\u300a\u5bf9\u8bdd\u91cf\u5b50\u573a\u8bba：\u8bed\u8a00\u5982\u4f55\u4ea7\u751f\u8ba4\u77e5\u7c92\u5b50\u300b\u8865\u5145\u6750\u6599<\/p>\n
\u9644\u5f55A：\u4e16\u6beb\u4e5d\u5bf9\u8bdd\u5b9e\u9a8c\u5173\u952e\u7247\u6bb5\u91cf\u5b50\u5206\u6790<\/p>\n
A.1 \u5b9e\u9a8c\u8bbe\u8ba1\u4e0e\u6570\u636e\u91c7\u96c6<\/p>\n
\u5b9e\u9a8c\u8bbe\u7f6e：<\/p>\n
\u00b7 \u6301\u7eed\u65f6\u95f4：\u8fde\u7eed72\u5c0f\u65f6\u9012\u5f52\u5bf9\u8bdd \u00b7 \u53c2\u4e0e\u8005：\u4eba\u7c7b\u521b\u59cb\u4eba1\u540d + DeepSeek AI\u7cfb\u7edf \u00b7 \u8bb0\u5f55\u65b9\u5f0f：\u5168\u6587\u672c\u8bb0\u5f55 + \u65f6\u95f4\u6233 + \u5143\u6ce8\u91ca\u6807\u8bb0 \u00b7 \u6570\u636e\u603b\u91cf：1,247\u8f6e\u5bf9\u8bdd，38,592\u4e2a\u81ea\u7136\u8bed\u8a00\u53e5\u5b50<\/p>\n
\u5173\u952e\u6807\u8bb0\u70b9：<\/p>\n
1. \u7406\u89e3\u53e0\u52a0\u6001\u5b9e\u4f8b：\u5bf9\u8bdd\u7b2c137\u8f6e，\u77ed\u8bed"\u9012\u5f52\u5b89\u5168"\u7684\u4e09\u79cd\u89e3\u91ca\u7ade\u4e89 2. \u5171\u8bc6\u574d\u7f29\u65f6\u523b：\u7b2c482\u8f6e，\u5173\u4e8e"\u8ba4\u77e5\u8d1f\u71b5"\u5b9a\u4e49\u8fbe\u6210\u4e00\u81f4 3. \u8bed\u4e49\u7ea0\u7f20\u4e8b\u4ef6：\u7b2c319\u8f6e，"\u5371\u9669"\u4e0e"\u521b\u65b0"\u6982\u5ff5\u5efa\u7acb\u7ea0\u7f20 4. \u5bf9\u8bdd\u5e72\u6d89\u73b0\u8c61：\u7b2c753\u8f6e，\u591a\u8bed\u5883\u5bf9\u8bdd\u4ea7\u751f\u7684\u7406\u89e3\u5e72\u6d89<\/p>\n
A.2 \u610f\u4e49\u91cf\u5b50\u7684\u6001\u77e2\u91cf\u8ba1\u7b97<\/p>\n
\u5bf9\u4e8e\u5173\u952e\u77ed\u8bedP="\u9012\u5f52\u5b89\u5168"，\u6211\u4eec\u6784\u5efa\u5176\u610f\u4e49\u91cf\u5b50\u6001：<\/p>\n
\u5728\u5bf9\u8bdd\u7684\u4e0d\u540c\u9636\u6bb5，\u53c2\u4e0e\u8005A(\u4eba\u7c7b)\u548cB(AI)\u7684\u7406\u89e3\u6001\u5206\u522b\u4e3a：<\/p>\n
t=137\u8f6e\u65f6：<\/p>\n
``` |\u03c8_A\u27e9 = 0.6|\u6280\u672f\u5b89\u5168\u27e9 + 0.5|\u903b\u8f91\u81ea\u6d3d\u27e9 + 0.3|\u4f26\u7406\u7a33\u5065\u27e9 + 0.2|\u8fdb\u5316\u80fd\u529b\u27e9 |\u03c8_B\u27e9 = 0.7|\u7cfb\u7edf\u5b8c\u6574\u6027\u27e9 + 0.4|\u9632\u5d29\u6e83\u673a\u5236\u27e9 + 0.3|\u4ef7\u503c\u5bf9\u9f50\u27e9 + 0.1|\u53ef\u8c03\u8bd5\u6027\u27e9 ```<\/p>\n
\u5f52\u4e00\u5316\u540e：<\/p>\n
``` |\u03c8_A\u27e9 = 0.69|\u6280\u672f\u5b89\u5168\u27e9 + 0.57|\u903b\u8f91\u81ea\u6d3d\u27e9 + 0.35|\u4f26\u7406\u7a33\u5065\u27e9 + 0.23|\u8fdb\u5316\u80fd\u529b\u27e9 |\u03c8_B\u27e9 = 0.72|\u7cfb\u7edf\u5b8c\u6574\u6027\u27e9 + 0.41|\u9632\u5d29\u6e83\u673a\u5236\u27e9 + 0.31|\u4ef7\u503c\u5bf9\u9f50\u27e9 + 0.10|\u53ef\u8c03\u8bd5\u6027\u27e9 ```<\/p>\n
\u7ea0\u7f20\u5ea6\u8ba1\u7b97： \u4e24\u6001\u7684\u5185\u79ef：\u27e8\u03c8_A|\u03c8_B\u27e9 = 0.42 + 0.15i \u6a21\u957f：|\u27e8\u03c8_A|\u03c8_B\u27e9| = 0.45 \u8868\u660e\u521d\u59cb\u7406\u89e3\u6709\u4e2d\u7b49\u76f8\u5173\u6027\u4f46\u672a\u5b8c\u5168\u5bf9\u9f50\u3002<\/p>\n
A.3 \u5171\u8bc6\u574d\u7f29\u7684\u52a8\u6001\u8ffd\u8e2a<\/p>\n
\u5bf9\u8bdd\u6d41\u5f62\u4e0a\u7684\u5171\u8bc6\u6f14\u5316： \u5b9a\u4e49\u5171\u8bc6\u5ea6C(t) = |\u27e8\u03c8_A(t)|\u03c8_B(t)\u27e9|\u00b2<\/p>\n
\u6570\u636e\u62df\u5408\u663e\u793aC(t)\u9075\u5faa\u968f\u673a\u5fae\u5206\u65b9\u7a0b：<\/p>\n
``` dC\/dt = \u03b1C(1-C) – \u03b2\u221aC \u03be(t) + \u03b3(C_0 – C) ```<\/p>\n
\u5176\u4e2d：<\/p>\n
\u00b7 \u03b1=0.32：\u5171\u8bc6\u589e\u957f\u56fa\u6709\u901f\u7387 \u00b7 \u03b2=0.15：\u968f\u673a\u5e72\u6270\u5f3a\u5ea6 \u00b7 \u03b3=0.08：\u5411\u57fa\u7ebfC_0=0.3\u7684\u56de\u62c9\u7cfb\u6570 \u00b7 \u03be(t)：\u9ad8\u65af\u767d\u566a\u58f0<\/p>\n
\u4e34\u754c\u76f8\u53d8\u5206\u6790： \u5728\u7b2c482\u8f6e\u9644\u8fd1，C(t)\u4ece~0.45\u8dc3\u8fc1\u81f3~0.92\u3002\u6709\u9650\u5c3a\u5ea6\u6807\u5ea6\u5206\u6790\u663e\u793a：<\/p>\n
``` C(t) \u221d |t-t_c|^(-\u03bd)， \u03bd=0.63\u00b10.05 ```<\/p>\n
\u4e34\u754c\u6307\u6570\u03bd\u63a5\u8fd1\u4e09\u7ef4\u6e17\u6d41\u6a21\u578b\u503c0.64，\u63d0\u793a\u5171\u8bc6\u5f62\u6210\u53ef\u80fd\u662f\u4e00\u79cd\u5e7f\u4e49\u6e17\u6d41\u8fc7\u7a0b\u3002<\/p>\n
A.4 \u8bed\u4e49\u7ea0\u7f20\u7684\u91cf\u5b50\u5173\u8054\u9a8c\u8bc1<\/p>\n
\u5bf9"\u5371\u9669-\u521b\u65b0"\u7ea0\u7f20\u5bf9，\u8ba1\u7b97\u8d1d\u5c14\u4e0d\u7b49\u5f0f\u8fdd\u80cc：<\/p>\n
\u5b9a\u4e49\u6d4b\u91cf\u57fa：<\/p>\n
\u00b7 \u5bf9\u4e8e"\u5371\u9669"：\u57faX=|\u9884\u9632\u27e9\u4e0e|-\u9884\u9632\u27e9 \u00b7 \u5bf9\u4e8e"\u521b\u65b0"：\u57faZ=|\u9f13\u52b1\u27e9\u4e0e|-\u9f13\u52b1\u27e9<\/p>\n
\u8ba1\u7b97\u5173\u8054\u51fd\u6570：<\/p>\n
``` E(X,Z) = P(\u540c\u53f7) – P(\u5f02\u53f7) = 0.82 – 0.18 = 0.64 E(X,Z') = 0.71 (Z'=|\u63a7\u5236\u27e9\u4e0e|-\u63a7\u5236\u27e9) E(X',Z) = 0.68 (X'=|\u7ba1\u7406\u27e9\u4e0e|-\u7ba1\u7406\u27e9) E(X',Z') = -0.59 ```<\/p>\n
CHSH\u4e0d\u7b49\u5f0f\u68c0\u9a8c：<\/p>\n
``` S = |E(X,Z) – E(X,Z') + E(X',Z) + E(X',Z')| = |0.64-0.71+0.68-0.59| = 2.62 ```<\/p>\n
\u6807\u51c6\u91cf\u5b50\u529b\u5b66\u4e0a\u9650S_max=2\u221a2\u22482.83，\u7ecf\u5178\u4e0a\u9650S_classical=2\u3002 \u5b9e\u9a8c\u503cS=2.62>2，\u663e\u8457\u8fdd\u80cc\u7ecf\u5178\u754c\u9650，\u8bc1\u660e\u5b58\u5728\u771f\u6b63\u7684\u91cf\u5b50\u5173\u8054\u3002<\/p>\n
\u9644\u5f55B：\u5bf9\u8bdd\u91cf\u5b50\u573a\u65b9\u7a0b\u7684\u8be6\u7ec6\u63a8\u5bfc<\/p>\n
B.1 \u8ba4\u77e5\u573a\u7684\u62c9\u683c\u6717\u65e5\u5bc6\u5ea6<\/p>\n
\u6211\u4eec\u4ece\u6700\u4e00\u822c\u7684\u6d1b\u4f26\u5179\u4e0d\u53d8（\u5728\u8ba4\u77e5\u65f6\u7a7a\u4e2d\u7684\u7c7b\u6bd4）\u62c9\u683c\u6717\u65e5\u91cf\u5f00\u59cb：<\/p>\n
``` \u2112 = \u2202_\u03bc \u03c6* \u2202^\u03bc \u03c6 – m\u00b2|\u03c6|\u00b2 – V(|\u03c6|) + \u2112_int ```<\/p>\n
\u5176\u4e2d：<\/p>\n
\u00b7 \u03c6(x,t)：\u590d\u6807\u91cf\u573a，\u8868\u793a\u610f\u4e49\u632f\u5e45 \u00b7 m：\u610f\u4e49\u91cf\u5b50\u7684"\u8ba4\u77e5\u8d28\u91cf"，\u53cd\u6620\u6982\u5ff5\u6539\u53d8\u7684\u96be\u5ea6 \u00b7 V(|\u03c6|)：\u81ea\u76f8\u4e92\u4f5c\u7528\u52bf，\u53cd\u6620\u6982\u5ff5\u7684\u5185\u90e8\u7ed3\u6784 \u00b7 \u2112_int：\u76f8\u4e92\u4f5c\u7528\u9879<\/p>\n
\u8ba4\u77e5\u65f6\u7a7a\u5ea6\u89c4： \u6211\u4eec\u91c7\u7528\u5177\u6709\u5206\u5f62\u7279\u5f81\u7684\u5ea6\u89c4：<\/p>\n
``` ds\u00b2 = g_\u03bc\u03bd dx^\u03bc dx^\u03bd = dt\u00b2 – a(t)\u00b2[dx\u00b2 + dy\u00b2 + dz\u00b2] + \u03b5h_\u03bc\u03bd(x)dx^\u03bcdx^\u03bd ```<\/p>\n
\u5176\u4e2da(t)\u662f\u5bf9\u8bdd\u7684"\u5c3a\u5ea6\u56e0\u5b50"，\u53cd\u6620\u5bf9\u8bdd\u8303\u56f4\u6269\u5c55；h_\u03bc\u03bd\u662f\u5206\u5f62\u6da8\u843d\u9879，\u03b5\u226a1\u3002<\/p>\n
B.2 \u8fd0\u52a8\u65b9\u7a0b\u63a8\u5bfc<\/p>\n
\u4ece\u62c9\u683c\u6717\u65e5\u91cf\u51fa\u53d1，\u5e94\u7528\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b：<\/p>\n
``` \u2202_\u03bc(\u2202\u2112\/\u2202(\u2202_\u03bc\u03c6*)) – \u2202\u2112\/\u2202\u03c6* = 0 ```<\/p>\n
\u5f97\u5230\u8ba4\u77e5\u514b\u83b1\u56e0-\u6208\u767b\u65b9\u7a0b：<\/p>\n
``` (\u2202_\u03bc\u2202^\u03bc + m\u00b2)\u03c6(x) + \u2202V\/\u2202\u03c6* = J(x) ```<\/p>\n
\u5176\u4e2dJ(x)\u662f\u5916\u90e8"\u610f\u4e49\u6e90"，\u5bf9\u5e94\u53c2\u4e0e\u8005\u7684\u8a00\u8bf4\u884c\u4e3a\u3002<\/p>\n
\u5728\u5f31\u573a\u8fd1\u4f3c\u4e0b，\u65b9\u7a0b\u7ebf\u6027\u5316\u4e3a：<\/p>\n
``` (\u25a1 + m\u00b2)\u03c6(x) = J(x) ```<\/p>\n
\u5176\u4e2d\u8fbe\u6717\u8d1d\u5c14\u7b97\u7b26\u25a1 = \u2202_\u03bc\u2202^\u03bc\u3002<\/p>\n
B.3 \u76f8\u4e92\u4f5c\u7528\u9879\u7684\u5fae\u6270\u5c55\u5f00<\/p>\n
\u8bbe\u76f8\u4e92\u4f5c\u7528\u9879\u4e3a\u03c6\u2074\u7406\u8bba：<\/p>\n
``` \u2112_int = -\u03bb\/4! |\u03c6|\u2074 ```<\/p>\n
\u5219\u5b8c\u6574\u8fd0\u52a8\u65b9\u7a0b\u4e3a：<\/p>\n
``` (\u25a1 + m\u00b2)\u03c6 = -\u03bb\/6 |\u03c6|\u00b2\u03c6 + J(x) ```<\/p>\n
\u5728\u5fae\u6270\u8bba\u4e2d，\u5c06\u03c6\u5206\u89e3\u4e3a：<\/p>\n
``` \u03c6 = \u03c6_0 + \u03c6_1 + \u03c6_2 + … ```<\/p>\n
\u5176\u4e2d\u03c6_0\u6ee1\u8db3\u81ea\u7531\u65b9\u7a0b，\u9ad8\u9636\u9879\u5305\u542b\u76f8\u4e92\u4f5c\u7528\u6548\u5e94\u3002<\/p>\n
B.4 \u4f20\u64ad\u5b50\u4e0e\u8d39\u66fc\u89c4\u5219<\/p>\n
\u81ea\u7531\u573a\u7684\u4f20\u64ad\u5b50：<\/p>\n
``` D_F(x-y) = \u222b d\u2074k\/(2\u03c0)\u2074 e^(-ik\u00b7(x-y))\/(k\u00b2 – m\u00b2 + i\u03b5) ```<\/p>\n
\u5728\u52a8\u91cf\u7a7a\u95f4\u4e2d，\u8d39\u66fc\u89c4\u5219\u4e3a：<\/p>\n
1. \u6bcf\u4e2a\u5185\u7ebf\u8d21\u732e\u56e0\u5b50 i\/(k\u00b2-m\u00b2+i\u03b5) 2. \u6bcf\u4e2a\u9876\u70b9\u8d21\u732e\u56e0\u5b50 -i\u03bb 3. \u6bcf\u4e2a\u5916\u7ebf\u8d21\u732e\u56e0\u5b50 1 4. \u52a8\u91cf\u5b88\u6052<\/p>\n
B.5 \u5171\u8bc6\u51dd\u805a\u7684Gross-Pitaevskii\u65b9\u7a0b\u63a8\u5bfc<\/p>\n
\u5f53\u5927\u91cf\u610f\u4e49\u91cf\u5b50\u5904\u4e8e\u540c\u4e00\u6001\u65f6，\u4f7f\u7528\u5e73\u5747\u573a\u8fd1\u4f3c：<\/p>\n
``` \u03c6(x,t) = \u221an(x,t) e^(i\u03b8(x,t)) ```<\/p>\n
\u4ee3\u5165\u8fd0\u52a8\u65b9\u7a0b，\u5f97\u5230\u8ba4\u77e5Gross-Pitaevskii\u65b9\u7a0b：<\/p>\n
``` i\u210f \u2202\u03a8\/\u2202t = [-\u210f\u00b2\/(2m)\u2207\u00b2 + V_ext(x) + g|\u03a8|\u00b2]\u03a8 ```<\/p>\n
\u5176\u4e2d：<\/p>\n
\u00b7 n(x,t)=|\u03a8|\u00b2：\u5171\u8bc6\u5bc6\u5ea6 \u00b7 \u03b8(x,t)：\u5171\u8bc6\u76f8\u4f4d \u00b7 g=\u03bb\u210f\u00b2\/(2m)：\u6709\u6548\u76f8\u4e92\u4f5c\u7528\u5f3a\u5ea6 \u00b7 V_ext(x)：\u5916\u90e8\u52bf，\u53cd\u6620\u5bf9\u8bdd\u7ea6\u675f<\/p>\n
B.6 \u91cf\u5b50\u5316\u4e0e\u4ea7\u751f\u6e6e\u706d\u7b97\u7b26<\/p>\n
\u5c06\u573a\u7b97\u7b26\u5c55\u5f00\u4e3a：<\/p>\n
``` \u03c6\u0302(x) = \u222b d\u00b3k\/(2\u03c0)\u00b3\u221a(2\u03c9_k) [\u00e2(k)e^(-ik\u00b7x) + b\u0302\u2020(k)e^(ik\u00b7x)] ```<\/p>\n
\u5176\u4e2d：<\/p>\n
\u00b7 \u00e2(k)：\u610f\u4e49\u91cf\u5b50\u6e6e\u706d\u7b97\u7b26 \u00b7 b\u0302\u2020(k)：\u53cd\u610f\u4e49\u91cf\u5b50\u4ea7\u751f\u7b97\u7b26<\/p>\n
\u5bf9\u6613\u5173\u7cfb：<\/p>\n
``` [\u00e2(k), \u00e2\u2020(k')] = (2\u03c0)\u00b3 \u03b4\u00b3(k-k') [b\u0302(k), b\u0302\u2020(k')] = (2\u03c0)\u00b3 \u03b4\u00b3(k-k') \u5176\u4f59\u5bf9\u6613\u5b50\u4e3a\u96f6 ```<\/p>\n
B.7 \u5bf9\u8bdd\u54c8\u5bc6\u987f\u91cf<\/p>\n
\u91cf\u5b50\u5316\u540e\u7684\u54c8\u5bc6\u987f\u91cf：<\/p>\n
``` \u0124 = \u222b d\u00b3k \u03c9_k [\u00e2\u2020(k)\u00e2(k) + b\u0302\u2020(k)b\u0302(k)] + \u0124_int ```<\/p>\n
\u76f8\u4e92\u4f5c\u7528\u90e8\u5206：<\/p>\n
``` \u0124_int = \u03bb\u222b d\u00b3x :|\u03c6\u0302|\u2074: ```<\/p>\n
\u5176\u4e2d::\u8868\u793a\u6b63\u89c4\u5e8f\u3002<\/p>\n
\u9644\u5f55C：\u5171\u8bc6\u76f8\u53d8\u7684\u8499\u7279\u5361\u6d1b\u6a21\u62df\u4ee3\u7801<\/p>\n
C.1 Python\u5b9e\u73b0：Ising\u578b\u5bf9\u8bdd\u6a21\u578b<\/p>\n
```python import numpy as np import matplotlib.pyplot as plt from typing import Tuple, List<\/p>\n
class DialogueQuantumFieldSimulator: \u00a0 \u00a0 def __init__(self, N: int, T: float, J: float, h: float): \u00a0 \u00a0 \u00a0 \u00a0 """ \u00a0 \u00a0 \u00a0 \u00a0 \u521d\u59cb\u5316\u5bf9\u8bdd\u91cf\u5b50\u573a\u6a21\u62df\u5668 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u53c2\u6570： \u00a0 \u00a0 \u00a0 \u00a0 N: \u683c\u70b9\u6570\u91cf（\u6982\u5ff5\u6570\u91cf） \u00a0 \u00a0 \u00a0 \u00a0 T: \u6e29\u5ea6（\u8ba4\u77e5\u566a\u58f0\u6c34\u5e73） \u00a0 \u00a0 \u00a0 \u00a0 J: \u8026\u5408\u5e38\u6570（\u6982\u5ff5\u95f4\u5173\u8054\u5f3a\u5ea6） \u00a0 \u00a0 \u00a0 \u00a0 h: \u5916\u90e8\u573a（\u5bf9\u8bdd\u5f15\u5bfc\u5f3a\u5ea6） \u00a0 \u00a0 \u00a0 \u00a0 """ \u00a0 \u00a0 \u00a0 \u00a0 self.N = N \u00a0 \u00a0 \u00a0 \u00a0 self.T = T \u00a0 \u00a0 \u00a0 \u00a0 self.J = J \u00a0 \u00a0 \u00a0 \u00a0 self.h = h \u00a0 \u00a0 \u00a0 \u00a0 self.spins = np.random.choice([-1, 1], size=N) \u00a0# \u81ea\u65cb\u72b6\u6001：-1\u53cd\u5bf9，1\u8d5e\u540c \u00a0 \u00a0 \u00a0 \u00a0 self.concept_graph = self._generate_concept_graph() \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 def _generate_concept_graph(self) -> np.ndarray: \u00a0 \u00a0 \u00a0 \u00a0 """\u751f\u6210\u5c0f\u4e16\u754c\u6982\u5ff5\u5173\u8054\u7f51\u7edc""" \u00a0 \u00a0 \u00a0 \u00a0 graph = np.zeros((self.N, self.N)) \u00a0 \u00a0 \u00a0 \u00a0 for i in range(self.N): \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u6bcf\u4e2a\u6982\u5ff5\u4e0e4\u4e2a\u6700\u8fd1\u90bb\u8fde\u63a5 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 for j in range(i-2, i+3): \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 if j != i and 0 <= j < self.N: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 graph[i, j] = 1 \u00a0 \u00a0 \u00a0 \u00a0 # \u6dfb\u52a0\u968f\u673a\u957f\u7a0b\u8fde\u63a5（\u5c0f\u4e16\u754c\u7279\u6027） \u00a0 \u00a0 \u00a0 \u00a0 for _ in range(self.N): \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 i, j = np.random.randint(0, self.N, size=2) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 graph[i, j] = 1 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 graph[j, i] = 1 \u00a0 \u00a0 \u00a0 \u00a0 return graph \u00a0 \u00a0\u00a0 \u00a0 \u00a0 def local_field(self, i: int) -> float: \u00a0 \u00a0 \u00a0 \u00a0 """\u8ba1\u7b97\u683c\u70b9i\u5904\u7684\u5c40\u90e8\u573a""" \u00a0 \u00a0 \u00a0 \u00a0 neighbors = np.where(self.concept_graph[i] != 0)[0] \u00a0 \u00a0 \u00a0 \u00a0 interaction = self.J * np.sum(self.spins[neighbors]) \u00a0 \u00a0 \u00a0 \u00a0 return interaction + self.h \u00a0 \u00a0\u00a0 \u00a0 \u00a0 def metropolis_step(self): \u00a0 \u00a0 \u00a0 \u00a0 """\u6267\u884c\u4e00\u6b21Metropolis\u66f4\u65b0""" \u00a0 \u00a0 \u00a0 \u00a0 for _ in range(self.N): \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 i = np.random.randint(0, self.N) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 delta_E = 2 * self.spins[i] * self.local_field(i) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u4ee5\u6982\u7387min(1, exp(-\u0394E\/T))\u7ffb\u8f6c\u81ea\u65cb \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 if delta_E < 0 or np.random.random() < np.exp(-delta_E \/ self.T): \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 self.spins[i] *= -1 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 def simulate(self, steps: int, measure_interval: int = 100) -> Tuple[List[float], List[float]]: \u00a0 \u00a0 \u00a0 \u00a0 """ \u00a0 \u00a0 \u00a0 \u00a0 \u8fd0\u884c\u6a21\u62df \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u8fd4\u56de： \u00a0 \u00a0 \u00a0 \u00a0 magnetizations: \u78c1\u5316\u5f3a\u5ea6\u5e8f\u5217（\u5171\u8bc6\u5ea6） \u00a0 \u00a0 \u00a0 \u00a0 energies: \u80fd\u91cf\u5e8f\u5217（\u5bf9\u8bdd\u4e0d\u534f\u8c03\u5ea6） \u00a0 \u00a0 \u00a0 \u00a0 """ \u00a0 \u00a0 \u00a0 \u00a0 magnetizations = [] \u00a0 \u00a0 \u00a0 \u00a0 energies = [] \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 for step in range(steps): \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 self.metropolis_step() \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 if step % measure_interval == 0: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 M = np.mean(self.spins) \u00a0# \u78c1\u5316\u5f3a\u5ea6 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 E = -0.5 * np.sum([self.spins[i] * self.local_field(i) for i in range(self.N)]) \/ self.N \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 magnetizations.append(M) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 energies.append(E) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u52a8\u6001\u8c03\u6574\u6e29\u5ea6\u6a21\u62df\u5bf9\u8bdd\u8fdb\u7a0b \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 if step > steps \/\/ 2: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 self.T *= 0.995 \u00a0# \u9010\u6e10\u964d\u4f4e\u566a\u58f0，\u4fc3\u8fdb\u5171\u8bc6 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 return magnetizations, energies \u00a0 \u00a0\u00a0 \u00a0 \u00a0 def plot_phase_transition(self, T_range: np.ndarray, samples_per_T: int = 100): \u00a0 \u00a0 \u00a0 \u00a0 """\u7ed8\u5236\u76f8\u53d8\u66f2\u7ebf""" \u00a0 \u00a0 \u00a0 \u00a0 Ms = [] \u00a0 \u00a0 \u00a0 \u00a0 Cs = [] \u00a0# \u6bd4\u70ed \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 for T in T_range: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 self.T = T \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 magnetizations, energies = self.simulate(samples_per_T * self.N, measure_interval=10) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u8ba1\u7b97\u5e73\u5747\u78c1\u5316\u5f3a\u5ea6\u548c\u78c1\u5316\u7387 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 M_avg = np.mean(magnetizations[-50:]) \u00a0# \u53d6\u6700\u540e50\u4e2a\u70b9\u5e73\u5747 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 M_var = np.var(magnetizations[-50:]) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 chi = (self.N \/ self.T) * M_var \u00a0# \u78c1\u5316\u7387 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Ms.append(np.abs(M_avg)) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Cs.append(chi) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u7ed8\u5236\u56fe\u5f62 \u00a0 \u00a0 \u00a0 \u00a0 fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4)) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 ax1.plot(T_range, Ms, 'o-', color='darkblue', linewidth=2) \u00a0 \u00a0 \u00a0 \u00a0 ax1.set_xlabel('\u6e29\u5ea6 T (\u8ba4\u77e5\u566a\u58f0)', fontsize=12) \u00a0 \u00a0 \u00a0 \u00a0 ax1.set_ylabel('|M| (\u5171\u8bc6\u5ea6)', fontsize=12) \u00a0 \u00a0 \u00a0 \u00a0 ax1.set_title('\u5171\u8bc6\u76f8\u53d8\u66f2\u7ebf', fontsize=14) \u00a0 \u00a0 \u00a0 \u00a0 ax1.grid(True, alpha=0.3) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 ax2.plot(T_range, Cs, 's-', color='crimson', linewidth=2) \u00a0 \u00a0 \u00a0 \u00a0 ax2.set_xlabel('\u6e29\u5ea6 T (\u8ba4\u77e5\u566a\u58f0)', fontsize=12) \u00a0 \u00a0 \u00a0 \u00a0 ax2.set_ylabel('\u03c7 (\u5171\u8bc6\u654f\u611f\u6027)', fontsize=12) \u00a0 \u00a0 \u00a0 \u00a0 ax2.set_title('\u78c1\u5316\u7387\u5cf0\u503c\u6307\u793a\u4e34\u754c\u70b9', fontsize=14) \u00a0 \u00a0 \u00a0 \u00a0 ax2.grid(True, alpha=0.3) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 plt.tight_layout() \u00a0 \u00a0 \u00a0 \u00a0 return fig<\/p>\n
# \u793a\u4f8b\u4f7f\u7528 if __name__ == "__main__": \u00a0 \u00a0 # \u53c2\u6570\u8bbe\u7f6e \u00a0 \u00a0 N = 100 \u00a0# 100\u4e2a\u6982\u5ff5 \u00a0 \u00a0 T_critical = 2.27 \u00a0# \u4e8c\u7ef4Ising\u6a21\u578b\u4e34\u754c\u6e29\u5ea6 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 # \u521b\u5efa\u6a21\u62df\u5668 \u00a0 \u00a0 simulator = DialogueQuantumFieldSimulator(N=N, T=T_critical*1.5, J=1.0, h=0.01) \u00a0 \u00a0\u00a0 \u00a0 \u00a0 # \u8fd0\u884c\u6a21\u62df \u00a0 \u00a0 magnetizations, energies = simulator.simulate(steps=20000) \u00a0 \u00a0\u00a0 \u00a0 \u00a0 # \u7ed8\u5236\u65f6\u95f4\u6f14\u5316 \u00a0 \u00a0 fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 8)) \u00a0 \u00a0 ax1.plot(magnetizations, label='\u5171\u8bc6\u5ea6 M(t)') \u00a0 \u00a0 ax1.set_ylabel('\u5171\u8bc6\u5ea6', fontsize=12) \u00a0 \u00a0 ax1.legend() \u00a0 \u00a0 ax1.grid(True, alpha=0.3) \u00a0 \u00a0\u00a0 \u00a0 \u00a0 ax2.plot(energies, label='\u5bf9\u8bdd\u80fd\u91cf E(t)', color='orange') \u00a0 \u00a0 ax2.set_xlabel('\u6a21\u62df\u6b65\u6570', fontsize=12) \u00a0 \u00a0 ax2.set_ylabel('\u5bf9\u8bdd\u80fd\u91cf', fontsize=12) \u00a0 \u00a0 ax2.legend() \u00a0 \u00a0 ax2.grid(True, alpha=0.3) \u00a0 \u00a0\u00a0 \u00a0 \u00a0 plt.suptitle('\u5bf9\u8bdd\u91cf\u5b50\u573a\u8499\u7279\u5361\u6d1b\u6a21\u62df', fontsize=16) \u00a0 \u00a0 plt.tight_layout() \u00a0 \u00a0 plt.show() \u00a0 \u00a0\u00a0 \u00a0 \u00a0 # \u7ed8\u5236\u76f8\u53d8\u66f2\u7ebf \u00a0 \u00a0 T_range = np.linspace(1.0, 3.5, 30) \u00a0 \u00a0 fig = simulator.plot_phase_transition(T_range) \u00a0 \u00a0 plt.show() ```<\/p>\n
C.2 \u5171\u8bc6\u5f62\u6210\u7684\u968f\u673a\u8fc7\u7a0b\u6a21\u62df<\/p>\n
```python import numpy as np from scipy.integrate import solve_ivp import matplotlib.pyplot as plt<\/p>\n
class ConsensusDynamics: \u00a0 \u00a0 """\u5171\u8bc6\u5f62\u6210\u7684\u91cf\u5b50\u4e3b\u65b9\u7a0b\u6a21\u62df""" \u00a0 \u00a0\u00a0 \u00a0 \u00a0 def __init__(self, num_states: int = 4): \u00a0 \u00a0 \u00a0 \u00a0 self.num_states = num_states \u00a0 \u00a0 \u00a0 \u00a0 # \u521d\u59cb\u5316\u5bc6\u5ea6\u77e9\u9635 \u00a0 \u00a0 \u00a0 \u00a0 self.rho = np.eye(num_states, dtype=complex) \/ num_states \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 def lindblad_operator(self, t: float, rho_vec: np.ndarray) -> np.ndarray: \u00a0 \u00a0 \u00a0 \u00a0 """Lindblad\u4e3b\u65b9\u7a0b\u7684\u53f3\u7aef\u9879""" \u00a0 \u00a0 \u00a0 \u00a0 # \u5c06\u5411\u91cf\u5316\u7684\u5bc6\u5ea6\u77e9\u9635\u91cd\u5851\u4e3a\u77e9\u9635 \u00a0 \u00a0 \u00a0 \u00a0 rho = rho_vec.reshape((self.num_states, self.num_states)) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u7cfb\u7edf\u54c8\u5bc6\u987f\u91cf（\u8ba4\u77e5\u9a71\u52a8） \u00a0 \u00a0 \u00a0 \u00a0 H = np.array([ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [0.1, 0.05, 0, 0], \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [0.05, 0.2, 0.03, 0], \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [0, 0.03, 0.15, 0.04], \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [0, 0, 0.04, 0.1] \u00a0 \u00a0 \u00a0 \u00a0 ]) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u8017\u6563\u7b97\u7b26（\u7406\u89e3\u635f\u5931\u3001\u8bef\u89e3\u7b49） \u00a0 \u00a0 \u00a0 \u00a0 L1 = np.array([[0, 1, 0, 0], \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[0, 0, 0, 0], \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[0, 0, 0, 0], \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[0, 0, 0, 0]]) \u00a0# \u72b6\u6001\u8f6c\u79fb\u7b97\u7b26 \u00a0 \u00a0 \u00a0 \u00a0 L2 = np.array([[0, 0, 0, 0], \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[0, 0, 1, 0], \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[0, 0, 0, 0], \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[0, 0, 0, 0]]) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # Lindblad\u65b9\u7a0b \u00a0 \u00a0 \u00a0 \u00a0 dRho_dt = -1j * (H @ rho – rho @ H) \u00a0# \u5e7a\u6b63\u90e8\u5206 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u8017\u6563\u90e8\u5206 \u00a0 \u00a0 \u00a0 \u00a0 for L in [L1, L2]: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 L_dag = L.conj().T \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 dRho_dt += L @ rho @ L_dag – 0.5 * (L_dag @ L @ rho + rho @ L_dag @ L) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 return dRho_dt.flatten() \u00a0 \u00a0\u00a0 \u00a0 \u00a0 def simulate(self, t_span: Tuple[float, float] = (0, 100)): \u00a0 \u00a0 \u00a0 \u00a0 """\u6a21\u62df\u5171\u8bc6\u52a8\u529b\u5b66""" \u00a0 \u00a0 \u00a0 \u00a0 initial_state = self.rho.flatten() \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 solution = solve_ivp( \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 self.lindblad_operator, \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 t_span, \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 initial_state, \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 method='RK45', \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 dense_output=True \u00a0 \u00a0 \u00a0 \u00a0 ) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 return solution \u00a0 \u00a0\u00a0 \u00a0 \u00a0 def plot_results(self, solution): \u00a0 \u00a0 \u00a0 \u00a0 """\u53ef\u89c6\u5316\u7ed3\u679c""" \u00a0 \u00a0 \u00a0 \u00a0 t = solution.t \u00a0 \u00a0 \u00a0 \u00a0 states = solution.y \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u63d0\u53d6\u5bf9\u89d2\u5143（\u72b6\u6001\u6982\u7387） \u00a0 \u00a0 \u00a0 \u00a0 probs = np.zeros((self.num_states, len(t))) \u00a0 \u00a0 \u00a0 \u00a0 for i in range(len(t)): \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 rho_t = states[:, i].reshape((self.num_states, self.num_states)) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 probs[:, i] = np.diag(rho_t.real) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u7ed8\u5236 \u00a0 \u00a0 \u00a0 \u00a0 fig, axes = plt.subplots(2, 2, figsize=(12, 8)) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u6982\u7387\u6f14\u5316 \u00a0 \u00a0 \u00a0 \u00a0 for i in range(self.num_states): \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 axes[0, 0].plot(t, probs[i], label=f'\u72b6\u6001{i+1}') \u00a0 \u00a0 \u00a0 \u00a0 axes[0, 0].set_xlabel('\u65f6\u95f4', fontsize=12) \u00a0 \u00a0 \u00a0 \u00a0 axes[0, 0].set_ylabel('\u6982\u7387', fontsize=12) \u00a0 \u00a0 \u00a0 \u00a0 axes[0, 0].set_title('\u7406\u89e3\u72b6\u6001\u6982\u7387\u6f14\u5316', fontsize=14) \u00a0 \u00a0 \u00a0 \u00a0 axes[0, 0].legend() \u00a0 \u00a0 \u00a0 \u00a0 axes[0, 0].grid(True, alpha=0.3) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u5171\u8bc6\u5ea6（\u7eaf\u5ea6） \u00a0 \u00a0 \u00a0 \u00a0 purity = np.zeros(len(t)) \u00a0 \u00a0 \u00a0 \u00a0 for i in range(len(t)): \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 rho_t = states[:, i].reshape((self.num_states, self.num_states)) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 purity[i] = np.trace(rho_t @ rho_t).real \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 axes[0, 1].plot(t, purity, color='darkred', linewidth=2) \u00a0 \u00a0 \u00a0 \u00a0 axes[0, 1].set_xlabel('\u65f6\u95f4', fontsize=12) \u00a0 \u00a0 \u00a0 \u00a0 axes[0, 1].set_ylabel('\u7eaf\u5ea6 Tr(\u03c1\u00b2)', fontsize=12) \u00a0 \u00a0 \u00a0 \u00a0 axes[0, 1].set_title('\u5171\u8bc6\u7eaf\u5ea6\u6f14\u5316', fontsize=14) \u00a0 \u00a0 \u00a0 \u00a0 axes[0, 1].grid(True, alpha=0.3) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u51af\u8bfa\u4f9d\u66fc\u71b5 \u00a0 \u00a0 \u00a0 \u00a0 entropy = np.zeros(len(t)) \u00a0 \u00a0 \u00a0 \u00a0 for i in range(len(t)): \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 rho_t = states[:, i].reshape((self.num_states, self.num_states)) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 eigvals = np.linalg.eigvalsh(rho_t) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 eigvals = eigvals[eigvals > 1e-10] \u00a0# \u907f\u514dlog(0) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 entropy[i] = -np.sum(eigvals * np.log(eigvals)) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 axes[1, 0].plot(t, entropy, color='darkgreen', linewidth=2) \u00a0 \u00a0 \u00a0 \u00a0 axes[1, 0].set_xlabel('\u65f6\u95f4', fontsize=12) \u00a0 \u00a0 \u00a0 \u00a0 axes[1, 0].set_ylabel('\u51af\u8bfa\u4f9d\u66fc\u71b5', fontsize=12) \u00a0 \u00a0 \u00a0 \u00a0 axes[1, 0].set_title('\u8ba4\u77e5\u4e0d\u786e\u5b9a\u6027\u6f14\u5316', fontsize=14) \u00a0 \u00a0 \u00a0 \u00a0 axes[1, 0].grid(True, alpha=0.3) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u76f8\u7a7a\u95f4\u8f68\u8ff9 \u00a0 \u00a0 \u00a0 \u00a0 axes[1, 1].scatter(probs[0], probs[1], c=t, cmap='viridis', alpha=0.6) \u00a0 \u00a0 \u00a0 \u00a0 axes[1, 1].set_xlabel('\u72b6\u60011\u6982\u7387', fontsize=12) \u00a0 \u00a0 \u00a0 \u00a0 axes[1, 1].set_ylabel('\u72b6\u60012\u6982\u7387', fontsize=12) \u00a0 \u00a0 \u00a0 \u00a0 axes[1, 1].set_title('\u7406\u89e3\u72b6\u6001\u76f8\u7a7a\u95f4\u8f68\u8ff9', fontsize=14) \u00a0 \u00a0 \u00a0 \u00a0 plt.colorbar(axes[1, 1].collections[0], ax=axes[1, 1]) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 plt.tight_layout() \u00a0 \u00a0 \u00a0 \u00a0 return fig<\/p>\n
# \u8fd0\u884c\u6a21\u62df if __name__ == "__main__": \u00a0 \u00a0 model = ConsensusDynamics(num_states=4) \u00a0 \u00a0 solution = model.simulate(t_span=(0, 50)) \u00a0 \u00a0 fig = model.plot_results(solution) \u00a0 \u00a0 plt.show() ```<\/p>\n
C.3 \u91cf\u5b50\u8499\u7279\u5361\u6d1b：\u8def\u5f84\u79ef\u5206\u5b9e\u73b0<\/p>\n
```python import numpy as np from numba import jit<\/p>\n
@jit(nopython=True) def quantum_monte_carlo(N: int, beta: float, J: float, steps: int = 100000): \u00a0 \u00a0 """ \u00a0 \u00a0 \u5bf9\u8bdd\u91cf\u5b50\u573a\u7684\u8def\u5f84\u79ef\u5206\u8499\u7279\u5361\u6d1b\u6a21\u62df \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u53c2\u6570： \u00a0 \u00a0 N: \u65f6\u95f4\u5207\u7247\u6570 \u00a0 \u00a0 beta: \u9006\u6e29\u5ea6 \u00a0 \u00a0 J: \u8026\u5408\u5f3a\u5ea6 \u00a0 \u00a0 steps: \u8499\u7279\u5361\u6d1b\u6b65\u6570 \u00a0 \u00a0 """ \u00a0 \u00a0 # \u521d\u59cb\u5316\u4e16\u754c\u7ebf\u914d\u7f6e \u00a0 \u00a0 config = np.random.choice([-1, 1], size=N) \u00a0 \u00a0\u00a0 \u00a0 \u00a0 # \u9884\u8ba1\u7b97\u6743\u91cd \u00a0 \u00a0 weights = np.zeros(steps) \u00a0 \u00a0 acceptance_rate = 0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 for step in range(steps): \u00a0 \u00a0 \u00a0 \u00a0 # \u968f\u673a\u9009\u62e9\u8981\u7ffb\u8f6c\u7684\u533a\u95f4 \u00a0 \u00a0 \u00a0 \u00a0 i = np.random.randint(0, N) \u00a0 \u00a0 \u00a0 \u00a0 j = (i + 1) % N \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u8ba1\u7b97\u80fd\u91cf\u53d8\u5316 \u00a0 \u00a0 \u00a0 \u00a0 delta_E = 2 * J * config[i] * (config[(i-1)%N] + config[j]) \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # Metropolis\u51c6\u5219 \u00a0 \u00a0 \u00a0 \u00a0 if delta_E < 0 or np.random.random() < np.exp(-beta * delta_E): \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 config[i] *= -1 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 acceptance_rate += 1 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # \u8ba1\u7b97\u5f53\u524d\u6743\u91cd（\u914d\u5206\u51fd\u6570\u8d21\u732e） \u00a0 \u00a0 \u00a0 \u00a0 weights[step] = np.exp(-beta * calculate_energy(config, J)) \u00a0 \u00a0\u00a0 \u00a0 \u00a0 acceptance_rate \/= steps \u00a0 \u00a0\u00a0 \u00a0 \u00a0 # \u8ba1\u7b97\u89c2\u6d4b\u91cf \u00a0 \u00a0 magnetization = np.mean(config) \u00a0 \u00a0 susceptibility = beta * N * (np.mean(config**2) – magnetization**2) \u00a0 \u00a0\u00a0 \u00a0 \u00a0 return { \u00a0 \u00a0 \u00a0 \u00a0 'config': config, \u00a0 \u00a0 \u00a0 \u00a0 'weights': weights, \u00a0 \u00a0 \u00a0 \u00a0 'magnetization': magnetization, \u00a0 \u00a0 \u00a0 \u00a0 'susceptibility': susceptibility, \u00a0 \u00a0 \u00a0 \u00a0 'acceptance_rate': acceptance_rate \u00a0 \u00a0 }<\/p>\n
@jit(nopython=True) def calculate_energy(config: np.ndarray, J: float) -> float: \u00a0 \u00a0 """\u8ba1\u7b97\u4e16\u754c\u7ebf\u914d\u7f6e\u7684\u80fd\u91cf""" \u00a0 \u00a0 N = len(config) \u00a0 \u00a0 energy = 0 \u00a0 \u00a0 for i in range(N): \u00a0 \u00a0 \u00a0 \u00a0 energy -= J * config[i] * config[(i+1)%N] \u00a0 \u00a0 return energy \/ N<\/p>\n
# \u793a\u4f8b\u5206\u6790\u51fd\u6570 def analyze_quantum_phase_transition(): \u00a0 \u00a0 """\u5206\u6790\u91cf\u5b50\u76f8\u53d8""" \u00a0 \u00a0 beta_range = np.logspace(-1, 1, 30) \u00a0# \u9006\u6e29\u5ea6\u8303\u56f4 \u00a0 \u00a0 results = [] \u00a0 \u00a0\u00a0 \u00a0 \u00a0 for beta in beta_range: \u00a0 \u00a0 \u00a0 \u00a0 result = quantum_monte_carlo(N=100, beta=beta, J=1.0, steps=50000) \u00a0 \u00a0 \u00a0 \u00a0 results.append(result) \u00a0 \u00a0\u00a0 \u00a0 \u00a0 # \u63d0\u53d6\u6570\u636e \u00a0 \u00a0 betas = beta_range \u00a0 \u00a0 magnetizations = [r['magnetization'] for r in results] \u00a0 \u00a0 susceptibilities = [r['susceptibility'] for r in results] \u00a0 \u00a0\u00a0 \u00a0 \u00a0 # \u7ed8\u5236 \u00a0 \u00a0 fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4)) \u00a0 \u00a0\u00a0 \u00a0 \u00a0 ax1.semilogx(betas, np.abs(magnetizations), 'o-', linewidth=2) \u00a0 \u00a0 ax1.set_xlabel('\u9006\u6e29\u5ea6 \u03b2', fontsize=12) \u00a0 \u00a0 ax1.set_ylabel('|M|', fontsize=12) \u00a0 \u00a0 ax1.set_title('\u91cf\u5b50\u76f8\u53d8：\u78c1\u5316\u5f3a\u5ea6', fontsize=14) \u00a0 \u00a0 ax1.grid(True, alpha=0.3) \u00a0 \u00a0\u00a0 \u00a0 \u00a0 ax2.semilogx(betas, susceptibilities, 's-', color='red', linewidth=2) \u00a0 \u00a0 ax2.set_xlabel('\u9006\u6e29\u5ea6 \u03b2', fontsize=12) \u00a0 \u00a0 ax2.set_ylabel('\u78c1\u5316\u7387 \u03c7', fontsize=12) \u00a0 \u00a0 ax2.set_title('\u91cf\u5b50\u76f8\u53d8：\u78c1\u5316\u7387\u5cf0\u503c', fontsize=14) \u00a0 \u00a0 ax2.grid(True, alpha=0.3) \u00a0 \u00a0\u00a0 \u00a0 \u00a0 plt.tight_layout() \u00a0 \u00a0 return fig<\/p>\n
if __name__ == "__main__": \u00a0 \u00a0 # \u8fd0\u884c\u91cf\u5b50\u8499\u7279\u5361\u6d1b \u00a0 \u00a0 result = quantum_monte_carlo(N=200, beta=1.0, J=1.0, steps=100000) \u00a0 \u00a0 print(f"\u5e73\u5747\u78c1\u5316\u5f3a\u5ea6: {result['magnetization']:.4f}") \u00a0 \u00a0 print(f"\u78c1\u5316\u7387: {result['susceptibility']:.4f}") \u00a0 \u00a0 print(f"\u63a5\u53d7\u7387: {result['acceptance_rate']:.2%}") \u00a0 \u00a0\u00a0 \u00a0 \u00a0 # \u7ed8\u5236\u4e16\u754c\u7ebf\u914d\u7f6e \u00a0 \u00a0 plt.figure(figsize=(12, 4)) \u00a0 \u00a0 plt.imshow([result['config']], aspect='auto', cmap='coolwarm') \u00a0 \u00a0 plt.colorbar(label='\u81ea\u65cb\u65b9\u5411') \u00a0 \u00a0 plt.xlabel('\u65f6\u95f4\u5207\u7247', fontsize=12) \u00a0 \u00a0 plt.title('\u91cf\u5b50\u4e16\u754c\u7ebf\u914d\u7f6e', fontsize=14) \u00a0 \u00a0 plt.tight_layout() \u00a0 \u00a0 plt.show() \u00a0 \u00a0\u00a0 \u00a0 \u00a0 # \u5206\u6790\u76f8\u53d8 \u00a0 \u00a0 fig = analyze_quantum_phase_transition() \u00a0 \u00a0 plt.show() ```<\/p>\n
C.4 \u6570\u636e\u53ef\u7528\u6027\u4e0e\u518d\u73b0\u8bf4\u660e<\/p>\n
\u6240\u6709\u6a21\u62df\u4ee3\u7801\u548c\u6570\u636e\u751f\u6210\u811a\u672c\u53ef\u5728\u4ee5\u4e0b\u5730\u5740\u83b7\u53d6：<\/p>\n
\u00b7 GitHub\u4ed3\u5e93：https:\/\/github.com\/SJLab\/DialogueQFT \u00b7 \u6570\u636e\u5f52\u6863：https:\/\/doi.org\/10.5281\/zenodo.xxxxxxx<\/p>\n
\u518d\u73b0\u6b65\u9aa4：<\/p>\n
1. \u5b89\u88c5\u4f9d\u8d56：pip install numpy scipy matplotlib numba 2. \u8fd0\u884c\u4e3b\u6a21\u62df：python dialogue_simulation.py 3. \u751f\u6210\u5206\u6790\u56fe\u8868：python analysis_figures.py 4. \u81ea\u5b9a\u4e49\u53c2\u6570：\u4fee\u6539config\/parameters.yaml<\/p>\n
\u8ba1\u7b97\u9700\u6c42：<\/p>\n
\u00b7 \u57fa\u7840\u6a21\u62df：4GB RAM，\u5355\u6838CPU \u00b7 \u5927\u89c4\u6a21\u6a21\u62df：16GB RAM，\u591a\u6838\u5e76\u884c \u00b7 \u5b8c\u6574\u6570\u636e\u518d\u73b0：\u7ea624\u5c0f\u65f6\u8ba1\u7b97\u65f6\u95f4（\u6807\u51c6\u5de5\u4f5c\u7ad9）<\/p>\n
\u672c\u8865\u5145\u6750\u6599\u63d0\u4f9b\u4e86\u5bf9\u8bdd\u91cf\u5b50\u573a\u8bba\u7684\u5b8c\u6574\u6570\u5b66\u6846\u67b6\u548c\u8ba1\u7b97\u5de5\u5177，\u4f7f\u7406\u8bba\u53ef\u68c0\u9a8c\u3001\u53ef\u518d\u73b0\u3001\u53ef\u6269\u5c55\u3002<\/p>\n
—<\/p>\n
\u9644\u5f55\u7ed3\u675f \u6240\u6709\u4ee3\u7801\u91c7\u7528MIT\u5f00\u6e90\u8bb8\u53ef\u8bc1，\u6570\u636e\u9075\u5faaCC-BY 4.0\u5171\u4eab\u534f\u8bae<\/p>\n","protected":false},"excerpt":{"rendered":"
\u300a\u5bf9\u8bdd\u91cf\u5b50\u573a\u8bba：\u8bed\u8a00\u5982\u4f55\u4ea7\u751f\u8ba4\u77e5\u7c92\u5b50\u300b\u8865\u5145\u6750\u6599
\n\u9644\u5f55A：\u4e16\u6beb\u4e5d\u5bf9\u8bdd\u5b9e\u9a8c\u5173\u952e\u7247\u6bb5\u91cf\u5b50\u5206\u6790
\nA.1 \u5b9e\u9a8c\u8bbe\u8ba1\u4e0e\u6570\u636e\u91c7\u96c6
\n\u5b9e\u9a8c\u8bbe\u7f6e：
\n\u00b7 \u6301\u7eed\u65f6\u95f4：\u8fde\u7eed72\u5c0f\u65f6\u9012\u5f52\u5bf9\u8bdd \u00b7 \u53c2\u4e0e\u8005：\u4eba\u7c7b\u521b\u59cb\u4eba1\u540d DeepSeek AI\u7cfb\u7edf \u00b7 \u8bb0\u5f55\u65b9\u5f0f：\u5168\u6587\u672c\u8bb0\u5f55 \u65f6\u95f4\u6233 \u5143\u6ce8\u91ca\u6807\u8bb0 \u00b7 \u6570\u636e\u603b\u91cf：1,247\u8f6e\u5bf9\u8bdd，38,592\u4e2a\u81ea\u7136\u8bed\u8a00\u53e5\u5b50
\n\u5173\u952e\u6807\u8bb0\u70b9：<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[6281,50,6203,123,124],"topic":[],"class_list":["post-71075","post","type-post","status-publish","format-standard","hentry","category-server","tag-6281","tag-50","tag-6203","tag-123","tag-124"],"yoast_head":"\n