{"id":9132,"date":"2025-02-18T02:10:38","date_gmt":"2025-02-17T18:10:38","guid":{"rendered":"https:\/\/www.andrewsun.net\/panta_rhei\/?p=9132"},"modified":"2025-02-20T01:25:49","modified_gmt":"2025-02-19T17:25:49","slug":"%e4%b8%8d%e5%9f%ba%e4%ba%8e%e7%b3%bb%e7%bb%bc%e6%a6%82%e5%bf%b5%e7%9a%84%e5%88%98%e7%bb%b4%e5%b0%94%e6%96%b9%e7%a8%8b%e6%8e%a8%e7%90%86","status":"publish","type":"post","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/9132","title":{"rendered":"\u4e0d\u57fa\u4e8e\u7cfb\u7efc\u6982\u5ff5\u7684\u5218\u7ef4\u5c14\u65b9\u7a0b\u63a8\u7406"},"content":{"rendered":"<div class=\"kcite-section\" kcite-section-id=\"9132\">\n\n<p>[latexpage]<\/p>\n\n\n\n<p>\u5218\u7ef4\u5c14\u65b9\u7a0b\u662f\u975e\u5e73\u8861\u6001\u7edf\u8ba1\u529b\u5b66\u4e2d\uff0c\u5173\u4e8e\u4efb\u610f\u7cfb\u7edf\u5fae\u72b6\u6001\u6982\u7387\u5bc6\u5ea6\u51fd\u6570$f\\left(\\bm\\Gamma,t\\right)$\u7684\u8fde\u7eed\u6027\u65b9\u7a0b\uff1a<\/p>\n\n\n\n<p>\\[\\frac{\\partial}{\\partial t}f+\\nabla_{\\bm\\Gamma}\\left(f\\dot{\\bm\\Gamma}\\right)=0\\]<\/p>\n\n\n\n<p>\u8fd9\u5b9e\u9645\u4e0a\u5728\u8bf4\uff0c\u4e0d\u4ec5\u4f5c\u4e3a\u6982\u7387\u5bc6\u5ea6\uff0c\u6709\u5f52\u4e00\u5316\u6761\u4ef6\u6240\u89c4\u5b9a\u7684\u5168\u57df\u6982\u7387\u5b88\u6052<br \/>\\[\\int_{\\Lambda_t}f\\left(\\bm\\Gamma,t\\right)\\mathrm{d}\\bm\\Gamma\\equiv1,\\quad\\forall t\\]<br \/>\u2014\u2014\u5176\u4e2d$\\Lambda_t\\in\\mathcal{B}$\u662f$t$\u65f6\u523b\u7cfb\u7edf\u53ef\u53d6\u7684\u6240\u6709\u72b6\u6001\u7684\u96c6\u5408\uff0c$\\mathcal{B}$\u8868\u793a$\\mathbb{R}^{6N}$\u4e0a\u7684Borel \u03c3-\u4ee3\u6570\u2014\u2014\u800c\u4e14\u5bf9\u4efb\u4e00\u5c40\u57df$\\omega_t\\subset\\Lambda_t,\\omega_t\\in\\mathcal{B}$\uff0c\u6982\u7387\u90fd\u5b88\u6052\u3002\u8fd9\u4ef6\u4e8b\uff0c\u5728\u4f20\u7edf\u6559\u79d1\u4e66\u5f53\u4e2d\u662f\u7528\u7cfb\u7efc\u7684\u6982\u5ff5\uff0c\u7136\u540e\u8bf4\u201c\u7cfb\u7efc\u7684\u7cfb\u7edf\u6570\u91cf\u4e0d\u53d8\u201d\u6765\u8ba8\u8bba\u7684\u3002\u4ee5\u4e0b\u6211\u4eec\u4ece\u6982\u7387\u8bba\u7684\u89d2\u5ea6\u5efa\u7acb\u7edf\u8ba1\u529b\u5b66\u7684\u57fa\u7840\u3002<\/p>\n\n\n\n<p>\u8003\u8651\u4e00\u4e2a\u7531$N$\u4e2a\u7c92\u5b50\u7ec4\u6210\u7684\u7cfb\u7edf\uff0c\u5728\u7edf\u8ba1\u529b\u5b66\u4e2d\uff0c\u8003\u8651\u7c92\u5b50\u6570$N$\u6052\u5b9a\u7684\u60c5\u51b5\u5df2\u7ecf\u8db3\u591f\u4e00\u822c\u3002\u5bf9\u4e8e\u7c92\u5b50\u6570\u53d8\u5316\u7684\u5f00\u653e\u7cfb\u7edf\uff0c\u7406\u8bba\u65b9\u6cd5\u4e0a\u603b\u662f\u901a\u8fc7\u6269\u5927\u7cfb\u7edf\u7684\u5212\u4efd\uff0c\u76f4\u81f3\u65b0\u7684\u7cfb\u7edf\u662f\u5c01\u95ed\u7684\uff08\u7c92\u5b50\u6570\u6052\u5b9a\uff09\uff0c\u7136\u540e\u57fa\u4e8e\u8fd9\u4e2a\u5927\u7cfb\u7edf\u7684\u6027\u8d28\u8ba8\u8bba\u5f53\u521d\u7684\u5c0f\u7cfb\u7edf\u4f5c\u4e3a\u5b50\u7cfb\u7edf\u7684\u6027\u8d28\u3002<\/p>\n\n\n\n<p>\u5728\u7ecf\u5178\u529b\u5b66\u5047\u5b9a\u4e0b\uff0c\u7cfb\u7edf\u7684\u72b6\u6001\u603b\u662f\u53ef\u4ee5\u7531\u6240\u6709\u7c92\u5b50\u7684\u4f4d\u7f6e\u548c\u52a8\u91cf\u6240\u786e\u5b9a\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c\u7cfb\u7edf\u7684\u72b6\u6001\u603b\u662f\u5bf9\u5e94\u7740$6N$\u7ef4\u5b9e\u6570\u7a7a\u95f4$\\mathbb{R}^{6N}$\u4e2d\u7684\u4e00\u4e2a\u70b9$\\bm\\Gamma\\equiv\\left(\\mathbf{r}^N,\\mathbf{p}^N\\right)$\uff0c\u5176\u4e2d\u6211\u4eec\u7b80\u8bb0$\\mathbf{r}^N\\equiv\\left(\\mathbf{r}_1,\\cdots,\\mathbf{r}_N\\right)$\u548c$\\mathbf{p}^N\\equiv\\left(\\mathbf{p}_1,\\cdots,\\mathbf{p}_N\\right)$\u3002<\/p>\n\n\n\n<p>\u7531\u4e8e\u6211\u4eec\u65e0\u6cd5\u786e\u5b9a\u7cfb\u7edf\u5728\u4efb\u4f55\u65f6\u523b\u7684\u5fae\u89c2\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u6bcf\u4e00\u65f6\u523b\uff0c\u90fd\u4e3a\u7cfb\u7edf\u53ef\u53d6\u7684\u6bcf\u4e00\u4e2a\u53ef\u80fd\u72b6\u6001\u8d4b\u4ee5\u6982\u7387\u3002\u5177\u4f53\u5730\uff0c\u8bbe$\\left(\\mathbb{R}^{6N},\\mathcal{B}\\right)$\u662f\u4e00\u4e2aBorel\u53ef\u6d4b\u7a7a\u95f4\uff0c$\\Lambda_t\\in\\mathcal{B}$\u8868\u793a\u7cfb\u7edf\u5728$t$\u65f6\u523b\u6240\u6709\u53ef\u53d6\u7684\u5fae\u89c2\u72b6\u6001\u7684\u96c6\u5408\uff0c$\\mu_t:\\mathcal{B}\\rightarrow\\left[0,1\\right]$\u662f\u5b9a\u4e49\u5728\u8fd9\u4e2a\u53ef\u6d4b\u7a7a\u95f4\u4e0a\u7684\u6982\u7387\u6d4b\u5ea6\uff0c$\\mu_t\\left[X\\right],X\\in\\mathcal{B}$\u8868\u793a$t$\u65f6\u523b\u7cfb\u7edf\u7684\u5fae\u89c2\u72b6\u6001\u5728$X$\u5185\u7684\u6982\u7387\uff0c\u5b9e\u9645\u4e0a<br \/>\\[\\mu_t\\left[X\\right]\\equiv\\int_Xf\\left(\\bm\\Gamma,t\\right)\\mathrm{d}\\bm\\Gamma\\]<br \/>\u5176\u4e2d$f\\left(\\bm\\Gamma,t\\right)\\mathrm{d}\\bm\\Gamma$\u662f\u7cfb\u7edf\u5728$t$\u65f6\u523b\u5fae\u89c2\u72b6\u6001\u5904\u4e8e$\\bm\\Gamma\\sim\\bm\\Gamma+\\mathrm{d}\\Gamma$\u7684\u6982\u7387\u3002\u5141\u8bb8\u6982\u7387\u5bc6\u5ea6$f\\left(\\bm\\Gamma,t\\right)$\u5f97\u5230\u5b9a\u4e49\u6240\u9700\u8981\u7684$\\mu_t$\u5173\u4e8e$\\mathbb{R}^{6N}$\u7684\u52d2\u8d1d\u683c\u6d4b\u5ea6\u7684\u7edd\u5bf9\u8fde\u7eed\u6027\uff0c\u5b9e\u9645\u4e0a\u662f\u7531\u7ecf\u5178\u529b\u5b66\u672c\u8eab\u4fdd\u8bc1\u7684\uff0c\u89c1<a href=\"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/8597\" data-type=\"post\" data-id=\"8597\">\u5173\u4e8e\u79bb\u6563\u6d4b\u5ea6\u6ca1\u6709\u5bc6\u5ea6\u7684\u95ee\u9898<\/a>\u3002<\/p>\n\n\n\n<p>\u5728\u7ecf\u5178\u529b\u5b66\u4e2d\uff0c\u7ed9\u5b9a\u67d0\u7cfb\u7edf\uff0c\u5b83\u7684\u8fd0\u52a8\u65b9\u7a0b\u603b\u662f\u628a\u4e00\u4e2a\u65f6\u523b$t$\u7684\u786e\u5b9a\u72b6\u6001$\\bm\\Gamma_t$\u6620\u5c04\u4e3a\u53e6\u4e00\u65f6\u523b$t+\\Delta t$\u7684\u552f\u4e00\u4e00\u4e2a\u786e\u5b9a\u72b6\u6001$\\bm\\Gamma_{t+\\Delta t}$\uff0c\u5373\u6211\u4eec\u53ef\u4ee5\u63d0\u53ca\u76f8\u5e94\u7684\u53cc\u5c04$\\chi_t:\\mathbb{R}^{6N}\\supset\\Lambda_t\\rightarrow\\mathbb{R}^{6N}$\u3002<\/p>\n\n\n\n<p>\u82e5$\\Lambda_t\\in\\mathcal{B},\\omega_t\\in\\mathcal{B},\\omega_t\\subset\\Lambda_t,\\omega_{t+\\Delta t}=\\chi_t\\left(\\omega_t\\right)$\uff0c\u5219\u9996\u5148\u770b\u5230\u56e0\u4e3a$\\chi_t$\u5fc5\u662f\u8fde\u7eed\u7684\uff08\u7ecf\u5178\u529b\u5b66\u8fd0\u52a8\uff09\uff0c\u6545$\\chi_t$\u603b\u628a\u5f00\u96c6\u6620\u5c04\u4e3a\u5f00\u96c6\uff0c\u5373$\\omega_{t+\\Delta t}\\in\\mathcal{B}$\uff0c\u6545$\\chi_t$\u662f\u53ef\u6d4b\u6620\u5c04\u3002\u5176\u6b21\uff0c\u6211\u4eec\u770b\u5230\uff1a<br \/>\\[\\mu_{t+\\Delta t}\\left(\\omega_{t+\\Delta t}\\right)=\\chi_t_*\\mu_t\\left(\\omega_t\\right)=\\mu_t\\left(\\chi_t^{-1}\\left(\\omega_{t+\\Delta t}\\right)\\right)=\\mu_t\\left(\\omega_t\\right)\\]<br \/>\u5176\u4e2d\u8bb0\u53f7$f_*\\mu$\u8868\u793a\u6d4b\u5ea6$\\mu_t$\u7684\u7531\u53ef\u6d4b\u6620\u5c04$f$\u5f15\u51fa\u7684\u524d\u63a8\u6d4b\u5ea6\uff08pushforward measrue\uff09\uff0c\u4e0a\u9762\u7b2c\u4e8c\u4e2a\u7b49\u5f0f\u7528\u5230\u4e86\u524d\u63a8\u6d4b\u5ea6\u7684\u5b9a\u4e49\uff0c\u7b2c\u4e09\u4e2a\u7b49\u5f0f\u7528\u5230\u4e86$\\chi_t$\u7684\u53cc\u5c04\u6027\u3002\u4ee5\u4e0a\u63a8\u7406\u5bf9\u4efb\u610f$t,\\Delta t\\in\\mathbb{R}$\u4ee5\u53ca\u4efb\u610f$\\omega_t$\u5747\u6210\u7acb\u3002\u56e0\u6b64\u6211\u4eec\u8bc1\u660e\u4e86\uff0c$\\mathbb{R}^{6N}$\u7a7a\u95f4\u4efb\u4e00\uff08\u53ef\u6d4b\uff09\u5c40\u57df\u4e0a\u7684\u6982\u7387\u662f\u6cbf\u533a\u57df\u968f\u6309\u7167\u5b9e\u9645\u7cfb\u7edf\u8fd0\u52a8\u7684\u8fd0\u52a8\u7684\u5b88\u6052\u91cf\u3002\u56e0\u6b64\u6709<br \/>\\[\\frac{\\mathrm{d}}{\\mathrm{d}t}\\mu_t\\left[\\omega_t\\right]\\equiv 0\\]<\/p>\n\n\n\n<p>\u540e\u7eed\u7531\u96f7\u8bfa\u4f20\u8f93\u5b9a\u7406\u5373\u53ef\u5f97\u5230\u5173\u4e8e\u6982\u7387\u5bc6\u5ea6\u51fd\u6570$f\\left(\\bm\\Gamma,t\\right)$\u7684\u5218\u7ef4\u5c14\u65b9\u7a0b\u3002<\/p>\n<!-- kcite active, but no citations found -->\n<\/div> <!-- kcite-section 9132 -->","protected":false},"excerpt":{"rendered":"<p>[latexpage] \u5218\u7ef4\u5c14\u65b9\u7a0b\u662f\u975e\u5e73\u8861\u6001\u7edf\u8ba1\u529b\u5b66\u4e2d\uff0c\u5173\u4e8e\u4efb\u610f\u7cfb\u7edf\u5fae\u72b6\u6001\u6982\u7387\u5bc6\u5ea6\u51fd\u6570$f\\left(\\bm\\Gamma,t\\right)$\u7684\u8fde\u7eed\u6027\u65b9\u7a0b\uff1a \\[\\frac{\\partial}{\\partial t}f+\\nabla_{\\bm\\Gamma}\\left(f\\dot{\\bm\\Gamma}\\right)=0\\] \u8fd9\u5b9e\u9645\u4e0a\u5728\u8bf4\uff0c\u4e0d\u4ec5\u4f5c\u4e3a\u6982\u7387\u5bc6\u5ea6\uff0c\u6709\u5f52\u4e00\u5316\u6761\u4ef6\u6240\u89c4\u5b9a\u7684\u5168\u57df\u6982\u7387\u5b88\u6052\\[\\int_{\\Lambda_t}f\\left(\\bm\\Gamma,t\\right)\\mathrm{d}\\bm\\Gamma\\equiv1,\\quad\\forall t\\]\u2014\u2014\u5176\u4e2d$\\Lambda_t\\in\\mathcal{B}$\u662f$t$\u65f6\u523b\u7cfb\u7edf\u53ef\u53d6\u7684\u6240\u6709\u72b6\u6001\u7684\u96c6\u5408\uff0c$\\mathcal{B}$\u8868\u793a$\\mathbb{R}^{6N}$\u4e0a\u7684Borel \u03c3-\u4ee3\u6570\u2014\u2014\u800c\u4e14\u5bf9\u4efb\u4e00\u5c40\u57df$\\omega_t\\subset\\Lambda_t,\\omega_t\\in\\mathcal{B}$\uff0c\u6982\u7387\u90fd\u5b88\u6052\u3002\u8fd9\u4ef6\u4e8b\uff0c\u5728\u4f20\u7edf\u6559\u79d1\u4e66\u5f53\u4e2d\u662f\u7528\u7cfb\u7efc\u7684\u6982\u5ff5\uff0c\u7136\u540e\u8bf4\u201c\u7cfb\u7efc\u7684\u7cfb\u7edf\u6570\u91cf\u4e0d\u53d8\u201d\u6765\u8ba8\u8bba\u7684\u3002\u4ee5\u4e0b\u6211\u4eec\u4ece\u6982\u7387\u8bba\u7684\u89d2\u5ea6\u5efa\u7acb\u7edf\u8ba1\u529b\u5b66\u7684\u57fa\u7840\u3002 \u8003\u8651\u4e00\u4e2a\u7531$N$\u4e2a\u7c92\u5b50\u7ec4\u6210\u7684\u7cfb\u7edf\uff0c\u5728\u7edf\u8ba1\u529b\u5b66\u4e2d\uff0c\u8003\u8651\u7c92\u5b50\u6570$N$\u6052\u5b9a\u7684\u60c5\u51b5\u5df2\u7ecf\u8db3\u591f\u4e00\u822c\u3002\u5bf9\u4e8e\u7c92\u5b50\u6570\u53d8\u5316\u7684\u5f00\u653e\u7cfb\u7edf\uff0c\u7406\u8bba\u65b9\u6cd5\u4e0a\u603b\u662f\u901a\u8fc7\u6269\u5927\u7cfb\u7edf\u7684\u5212\u4efd\uff0c\u76f4\u81f3\u65b0\u7684\u7cfb\u7edf\u662f\u5c01\u95ed\u7684\uff08\u7c92\u5b50\u6570\u6052\u5b9a\uff09\uff0c\u7136\u540e\u57fa\u4e8e\u8fd9\u4e2a\u5927\u7cfb\u7edf\u7684\u6027\u8d28\u8ba8\u8bba\u5f53\u521d\u7684\u5c0f\u7cfb\u7edf\u4f5c\u4e3a\u5b50\u7cfb\u7edf\u7684\u6027\u8d28\u3002 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\u7531\u4e8e\u6211\u4eec\u65e0\u6cd5\u786e\u5b9a\u7cfb\u7edf\u5728\u4efb\u4f55\u65f6\u523b\u7684\u5fae\u89c2\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u6bcf\u4e00\u65f6\u523b\uff0c\u90fd\u4e3a\u7cfb\u7edf\u53ef\u53d6\u7684\u6bcf\u4e00\u4e2a\u53ef\u80fd\u72b6\u6001\u8d4b\u4ee5\u6982\u7387\u3002\u5177\u4f53\u5730\uff0c\u8bbe$\\left(\\mathbb{R}^{6N},\\mathcal{B}\\right)$\u662f\u4e00\u4e2aBorel\u53ef\u6d4b\u7a7a\u95f4\uff0c$\\Lambda_t\\in\\mathcal{B}$\u8868\u793a\u7cfb\u7edf\u5728$t$\u65f6\u523b\u6240\u6709\u53ef\u53d6\u7684\u5fae\u89c2\u72b6\u6001\u7684\u96c6\u5408\uff0c$\\mu_t:\\mathcal{B}\\rightarrow\\left[0,1\\right]$\u662f\u5b9a\u4e49\u5728\u8fd9\u4e2a\u53ef\u6d4b\u7a7a\u95f4\u4e0a\u7684\u6982\u7387\u6d4b\u5ea6\uff0c$\\mu_t\\left[X\\right],X\\in\\mathcal{B}$\u8868\u793a$t$\u65f6\u523b\u7cfb\u7edf\u7684\u5fae\u89c2\u72b6\u6001\u5728$X$\u5185\u7684\u6982\u7387\uff0c\u5b9e\u9645\u4e0a\\[\\mu_t\\left[X\\right]\\equiv\\int_Xf\\left(\\bm\\Gamma,t\\right)\\mathrm{d}\\bm\\Gamma\\]\u5176\u4e2d$f\\left(\\bm\\Gamma,t\\right)\\mathrm{d}\\bm\\Gamma$\u662f\u7cfb\u7edf\u5728$t$\u65f6\u523b\u5fae\u89c2\u72b6\u6001\u5904\u4e8e$\\bm\\Gamma\\sim\\bm\\Gamma+\\mathrm{d}\\Gamma$\u7684\u6982\u7387\u3002\u5141\u8bb8\u6982\u7387\u5bc6\u5ea6$f\\left(\\bm\\Gamma,t\\right)$\u5f97\u5230\u5b9a\u4e49\u6240\u9700\u8981\u7684$\\mu_t$\u5173\u4e8e$\\mathbb{R}^{6N}$\u7684\u52d2\u8d1d\u683c\u6d4b\u5ea6\u7684\u7edd\u5bf9\u8fde\u7eed\u6027\uff0c\u5b9e\u9645\u4e0a\u662f\u7531\u7ecf\u5178\u529b\u5b66\u672c\u8eab\u4fdd\u8bc1\u7684\uff0c\u89c1\u5173\u4e8e\u79bb\u6563\u6d4b\u5ea6\u6ca1\u6709\u5bc6\u5ea6\u7684\u95ee\u9898\u3002 \u5728\u7ecf\u5178\u529b\u5b66\u4e2d\uff0c\u7ed9\u5b9a\u67d0\u7cfb\u7edf\uff0c\u5b83\u7684\u8fd0\u52a8\u65b9\u7a0b\u603b\u662f\u628a\u4e00\u4e2a\u65f6\u523b$t$\u7684\u786e\u5b9a\u72b6\u6001$\\bm\\Gamma_t$\u6620\u5c04\u4e3a\u53e6\u4e00\u65f6\u523b$t+\\Delta t$\u7684\u552f\u4e00\u4e00\u4e2a\u786e\u5b9a\u72b6\u6001$\\bm\\Gamma_{t+\\Delta t}$\uff0c\u5373\u6211\u4eec\u53ef\u4ee5\u63d0\u53ca\u76f8\u5e94\u7684\u53cc\u5c04$\\chi_t:\\mathbb{R}^{6N}\\supset\\Lambda_t\\rightarrow\\mathbb{R}^{6N}$\u3002 \u82e5$\\Lambda_t\\in\\mathcal{B},\\omega_t\\in\\mathcal{B},\\omega_t\\subset\\Lambda_t,\\omega_{t+\\Delta t}=\\chi_t\\left(\\omega_t\\right)$\uff0c\u5219\u9996\u5148\u770b\u5230\u56e0\u4e3a$\\chi_t$\u5fc5\u662f\u8fde\u7eed\u7684\uff08\u7ecf\u5178\u529b\u5b66\u8fd0\u52a8\uff09\uff0c\u6545$\\chi_t$\u603b\u628a\u5f00\u96c6\u6620\u5c04\u4e3a\u5f00\u96c6\uff0c\u5373$\\omega_{t+\\Delta t}\\in\\mathcal{B}$\uff0c\u6545$\\chi_t$\u662f\u53ef\u6d4b\u6620\u5c04\u3002\u5176\u6b21\uff0c\u6211\u4eec\u770b\u5230\uff1a\\[\\mu_{t+\\Delta t}\\left(\\omega_{t+\\Delta t}\\right)=\\chi_t_*\\mu_t\\left(\\omega_t\\right)=\\mu_t\\left(\\chi_t^{-1}\\left(\\omega_{t+\\Delta t}\\right)\\right)=\\mu_t\\left(\\omega_t\\right)\\]\u5176\u4e2d\u8bb0\u53f7$f_*\\mu$\u8868\u793a\u6d4b\u5ea6$\\mu_t$\u7684\u7531\u53ef\u6d4b\u6620\u5c04$f$\u5f15\u51fa\u7684\u524d\u63a8\u6d4b\u5ea6\uff08pushforward measrue\uff09\uff0c\u4e0a\u9762\u7b2c\u4e8c\u4e2a\u7b49\u5f0f\u7528\u5230\u4e86\u524d\u63a8\u6d4b\u5ea6\u7684\u5b9a\u4e49\uff0c\u7b2c\u4e09\u4e2a\u7b49\u5f0f\u7528\u5230\u4e86$\\chi_t$\u7684\u53cc\u5c04\u6027\u3002\u4ee5\u4e0a\u63a8\u7406\u5bf9\u4efb\u610f$t,\\Delta 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https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2023\/08\/20230808_012812896_iOS-%E5%A4%A7.jpg?resize=1400%2C800&ssl=1 4x"},"classes":[]},{"id":8597,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/8597","url_meta":{"origin":9132,"position":1},"title":"\u5173\u4e8e\u79bb\u6563\u6d4b\u5ea6\u6ca1\u6709\u5bc6\u5ea6\u7684\u95ee\u9898","author":"Andrew","date":"2022\u5e748\u670813\u65e5","format":false,"excerpt":"[latexpage] \u5bf9\u4e8e\u82e5\u5e72\u4e2a\u79bb\u6563\u8d28\u70b9\u7ec4\u6210\u7684\u7269\u4f53\uff0c\u80fd\u5426\u5b9a\u4e49\u5176\u8d28\u91cf\u5bc6\u5ea6\uff1f\u4e0d\u53ef\u4ee5\u3002\u8003\u8651\u4ee5\u4e0b\u7b80\u5316\u7684\u95ee\u9898\u3002 \u8bbe$x_1,\\cdots,x_n\\in\\mathbb{R}$\uff0c\u4e14\u5404\u4e0d\u76f8\u7b49\u3002\u5bf9\u4efb\u4e00\u52d2\u8d1d\u683c\u53ef\u6d4b\u96c6$A\\subset\\mathbb{R}$\uff0c\u5b9a\u4e49\u4ee5\u4e0b\u6d4b\u5ea6\uff1a \\[ \\delta_i\\left(A\\right)=\\left\\{\\begin{array}{cc} 1,&x_i\\in A\\\\ 0,&x_i\\notin A \\end{array} \\right.,\\quad \\mu\\left(A\\right)=\\sum_i\\delta_i\\left(A\\right),\\quad\\nu\\left(A\\right)=\\sum_i p_i \\delta_i\\left(A\\right) \\] \u5176\u4e2d $p_1,\\cdots,p_n\\in\\mathbb{R}^+$\u3002\u82e5\u662f\u5728\u8ba8\u8bba\u6982\u7387\u5219\u8fd8\u9700$\\sum_i p_i=1$\u3002\u867d\u7136$\\nu$\u5bf9$\\mu$\u7edd\u5bf9\u8fde\u7eed\uff08\u53ef\u9a8c\uff09\uff0c\u4f46\u7531\u4e8e$\\mu$\u4e0d\u662f$\\sigma$-\u6709\u9650\u7684\uff0c\u56e0\u6b64\u4e0d\u9002\u7528\u4e8e\u62c9\u4e1c-\u5c3c\u79d1\u8fea\u59c6\u5b9a\u7406\u4ee5\u627e\u5230\u6ee1\u8db3 \\[\\nu\\left(A\\right)=\\int_A \\rho d\\mu\\] \u7684\u201c\u5bc6\u5ea6\u573a\u201d$\\rho$\u3002Truesdell & Toupin\u7684CFT\u53ea\u5bf9\u8fde\u7eed\u7269\u8d28\u5b9a\u4e49\u4e86\u5bc6\u5ea6\uff0c\u79bb\u6563\u8d28\u70b9\u53ea\u6709\u8d28\u91cf\uff0c\u6ca1\u6709\u4f53\u79ef\u548c\u5bc6\u5ea6\u3002\u5728Truesdell & Noll\u7684NFT\u4e2d\u8fd9\u4e00\u70b9\u66f4\u660e\u786e\u3002 \u5728\u8ba8\u8bba\u7269\u4f53\u7684\u8d28\u91cf\u65f6\uff0c\u5728\u8fd9\u6837\u4e00\u4e2a\u5b9a\u4e49\u4e0b\uff0c\u4e00\u4e2a\u7531\u65e0\u9650\u4e2a\u79bb\u6563\u8d28\u70b9\u7ec4\u6210\u7684\u7269\u4f53\u8d28\u91cf\u5c06\u4f1a\u662f\u65e0\u7a77\u5927\u7684\u3002\u7406\u8bba\u4e0a\u6211\u4eec\u63a5\u53d7\u8fd9\u4e00\u8bbe\u5b9a\u3002\u76f8\u5e94\u5730\uff0c\u5728\u8ba8\u8bba\u6982\u7387\u7684\u65f6\u5019\uff0c\u5982\u4f55\u8ba8\u8bba\u4e00\u4e2a\u65e0\u7a77\u79bb\u6563\u96c6\u4e0a\u7684\u6982\u7387\uff1f\u53ef\u4ee5\u4ece\u6709\u9650\u96c6\u7684\u60c5\u51b5\u5411\u65e0\u7a77\u8fdb\u884c\u63a8\u5e7f\u800c\u4e0d\u4f1a\u4ea7\u751f\u6982\u5ff5\u4e0a\u7684\u5371\u673a\uff0c\u4f8b\u5982\u8fd9\u4e00\u94fe\u63a5\u4e2d\u7684\u4f8b\u5b50\u3002 \u4e00\u4e2a\u7531\u82e5\u5e72\u4e2a\u79bb\u6563\u8d28\u70b9\u7684\u7ec4\u6210\u7684\u7269\u4f53\u65e2\u7136\u6ca1\u6709\u8d28\u91cf\u5bc6\u5ea6\u573a\uff0c\u90a3\u4e48\u5b83\u7684\u201c\u8fde\u7eed\u6027\u65b9\u7a0b\u201d\u662f\u600e\u6837\u7684\u5462\uff1f\u5982\u4f55\u8ba8\u8bba\u5176\u201c\u8d28\u91cf\u5b88\u6052\u201d\u5f8b\u5462\uff1fCFT\u5728\u5b9a\u4e49\u4e86\u79bb\u6563\u8d28\u70b9\u7684\u8d28\u91cf\u540e\u6ca1\u6709\u518d\u8ba8\u8bba\u8fd9\u4e2a\u95ee\u9898\uff0c\u800cNFT\u4ece\u5b9a\u4e49\u7269\u4f53\u8d28\u91cf\u7684\u65f6\u5019\u5c31\u660e\u786e\u4e0d\u8003\u8651\u79bb\u6563\u8d28\u70b9\u7684\u60c5\u51b5\u3002\u6211\u5728\u8fd9\u91cc\u7ed9\u51fa\u7684\u610f\u89c1\u662f\uff1a\u8fde\u7eed\u6027\u65b9\u7a0b\u53ea\u662f\u5728\u9700\u8981\u7528\u8fde\u7eed\u4ecb\u8d28\u529b\u5b66\u65f6\u624d\u9700\u8981\u9996\u5148\u5217\u51fa\u7684\u3002\u5bf9\u4e8e\u79bb\u6563\u8d28\u70b9\u7ec4\u6210\u7684\u52a8\u529b\u5b66\u7cfb\u7edf\uff0c\u5df2\u7ecf\u53ef\u4ee5\u4f7f\u7528\u5e38\u89c4\u7ecf\u5178\u529b\u5b66\u8bfe\u672c\u4e2d\u4ecb\u7ecd\u7684\u725b\u987f\u529b\u5b66\u6216\u62c9\u683c\u6717\u65e5\u529b\u5b66\u6765\u523b\u5212\u3002\u5982\u679c\u975e\u8981\u8ba8\u8bba\u79bb\u6563\u8d28\u70b9\u4f53\u7cfb\u7684\u8fde\u7eed\u6027\u65b9\u7a0b\uff0c\u90a3\u5b8c\u5168\u4e0d\u59a8\u5f15\u5165Irving-Kirkwood-Noll procedure\uff0c\u6b64\u65f6\u6240\u6709\u8fde\u7eed\u4ecb\u8d28\u7684\u516c\u8bbe\u4ecd\u7136\u6210\u7acb\uff08see also \uff09\u3002 \u73b0\u5728\u6211\u4eec\u8f6c\u800c\u8003\u8651\u76f8\u7a7a\u95f4\u6d41\u4f53\u7531\u79bb\u6563\u70b9\u7ec4\u6210\u65f6\u7684\u76f8\u5bc6\u5ea6\u573a\u95ee\u9898\u3002\u672c\u6587\u4e00\u5f00\u59cb\u5df2\u7ecf\u5426\u51b3\u4e86\uff0c\u5bf9\u82e5\u5e72\u4e2a\u79bb\u6563\u70b9\u662f\u65e0\u6cd5\u5b9a\u4e49\u5bc6\u5ea6\u573a\u7684\uff0c\u4e5f\u6ca1\u6709\u8d44\u6599\u8ba8\u8bba\u8fd9\u79cd\u4f53\u7cfb\u7684\u201c\u8fde\u7eed\u6027\u65b9\u7a0b\u201d\u3002\u56e0\u6b64\uff0c\u5982\u679c\u4e00\u4e2a\u52a8\u529b\u5b66\u7cfb\u7edf\u5728\u67d0\u79cd\u7ea6\u675f\u4e0b\uff0c\u5176\u6240\u6709\u53ef\u53d6\u7684\u72b6\u6001\uff08all admissible states\uff09\u5728\u76f8\u7a7a\u95f4\u662f\u79bb\u6563\u7684\u72b6\u6001\u70b9\uff08\u591a\u4e8e1\u4e2a\uff09\uff0c\u90a3\u5c06\u65e0\u6cd5\u5b9a\u4e49\u201c\u5bc6\u5ea6\u201d\u5e76\u8ba8\u8bba\u5176\u5218\u7ef4\u5c14\u65b9\u7a0b\u3002\u95ee\u9898\u662f\uff0c\u8fd9\u79cd\u60c5\u51b5\u662f\u5426\u53ef\u80fd\uff1f\u6211\u7684\u63a8\u65ad\u662f\u4e0d\u53ef\u80fd\u3002\u4ee5\u4e0b\u662f\u4e00\u4e9b\u4e0d\u592a\u5145\u4efd\u7684\u7406\u7531\u3002 \u5982\u679c\u4e00\u4e2a\u52a8\u529b\u5b66\u4f53\u7cfb\u5728\u67d0\u7ea6\u675f\u4e0b\u53ef\u4ee5\u53d6\u4e24\u4e2a\u79bb\u6563\u7684\u72b6\u6001\u70b9\uff0c\u90a3\u4e48\u65e2\u7136\u5b83\u4eec\u90fd\u201c\u53ef\u53d6\u201d\uff0c\u5c31\u5e94\u5f53\u5b58\u5728\u4e00\u4e2a\u865a\u62df\u8fd0\u52a8\u7531\u5176\u4e2d\u4e00\u4e2a\u72b6\u6001\u8fbe\u5230\u53e6\u4e00\u4e2a\u72b6\u6001\u3002\u4f46\u8fd9\u5c06\u8981\u6c42\u8fd9\u4e2a\u52a8\u529b\u5b66\u4f53\u7cfb\u8981\u4e48\u901f\u5ea6\u65e0\u7a77\u5927\uff08\u6784\u578b\u7a7a\u95f4\u4e2d\u7684\u77ac\u79fb\uff09\uff0c\u8981\u4e48\u52a0\u901f\u5ea6\u65e0\u7a77\u5927\uff08\u52a8\u91cf\u7a81\u53d8\uff09\u3002\u80fd\u5426\u8bf4\u5728\u7ecf\u5178\u529b\u5b66\u4e2d\u8fd9\u4e24\u8005\u90fd\u4e0d\u88ab\u5141\u8bb8\uff1f\u5982\u679c\u901f\u5ea6\uff08\u52a8\u91cf\uff09\u65e0\u7a77\u5927\uff0c\u5c06\u5bfc\u81f4\u65e0\u7a77\u5927\u7684\u80fd\u91cf\u3002\u53ea\u80fd\u8bf4\uff0c\u5982\u679c\u8ba4\u4e3a\u6574\u4e2a\u7ecf\u5178\u5b87\u5b99\u662f\u4e00\u4e2a\u5b64\u7acb\u4f53\u7cfb\u7684\u8bdd\u90a3\u4e48\u6574\u4e2a\u5b87\u5b99\u7684\u603b\u80fd\u91cf\u662f\u6709\u9650\u7684\u5e38\u6570\uff08\u5426\u5219\u6ca1\u6709\u80fd\u91cf\u5b88\u6052\u7684\u8ba8\u8bba\uff09\uff0c\u4efb\u4e8e\u8fd9\u4e2a\u5b87\u5b99\u5185\u7684\u52a8\u529b\u5b66\u4f53\u7cfb\u4e0d\u53ef\u4ee5\u5177\u6709\u65e0\u7a77\u5927\u7684\u80fd\u91cf\u3002\u56e0\u6b64\uff0c\u4e5f\u4e0d\u53ef\u80fd\u6709\u65e0\u7a77\u5927\u7684\u52a0\u901f\u5ea6\uff0c\u56e0\u4e3a\u8fd9\u9700\u8981\u6d88\u8017\u65e0\u7a77\u5927\u7684\u80fd\u91cf\uff0c\u7ecf\u5178\u5b87\u5b99\u4e0d\u5177\u5907\u3002\u4f46\u662f\u7ecf\u5178\u529b\u5b66\u5bf9\u5b87\u5b99\u662f\u5426\u5fc5\u987b\u6709\u8fd9\u79cd\u5b64\u7acb\u7cfb\u7edf\u7684\u89c4\u5b9a\uff0c\u6211\u4e0d\u592a\u6e05\u695a\u3002\u800c\u4e14\u5982\u679c\u771f\u7684\u4e00\u5b9a\u8981\u4f9d\u8d56\u8fd9\u4e00\u89c4\u5b9a\uff0c\u624d\u80fd\u7981\u6b62\u52a0\u901f\u5ea6\u65e0\u7a77\u5927\uff0c\u90a3\u53ef\u80fd\u5b64\u7acb\u5b87\u5b99\u5c31\u662f\u7ecf\u5178\u529b\u5b66\u7684\u5176\u4e2d\u4e00\u4e2a\u5fc5\u8981\u7684\u57fa\u7840\u3002\u603b\u4e4b\uff0c\u6211\u76ee\u524d\u4f9d\u8d56\u4e8e\u901f\u5ea6\u65e0\u7a77\u5927\u548c\u52a0\u901f\u5ea6\u65e0\u7a77\u5927\u7684\u7edd\u5bf9\u7981\u6b62\u3002 \u4e3a\u4f55\u4e24\u4e2a\u201c\u53ef\u53d6\u201d\u72b6\u6001\u95f4\u4e00\u5b9a\u8981\u5b58\u5728\u81f3\u5c11\u4e00\u79cd\u8fde\u7ed3\u5b83\u4eec\u7684\u865a\u62df\u8fd0\u52a8\u5462\uff1f\u4e3a\u4f55\u4e0d\u80fd\u901a\u8fc7\u8bbe\u7f6e\u521d\u6761\u4ef6\u4e3a\u4efb\u4e00\u4e2a\u53ef\u53d6\u72b6\u6001\uff0c\u7136\u540e\u4f53\u7cfb\u4e4b\u540e\u5c31\u7559\u5728\u90a3\u4e2a\u72b6\u6001\u4e0d\u52a8\uff1f\u2014\u2014\u8fd9\u6837\u4e5f\u4e0d\u8fdd\u53cd\u201c\u4e24\u4e2a\u72b6\u6001\u90fd\u662f\u8be5\u7ea6\u675f\u4e0b\u53ef\u53d6\u7684\u201d\uff0c\u4f46\u4e4b\u95f4\u6ca1\u6709\u4efb\u4e00\u79cd\u865a\u62df\u8fd0\u52a8\u53ef\u4ee5\u8054\u7cfb\u3002\u6211\u7b54\u6848\u662f\uff0c\u8fd9\u79cd\u60c5\u51b5\u8fdd\u53cd\u56e0\u679c\u5f8b\u6216\u4fe1\u606f\u5b88\u6052\u3002\u56e0\u4e3a\u529b\u5b66\u95ee\u9898\u4e2d\u7684\u201c\u521d\u6761\u4ef6\u201d\uff0c\u4e5f\u5fc5\u987b\u662f\u5728\u540c\u4e00\u52a8\u529b\u5b66\u89c4\u5f8b\u4e0b\u53ef\u4ee5\u4ece\u5176\u4ed6\u72b6\u6001\u201c\u8fbe\u5230\u201d\u7684\u7ec8\u70b9\uff0c\u4e0d\u80fd\u662f\u201c\u65e0\u56e0\u4e4b\u679c\u201d\u3002\u5982\u679c\u4e00\u4e2a\u521d\u6761\u4ef6\u72b6\u6001\uff0c\u9664\u4e86\u4eba\u4e3a\u8bbe\u5b9a\u5b83\u5728\u8fd9\u4e00\u72b6\u6001\u4e4b\u5916\uff0c\u65e0\u6cd5\u4ece\u5176\u4ed6\u72b6\u6001\u8fbe\u5230\uff0c\u90a3\u8fd9\u76f8\u5f53\u4e8e\u201c\u4e0a\u5e1d\u7684\u7b2c\u4e00\u63a8\u52a8\u201d\u95ee\u9898\u3002\u800c\u8fd9\u4e00\u95ee\u9898\u9002\u7528\u4e14\u53ea\u9002\u7528\u4e8e\u4e00\u4e2a\u52a8\u529b\u5b66\u7cfb\u7edf\u2014\u2014\u6574\u4e2a\u5b87\u5b99\u3002\u6240\u4ee5\u5728\u4efb\u4f55\u5177\u4f53\u52a8\u529b\u5b66\u7cfb\u7edf\u7684\u95ee\u9898\u4e2d\u8fd9\u662f\u4e0d\u5141\u8bb8\u7684\u3002 \u6240\u4ee5\uff0c\u52a8\u529b\u5b66\u7cfb\u7edf\u7684\u76f8\u7a7a\u95f4\u6d41\u4f53\u53ef\u4ee5\u4e00\u822c\u5730\u907f\u514d\u201c\u79bb\u6563\u70b9\u6ca1\u6709\u5bc6\u5ea6\u201d\u7684\u95ee\u9898\u3002\u5b83\u603b\u662f\u201c\u8fde\u7eed\u4ecb\u8d28\u201d\uff08\u6216\u8bf4\u662f\u5355\u8fde\u901a\u7d27\u96c6\uff09\u3002 2022\/10\/4\u8865\u5145\uff1a\u5728R. Tolman\u6559\u6750\u4e2d\u662f\u8fd9\u6837\u5f15\u5165\u7684\uff1a ..., it is also to be noted for statistical purposes that\u2026","rel":"","context":"In &quot;\u6211\u7684\u7814\u7a76\u7b14\u8bb0&quot;","block_context":{"text":"\u6211\u7684\u7814\u7a76\u7b14\u8bb0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/rheo"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":8145,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/8145","url_meta":{"origin":9132,"position":2},"title":"\u9ad8\u5206\u5b50\u4e13\u4e1a\u7684\u540c\u5b66\u600e\u4e48\u5b66\u6d41\u53d8\u5b66","author":"Andrew","date":"2021\u5e743\u670825\u65e5","format":false,"excerpt":"\u8fde\u7eed\u4ecb\u8d28\u529b\u5b66\u662f\u6d41\u53d8\u5b66\u7684\u57fa\u7840\u3002\u800c\u5fc5\u8981\u7684\u7ebf\u6027\u4ee3\u6570\u548c\u5411\u91cf\u51fd\u6570\u5fae\u79ef\u5206\u53c8\u662f\u8fde\u7eed\u4ecb\u8d28\u529b\u5b66 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of Polymeric 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Hansen\u7684\u4e66\u80f6\u4f53\u60ac\u6d6e\u6db2\u4f53\u7cfb\u7684\u7406\u8bba\u5f80\u5f80\u5ef6\u7528\u7b80\u5355\u6db2\u4f53\u7406\u8bba\u3002Hansen and McDonald\u7684Theory of Simple Liquids\u662f\u80f6\u4f53\u7269\u7406\u8bba\u6587\u7ecf\u5e38\u88ab\u5f15\u7528\u7684\u7ecf\u5178\u30022013\u5e74\u51fa\u7248\u7684\u7b2c4\u7248\u8fd8\u4e3a\u8fd1\u5e74\u6765\u201c\u8f6f\u7269\u8d28\u7269\u7406\u201d\u7684\u5174\u8d77\u589e\u52a0\u4e86\u4e00\u4e9b\u76f8\u5173\u7684\u5185\u5bb9\uff0c\u4f8b\u5982\u6709\u5173\u6db2\u4f53\u8868\u3001\u754c\u9762\u5904\u7684\u7ed3\u6784\u4e0e\u52a8\u6001\uff0c\u79bb\u5b50\u6db2\u4f53\u7684\u7406\u8bba\u7b49\u3002\u6700\u540e\u8fd8\u7279\u522b\u52a0\u4e86\u4e00\u7ae0\u8f6f\u7269\u8d28\u5e94\u7528\uff0c\u4ecb\u7ecd\u4e86\u805a\u5408\u7269\u548c\u80f6\u4f53\u4f53\u7cfb\u7684\u57fa\u672c\u7406\u8bba\u3002\u6211\u662f\u6700\u8fd1\u624d\u770b\u5230\u7b2c4\u7248\u7684\uff0c\u4e4b\u524d\u6211\u7ffb\u9605\u7684\u90fd\u662f\u7b2c3\u7248\uff082006\uff09\u3002\u4ee5\u4e0b\u6211\u5b66\u4e60van Hove\u51fd\u6570\u7684\u8bb0\u5f55\u4e2d\uff0c\u5c31\u987a\u4fbf\u5bf9\u8fd9\u4e24\u7248\u4f5c\u4e86\u4e00\u4e0b\u5bf9\u6bd4\uff0c\u53d1\u73b0\u7b2c4\u7248\u5728\u8fd9\u4e00\u90e8\u5206\u5176\u5b9e\u589e\u52a0\u4e86\u8fc7\u51b7\u6db2\u4f53\u7684\u5185\u5bb9\uff0c\u9664\u4e86van Hove\u51fd\u6570\u7684\u975e\u9ad8\u65af\u5f62\u5f0f\u3001\u52a8\u6001\u4e0d\u5747\u5300\u6027\u4e4b\u5916\uff0c\u8fd8\u6709\u6df1\u5ea6\u8fc7\u51b7\u6db2\u4f53\u8fdc\u79bb\u5e73\u8861\u6001\u7684\u8001\u5316\u884c\u4e3a\uff0c\u7279\u522b\u662ftime-aging time superpoaition\u548c\u8fdd\u53cdFDT\u7684\u6027\u8d28\u3002\u5bf9\u6bd4\u7b2c3\u548c\u7b2c4\u7248\u7684\u5173\u952e\u8bcd\u7d22\u5f15\u4e5f\u80fd\u5f88\u5feb\u53d1\u73b0D\u6253\u5934\u7684\u8bcd\u4e0b\u9762\u7b2c4\u7248\u6bd4\u7b2c3\u7248\u591a\u4e86\u4e00\u6761Dynamical heterogeneity\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u4e66\u4e2d\u5199\u6210time\/aging-time superposition\uff0c\u8fd9\u4e2a\u5199\u6cd5\u6bd4\u5e38\u89c1\u7684time-aging time superposition\u597d\uff0c\u5b83\u5f88\u6e05\u6670\u5730\u8868\u660e\u4e86time\u662f\u4e00\u4e2a\u91cf\u3001aging time\u662f\u53e6\u4e00\u4e2a\u91cf\uff0c\u662f\u5b83\u4eec\u4e4b\u95f4\u7684\u53e0\u52a0\u6027\u3002\u539f\u6765\u5730\u5199\u6cd5\u5f88\u5bb9\u610f\u8ba9\u4eba\u8ba4\u4e3a\u5f00\u59cb\u7684time-aging\u662f\u4e00\u4e2a\u4e1c\u897f\u3001\u540e\u9762\u7684time\u53c8\u662f\u53e6\u4e00\u4e2a\u4e1c\u897f\u3002\u6211\u6253\u7b97\u4ee5\u540e\u90fd\u7528\u7c7b\u4f3c\u7684\u5199\u6cd5\uff0c\u4f8b\u5982straing-rate\/frequency superposition\u3001time\/temperature superposition\u7b49\u30023. van Hove\u51fd\u65703.1 van Hove\u51fd\u6570\u7684\u5b9a\u4e49\u53ca\u4e09\u7ef4\u5f62\u5f0f\u8bba\u6587\u4e2d\u7ecf\u5e38\u4f1a\u4f7f\u7528van Hove\u51fd\u6570\u3001\u9ad8\u65af\u5206\u5e03\u7b49\u5b57\u773c\uff0c\u4f46\u662f\u4ed4\u7ec6\u7684\u5148\u540e\u5173\u7cfb\u6587\u7ae0\u91cc\u5f88\u5c11\u4f53\u73b0\uff0c\u5149\u770b\u6587\u7ae0\u662f\u7ecf\u4e0d\u8d77\u63a8\u6572\u7684\u3002\u4ee5\u4e0b\u6211\u628a\u4e00\u4e9b\u57fa\u672c\u5173\u7cfb\u8bb0\u5f55\u4e0b\u6765\u3002van Hove\u51fd\u6570\u7684\u5b9a\u4e49\u662f \\begin{equation}\\label{eq:1} \\begin{aligned} G\\left(\\mathbf{r},t\\right)&\\overset{\\mathrm{def}}{=}\\langle\\frac{1}{N}\\sum_{i=1}^N\\sum_{j=1}^N\\int d\\mathbf{r}\\delta\\left[\\mathbf{r}-\\mathbf{r}_j\\left(t\\right)+\\mathbf{r}_i\\left(0\\right)\\right]\\rangle \\\\&=\\frac{1}{\\rho}\\langle\\rho\\left(\\mathbf{r},t\\right)\\rho\\left(\\mathbf{0},0\\right)\\rangle \\\\&=G_\\mathrm{d}+G_\\mathrm{s} \\end{aligned} \\end{equation} \u5176\u4e2d \\begin{equation}\\label{eq:2} G_\\mathrm{s}\\left(\\mathbf{r},t \\right )=\\langle\\frac{1}{N}\\sum_{i=1}^N\\delta\\left[\\mathbf{r}-\\mathbf{r}_i\\left(t \\right )+\\mathbf{r}_i\\left(0 \\right ) \\right]\\rangle \\end{equation} \\begin{equation}\\label{eq:3} G_\\mathrm{d}\\left(\\mathbf{r},t\u2026","rel":"","context":"In &quot;\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0&quot;","block_context":{"text":"\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/tagged"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":8462,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/8462","url_meta":{"origin":9132,"position":4},"title":"\u6db2\u4f53\u7269\u7406\u3001\u6d41\u53d8\u5b66\u4e0e\u6d41\u4f53\u529b\u5b66","author":"Andrew","date":"2022\u5e746\u670816\u65e5","format":false,"excerpt":"\u6211\u5728\u77e5\u4e4e\u7f51\u7ad9\u5199\u4e86\u4e00\u4e2a\u56de\u7b54\u4ecb\u7ecd\u6db2\u4f53\u7269\u7406\u3002\u4f46\u7531\u4e8e\u539f\u95ee\u9898\u662f\u5728\u95ee\u201c\u80fd\u5426\u4ece\u7edf\u8ba1\u529b\u5b66\u63a8\u51fa\u6d41\u4f53\u529b\u5b66\u201d\uff0c\u56e0\u6b64\u9020\u6210\u4e86\u4e00\u4e9b\u4eba\u7684\u7591\u95ee\u5c31\u662f\uff0c\u4e3a\u4f55\u8fd9\u4e9b\u8003\u8651\u5e73\u8861\u6001\u6216\u8fd1\u5e73\u8861\u6001\u5f3a\u5173\u8054\u6027\u8d28\u7406\u8bba\u4f1a\u662f\u5bf9\u8fd9\u4e2a\u95ee\u9898\u7684\u4e00\u4e2a\u56de\u7b54\u3002\u4ed6\u4eec\u8ba4\u4e3a\u8fd9\u4e9b\u4e0d\u662f\u201c\u6d41\u4f53\u529b\u5b66\u201d\uff0c\u6d41\u4f53\u529b\u5b66\u5173\u5fc3\u8fde\u7eed\u4ecb\u8d28\u5728\u5b88\u6052\u5f8b\u4e0b\u7684\u6d41\u52a8\u95ee\u9898\uff0c\u8bf4\u767d\u4e86\u6d41\u4f53\u529b\u5b66\u5173\u5fc3\u7684\u5c31\u662fNS\u65b9\u7a0b\u3002\u8fd9\u6837\u7684\u7591\u95ee\u5df2\u7ecf\u4e0d\u51fa\u6211\u6240\u6599\u3002\u957f\u671f\u4ee5\u6765\uff0c\u4f5c\u4e3a\u8bfe\u7a0b\u7684\u201c\u6d41\u4f53\u529b\u5b66\u201d\u4e3b\u8981\u5185\u5bb9\u786e\u5b9e\u5c31\u662f\u8fd9\u6837\u7684\u3002\u901a\u8fc7\u201c\u987e\u540d\u601d\u4e49\u201d\u628a\u6d41\u4f53\u529b\u5b66\u76f4\u63a5\u7406\u89e3\u4e3a\u201c\u6d41\u4f53\u7684\u529b\u5b66\u201d\u7684\u4e3b\u5f20\u5e76\u4e0d\u4e3b\u6d41\uff0c\u66f4\u4e0d\u662f\u9ed8\u8ba4\u7684\u3002 \u76f4\u63a5\u89e3\u91ca\u8fd9\u4ef6\u4e8b\u60c5\u4e0d\u662f\u672c\u6587\u7684\u76ee\u7684\u3002\u672c\u6587\u5173\u5fc3\u7684\u662f\u4e3a\u4ec0\u4e48\u6d41\u4f53\u529b\u5b66\u4f1a\u81f3\u4eca\u4fdd\u6301\u5728\u8fd9\u4e48\u7a84\u7684\u5173\u6ce8\u8303\u56f4\uff0c\u6211\u731c\u6d4b\u8fd9\u662f\u6d41\u53d8\u5b66\u7684\u8bde\u751f\u9020\u6210\u7684\u3002 \u6d41\u53d8\u5b66\uff08rheology\uff09\u53e6\u7acb\u95e8\u6237\u662fBingham\u7684\u4e3b\u610f\uff0c\u8fd9\u5728 Reiner\u8457\u540d\u7684\u6563\u6587\u4e2d\u5c31\u8bb0\u8f7d\u4e86\u3002 When I arrived, Bingham said to me, \u201cHere you, a civil engineer, and I, a chemist, are working together at joint problems. With the development of colloid chemistry, such a situation will be more and more common. We therefore must establish a branch of physics where\u2026","rel":"","context":"In &quot;\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0&quot;","block_context":{"text":"\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/tagged"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":8007,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/8007","url_meta":{"origin":9132,"position":5},"title":"\u6b27\u51e0\u91cc\u5fb7\u7a7a\u95f4\u7684\u73b0\u4ee3\u5f15\u5165\u4e2d\u7684\u4e00\u4e2a\u95ee\u9898","author":"Andrew","date":"2020\u5e748\u670831\u65e5","format":"aside","excerpt":"[latexpage] $\\mathcal{E}$\u662f\u4e00\u4e2a\u975e\u7a7a\u96c6\u5408\uff0c\u91cc\u9762\u7684\u5143\u7d20\u79f0\u4e3a\u201c\u70b9\u201d\u3002\u5bf9\u4e8e\u96c6\u5408\u4e2d\u7684\u8fd9\u4e9b\u70b9\uff0c\u6211\u4eec\u53ef\u4ee5\u505a\u300a\u51e0\u4f55\u539f\u672c\u300b\u8bf4\u80fd\u505a\u7684\u6240\u6709\u4e8b\uff0c\u4ece\u800c\u5f97\u5230\u4e00\u5207\u300a\u51e0\u4f55\u539f\u672c\u300b\u80fd\u5f97\u5230\u7684\u7ed3\u8bba\u3002\u5176\u4e2d\u5c31\u5305\u62ec\u5141\u8bb8\u6211\u4eec\u8c08\u53ca\u201c\u7531\u70b9$a$\u5230\u70b9$b$\u7684\u6709\u5411\u7ebf\u6bb5\u201d\uff0c$a,b\\in\\mathcal{E}$\u3002\u6211\u4eec\u5efa\u7acb\u4e00\u4e2a\u201c\u4ece\u70b9$a$\u5230\u70b9$b$\u7684\u6709\u5411\u7ebf\u6bb5\u201d\u5230\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4$\\mathcal{V}$\u7684\u6620\u5c04$\\Phi:\\mathcal{E}\\times\\mathcal{E}\\rightarrow\\mathcal{V}$\uff0c\u4e14$\\Phi\\left(x,x\\right)\\equiv\\mathbf{0}\\forall x\\in\\mathcal{E}$\uff0c\u5176\u4e2d$\\mathbf{0}$\u662f$\\mathcal{V}$\u7684\u96f6\u5411\u91cf\u3002\u9009\u5b9a\u4e00\u4e2a\u70b9$o\\in\\mathcal{E}$\uff0c\u53c8\u53ef\u8bb0$\\Phi_o:\\mathcal{E}\\rightarrow\\mathcal{V}$\uff0c$\\Phi_o\\left(x\\right)\\equiv\\Phi\\left(o,x\\right)$\u3002\u5728\u5f88\u591a\u4e66\u4e2d\u200b(Berger 1987; Audin 2002)\u200b\uff0c\u76f4\u63a5\u9884\u5148\u628a$\\Phi_o$\u89c4\u5b9a\u4e3a\u53cc\u5c04\u3002\u4f46\u6211\u5728\u8fd9\u91cc\u60f3\u5148\u4e0d\u4f5c\u6b64\u89c4\u5b9a\u3002\u6700\u591a\uff0c\u7531\u4e8e\u6211\u4eec\u5e0c\u671b$\\mathcal{E}$\u4e2d\u7684\u6bcf\u4e2a\u539f\u7d20\u90fd\u80fd\u5728\u7ed9\u5b9a\u539f\u70b9\u7684\u524d\u63d0\u4e0b\u5728$\\mathcal{V}$\u4e2d\u627e\u5230\u552f\u4e00\u5bf9\u5e94\uff08\u53cd\u4e4b\u5e76\u4e0d\u5fc5\u4ea6\u7136\uff09\uff0c\u4ece\u800c\u89c4\u5b9a$\\Phi_o$\u4e3a\u5355\u5c04\u975e\u6ee1\u5c04\u3002 \u6211\u4e0d\u786e\u5b9a\u79bb\u5f00\u67d0\u6570\u96c6\uff0c\u5149\u9760\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u7684\u516c\u8bbe\u4f53\u7cfb\u80fd\u5426\u72ec\u7acb\u5b9a\u4e49\u201c\u7ebf\u6bb5\u7684\u957f\u5ea6\u201d\u3001\u201c\u4e24\u7ebf\u6bb5\u957f\u5ea6\u76f8\u7b49\u201d\u3001\u201c\u4e00\u6761\u7ebf\u6bb5\u7684\u957f\u5ea6\u5927\u4e8e\u53e6\u4e00\u6761\u7ebf\u6bb5\u7684\u957f\u5ea6\u201d\u3002\u6211\u4f3c\u61c2\u975e\u61c2\u5730\u770b\u4e86\u4e00\u4e0bWikipedia\uff0c\u5174\u8bb8Tarski\u7684\u516c\u8bbe\u4f53\u7cfb\u5b9e\u73b0\u4e86\u8fd9\u4ef6\u4e8b\u3002\u4f46\u4e0d\u7ba1\u5982\u4f55\uff0c\u6211\u4eec\u5728\u201c\u957f\u5ea6\u201d\u95ee\u9898\u4e0a\u7684\u91cd\u65b0\u5b9a\u4e49\uff0c\u5e94\u8be5\u4e0d\u4f1a\u9020\u6210\u6574\u4e2a\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u7684\u91cd\u65b0\u5b9a\u4e49\uff0c\u56e0\u4e3a\u201c\u76f8\u7b49\u201d\u3001\u201c\u5927\u5c0f\u201d\u4e4b\u7c7b\u7684\u903b\u8f91\uff0c\u5728\u6570\u96c6\u91cc\u8ddf\u5728\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u91cc\u662f\u76f8\u540c\u7684\u3002\u6240\u4ee5\u4e0b\u4e00\u6b65\u6211\u4eec\u5b9a\u4e49\uff0c$\\mathcal{E}$\u4e2d\u6709\u5411\u7ebf\u6bb5\uff08\u8bb0\u4e3a$\\overline{xy},x,y\\in\\mathcal{E}$\uff09\u7684\u957f\u5ea6\u5c31\u662f\u6240\u5bf9\u5e94\u7684$\\mathcal{V}$\u4e2d\u5411\u91cf\u7684\u8303\uff1a $$d:\\mathcal{E}\\times\\mathcal{E}\\rightarrow\\mathbb{R},\\overline{ab}\\equiv d\\left(a,b\\right)\\equiv\\left\\|\\Phi\\left(a,b\\right)\\right\\|$$ \u5176\u4e2d$\\left\\|\\mathbf{u}\\right\\|\\equiv\\sqrt{\\mathbf{u}\\cdot\\mathbf{u}},\\mathbf{u}\\in\\mathcal{V}$\u3002 \u540c\u6837\u7684\u9053\u7406\uff0c\u5982\u679c\u80fd\u8ba4\u4e3a\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u539f\u672c\u7684\u201c\u89d2\u5ea6\u5927\u5c0f\u201d\u4ee5\u53ca\u5176\u4e8e\u8fd9\u4e2a\u6982\u5ff5\u7684\u6210\u7acb\u800c\u8bc1\u5f97\u7684\u4e00\u5207\u63a8\u8bba\u4e5f\u662f\u901a\u8fc7\u201c\u76f8\u7b49\u201d\u3001\u201c\u5927\u5c0f\u201d\u7b49\u903b\u8f91\u72ec\u7acb\u4e8e\u6570\u96c6\u800c\u88ab\u89c4\u5b9a\u7684\u200b*\u200b\uff0c\u90a3\u4e48\u91c7\u7528\u5b9e\u6570\u96c6\u6765\u91cd\u65b0\u89c4\u5b9a\u5c31\u4e0d\u4f1a\u6539\u53d8\u539f\u6709\u903b\u8f91\u57fa\u7840\uff0c\u4ece\u800c\u539f\u6709\u7684\u51e0\u4f55\u63a8\u8bba\u4e5f\u90fd\u6210\u7acb\uff08\u53ef\u7528\uff09\u3002\u73b0\u5728\u6211\u4eec\u901a\u8fc7\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f\u6765\u5b9a\u4e49$\\mathcal{V}$\u7531\u4e09\u4e2a\u4e0d\u540c\u7684\u70b9\u6240\u6784\u6210\u7684\u89d2\u7684\u5ea6\u6570\u3002 \u5728$\\mathcal{V}$\u4e2d\uff0c\u7531\u5185\u79ef\u8fd0\u7b97\u7684\u6027\u8d28\uff0c\u6709\u5982\u4e0b\u5173\u7cfb\uff1a \\begin{equation}\\left\\|\\Phi_o\\left(x\\right)\\right\\|^2+\\left\\|\\Phi_o\\left(y\\right)\\right\\|^2=\\left\\|\\Phi_o\\left(x\\right)-\\Phi_o\\left(y\\right)\\right\\|+2\\Phi_o\\left(x\\right)\\cdot\\Phi_o\\left(y\\right)\\label{eq1}\\end{equation} \u7531\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f \\begin{align*}&\\left\\|\\Phi_o\\left(x\\right)\\right\\|^2+\\left\\|\\Phi_o\\left(y\\right)\\right\\|^2\\geq\\left|\\Phi_o\\left(x\\right)\\cdot\\Phi_o\\left(y\\right)\\right|^2\\\\\\Leftrightarrow&-1\\leq\\frac{\\Phi_o\\left(x\\right)\\cdot\\Phi_o\\left(y\\right)}{\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|}\\leq1\\end{align*} \u7531\u4e8e$\\cos:\\left[0,\\pi\\right]\\rightarrow\\left[-1,1\\right]$\u662f\u53cc\u5c04\uff0c\u6545\u53ef\u5b9a\u4e49\u201c\u4e24\u5411\u91cf\u7684\u5939\u89d2\u201d\uff1a $$\\theta:\\mathcal{V}\\times\\mathcal{V}\\rightarrow\\left[0,\\pi\\right],\\theta\\left(\\mathbf{u},\\mathbf{v}\\right)\\equiv\\mathrm{arccos}\\frac{\\mathbf{u}\\cdot\\mathbf{v}}{\\left\\|\\mathbf{u}\\right\\|\\left\\|\\mathbf{v}\\right\\|}$$ \u5f0f(\\ref{eq1})\u53d8\u4e3a\uff1a \\begin{align}&\\left\\|\\Phi_o\\left(x\\right)\\right\\|^2+\\left\\|\\Phi_o\\left(y\\right)\\right\\|^2\\\\=&\\left\\|\\Phi_o\\left(x\\right)-\\Phi_o\\left(y\\right)\\right\\|^2+2\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|\\cos\\theta\\left(\\Phi_o\\left(x\\right),\\Phi_o\\left(y\\right)\\right)\\label{eq1a}\\end{align} \u7531\u5185\u79ef\u5b9a\u4e49\u53ef\u77e5$\\theta\\left(\\mathbf{u},\\mathbf{v}\\right)=\\theta\\left(\\mathbf{v},\\mathbf{u}\\right)\\forall\\mathbf{u},\\mathbf{v}\\in\\mathcal{V}$\u3002 \u5728$\\mathcal{E}$\u4e2d\uff0c\u7531\u4f59\u5f26\u5b9a\u7406\uff08\u300a\u51e0\u4f55\u539f\u672c\u300b\u547d\u989812\u300113\uff09\uff0c \\begin{align}&\\overline{ox}^2+\\overline{oy}^2=\\overline{xy}^2+2\\overline{ox}\\times\\overline{oy}\\cos\\angle xoy\\nonumber\\\\\\Leftrightarrow&\\left\\|\\Phi_o\\left(x\\right)\\right\\|^2+\\left\\|\\Phi_o\\left(y\\right)\\right\\|^2=\\left\\|\\Phi_x\\left(y\\right)\\right\\|^2+2\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|\\cos\\angle xoy\\label{eq2}\\end{align} \u5176\u4e2d$\\angle xoy,x,o,y\\in\\mathcal{E}$\u8868\u793a\u7531\u70b9$x$\u3001$o$\u3001$y$\u6784\u6210\u7684\u89d2\u3002\u5c06\u5f0f(\\ref{eq1a})\u4ee3\u5165\u5f0f(\\ref{eq2})\u5f97\uff1a \\begin{align*}&\\left\\|\\Phi_o\\left(x\\right)-\\Phi_o\\left(y\\right)\\right\\|^2+2\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|\\cos\\theta\\left(\\Phi_o\\left(x\\right),\\Phi_o\\left(y\\right)\\right)\\\\=&\\left\\|\\Phi_x\\left(y\\right)\\right\\|+2\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|\\cos\\angle xoy\\end{align*} \u6bd4\u8f83\u4e0a\u5f0f\u7b49\u53f7\u5de6\u53f3\u4e24\u8fb9\uff0c\u53ef\u77e5\u4ee5\u4e0b\u4e24\u547d\u9898\u4e92\u4e3a\u5145\u8981\u6761\u4ef6\uff1a \\[\\Phi_o\\left(x\\right)-\\Phi_o\\left(y\\right)=\\pm\\Phi_x\\left(y\\right)\\Leftrightarrow\\cos\\angle xoy=\\cos\\theta\\left(\\Phi_o\\left(x\\right),\\Phi_o\\left(y\\right)\\right)\\] \u6211\u4eec\u4e0d\u59a8\u5bf9\u6620\u5c04$\\Phi$\u589e\u52a0\u89c4\u5b9a\uff1a$\\Phi\\left(o,y\\right)-\\Phi\\left(o,x\\right)=\\Phi\\left(x,y\\right)=-\\Phi\\left(y,x\\right),\\forall o,x,y\\in\\mathcal{E}$\uff0c\u5219\u7531\u4e0a\u8ff0\u7b49\u4ef7\u5173\u7cfb\u6211\u4eec\u540c\u65f6\u83b7\u5f97\u4e86$\\mathcal{E}$\u4e2d\u7684\u89d2$\\angle xoy,x,o,y\\in\\mathcal{E}$\u7684\u5927\u5c0f\u4e0e\u5b9e\u6570\u57df$\\left[0,\\pi\\right]$\u7684\u6620\u5c04\uff1a$\\measuredangle:\\mathcal{E}^3\\rightarrow\\left[0,\\pi\\right],\\measuredangle\\left(xoy\\right)\\equiv\\measuredangle xoy\\equiv\\theta\\left(\\Phi_o\\left(x\\right),\\Phi_o\\left(y\\right)\\right)$\u3002 \u81f3\u6b64\uff0c\u6211\u505a\u7684\u4e8b\u60c5\u662f\u662f\uff1a1\uff09\u89c4\u5b9a\u4e86$\\mathcal{E}$\u4e2d\u7684\u6709\u5411\u7ebf\u6bb5\u4e0e$\\mathcal{V}$\u4e2d\u7684\u5411\u91cf\u7684\u6620\u5c04\uff08\u672a\u89c4\u5b9a\u662f\u4ec0\u4e48\u5c04\uff09\uff1b2\uff09\u5229\u7528$\\mathcal{V}$\u7684\u5185\u79ef\u5b9a\u4e49\uff0c\u89c4\u5b9a\u4e86$\\mathcal{E}$\u4e2d\u7ebf\u6bb5\u7684\u957f\u5ea6\u548c\u89d2\u7684\u5927\u5c0f\u4e0e\u5b9e\u6570\u96c6\u7684\u6620\u5c04\u3002\u5728\u505a\u8fd9\u4e9b\u89c4\u5b9a\u7684\u65f6\u5019\uff0c\u6211\u5229\u7528\u4e86\u5df2\u77e5\u7684\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u63a8\u8bba\uff08\u4f59\u5f26\u5b9a\u7406\uff09\u3002\u8fd9\u4e00\u89c4\u5b9a\u6027\u81ea\u7136\u800c\u7136\u5e26\u6765\u4e00\u4e9b\u9700\u8981\u91cd\u65b0\u8bc1\u660e\u7684\u547d\u9898\u3002\u4f8b\u5982\uff0c\u539f\u51e0\u4f55\u89c4\u5b9a\uff08Tarski\uff09\u7684\u201c\u4e24\u7ebf\u6bb5\u76f8\u7b49\u201d\uff0c\u5219\u5b83\u4eec\u6240\u5bf9\u5e94\u7684\u5b9e\u6570\u503c\u76f8\u7b49\uff1b\u539f\u51e0\u4f55\u5b9a\u4e49\u7684\u76f4\u89d2\uff0c\u5728\u73b0\u6709\u7684\u89c4\u5b9a\u91cc\u5bf9\u5e94\u5b9e\u6570$\\pi\/2$\u7b49\u7b49\u3002\u8fd9\u91cc\u6211\u672a\u80fd\u5b8c\u5907\u5730\u5217\u51fa\u4e0a\u8ff0\u91cd\u65b0\u89c4\u5b9a\u6240\u5f15\u8d77\u7684\u8fd9\u7c7b\u91cd\u65b0\u8bc1\u660e\u4efb\u52a1\uff0c\u4f46\u6211\u76f8\u4fe1\u8fd9\u4e9b\u547d\u9898\u662f\u6709\u9650\u7684\uff0c\u6613\u8bc1\u7684\u3002\u8fd9\u4e3a\u6211\u4eec\u5efa\u7acb$\\mathcal{E}$\u4e0e$\\mathcal{V}$\u7684\u7ef4\u6570\u5bf9\u5e94\u6027\u6253\u4e0b\u4e86\u90e8\u5206\u57fa\u7840\uff0c\u5373$\\mathcal{E}$\u7684\u201c\u76f4\u89d2\u201d\u4e0e$\\mathcal{V}$\u7684\u201c\u6b63\u4ea4\u5411\u91cf\u201d\u5bf9\u5e94\u4e0a\u4e86\u3002 \u5c1a\u7f3a\u5c11\u7684\u6982\u5ff5\u5bf9\u5e94\u662f\u201c\u8fc7\u4e00\u70b9\u7684\u76f4\u7ebf\u201d\u3002\u4e5f\u6b63\u662f\u5728\u8fd9\u4e2a\u95ee\u9898\u91cc\uff0c\u6211\u53d1\u73b0$\\Phi_o$\u4e0d\u5f97\u4e0d\u5982\u73b0\u6709\u7684\u4e66\u4e2d\u90a3\u822c\u8fdb\u4e00\u6b65\u89c4\u5b9a\u4e3a\u53cc\u5c04\u3002\u5148\u770b\u4e00\u4e2a\u51fa\u53d1\u70b9\u6bd4\u8f83\u597d\uff0c\u4f46\u5931\u8d25\u7684\u4f8b\u5b50\u2014\u2014\u4ece$\\mathcal{E}$\u7684\u5df2\u77e5\u6982\u5ff5\u51fa\u53d1\uff0c\u901a\u8fc7\u5df2\u89c4\u5b9a\u7684\uff08\u672a\u5fc5\u53cc\u5c04\u7684\uff09\u6620\u5c04$\\Phi_o$\u6765\u627e\u5230$\\mathcal{V}$\u4e2d\u7684\u5bf9\u5e94\u3002 \u7531\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u6613\u77e5\uff0c\u8fc7\u70b9$a$\u3001$b$\u7684\u76f4\u7ebf\u4e0a\u7684\u4efb\u4e00\u70b9$c$\uff0c$a,b,c\\in\\mathcal{E},a\\neq b$\uff0c\u5fc5\u6ee1\u8db3\u4ee5\u4e0b\u56db\u79cd\u60c5\u51b5\u4e4b\u4e00\uff1a1\uff09$\\measuredangle cab=0$\uff1b2\uff09$\\measuredangle cab=\\pi$\uff1b3\uff09$c=a$\uff1b4\uff09$c=b$\u3002\u6839\u636e\u524d\u6587\u7684\u89c4\u5b9a\uff0c\u53c8\u6709$\\overline{ac}=\\left\\|\\Phi_a\\left(c\\right)\\right\\|$\u3001$\\overline{ab}=\\left\\|\\Phi_a\\left(b\\right)\\right\\|$\u3002\u5bf9\u4e8e\u60c5\u51b51\uff09\u548c2\uff09\uff0c\u7531 $$\\cos\\measuredangle cab =\\frac{ \\Phi_a\\left(c\\right)\\cdot\\Phi_a\\left(b\\right)}{\\left\\|\\Phi_a\\left(c\\right)\\right\\|\\left\\|\\Phi_a\\left(b\\right)\\right\\|}=\\pm 1$$ \u5de6\u53f3\u4e24\u8fb9\u540c\u9664$\\left\\|\\Phi_a\\left(b\\right)\\right\\|$\u6709 $$\\frac{\\Phi_a\\left(c\\right)\\cdot\\Phi_a\\left(b\\right)}{\\Phi_a\\left(b\\right)\\cdot\\Phi_a\\left(b\\right)}=\\pm\\frac{\\left\\|\\Phi_a\\left(c\\right)\\right\\|}{\\left\\|\\Phi_a\\left(b\\right)\\right\\|}\\equiv\\alpha^\\prime$$ \u5176\u4e2d\u60c5\u51b51\uff09\u53d6\u6b63\u53f7\uff0c\u60c5\u51b52\uff09\u53d6\u8d1f\u53f7\u3002\u53e6\u5916\uff0c\u663e\u7136\u60c5\u51b53\uff09\u548c4\uff09$\\alpha^\\prime$\u7b49\u4e8e0\u6216\u65e0\u7a77\u3002\u6211\u4eec\u53ef\u4ee5\u91cd\u65b0\u6574\u7406\u56db\u79cd\u60c5\u51b5\uff0c\u5199\u6210\\begin{align*}&\\Phi_a\\left(c\\right)\\cdot\\Phi_a\\left(b\\right)=\\alpha\\left(\\Phi_a\\left(b\\right)\\cdot\\Phi_a\\left(b\\right)\\right),\\alpha\\in\\mathbb{R}\\\\\\Leftarrow&\\Phi_a\\left(c\\right)=\\alpha\\Phi_a\\left(b\\right)\\end{align*} \u4e0a\u5f0f\u4e2d\uff0c\u524d\u8005\u4ec5\u662f\u540e\u8005\u7684\u5fc5\u8981\u975e\u5145\u4efd\u6761\u4ef6\u3002\u5982\u679c\u8fd9\u4e00\u6b65\u662f\u5145\u8981\u6761\u4ef6\uff0c\u4ece\u800c\u5f97\u5230\u6700\u540e\u7684\u7ed3\u8bba\uff0c\u90a3\u4e48\u6211\u4eec\u65e0\u9700\u89c4\u5b9a$\\Phi_a$\u662f\u53cc\u5c04\uff0c\u5c31\u80fd\u628a$\\mathcal{E}$\u4e2d\u8fc7\u4efb\u610f\u4e24\u70b9$a,b$\u7684\u76f4\u7ebf\u4e0a\u7684\u70b9\u7684\u96c6\u5408\uff0c\u6620\u5c04\u4e3a$\\mathcal{V}$\u7684\u5b50\u96c6$\\mathcal{L}_{ab}=\\left\\{\\mathbf{c}|\\Phi_a\\left(c\\right)=\\alpha\\Phi_a\\left(b\\right),\\alpha\\in\\mathbb{R}\\}$\u3002\u4e8b\u5b9e\u4e0a\uff0c\u6613\u8bc1\u8fd9\u4e00\u5b50\u96c6\u662f$\\mathcal{V}$\u7684\u5b50\u7a7a\u95f4\u4e14\u7ef4\u6570\u662f1\u3002\u82e5\u6709\u4e86\u6b64\u57fa\u7840\uff0c\u5c31\u53ef\u8fdb\u4e00\u6b65\u63a8\u6f14\u51fa$\\mathcal{E}$\u7684\u7ef4\u6570\u5c31\u662f$\\mathcal{V}$\u7684\u7ef4\u6570\uff0c\u4ece\u5934\u5230\u5c3e\u65e0\u9700\u8bf4\u660e$\\Phi_a$\u662f\u53cc\u5c04\u3002\u53e6\u5916\uff0c\u8fd9\u4e5f\u540c\u65f6\u8bc1\u660e\u4e86\u53e6\u4e00\u4e2a\u91cd\u8981\u7684\u7ed3\u679c\uff0c\u5c31\u662f\u628a$\\mathcal{E}$\u4e2d\u7684\u76f4\u7ebf\u4e0e\u5b9e\u6570\u8f74\u5efa\u7acb\u4e86\u5bf9\u5e94\uff0c\u4e3a\u5efa\u7acb\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u5960\u5b9a\u4e86\u57fa\u7840\u3002\u53ef\u60dc\uff0c\u5728\u4e0a\u9762\u6211\u4eec\u53ea\u80fd\u5f97\u5230\u5fc5\u8981\u975e\u5145\u4efd\u6761\u4ef6\u7684\u5173\u7cfb\uff0c\u6240\u4ee5\u6211\u4eec\u5c31\u65e0\u6cd5\u62e5\u6709\u540e\u9762\u7684\u8fd9\u4e9b\u4f18\u7f8e\u53d9\u8ff0\u3002\u2026","rel":"","context":"In 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