{"id":8597,"date":"2022-08-13T12:03:16","date_gmt":"2022-08-13T04:03:16","guid":{"rendered":"https:\/\/www.andrewsun.net\/panta_rhei\/?p=8597"},"modified":"2022-10-04T11:51:34","modified_gmt":"2022-10-04T03:51:34","slug":"%e5%85%b3%e4%ba%8e%e7%a6%bb%e6%95%a3%e6%b5%8b%e5%ba%a6%e6%b2%a1%e6%9c%89%e5%af%86%e5%ba%a6%e7%9a%84%e9%97%ae%e9%a2%98","status":"publish","type":"post","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/8597","title":{"rendered":"\u5173\u4e8e\u79bb\u6563\u6d4b\u5ea6\u6ca1\u6709\u5bc6\u5ea6\u7684\u95ee\u9898"},"content":{"rendered":"
\n\n

[latexpage]<\/p>\n\n\n\n

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

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

\\[\n\\delta_i\\left(A\\right)=\\left\\{\\begin{array}{cc}\n1,&x_i\\in A\\\\\n0,&x_i\\notin A\n\\end{array}\n\\right.,\\quad\n\\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)\n\\]<\/pre>\n\n\n\n
\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<\/p>\n\n\n\n
\\[\\nu\\left(A\\right)=\\int_A \\rho d\\mu\\]<\/pre>\n\n\n\n
\u7684\u201c\u5bc6\u5ea6\u573a\u201d$\\rho$\u3002Truesdell & Toupin\u7684CFT[1]<\/a><\/span>\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[2]<\/a><\/span>\u4e2d\u8fd9\u4e00\u70b9\u66f4\u660e\u786e\u3002<\/p>\n\n\n\n
\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[1]<\/a><\/span>\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<\/a>\u4e2d\u7684\u4f8b\u5b50\u3002<\/p>\n\n\n\n
\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[3]<\/a><\/span>\uff0c\u6b64\u65f6\u6240\u6709\u8fde\u7eed\u4ecb\u8d28\u7684\u516c\u8bbe\u4ecd\u7136\u6210\u7acb\uff08see also [4]<\/a><\/span>\uff09\u3002<\/p>\n\n\n\n
\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<\/p>\n\n\n\n
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\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<\/p>\n\n\n\n
\n\n\n\n
2022\/10\/4\u8865\u5145\uff1a\u5728R. Tolman\u6559\u6750\u4e2d\u662f\u8fd9\u6837\u5f15\u5165\u7684\uff1a<\/p>\n\n\n\n
…, it is also to be noted for statistical purposes that we shall wish to use ensembles containing a large enough population of separate members so that the numbers of systems in such different states can be regarded as changing continuously as we pass from the states lying in one region of the phase space to those in another. Hence, for the purposes in view, it is evident that the condition of an ensemble at any time can be regarded as appropriately specified by the density<\/em> $\\rho$ with which representative points are distributed over the phase space.<\/p>
…<\/p>
The quantity $\\rho$ is then to be understood as determining the number of systems $\\delta N$, which would be found at time $t$ to have coordinates and momenta lying in any selected infinitesimal range $\\delta q_1\\cdots\\delta q_f\\delta p_1\\cdots\\delta p_f$, in accordance with the equation<\/p>
$$\\delta N=\\rho\\left(q,p,t\\right)\\delta q_1\\cdot\\delta q_f\\delta p_1\\cdots\\delta p_f$$<\/p>
We assume a large enough total population of systems so that $\\rho$ and $\\delta N$ can be regarded with sufficient approximation as changing continuously as we go from one region in the phase space to another.<\/p>
By integrating over the whole of phase space, we can write<\/p>
$$N=\\int\\cdots\\int\\rho\\left(q,p,t\\right)dq_1\\cdots dp_f$$<\/p>
as an expression for the total number of systems $N$ in the ensemble or phase points in the phase space.<\/p>p. 46~47<\/cite><\/blockquote>\n\n\n\n
\u8fd9\u6bb5\u5f15\u5165\u5b57\u9762\u4e0a\u662f\u5e0c\u671b\u5f53$N$\u8db3\u591f\u5927\u65f6\uff0c\u5fae\u3001\u79ef\u5206\u8868\u8fbe\u5f0f\u80fd\u4f5c\u4e3a\u76f8\u7a7a\u95f4\u5bc6\u5ea6\u7684\u4e00\u79cd\u8fd1\u4f3c\u3002\u4f46\u4f5c\u8005\u4ecd\u7136\u575a\u6301\u79ef\u5206\u540e\u7684\u662f\u201c\u7cfb\u7edf\u7684\u603b\u4e2a\u6570\u201d\uff08total number of systems\uff09\uff0c\u662f\u53ef\u6570\u7684\u6982\u5ff5\u3002\u8fd9\u5176\u5b9e\u6697\u793a\u4e86\u79cd\u7528Dirac delta\u51fd\u6570\u6765\u8868\u793a\u79bb\u6563\u70b9\u5bc6\u5ea6\u7684\u65b9\u6cd5\u3002\u5982\u679c\u8fd9\u4e48\u5904\u7406\uff0c\u4e0a\u8ff0\u7684\u8868\u793a\u5c31\u4e0d\u662f\u4e00\u79cd$N$\u5f88\u5927\u65f6\u7684\u8fd1\u4f3c\uff0c\u800c\u662f\u5bf9\u4efb\u610f$N$\u7684\u7cbe\u786e\u8868\u793a\u3002 <\/p>\n
References<\/h2>\n    
    \n    
  1. <\/a>\nC. Truesdell, and R. Toupin, \"The Classical Field Theories\", Encyclopedia of Physics \/ Handbuch der Physik<\/i>, pp. 226-858, 1960. http:\/\/dx.doi.org\/10.1007\/978-3-642-45943-6_2<\/a>\n\n\n<\/li>\n
  2. <\/a>\nC. Truesdell, and W. Noll, \"The Non-Linear Field Theories of Mechanics\", The Non-Linear Field Theories of Mechanics<\/i>, pp. 1-579, 2004. http:\/\/dx.doi.org\/10.1007\/978-3-662-10388-3_1<\/a>\n\n\n<\/li>\n
  3. <\/a>\nP. Podio-Guidugli, \"On the Mechanical Modeling of Matter, Molecular and Continuum\", Journal of Elasticity<\/i>, vol. 135, pp. 435-456, 2018. http:\/\/dx.doi.org\/10.1007\/s10659-018-9709-y<\/a>\n\n\n<\/li>\n
  4. <\/a>\nA. DiCarlo, . , P. Podio-Guidugli, and . , \"From point particles to body points\", Mathematics in Engineering<\/i>, vol. 4, pp. 1-29, 2021. http:\/\/dx.doi.org\/10.3934\/mine.2022007<\/a>\n\n\n<\/li>\n<\/ol>\n\n<\/div> ","protected":false},"excerpt":{"rendered":"
    [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 we shall wish to use […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[303],"tags":[],"class_list":["post-8597","post","type-post","status-publish","format-standard","hentry","category-rheo"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p1aPvF-2eF","jetpack-related-posts":[{"id":9132,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/9132","url_meta":{"origin":8597,"position":0},"title":"\u4e0d\u57fa\u4e8e\u7cfb\u7efc\u6982\u5ff5\u7684\u5218\u7ef4\u5c14\u65b9\u7a0b\u63a8\u7406","author":"Andrew","date":"2025\u5e742\u670818\u65e5","format":false,"excerpt":"[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 \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 \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$t,\\Delta t\\in\\mathbb{R}$\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\\[\\frac{\\mathrm{d}}{\\mathrm{d}t}\\mu_t\\left[\\omega_t\\right]\\equiv 0\\] \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","rel":"","context":"In "\u6211\u7684\u7814\u7a76\u7b14\u8bb0"","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":6836,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/6836","url_meta":{"origin":8597,"position":1},"title":"Hansen\u7684\u4e66\u3001van Hove\u51fd\u6570\u4ee5\u53ca\u5e03\u6717\u8fd0\u52a8","author":"Andrew","date":"2016\u5e742\u670821\u65e5","format":"aside","excerpt":"[latexpage] 1. \u77e5\u8bc6\u7684\u6f0f\u6d1e\u6211\u672c\u79d1\u6ca1\u6709\u5b66\u8fc7\u6982\u7387\u8bba\uff0c\u8003\u7814\u4e0d\u8003\u6570\u5b66\uff0c\u5230\u73b0\u5728\u90fd\u6ca1\u6709\u5b8c\u6574\u5730\u9605\u8bfb\u8fc7\u6982\u7387\u8bba\u4e0e\u6570\u7406\u7edf\u8ba1\u8bfe\u672c\uff0c\u53ea\u662f\u4e4b\u524d\u6d4f\u89c8\u8fc7\u4e00\u4e9b\u7edf\u8ba1\u7269\u7406\uff0c\u5176\u5b9e\u57fa\u7840\u7684thermodynamics\u4e3b\u8981\u4f9d\u8d56Boltzmann\u5206\u5e03\uff0c\u8bb0\u4f4f\u4e00\u4e9b\u516c\u5f0f\u5c31\u884c\u4e86\u3002\u6240\u4ee5\u5f88\u591a\u7edf\u8ba1\u6570\u5b66\u57fa\u672c\u7684fact\u4e0d\u77e5\u9053\uff0c\u73b0\u5728\u4ece\u4e8b\u9700\u8981\u7edf\u8ba1\u57fa\u7840\u7684\u8bfe\u9898\uff0c\u5c31\u5176\u5b9e\u5904\u4e8e\u4e00\u79cd\u65e0\u77e5\u7684\u72b6\u6001\u3002\u6709\u65f6paper\u4e0a\u5c55\u793a\u7684\u77e5\u8bc6\u4e0d\u4f1a\u4f53\u73b0\u5168\u8c8c\uff0c\u9047\u5230\u56f0\u60d1\u8981\u82b1\u5f88\u591a\u65f6\u95f4\u8d70\u5f2f\u8def\uff0c\u624d\u80fd\u641e\u6e05\u695a\u5bfc\u81f4\u56f0\u60d1\u7684\u77e5\u8bc6\u7a7a\u767d\u5230\u5e95\u5728\u54ea\u91cc\uff0c\u7136\u540e\u624d\u80fd\u53bb\u6076\u8865\u3002\u8fd9\u8fd8\u662f\u9532\u800c\u4e0d\u820d\u5730\u53bb\u641c\u5bfb\u624d\u80fd\u8fbe\u5230\u7684\u7ed3\u679c\uff0c\u4e5f\u8bb8\u5f88\u591a\u65f6\u5019\u751a\u81f3\u515c\u6765\u515c\u53bb\u8fd8\u627e\u4e0d\u5230\u81ea\u5df1\u7684\u77e5\u8bc6\u7a7a\u767d\uff0c\u5c31\u8fd9\u4e48\u653e\u8fc7\u4e86\uff0c\u8fd9\u4e00\u5757\u4e5f\u8bb8\u5c31\u5230\u8001\u90fd\u4e00\u76f4\u662f\u65e0\u77e5\u7684\u3002\u6240\u4ee5\uff0c\u5728\u4f60\u8fd8\u65e0\u77e5\u7684\u65f6\u5019\uff0c\u4e0d\u7ba1\u4e09\u4e03\u4e8c\u5341\u4e00\u5148\u7cfb\u7edf\u5730\u5b66\u4e60\u77e5\u8bc6\uff0c\u4e0d\u8981\u592a\u529f\u5229\u5730\u8003\u8651\u54ea\u4e9b\u90e8\u5206\u5c06\u6765\u6709\u6ca1\u6709\u7528\uff0c\u5176\u5b9e\u662f\u6700\u7701\u65f6\u95f4\u56de\u62a5\u4e5f\u6700\u5927\u7684\u505a\u6cd5\u3002\u5c3d\u7ba1\u8ba4\u8bc6\u5230\u8fd9\u4e00\u70b9\uff0c\u6211\u73b0\u5728\u8fd8\u662f\u4f1a\u53bb\u627e\u4e00\u4e9b\u80fd\u591f\u8ba9\u6211\u4e0d\u4ece\u5934\u9605\u8bfb\u57fa\u7840\u8bfe\u672c\u53c8\u8fbe\u5230\u8865\u4e60\u76ee\u7684\u7684\u4e66\u3002\u56e0\u4e3a\u5047\u8bbe\u8981\u8865\u4e60\u7684\u8bdd\uff0c\u6211\u6700\u6068\u4e0d\u5f97\u53bb\u8865\u7684\u662f\u6574\u4e2a\u7269\u7406\u7cfb\u672c\u79d1\u5e94\u8be5\u5b66\u7684\u6240\u6709\u8bfe\uff0c\u751a\u81f3\u60f3\u91cd\u8bfb\u9ad8\u4e09\u51b2\u51fb\u6e05\u534e\u5317\u5927\u3002\u8bf4\u5230\u5e95\uff0c\u90fd\u8fd9\u5c81\u6570\u4e86\uff0c\u4e0d\u7ba1\u662f\u56e0\u4e3a\u5148\u5929\u667a\u5546\u4e0d\u8db3\u8fd8\u662f\u4ec0\u4e48\u4e3b\u5ba2\u89c2\u4e0a\u7684\u5931\u8bef\uff0c\u73b0\u5728\u53ea\u80fd\u63a5\u53d7\u81ea\u5df1\u662f\u4e00\u4e2a\u5eb8\u624d\u8fd9\u4e2a\u4e8b\u5b9e\uff0c\u4eba\u8fd8\u8981\u5411\u524d\u770b\uff0c\u4e0d\u80fd\u65e0\u7a77\u65e0\u5c3d\u5730\u8865\u507f\u8fc7\u5f80\u4ee5\u6b64\u7ec8\u8001\u3002\u5f53\u524d\u7684\u9700\u6c42\u9002\u5f53\u8865\u4e00\u4e0b\u5c31\u884c\u4e86\u3002\u505a\u6cd5\u5c31\u662f\uff0c\u76f4\u63a5\u53bb\u770b\u4e00\u4e9b\u76f4\u63a5\u8ddf\u6211\u7684\u95ee\u9898\u76f8\u5173\u7684\u7814\u7a76\u751f\u6559\u6750\u6216\u8005\u63a5\u8fd1\u7684\u4e13\u8457\u3002\u91cc\u9762\u6d89\u53ca\u5230\u7684\u4e00\u4e9b\u57fa\u7840\u77e5\u8bc6Google\u5feb\u901f\u89e3\u51b3\uff0c\u4e0d\u5f71\u54cd\u4e13\u8457\u9605\u8bfb\u7684\u8fde\u7eed\u6027\u5373\u53ef\u3002\u751a\u81f3\u4e13\u8457\u672c\u8eab\u4e5f\u662f\u626b\u8bfb\u3002\u6211\u8bfb\u8fc7\u7684\u6700\u57fa\u7840\u7684\u4e13\u8457\u7b97\u662fBird\u7684Dynamics of Polymeric Liquids\u7b2c1\u5377\uff0c\u800c\u4e14\u4e5f\u662f\u5f53\u573a\u7406\u89e3\uff0c\u4e8b\u540e\u5fd8\u8bb0\uff0c\u53cd\u6b63\u201c\u66fe\u7ecf\u7406\u89e3\u8fc7\u201d\u3002\u8bf4\u767d\u4e86\uff0c\u770b\u4e66\u53ea\u662f\u53bb\u786e\u8ba4\u4e66\u4e0a\u54ea\u4e9b\u5185\u5bb9\u662f\u6211\u4ec0\u4e48\u65f6\u5019\u56de\u5934\u770b\u90fd\u4e0d\u81f3\u4e8e\u770b\u4e0d\u61c2\u7684\u3002\u7136\u540e\u5c31\u5fd8\u6389\u7ec6\u8282\uff0c\u8ba9\u65e5\u540e\u7684\u6211\u56de\u5934\u770b\u5427\u2014\u2014\u53cd\u6b63\u5230\u65f6\u6211\u4f1a\u770b\u5f97\u61c2\u7684\u3002\u5c11\u6570\uff08\u6216\u591a\u6570\uff0c\u4f46\u6211\u4ecd\u7136\u9009\u4e00\u4e9b\u91cd\u8981\u7684\u5c11\u6570\uff09\u4e66\u4e0a\u6211\u5f53\u573a\u770b\u4e0d\u61c2\u7684\u5730\u65b9\uff0cGoogle\u4e00\u4e0b\u8bd5\u8bd5\uff0c\u4e5f\u4e0d\u591a\u82b1\u4ec0\u4e48\u65f6\u95f4\uff0c\u4e0d\u884c\u5c31\u6253\u4e2a\u8bb0\u53f7\uff0c\u5bc4\u5e0c\u671b\u4e8e\u6211\u8fd9\u8f88\u5b50\u4e0d\u4f1a\u5361\u5728\u8fd9\u4ef6\u4e8b\u4e0a\u3002\u771f\u5361\u5728\u8fd9\u4ef6\u4e8b\u4e0a\u7684\u65f6\u5019\uff0c\u6211\u4e5f\u4f1a\u77e5\u9053\u8fd9\u4e2a\u6211\u4ee5\u524dGoogle\u8fc7\u884c\u4e0d\u901a\uff0c\u53ef\u4ee5\u7acb\u5373\u53bb\u627e\u4eba\u95ee\uff0c\u4e0d\u4f1a\u82b1\u91cd\u590d\u7684\u65f6\u95f4\u53c8\u53bbGoogle\u904d\u3002\u6240\u4ee5\uff0c\u5f88\u591a\u6211\u8ba4\u4e3a\u81ea\u5df1\u61c2\u7684\u77e5\u8bc6\uff0c\u5176\u5b9e\u53ea\u662f\u201c\u6211\u77e5\u9053\u54ea\u672c\u4e66\u4e0a\u6709\uff0c\u6211\u53bb\u770b\u7684\u8bdd\u4f1a\u770b\u61c2\uff0c\u4f46\u73b0\u5728\u6211\u4e0d\u77e5\u9053\u201d\u3002\u5982\u679c\u6709\u4eba\u62ff\u4e00\u672c\u4e66\u4e0a\u7684\u4efb\u4f55\u7ec6\u8282\u6765\u95ee\u6211\uff0c\u6211\u4f1a\u5f88\u50cf\u5b8c\u5168\u6ca1\u5b66\u8fc7\u8fd9\u672c\u4e66\u3002\u4f8b\u5982\uff0c\u4f5c\u4e3arheologist\uff0c\u6211\u5f88\u50cf\u5b8c\u5168\u6ca1\u770b\u8fc7Bird\u7684\u4e66\u3002\u8fd9\u5f88\u4e22\u4eba\uff0c\u5176\u5b9e\u4e0d\u662f\u8fd9\u6837\uff0c\u6240\u4ee5\u5728\u4ea4\u6d41\u4e0a\u5f88\u201c\u5403\u4e8f\u201d\uff0c\u53ea\u6c42\u80fd\u53d1paper\uff0c\u80fd\u62ff\u7ecf\u8d39\u5427\u30022. 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 "\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0"","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":9108,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/9108","url_meta":{"origin":8597,"position":2},"title":"Leon B. Lucy","author":"Andrew","date":"2024\u5e7412\u670814\u65e5","format":false,"excerpt":"[latexpage] \u80cc\u666f \u5728\u6570\u503c\u8ba1\u7b97\u9886\u57df\u6709\u4e00\u4e2a\u6bd4\u8f83\u77e5\u540d\u7684\u53bb\u5377\u79ef\u8fed\u4ee3\u7b97\u6cd5\u2014\u2014Lucy--Richardson\u7b97\u6cd5\uff0c\u5b83\u88ab\u540e\u4e16\u96c6\u4e2d\u7528\u4e8e\u56fe\u50cf\u53bb\u566a\u3002\u4f8b\u5982\uff0cMATLAB\u7684Image Processing Toolbox\u6709\u4e00\u4e2adeconvlucy\u547d\u4ee4\uff0c\u58f0\u79f0\u5c31\u662f\u7528Lucy--Richardson\u7b97\u6cd5\u5bf9\u7ed9\u5b9a\u56fe\u7247\uff08\u50cf\u7d20\u77e9\u9635\uff09\u4f5c\u7ed9\u5b9a\u70b9\u6269\u6563\u51fd\u6570\u7684\u53bb\u566a\u3002 \u4f46\u662f\uff0cLucy\u7684\u539f\u6587\u6240\u9488\u5bf9\u7684\u95ee\u9898\uff0c\u6bd4\u73b0\u5728\u4e00\u822c\u5e94\u7528\u66f4\u5e7f\u4e49\u3002\u5047\u5b9a$X$\u662f\u4e00\u4e2a\u8fde\u7eed\u53d6\u503c\u968f\u673a\u53d8\u91cf\u3002\u5b83\u7406\u5e94\u6309\u7167\u5206\u5e03\u5bc6\u5ea6\u51fd\u6570$\\phi\\left(x\\right)$\u3002\u6211\u4eec\u60f3\u628a$\\phi\\left(x\\right)$\u89c6\u4e3a\u67d0\u79cd\u7b80\u5355\u5206\u5e03$P\\left(x\\middle|\\xi\\right)$\u6309\u6743\u91cd\u8c31$\\psi\\left(\\xi\\right)$\u7684\u53e0\u52a0\u7ed3\u679c\uff1a \\begin{equation}\\label{eq:original_phi_expression}\\phi\\left(x\\right)=\\int P\\left(x\\middle|\\xi\\right)\\psi\\left(\\xi\\right)\\mathrm{d}\\xi\\end{equation} \u800c\u6211\u4eec\u60f3\u5f97\u77e5\u7ed9\u5b9a\u5f62\u5f0f\u7684\u6838$P\\left(x\\middle|\\xi\\right)$\u6240\u5bf9\u5e94\u7684\u6743\u91cd\u8c31$\\psi\\left(\\xi\\right)$\u3002\u5728\u8fd9\u91cc\uff0c$\\xi$\u662f\u6838\u51fd\u6570$P$\u7684\u53c2\u6570\u3002\u6bd4\u5982\uff0c\u6211\u4eec\u5173\u5fc3\u9ad8\u65af\u6838\u51fd\u6570\u7684\u60c5\u51b5\uff0c\u90a3\u4e48$P\\left(x\\middle|\\xi\\right)$\u53ef\u80fd\u662f\u4ee5$\\xi$\u4e3a\u6807\u51c6\u5dee\u7684\u9ad8\u65af\u51fd\u6570 \\[P\\left(x\\middle|\\xi\\right)=\\frac{1}{\\sqrt{2\\pi\\xi^2}}\\exp\\left[-\\frac{\\left(x-\\mu\\right)^2}{2\\xi^2}\\right]\\] 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Noll (1966), The Foundations of Mechanics, In: G. Grioli, C. Truesdell (eds), Non-linear Continuum Theories, Springer-Verlag \u8fd9\u4e66\u7684\u5370\u5237\u662f\u7528\u6253\u5b57\u673a\u7684\uff0c\u6709\u5f88\u82b1\u4f53notation\u662f\u540e\u9762\u518d\u7528\u624b\u5199\u8865\u4e0a\u7684\u3002\u6211\u4e0d\u77e5\u9053\u8fd9\u4e9b\u7b14\u8ff9\u662f\u4e0d\u662fNoll\u672c\u4eba\u7684\u3002\u4f46\u4ece\u6392\u7248\u4e0a\u53ef\u4ee5\u8bf4\u662f\u975e\u5e38\u4e0d\u53cb\u597d\u3002\u6211\u5bf9\u65f6\u7a7a\u6784\u5efa\u7684\u5185\u5bb9\u4e5f\u5b8c\u5168\u964c\u751f\u3002\u4f46\u662f\u4e0d\u77e5\u9053\u4e3a\u4ec0\u4e48\uff0c\u6211\u4e00\u5f00\u59cb\u5c31\u611f\u53d7\u5230\u8fd9\u4e2a\u8bdd\u9898\u662f\u5f88\u53ef\u80fd\u611f\u5174\u8da3\u7684\uff0c\u9b3c\u4f7f\u795e\u5dee\u5730\u628a\u5b83\u786c\u5543\u5b8c\u4e86\uff0c\u679c\u7136\u611f\u53d7\u5230\u4e86\u65e0\u6cd5\u7528\u8a00\u8bed\u5f62\u5bb9\u7684\u7f8e\u5b66\u4eab\u53d7\u3002\u9664\u4e86\u7f8e\u7684\u4eab\u53d7\u4e4b\u5916\uff0c\u65f6\u7a7a\u6784\u5efa\u7684\u4e25\u5bc6\u6027\uff0c\u53ef\u4ee5\u8ba9\u540e\u7eed\u7684\u5f88\u591a\u8fde\u7eed\u4ecb\u8d28\u529b\u5b66\u5f53\u4e0b\u6559\u6750\u4e2d\u5bb9\u6613\u8ba9\u5b66\u751f\u641e\u6df7\u7684\u8bdd\u9898\u53d8\u5f97\u6839\u672c\u4e0d\u53ef\u80fd\u641e\u6df7\u3002 \u65e9\u671f\uff0c\u6211\u8865\u4e60\u6570\u5b66\u7684\u552f\u4e00\u53c2\u8003\u5c31\u662f S. Hassani (1999), Mathematical Physics: A Modern Introduction to Its Foundations, Springer \u6211\u77e5\u9053\u8fd9\u672c\u4e66\uff0c\u4e5f\u662f\u4ece\u65e9\u5e74\u6d41\u4f20\u7684\u201cFang\u7684\u4e66\u5355\u201d\u4e2d\u77e5\u9053\u7684\u3002\u8fd9\u672c\u4e66\u5bf9Fang\u800c\u8a00\u662f\u65b0\u4e66\uff0c\u4ed6\u53ea\u662f\u8bf4\u597d\u50cf\u8fd8\u53ef\u4ee5\u3002\u73b0\u5728\u56de\u770b\uff0c\u8fd9\u672c\u4e66\u7ed9\u4e86\u6211\u4e24\u4e2a\u5f71\u54cd\uff1a1\uff09\u662f\u8fd9\u672c\u4e66\u5185\u5bb9\u5f88\u5168\uff0c\u4e14\u5728\u4e00\u672c\u4e66\u4e4b\u5185Notation\u548c\u6982\u5ff5\u4f53\u7cfb\u662f\u7edf\u4e00\u7684\u3002\u8ba9\u6211\u5728\u521a\u5b66\u6570\u5b66\u5c31\u611f\u53d7\u5230\u4e86\u4e16\u4e0a\u53ea\u6709\u4e00\u4e2a\u6570\u5b66\uff0c\u800c\u4e0d\u662f\u5272\u88c2\u770b\u4f3c\u65e0\u5173\u7684\u4e0d\u540c\u8bfe\u7a0b\u3002\u540e\u8005\u53ef\u80fd\u662f\u5f88\u591a\u6570\u5b66\u6216\u7269\u7406\u4e13\u4e1a\u7684\u672c\u79d1\u751f\u5728\u73b0\u5728\u7684\u6559\u80b2\u4f53\u7cfb\u4e2d\u83b7\u5f97\u7684\u5370\u8c61\u30022\uff09\u7531\u4e8e\u8981\u505a\u52301\uff09\u8fd9\u4e66\u4e0d\u53ef\u907f\u514d\u5730\u8981\u5982\u5176\u526f\u6807\u9898\u6240\u8a00\u6bd4\u8f83\u73b0\u4ee3\u3002\u4f8b\u5982\u6240\u6709\u6784\u5efa\u90fd\u4ece\u96c6\u5408\u51fa\u53d1\uff0ccoordinate-free\u7684\u7ebf\u6027\u4ee3\u6570\uff0c\u628a\u5f20\u91cf\u5f15\u5165\u4e3a\u591a\u7ebf\u6027\u5f62\u5f0f\u7b49\u3002\u4e4b\u524d\u4e00\u5ea6\u6709\u4e00\u70b9\u8ba9\u6211\u5f88\u8be7\u5f02\u7684\u662f\uff0c\u4e3a\u4ec0\u4e48\u6211\u770b\u8fc7\u4e00\u4e9b\u4e66\uff0c\u5728\u77e5\u4e4e\u4e0a\u5c31\u4ea7\u751f\u4e86\u4e00\u4e2a\u611f\u89c9\u5c31\u662f\u6211\u770b\u5f97\u61c2\u90a3\u4e9b\u79d1\u73ed\u51fa\u8eab\u7684\u4eba\u8bf4\u4ec0\u4e48\uff0c\u8fd8\u597d\u50cf\u80fd\u5bf9\u4e00\u4e9b\u8d85\u8d8a\u672c\u79d1\u7684\u8bdd\u9898\u8fdb\u884c\u8bc4\u8bba\u3002\u5982\u679c\u4e0d\u5b8c\u5168\u662f\u9519\u89c9\uff0c\u90a3\u53ef\u80fd\u662f\u56e0\u4e3a\u6211\u4e00\u5f00\u59cb\u5c31\u63a5\u89e6\u73b0\u4ee3\u7684\u6570\u5b66\u6587\u672c\u3002 \u540e\u6765\u6211\u4eceHassani\u7684\u4e66\u540e\u53c2\u8003\u4e66\u76ee\u53c8\u76f4\u63a5\u6216\u95f4\u63a5\u5730\u77e5\u9053\u4e86\u5f88\u591a\u4e66\u548c\u4f5c\u8005\u3002\u4f8b\u5982Halmos\u548c Hoffman and Kunze\u7684\u7ebf\u6027\u4ee3\u6570\u3002\u540e\u8005\u6210\u4e3a\u4e86\u6211\u7b2c\u4e00\u672c\u4e3b\u8981\u5b66\u4e60\u7684\u4e66\u3002Hoffman and Kunze\u5176\u5b9e\u662f\u4e00\u672c\u5904\u5904\u900f\u7740\u62bd\u8c61\u4ee3\u6570\u7684\u666e\u901a\u7ebf\u6027\u4ee3\u6570\uff0ccoordinate-free\u7684\u8272\u5f69\u66f4\u6d53\u4e86\u3002\u4e00\u4e2a\u8ba9\u6211\u66fe\u7ecf\u56f0\u60d1\uff0c\u5374\u53cd\u800c\u662f\u6781\u5176\u5177\u6709\u542f\u53d1\u6027\u7684\u7279\u70b9\u662f\uff0c\u4f5c\u4e3a\u4e00\u4e2a\u7ebf\u6027\u4ee3\u6570\uff0c\u5168\u4e66\u6ca1\u6709\u201ccross product\u201d\u8fd9\u4e2a\u8bcd\u3002\u7ffb\u904d\u5168\u4e66\u53ea\u6709tensor product\u6ca1\u6709tensor\uff0c\u8fdb\u884ctensor product\u7684\u5bf9\u8c61\u53eb\u591a\u7ebf\u6027\u5f62\u5f0f\u3002\u73b0\u5728\u56de\u60f3\u8d77\u6765\uff0c\u6211\u6562\u8bf4\u5f88\u591a\u529b\u5b66\u8bfe\u7a0b\u7684\u8bb2\u8ff0\u8005\u4e0e\u8bb8\u672a\u5fc5\u80fd\u89e3\u7b54\u5b66\u751f\u5728\u4e00\u5207\u5176\u4ed6\u4e0d\u540c\u5730\u65b9\u770b\u5230\u7528\u201c\u5f20\u91cf\u201d\u4e8c\u5b57\u79f0\u547c\u7684\u5bf9\u8c61\u662f\u4e0d\u662f\u540c\u4e00\u4e2a\u4e1c\u897f\uff0c\u6709\u4ec0\u4e48\u8054\u7cfb\u3002 \u8d2f\u7a7f\u6211\u6559\u5b66\u7684\u4e00\u79cd\u503e\u5411\u5c31\u662f\u5e0c\u671b\u4f20\u8fbe\u77e5\u8bc6\u7684\u7edf\u4e00\u6027\uff0c\u4e00\u79cd\u5173\u4e8e\u4e16\u754c\u7684\u4e00\u5143\u8bba\u3002\u6211\u5f88\u538c\u70e6\u7684\u4e00\u79cd\u770b\u6cd5\u5c31\u662f\uff0c\u5728\u8fd9\u4e2a\u8bfe\u4f60\u5c31\u8fd9\u6837\u6765\uff0c\u5230\u4e86\u90a3\u4e2a\u8bfe\u4e86\u4f60\u53c8\u5c31\u90a3\u6837\u6765\u3002\u524d\u9762\u5b66\u8fc7\u7684\u77e5\u8bc6\uff0c\u597d\u50cf\u6709\u4e9b\u9053\u7406\uff0c\u4f46\u5728\u540e\u9762\u7684\u8bfe\u603b\u662f\u65e0\u6cd5\u76f4\u63a5\u4f7f\u7528\u3002\u4e00\u4e2a\u4eba\u5728\u5b66\u751f\u65f6\u671f\u8981\u540c\u65f6\u638c\u63e1\u5f88\u591a\u5272\u88c2\u7684\u8bfe\u7a0b\u77e5\u8bc6\uff0c\u800c\u5b83\u4eec\u7684\u878d\u6c47\u8d2f\u901a\u5219\u8981\u9760\u6bcf\u4e00\u4e2a\u4eba\u5728\u65e5\u540e\u804c\u4e1a\u751f\u6daf\u4e2d\u7684\u4e2a\u4eba\u7684\u3001\u79c1\u4e0b\u7684\u3001\u4e5f\u8bb8\u5f88\u6c11\u79d1\u7684\u603b\u7ed3\uff0c\u6216\u8005\u66f4\u591a\u60c5\u51b5\u4e0b\u5e72\u8106\u662f\u5fd8\u5374\u6216\u653e\u5f03\u3002 coordinate-free\u5176\u5b9e\u662f\u7ebf\u6027\u4ee3\u6570\u7684\u5176\u4e2d\u4e00\u79cd\u54c1\u5473\u3002\u73b0\u5728\u5728\u77e5\u4e4e\u4e0a\u6d41\u884c\u7740\u4e00\u79cd\u6279\u8bc4\u4ee5\u77e9\u9635\u4ee3\u6570\u4e3a\u4e3b\u7684\u7ebf\u6027\u4ee3\u6570\u6559\u6750\u3002\u7531\u4e8e\u8fd9\u662f\u56fd\u5185\u6559\u6750\u6d41\u884c\u7684\u4f53\u4f8b\uff0c\u8fd9\u79cd\u6279\u8bc4\u53c8\u9644\u5e26\u7740\u5305\u62ec\u5bf9\u56fd\u5185\u6559\u5b66\u843d\u540e\u7684\u6279\u8bc4\u3002\u4f46\u4e8b\u5b9e\u4e0a\u7ebf\u6027\u4ee3\u6570\u6559\u6750\u4e00\u76f4\u6709\u4e24\u79cd\u4f20\u7edf\u3002\u77e9\u9635\u4ee3\u6570\u8fd9\u79cd\u662f\u5341\u4e5d\u4e16\u7eaa\u672b\u4e8c\u5341\u4e16\u7eaa\u521d\u552f\u4e00\u7684\u4e3b\u6d41\u3002coordinate-free\u662f\u6bd4\u8f83\u73b0\u4ee3\u7684\u98ce\u683c\uff0c\u662f\u968f\u7740\u62bd\u8c61\u4ee3\u6570\u7684\u53d1\u5c55\u4e4b\u540e\u53cd\u4f5c\u7528\u4e8e\u672c\u79d1\u7ebf\u6027\u4ee3\u6570\u6559\u5b66\u4e0a\u7684\u7ed3\u679c\u3002\u800c\u5c31\u7b97\u540e\u8005\u66f4\u73b0\u4ee3\uff0c\u5728\u540e\u8005\u51fa\u73b0\u540e\u4e5f\u8fd8\u6709\u4eba\u559c\u6b22\u6309\u524d\u8005\u7684\u65b9\u5f0f\u5199\u4e66\uff0c\u6d41\u884c\u5168\u7403\uff0c\u90a3\u4f8b\u5982\u90a3\u672c\u201cLinear\u2026","rel":"","context":"In "\u4e2a\u4eba\u6587\u96c6"","block_context":{"text":"\u4e2a\u4eba\u6587\u96c6","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/featured"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":5675,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/5675","url_meta":{"origin":8597,"position":5},"title":"\u7a33\u6001\u526a\u5207\u4e0b\u7684\u7ed3\u6784\u677e\u5f1b","author":"Andrew","date":"2013\u5e749\u670819\u65e5","format":false,"excerpt":"\u4e2d\u79cb\u8fd9\u4e24\u5929\u5047\u671f\uff0c\u6211\u90fd\u5728\u770bmode coupling theory\u7684\u6587\u732e\u3002\u82b1\u4e86\u5f88\u591a\u529f\u592b\uff0c\u7ed3\u679c\u5565\u4e5f\u6ca1\u771f\u6b63\u770b\u61c2\u3002\u4ee5\u540e\u53ef\u80fd\u4e0d\u4f1a\u518d\u50cf\u8fd9\u4e24\u5929\u8fd9\u4e48\u8ba4\u771f\u5730\u53bb\u770b\u8fd9\u73a9\u610f\u4e86\uff0c\u6240\u4ee5\u597d\u6b79\u8bb0\u5f55\u4e00\u4e0b\u3002 Time-shear superposition \u5bf9\u8fc7\u51b7\u6db2\u4f53\u7684\u6a21\u62df\u7814\u7a76\u53d1\u73b0\uff0c\u5728\u65bd\u52a0\u4e00\u4e2a\u7a33\u6001\u526a\u5207\u65f6\uff0c\u5176\u5bc6\u5ea6\u6216\u6563\u5c04\u81ea\u76f8\u5173\u51fd\u6570\u7684\u03b1\u677e\u5f1b\u968f\u589e\u5927\u5411\u77ed\u65f6\u95f4\u65b9\u5411\u79fb\u52a8\uff0c\u677e\u5f1b\u65f6\u95f4\u51cf\u5c0f\u3002\u5982\u679c\u628a\u4e0d\u540c\u7684\u81ea\u76f8\u5173\u51fd\u6570\u6539\u5bf9\u4e00\u4e2a\u7ea6\u5316\u7684\u65f6\u95f4t\/\u03c4\u4f5c\u56fe\uff0c\u5404\u66f2\u7ebf\u7684\u03b1\u677e\u5f1b\u90e8\u5206\u80fd\u591f\u53e0\u52a0\u6210\u4e00\u6761\u4e3b\u66f2\u7ebf\u3002\u03c4\u8ddf\u7684\u5173\u7cfb\u5f80\u5f80\u5177\u6709\u7684\u5f62\u5f0f\uff0ca\u5f88\u63a5\u8fd11\u3002\u8fd9\u4e2a\u73b0\u8c61\u8bf4\u660e\uff0c\u7a33\u6001\u526a\u5207\u80fd\u591f\u7f29\u77ed\u03b1\u677e\u5f1b\u7684\u65f6\u95f4\u5c3a\u5ea6\uff0c\u4f46\u4e0d\u6539\u53d8\u677e\u5f1b\u66f2\u7ebf\u7684\u5f62\u72b6\uff0c\u8fd9\u4e00\u73b0\u8c61\u53c8\u88ab\u79f0\u4e3atime-shear superposition\u3002 Strain rate-frequency superposition 2007\u5e74Wyss\u7b49\u62a5\u9053\uff0c\u5728\u51e0\u79cd\u4e0d\u540c\u7684\u6d53\u80f6\u4f53\u4f53\u7cfb\u90fd\u5448\u73b0\u6240\u8c13strain rate-frequency superposition\uff08SRFS\uff09\uff1a\u5bf9\u6837\u54c1\u65bd\u52a0\u632f\u8361\u526a\u5207\uff0c\u5728\u6052\u5b9a\u5e94\u53d8\u901f\u7387\u5e45\u503c\u6761\u4ef6\u4e0b\u626b\u9891\u7387\uff0c\u5f97\u5230G'(\u03c9)\u3001G''(\u03c9)\u66f2\u7ebf\u3002\u6052\u5b9a\u4e0d\u540c\u5927\u5c0f\u7684\u7684\u66f2\u7ebf\u53ef\u5e73\u79fb\u53e0\u52a0\u6210\u4e00\u6761\u4e3b\u66f2\u7ebf\u3002\u5e73\u79fb\u56e0\u5b50\u03c4\u4e0e\u7684\u5173\u7cfb\u4e5f\u6709\u7684\u89c4\u5f8b\u3002\u800c\u4e14\u4e3b\u66f2\u7ebf\u7684\u5f62\u72b6\u8ddf\u5178\u578b\u7684\u7ebf\u6027\u626b\u9891\u8c31\u56fe\u76f8\u4f3c\uff0c\u53ef\u4ee5\u770b\u5230\u201c\u672b\u7aef\u533a\u201d\u3002Wyss\u8ba4\u4e3a\u8fd9\u5c31\u662f\u5916\u90e8\u526a\u5207\u52a0\u901f\u4e86\u6837\u54c1\u7684\u7ed3\u6784\u677e\u5f1b\uff0c\u5e73\u79fb\u56e0\u5b50\u03c4\u5c31\u662f\u677e\u5f1b\u65f6\u95f4\uff0c\u5b83\u8ddf\u7684\u5173\u7cfb\u8ddf\u4e0a\u8ff0\u7684\u7a33\u6001\u526a\u5207\u7684\u62a5\u9053\u7ed3\u679c\u662f\u6765\u81ea\u540c\u4e00\u4e2a\u7269\u7406\u673a\u5236\u3002\u5f88\u591a\u4eba\u8ffd\u968fWyss\u7684\u8bf4\u6cd5\uff0c\u5bf9\u66f4\u591a\u7684\u4f53\u7cfb\u8fdb\u884c\u4e86\u76f8\u540c\u7684\u5b9e\u9a8c\uff0c\u6700\u8fd1\u7684\u4e00\u4e2a\u62a5\u9053\u662f\uff0c\u8bf4\u660eSRFS\u73b0\u8c61\u662f\u5177\u6709\u666e\u904d\u6027\u7684\u3002 \u9996\u5148\uff0c\u6211\u89c9\u5f97\u6240\u8c13\u7684strain rate-frequency superposition\u4e25\u683c\u6765\u8bb2\u5e94\u8be5\u79f0\u4e3atime-shear rate superposition\u3002\u5047\u5982\u6309\u7167Wyss\u7684\u610f\u89c1\uff0c\u8fd9\u6837\u7684\u6052\u5b9a\u7684\u626b\u9891\u7ed3\u679c\u786e\u5b9e\u662f\u4e00\u79cd\u677e\u5f1b\u8c31\u7684\u8bdd\uff0c\u90a3\u4e48\u9891\u7387\u53ea\u662f\u65f6\u95f4\u5728Fourier\u53d8\u6362\u4e4b\u540e\u7684\u5750\u6807\u800c\u5df2\u3002\u201c\u65f6\u6e29\u53e0\u52a0\u539f\u7406\u201d\u4e0d\u4f1a\u56e0\u4e3a\u505a\u4e86\u5c0f\u5e45\u626b\u9891\u5b9e\u9a8c\u5c31\u6539\u79f0\u201c\u6e29\u9891\u53e0\u52a0\u201d\u3002 \u5176\u6b21\uff0c\u867d\u7136\u8bf4SRFS\u7684\u73b0\u8c61\u5927\u4f53\u4e0a\u53ef\u4ee5\u8bf4\u662f\u526a\u5207\u52a0\u901f\u677e\u5f1b\uff0c\u770b\u5230\u4e86\u6837\u54c1\u7684\u6d41\u52a8\u3002\u4f46\u662f\u662f\u5426\u80fd\u591f\u76f4\u63a5\u628a\u6052\u5b9a\u7684\u626b\u9891\u7ed3\u679c\u5f53\u4f5c\u6052\u5b9a\u7684\u7ebf\u6027\u626b\u9891\u6765\u770b\u5f85\uff0c\u628a\u5e73\u79fb\u56e0\u5b50\u03c4\u5f53\u6210\u7ebf\u6027\u54cd\u5e94\u7406\u8bba\u6846\u67b6\u5185\u7684\u677e\u5f1b\u65f6\u95f4\uff1f\u6211\u89c9\u5f97\u8fd9\u4e48\u505a\u662f\u6709\u95ee\u9898\u7684\u3002\u4e00\u7ec4SRFS\u4e3b\u66f2\u7ebf\u4e2d\uff0c\u4f4e\u9891\u90e8\u5206\u7c7b\u4f3c\u201c\u672b\u7aef\u201d\u7684\u533a\u57df\uff0c\u4e8b\u5b9e\u4e0a\u5bf9\u5e94\u7684\u5e94\u53d8\u5e45\u5ea6\u65e9\u5df2\u8d85\u8fc7\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u8303\u56f4\uff0c\u662f\u5927\u5e45\u632f\u8361\u526a\u5207\uff08LAOS\uff09\u7684\u7ed3\u679c\u3002\u7b2c\u4e00\u4e2a\u62c5\u5fe7\u662f\uff0c\u6837\u54c1\u7684\u54cd\u5e94\u542b\u6709\u591a\u4e2a\u9ad8\u6b21\u8c10\u6ce2\u5206\u91cf\uff0cG'\u548cG''\u53ea\u662f\u57fa\u9891\u8c10\u6ce2\u5206\u91cf\u3002\u56e0\u6b64\u81f3\u5c11\u5b83\u4eec\u6ca1\u6709\u53cd\u6620\u6837\u54c1\u54cd\u5e94\u7684\u5168\u90e8\u4fe1\u606f\uff0c\u4e0d\u8db3\u4ee5\u4f5c\u4efb\u4f55\u8ba8\u8bba\u3002\u4f46\u8fd9\u4e00\u62c5\u5fe7\u5df2\u7ecf\u53ef\u4ee5\u6d88\u9664\uff0c\u6709\u4eba\u62a5\u9053\u5728\u8fd9\u4e00\u533a\u57df\uff0cLAOS\u7684\u4e00\u4e9b\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u53c2\u6570\u4e5f\u662f\u7b26\u5408SRFS\u7684\uff0c\u6211\u6700\u8fd1\u505a\u4e86\u4e00\u4e9b\u5b9e\u9a8c\u518d\u6b21\u8bc1\u5b9e\u4e86\u8fd9\u4e00\u70b9\u3002\u8fd9\u8bf4\u660eSRFS\u7684\u6027\u8d28\u662f\u8d2f\u7a7f\u4e86\u7ebf\u6027\u4e0e\u975e\u7ebf\u6027\u533a\u57df\u7684\u3002\u7b2c\u4e8c\u62c5\u5fe7\u662f\uff0c\u901a\u8fc7LAOS\u89c2\u5bdf\u5230\u7684\u672b\u7aef\uff0c\u662f\u5426\u5bf9\u5e94\u4f53\u7cfb\u7684\u7ed3\u6784\u677e\u5f1b\uff1f\u8fd9\u79cd\u5bf9\u5e94\u662f\u7f3a\u4e4f\u7406\u8bba\u57fa\u7840\u7684\uff0c\u8be6\u7ec6\u8ba8\u8bba\u4e86\u8fd9\u4e2a\u95ee\u9898\u3002 \u95ee\u9898\u7684\u63d0\u51fa \u4e8e\u662f\u6211\u60f3\uff0c1) SRFS\u770b\u5230\u7684\u5b9e\u8d28\u5230\u5e95\u662f\u4ec0\u4e48\uff1f\u5982\u679c\u5b83\u771f\u7684\u770b\u5230\u4e86\u4f53\u7cfb\u7684\u7ed3\u6784\u677e\u5f1b\uff0c\u5982\u4f55\u8bc1\u660e\u8fd9\u4e2a\u5bf9\u5e94\u5173\u7cfb\uff1f2) \u5982\u679c\u4e0d\u662f\uff0c\u90a3\u4e48\u6709\u4ec0\u4e48\u522b\u7684\u6d41\u53d8\u5b66\u65b9\u6cd5\uff0c\u80fd\u591f\u770b\u5230\u4f53\u7cfb\u5728\u975e\u7ebf\u6027\u6761\u4ef6\u4e0b\u7684\u7ed3\u6784\u677e\u5f1b\u5417\uff1f \u5bf9\u7b2c\u4e00\u4e2a\u95ee\u9898\uff0c\u53ef\u4ee5\u5148\u5148\u501f\u52a9\u6a21\u578b\u6765\u5bf9SRFS\u73b0\u8c61\u52a0\u6df1\u7406\u89e3\u3002\u6d41\u53d8\u5b9e\u9a8c\u89c2\u5bdf\u5230\u7684SRFS\u4e5f\u597d\uff0c\u8fc7\u51b7\u6db2\u4f53\u7684\u6a21\u62df\u4e2d\u89c2\u5bdf\u5230\u7684\u7ed3\u6784\u677e\u5f1b\u4e5f\u597d\uff0c\u4ece\u76ee\u524d\u62a5\u9053\u6765\u770b\u8fd9\u4e9b\u73b0\u8c61\u4f3c\u4e4e\u90fd\u662f\u73bb\u7483\u4f53\u7cfb\u7684\u7279\u5f81\u3002\uff08\u9ad8\u5206\u5b50\uff09\u7f20\u7ed3\u4f53\u7cfb\u6709\u6ca1\u6709\u8fd9\u6837\u7684\u7279\u5f81\u5462\uff1f\u4e3a\u4ec0\u4e48\u5728\u5386\u53f2\u4e0d\u77ed\u7684\u805a\u5408\u7269\u6d41\u53d8\u5b66\u7814\u7a76\u4e2d\u6ca1\u4eba\u62a5\u9053\u8fc7\u7c7b\u4f3c\u7684\u73b0\u8c61\u5462\uff1f\u6211\u5b9e\u9a8c\u5ba4\u7684\u6d41\u53d8\u4eea\u6ca1\u6709\u914doven\uff0c\u4e0d\u80fd\u505a\u9ad8\u5206\u5b50\u7194\u4f53\u3002\u6211\u5c31\u914d\u5236\u4e86\u4e00\u70b9worm-like micelle\u6765\u5f53\u4f5c\u7f20\u7ed3\u7684\u6a21\u578b\u4f53\u7cfb\u8bd5\u9a8c\uff0c\u53d1\u73b0\u662f\u5b8c\u5168\u4e0d\u7b26\u5408SRFS\u7684\u3002\u6211\u53c8\u7528Giesekus\u6a21\u578b\u505a\u4e86\u4e00\u4e0b\uff0c\u4e5f\u4e0d\u7b26\u5408\u3002\u8fd9\u4e5f\u8bb8\u53ef\u4ee5\u521d\u6b65\u8bf4\u660e\uff0c\u7f20\u7ed3\u4f53\u7cfb\u6ca1\u6709\u8fd9\u6837\u7684\u6027\u8d28\u3002SRFS\u65e2\u7136\u4f1a\u4f9d\u8d56\u8fd9\u79cddynamics\u7684\u5177\u4f53\u673a\u7406\uff0c\u53ef\u80fd\u771f\u7684\u662f\u53cd\u6620\u7740\u4ec0\u4e48\u7ed3\u6784\u677e\u5f1b\u7684\uff0c\u4f46\u662f\u8fd8\u4e0d\u597d\u4e0b\u4ec0\u4e48\u7ed3\u8bba\u3002 \u90a3\u4e48\uff0c\u9002\u7528\u4e8e\u80f6\u4f53\u4f53\u7cfb\u7684\u6d41\u53d8\u5b66\u6a21\u578b\uff0c\u662f\u5426\u5c31\u90fd\u4e00\u5b9a\u80fd\u7b26\u5408SRFS\u5462\uff1f\u6211\u770b\u4e0d\u4e00\u5b9a\u3002\u4e5f\u8bb8\u4ece\u8fd9\u4e2a\u89d2\u5ea6\u51fa\u53d1\uff0c\u53ef\u4ee5\u53bb\u8003\u5bdf\u8981\u6ee1\u8db3SRFS\u6d41\u53d8\u5b66\u6a21\u578b\u9700\u8981\u7b26\u5408\u4ec0\u4e48\u6837\u7684\u7279\u5f81\uff0c\u4ece\u800c\u53ef\u4ee5\u731c\u5230\u4e00\u4e9bSRFS\u7684\u672c\u8d28\uff0c\u540c\u65f6\u4e5f\u53ef\u4ee5\u770b\u770bSRFS\u53cd\u6620\u7684\u662f\u4ec0\u4e48\u201c\u7ed3\u6784\u677e\u5f1b\u201d\u3002\u8fd9\u9879\u5de5\u4f5c\u6211\u8fd8\u6ca1\u6709\u505a\uff0c\u56e0\u4e3a\u76ee\u524d\u63d0\u51fa\u7684\u80fd\u591f\u6bd4\u8f83\u5168\u9762\u5730\u63cf\u8ff0\u5c48\u670d\u5e94\u529b\u6d41\u4f53\u7684\u77ac\u6001\u6a21\u578b\u5e76\u4e0d\u591a\u3002\u8fd9\u6837\u7684\u672c\u6784\u65b9\u7a0b\u5f80\u5f80\u5c5e\u4e8estructural kinetic model\u7684\u7c7b\u578b\uff0c\u8bbe\u8ba1\u4e86\u5f88\u591a\u6ca1\u6709\u76f4\u63a5\u7269\u7406\u610f\u4e49\u7684\u8868\u8fbe\u5f0f\u51d1\u6765\u51d1\u53bb\u3002\u5982\u679c\u7528\u8fd9\u4e9b\u6a21\u578b\u53bb\u5c1d\u8bd5SRFS\uff0c\u65e0\u8bba\u662f\u6210\u8fd8\u662f\u4e0d\u6210\uff0c\u4f60\u90fd\u5f88\u96be\u83b7\u5f97\u4ec0\u4e48\u7269\u7406\u4e0a\u7684insight\u3002\u6240\u4ee5\uff0c\u6211\u5c31\u60f3\u6539\u7528\u4e00\u4e9b\u80f6\u4f53\u4f53\u7cfb\u7684first principle\u7684\u6d41\u53d8\u7406\u8bba\u3002\u9996\u5148\u60f3\u5230\u7684\u5c31\u662fMCT\u5411\u6d41\u53d8\u5b66\u7684\u63a8\u5e7f\uff0c\u5373Fuchs\u7b49\u4eba\u7684\u5de5\u4f5c\u3002\u5177\u4f53\u5730\uff0c\u662f\u5e0c\u671b\u91c7\u7528MCT-ITT\u7684\u4e00\u4e9b\u7b80\u5316\u7248schematic -model\u3002\u4f7f\u7528\u8fd9\u4e2a\u6a21\u578b\u6765\u9884\u6d4bLAOS\u7ed3\u679c\u7684\u62a5\u9053\u5df2\u7ecf\u6709\u4e86\u3002MCT-ITT\u7684\u6570\u5b66\u8fdc\u8fdc\u8d85\u8fc7\u4e86\u6211\u7684\u80fd\u529b\uff0c\u6211\u662f\u6700\u8fd1\u627e\u5230\u4e86\u4e00\u4e2a\u61c2\u6570\u5b66\u7684\u670b\u53cb\u5e2e\u5fd9\u8ba1\u7b97MCT\uff0c\u624d\u8003\u8651\u8fd9\u4e2a\u95ee\u9898\u7684\u3002\u8fd9\u9879\u5de5\u4f5c\u6b63\u5728\u8fdb\u884c\u4e2d\uff0c\u5b58\u5728\u4e00\u4e9b\u95ee\u9898\uff0c\u7a0d\u540e\u518d\u8bf4\u3002 \u8981\u56de\u7b54\u7b2c\u4e8c\u4e2a\u95ee\u9898\uff0c\u5373\u6709\u4ec0\u4e48\u6d41\u53d8\u5b66\u65b9\u6cd5\u80fd\u591f\u770b\u5230\u4f53\u7cfb\u5728\u975e\u7ebf\u6027\u6761\u4ef6\u4e0b\u7684\u7ed3\u6784\u677e\u5f1b\uff0c\u6211\u4eec\u5e94\u8be5\u9996\u5148\u628a\u60c5\u51b5\u7b80\u5316\u56de\u7a33\u6001\u526a\u5207\u3002\u8fb9\u8fdb\u884c\u7a33\u6001\u526a\u5207\uff0c\u8fb9\u8fdb\u884c\u677e\u5f1b\u6d4b\u8bd5\u7684\u6d41\u53d8\u5b66\u65b9\u6cd5\u662f\u6709\u7684\uff0c\u53eb\u505a\u201c\u53e0\u52a0\u6d41\u53d8\u5b66\u201d\uff0csuperposition rheology\u3002\u8fd9\u91cc\u7684superposition\u662f\u6307\u5728\u4e00\u4e2a\u7a33\u6001\u526a\u5207\u7684\u6d41\u573a\u4e0a\u53e0\u52a0\u4e00\u4e2a\u5c0f\u5e45\u632f\u8361\u7684\u6d41\u573a\u3002\u6837\u54c1\u7684\u54cd\u5e94\u53ef\u4ee5\u5206\u79bb\u6210\u7a33\u6001\u526a\u5207\u7684\u90e8\u5206\u548c\u5c0f\u5e45\u632f\u8361\u7684\u90e8\u5206\uff0c\u540e\u8005\u53ef\u4ee5\u6309\u7167\u7ebf\u6027\u54cd\u5e94\u7684\u505a\u6cd5\u5f97\u5230\u677e\u5f1b\u8c31\u3002\u4e00\u5171\u6709\u5e73\u884c\u548c\u6b63\u4ea4\u4e24\u79cd\u53e0\u52a0\u65b9\u6cd5\u3002\u6709\u62a5\u9053\u79f0\uff0c\u5e73\u884c\u53e0\u52a0\u7531\u4e8e\u8026\u5408\u7684\u56e0\u7d20\u5e76\u4e0d\u80fd\u83b7\u5f97\u771f\u6b63\u7684\u677e\u5f1b\u8c31\uff0c\u5fc5\u9762\u8981\u6b63\u4ea4\u53e0\u52a0\u7684\u624d\u53ef\u4ee5\u3002\u65e0\u8bba\u662f\u54ea\u79cd\u53e0\u52a0\uff0c\u90fd\u4e0d\u5bb9\u6613\u5728\u5546\u7528\u6d41\u53d8\u4eea\u4e0a\u5b9e\u73b0\u7684\uff0c\u9700\u8981\u8fdb\u884c\u6539\u9020\uff0c\u6240\u4ee5\u6211\u6682\u65f6\u4e0d\u80fd\u4eb2\u81ea\u505a\u8fd9\u6837\u7684\u5b9e\u9a8c\u3002\u6587\u732e\u62a5\u9053\u4e2d\u503c\u5f97\u7559\u610f\u7684\u6709\u5982\u4e0b\u8fd9\u4e9b\u3002\u5bf9\u9ad8\u5206\u5b50\u6eb6\u6db2\u505a\u4e86\u53e0\u52a0\u6d41\u53d8\u6d4b\u8bd5\uff0c\u53d1\u73b0\u6240\u5f97\u5230\u7684\u7ed3\u679c\u65e0\u6cd5\u8fdb\u884ctime-shear 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superposition\u4e86\uff0c\u4f5c\u8005\u7ed9\u51fa\u4e86\u4e0d\u540c\u7a33\u6001\u5e94\u529b\u53e0\u52a0\u7684\u5c0f\u5e45\u626b\u9891\u66f2\u7ebf\uff0c\u5e76\u53e0\u52a0\u6210\u4e86\u4e3b\u66f2\u7ebf\uff0c\u4f46\u662f\u6ca1\u6709\u7ed9\u51fa\u76f8\u5e94\u7684\u526a\u5207\u901f\u7387\uff0c\u4e5f\u6ca1\u6709\u5bf9\u5e73\u79fb\u56e0\u5b50\u8fdb\u884c\u8ba8\u8bba\u3002\u4f46\u8fd9\u4e00\u62a5\u9053\u8bf4\u660e\uff0c\u80f6\u4f53\u4f53\u7cfb\u5728\u6d41\u53d8\u5b66\u884c\u4e3a\u4e0a\u4e5f\u662f\u7b26\u5408time-shear superposition\u7684\uff0c\u8ddf\u524d\u6587\u4ecb\u7ecd\u7684\u8fc7\u51b7\u6db2\u4f53\u7684\u6a21\u62df\u7ed3\u679c\u76f8\u547c\u5e94\u3002\u5728\u7ebf\u6027\u54cd\u5e94\u6761\u4ef6\u4e0b\uff0c\u6d41\u53d8\u5b66\u4e0a\u7684\u529b\u5b66\u677e\u5f1b\u8ddf\u7ed3\u6784\u677e\u5f1b\u4e4b\u95f4\u5c31\u5dee\u4e2aGreen-Kubo\u5173\u7cfb\u3002\u4f46\u662f\u5728\u975e\u7ebf\u6027\u6761\u4ef6\u4e0b\uff0c\u662f\u5426\u4ecd\u6709\u67d0\u79cd\u5e7f\u4e49\u7684Green-Kubo\u5173\u7cfb\uff1f\u8fd9\u81f3\u5c11\u662f\u7406\u8bba\u5bb6\u4eec\u5e38\u5e38\u91c7\u7528\u7684\u4e00\u79cd\u5047\u8bbe\u3002 Schematic -model MCT\u5411\u6d41\u53d8\u5b66\u7684\u63a8\u5e7f\uff0c\u6709Fuchs\u7b49\u548cMiyazaki\u7b49\u7684\u4e24\u79cd\u505a\u6cd5\u3002Fuchs\u7b49\u4eba\u7684MCT-ITT\u53c8\u6709\u4e24\u79cd\u7b80\u5316\uff0c\u4e00\u79cd\u662fisotropically sheared hard sphere model\uff08ISHSM\uff09\uff0c\u53e6\u4e00\u79cd\u5c31\u662fschematic -model\u3002 Time-shear 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