N<\/annotation><\/semantics><\/math>m=M\/N \u4e3a\u5355\u4f4d\u78c1\u5316<\/li>\n\n\n\n
                    • U<\/mi>=<\/mo>\u27e8<\/mo>H<\/mi>\u27e9<\/mo><\/mrow>U=\\langle H\\rangle<\/annotation><\/semantics><\/math>U=\u27e8H\u27e9\uff1a\u5185\u80fd<\/li>\n\n\n\n
                    • c<\/mi><\/mrow>c<\/annotation><\/semantics><\/math>c\uff1a\u6bd4\u70ed\uff08\u5355\u4f4d\u81ea\u65cb\uff09<\/li>\n\n\n\n
                    • \u03c7<\/mi><\/mrow>\\chi<\/annotation><\/semantics><\/math>\u03c7\uff1a\u78c1\u5316\u7387<\/li>\n\n\n\n
                    • C<\/mi>(<\/mo>r<\/mi>)<\/mo><\/mrow>C(r)<\/annotation><\/semantics><\/math>C(r)\uff1a\u4e24\u70b9\u5173\u8054\u51fd\u6570\uff1b\u03be<\/mi><\/mrow>\\xi<\/annotation><\/semantics><\/math>\u03be\uff1a\u5173\u8054\u957f\u5ea6<\/li>\n\n\n\n
                    • d<\/mi><\/mrow>d<\/annotation><\/semantics><\/math>d\uff1a\u7a7a\u95f4\u7ef4\u5ea6\uff1bz<\/mi><\/mrow>z<\/annotation><\/semantics><\/math>z\uff1a\u914d\u4f4d\u6570\uff08\u6700\u8fd1\u90bb\u6570\uff09<\/li>\n<\/ul>\n\n\n\n
                      \n\n\n\n

                      \u53c2\u8003\u9605\u8bfb\u5efa\u8bae\uff08\u53ef\u68c0\u7d22\u5173\u952e\u8bcd\uff09<\/h2>\n\n\n\n
                        \n
                      • \u201cIsing model\u201d \u7edf\u8ba1\u7269\u7406\u6559\u6750\u7ae0\u8282\uff08\u5982 Pathria\u3001Reichl\u3001Kardar \u7b49\uff09<\/li>\n\n\n\n
                      • \u201cOnsager solution 2D Ising model\u201d<\/li>\n\n\n\n
                      • \u201cMetropolis algorithm Ising model\u201d<\/li>\n\n\n\n
                      • \u201cWolff cluster algorithm\u201d<\/li>\n\n\n\n
                      • \u201cFinite size scaling Ising model\u201d<\/li>\n\n\n\n
                      • \u201cRenormalization group Ising universality class\u201d<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"

                        \u6458\u8981 \u4f0a\u8f9b\u6a21\u578b\uff08Ising model\uff09\u662f\u7edf\u8ba1\u7269\u7406\u4e0e\u76f8\u53d8\u7406\u8bba\u4e2d\u6700\u7ecf\u5178\u7684\u683c\u70b9\u81ea\u65cb\u6a21\u578b\u4e4b\u4e00\uff0c\u7528\u6700\u5c11\u7684\u81ea\u7531\u5ea6\uff08\u6bcf\u4e2a\u683c\u70b9\u53ea\u6709\u4e24\u6001\u81ea\u65cb si=\u00b11s_i=\\pm1si\u200b=\u00b11\uff09\u523b\u753b\u4e86\u201c\u5c40\u57df\u76f8\u4e92\u4f5c\u7528 + \u70ed\u6da8\u843d\u201d\u5982\u4f55\u4ea7\u751f\u5b8f\u89c2\u6709\u5e8f\uff08\u78c1\u5316\uff09\u4ee5\u53ca\u4e34\u754c\u76f8\u53d8\u3002\u672c\u6587\u4ee5\u8bba\u6587\u7efc\u8ff0\u7684\u5199\u6cd5\u7cfb\u7edf\u4ecb\u7ecd\u4f0a\u8f9b\u6a21\u578b\u7684\u5b9a\u4e49\u3001\u7edf\u8ba1\u529b\u5b66\u6846\u67b6\u3001\u5178\u578b\u53ef\u89c2\u6d4b\u91cf\u30011D\/2D \u7684\u5173\u952e\u7ed3\u8bba\u3001\u5e38\u7528\u89e3\u6790\u4e0e\u8fd1\u4f3c\u65b9\u6cd5\u3001\u6570\u503c\u6a21\u62df\u8981\u70b9\u3001\u4e34\u754c\u73b0\u8c61\u4e0e\u666e\u9002\u6027\uff0c\u5e76\u8bf4\u660e\u5176\u4e0e\u6982\u7387\u56fe\u6a21\u578b\/\u73bb\u5c14\u5179\u66fc\u5206\u5e03\u7684\u5bf9\u5e94\u5173\u7cfb\u3002 \u5173\u952e\u8bcd\uff1a\u4f0a\u8f9b\u6a21\u578b\uff1b\u914d\u5206\u51fd\u6570\uff1b\u76f8\u53d8\uff1b\u4e34\u754c\u73b0\u8c61\uff1bMonte Carlo\uff1b\u91cd\u6574\u5316\u7fa4\uff1b\u666e\u9002\u6027 1. \u5f15\u8a00 \u5728\u8bb8\u591a\u7269\u7406\u7cfb\u7edf\u4e2d\uff0c\u5b8f\u89c2\u6027\u8d28\u5e76\u975e\u7b80\u5355\u53e0\u52a0\uff0c\u800c\u662f\u7531\u5927\u91cf\u81ea\u7531\u5ea6\u5728\u76f8\u4e92\u4f5c\u7528\u4e0e\u6da8\u843d\u4e4b\u95f4\u7ade\u4e89\u6240\u6d8c\u73b0\u3002\u94c1\u78c1\u4f53\u7684\u201c\u81ea\u53d1\u78c1\u5316\u201d\u3001\u6db2-\u6c14\u76f8\u53d8\u3001\u4e8c\u5143\u5408\u91d1\u6709\u5e8f\u5316\u7b49\u73b0\u8c61\u90fd\u5177\u6709\u5171\u540c\u7684\u7edf\u8ba1\u7ed3\u6784\uff1a\u5c40\u57df\u8026\u5408\u503e\u5411\u6709\u5e8f\uff0c\u6e29\u5ea6\u5f15\u5165\u65e0\u5e8f\u3002\u4f0a\u8f9b\u6a21\u578b\u4f5c\u4e3a\u8fd9\u7c7b\u73b0\u8c61\u7684\u6700\u5c0f\u8303\u5f0f\uff0c\u5177\u6709\u4ee5\u4e0b\u4ef7\u503c\uff1a 2. \u6a21\u578b\u5b9a\u4e49 2.1 \u683c\u70b9\u4e0e\u81ea\u65cb\u53d8\u91cf \u5728 ddd \u7ef4\u6676\u683c\uff08\u5982\u6b63\u65b9\u683c\u3001\u7acb\u65b9\u683c\uff09\u4e0a\uff0c\u6bcf\u4e2a\u683c\u70b9 i\u2208\u039bi\\in\\Lambdai\u2208\u039b \u8d4b\u4e88\u4e8c\u503c\u81ea\u65cbsi\u2208{+1,\u22121}.s_i\\in\\{+1,-1\\}.si\u200b\u2208{+1,\u22121}. \u7cfb\u7edf\u89c4\u6a21\u4e3a N=\u2223\u039b\u2223N=|\\Lambda|N=\u2223\u039b\u2223\u3002 2.2 \u54c8\u5bc6\u987f\u91cf\uff08\u80fd\u91cf\u51fd\u6570\uff09 \u6700\u5e38\u7528\u7684\u6700\u8fd1\u90bb\u4f0a\u8f9b\u6a21\u578b\u54c8\u5bc6\u987f\u91cf\u4e3aH({s})=\u2212J\u2211\u27e8i,j\u27e9sisj\u2212h\u2211isi,H(\\{s\\})=-J\\sum_{\\langle i,j\\rangle}s_is_j-h\\sum_i s_i,H({s})=\u2212J\u27e8i,j\u27e9\u2211\u200bsi\u200bsj\u200b\u2212hi\u2211\u200bsi\u200b, \u5176\u4e2d\uff1a \u7269\u7406\u89e3\u91ca 3. \u7edf\u8ba1\u529b\u5b66\u6846\u67b6 3.1 \u73bb\u5c14\u5179\u66fc\u6743\u91cd\u4e0e\u914d\u5206\u51fd\u6570 \u5728\u6e29\u5ea6 TTT \u4e0b\uff0c\u7cfb\u7edf\u5904\u4e8e\u914d\u7f6e {s}\\{s\\}{s} \u7684\u6982\u7387\u4e3aP({s})=1Zexp\u2061(\u2212\u03b2H({s})),\u03b2=1kBT,P(\\{s\\})=\\frac{1}{Z}\\exp(-\\beta H(\\{s\\})), \\quad \\beta=\\frac{1}{k_BT},P({s})=Z1\u200bexp(\u2212\u03b2H({s})),\u03b2=kB\u200bT1\u200b, \u914d\u5206\u51fd\u6570Z=\u2211{s}exp\u2061(\u2212\u03b2H({s})),Z=\\sum_{\\{s\\}}\\exp(-\\beta H(\\{s\\})),Z={s}\u2211\u200bexp(\u2212\u03b2H({s})), \u5176\u4e2d\u6c42\u548c\u904d\u5386\u5168\u90e8 2N2^N2N \u4e2a\u914d\u7f6e\u3002\u5bf9\u5927 NNN \u800c\u8a00\uff0cZZZ \u7684\u7cbe\u786e\u8ba1\u7b97\u901a\u5e38\u4e0d\u53ef\u884c\uff0c\u8fd9\u4e5f\u662f\u76f8\u53d8\u7406\u8bba\u4e0e\u6570\u503c\u6a21\u62df\u7684\u91cd\u8981\u52a8\u673a\u3002 …<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[185],"tags":[],"class_list":["post-5432","post","type-post","status-publish","format-standard","hentry","category-n19raczg26ymrrw"],"_links":{"self":[{"href":"https:\/\/eternalsphere.net\/echoes\/index.php\/wp-json\/wp\/v2\/posts\/5432","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/eternalsphere.net\/echoes\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/eternalsphere.net\/echoes\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/eternalsphere.net\/echoes\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/eternalsphere.net\/echoes\/index.php\/wp-json\/wp\/v2\/comments?post=5432"}],"version-history":[{"count":1,"href":"https:\/\/eternalsphere.net\/echoes\/index.php\/wp-json\/wp\/v2\/posts\/5432\/revisions"}],"predecessor-version":[{"id":5433,"href":"https:\/\/eternalsphere.net\/echoes\/index.php\/wp-json\/wp\/v2\/posts\/5432\/revisions\/5433"}],"wp:attachment":[{"href":"https:\/\/eternalsphere.net\/echoes\/index.php\/wp-json\/wp\/v2\/media?parent=5432"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/eternalsphere.net\/echoes\/index.php\/wp-json\/wp\/v2\/categories?post=5432"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/eternalsphere.net\/echoes\/index.php\/wp-json\/wp\/v2\/tags?post=5432"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}