{"id":7088,"date":"2016-07-08T14:48:02","date_gmt":"2016-07-08T06:48:02","guid":{"rendered":"http:\/\/www.andrewsun.net\/panta_rhei\/?p=7088"},"modified":"2020-02-04T23:33:14","modified_gmt":"2020-02-04T15:33:14","slug":"rice%e5%88%86%e5%b8%83%e7%9a%84%e6%95%b0%e5%80%bc%e8%ae%a1%e7%ae%97","status":"publish","type":"post","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/7088","title":{"rendered":"Rice\u5206\u5e03\u7684\u6570\u503c\u8ba1\u7b97"},"content":{"rendered":"<div class=\"kcite-section\" kcite-section-id=\"7088\">\n<p>[latexpage]<\/p>\n<p>\u6211\u5728<a href=\"http:\/\/www.andrewsun.net\/panta_rhei\/archives\/7025\">\u6781\u5750\u6807\u4e0b\u7684\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70<\/a>\u4e2d\u8ba8\u8bba\u5230Rice\u5206\u5e03\uff1a<\/p>\n<p>\\begin{equation}\\label{eq:1}<br \/>\np_R\\left(r\\right)=\\frac{r}{\\sigma^2}I_0\\left(\\frac{r\\mu_r}{\\sigma^2}\\right)\\exp\\left(-\\frac{r^2+\\mu_r^2}{2\\sigma^2}\\right)<br \/>\n\\end{equation}<\/p>\n<p>\u5f53\u65e0\u89c4\u884c\u8d70\u53d1\u751f\u7684\u201c\u4f4d\u7f6e\u201d\uff08$\\mu_r$\uff09\u79bb\u539f\u70b9\u5f88\u8fdc\u65f6\uff0c$\\mu_r$\u503c\u5f88\u5927\uff0c\u800c\u5206\u5e03\u7684\u6781\u503c\u4e5f\u5927\u81f3\u5904\u4e8e$r=\\mu_r$\u5904\uff0c\u56e0\u6b64\uff0c\u80fd\u770b\u5230\u5206\u5e03\u4e3b\u8981\u7279\u5f81\u7684$r$\u8303\u56f4\u4e5f\u5728$\\mu_r$\u9644\u8fd1\u3002\u8fd9\u65f6\uff0c\u5f0f(\\ref{eq:1})\u4e2d\u7684\u7b2c\u4e00\u7c7b\u4fee\u6b63\u8d1d\u585e\u5c14\u51fd\u6570$I_0\\left(\\frac{r\\mu_r}{\\sigma^2}\\right)$\u7684\u503c\u4f1a\u5f88\u5927\uff0c\u800c\u81ea\u7136\u6307\u6570\u9879$\\exp\\left(-\\frac{r^2+\\mu_r^2}{2\\sigma^2}\\right)$\u4f1a\u5f88\u5c0f\uff0c\u4f46\u6574\u4e2a\u5f0f(\\ref{eq:1})\u7684\u503c\u4ecd\u7136\u662f\u9002\u4e2d\u7684\u3002\u5728MATLAB\u8f93\u5165\u4e0a\u5f0f\u8ba1\u7b97\u65f6\uff0c\u4f1a\u56e0\u4e3a\u8ba1\u7b97\u8fc7\u7a0b\u4e2d\u6d89\u53ca\u5230\u5f88\u5927\u7684\u548c\u5f88\u5c0f\u7684\u503c\uff0c\u6240\u4ee5\u5c3d\u7ba1\u6700\u7ec8\u7ed3\u679c\u7684\u503c\u662f\u9002\u4e2d\u7684\uff0c\u8ba1\u7b97\u4e5f\u4f1a\u6ea2\u51fa\u3002\u8fd9\u65f6\u53ef\u4ee5\u5229\u7528\u7b2c\u4e00\u7c7b\u4fee\u6b63\u8d1d\u585e\u5c14\u51fd\u6570\u7684\u6e10\u8fd1\u5c55\u5f00\u3002\u5f53$\\alpha$\u4e3a\u5b9a\u503c\u3001$\\left|z\\right|$\u5f88\u5927\u4e14$\\left|\\mathrm{arg}z\\right|<\\frac{\\pi}{2}$\u65f6\uff0c\n\n\\begin{equation}\\label{eq:2}\n\\begin{aligned}\nI_\\alpha\\left(z\\right)&#038;=\\frac{\\exp\\left(z\\right)}{\\sqrt{2\\pi z}}\\left[1-\\frac{4\\alpha^2-1}{8z}-\\frac{\\left(4\\alpha^2-1\\right)\\left(4\\alpha^2-9\\right)}{2!\\left(8z\\right)^2}\\right.\\\\\n&#038;\\left.-\\frac{\\left(4\\alpha^2-1\\right)\\left(4\\alpha^2-9\\right)\\left(4\\alpha^2-25\\right)}{3!\\left(8z\\right)^3}+\\cdots\\right]\\\\\n&#038;\\approx\\frac{\\exp\\left(z\\right)}{\\sqrt{2\\pi z}}\\left[1+O\\left(z^{-1}\\right)\\right]\n\\end{aligned}\n\\end{equation}\n\n$z$\u8d8a\u5927\uff0c\u7ea7\u6570\u4f1a\u8870\u51cf\u5f97\u8d8a\u5feb.\u5982\u679c$z$\u8db3\u591f\u5927\uff0c$I_0$\u53ef\u8fd1\u4f3c\u4e3a$\\frac{\\exp\\left(z\\right)}{\\sqrt{2\\pi z}}$\u3002\u8fd9\u4e2a\u8fd1\u4f3c\u5f0f\u5e76\u4e0d\u6539\u53d8$I_0$\u5f88\u5927\u7684\u4e8b\u5b9e\u3002\u4f46\u5982\u679c\u5c06\u5f0f(\\ref{eq:2})\u4ee3\u5165\u5f0f(\\ref{eq:1})\uff0c\n\n\\begin{equation}\\label{eq:3}\np_R\\left(r\\right)=\\frac{r\\exp\\left[-\\frac{\\left(r-\\mu_r\\right)^2}{2\\sigma^2}\\right]}{\\sqrt{2\\pi r\\mu_r\\sigma^2}}\\left[1+\\frac{\\sigma^2}{8r\\mu_r}-\\frac{9\\sigma^4}{128r^2\\mu_r^2}+\\cdots\\right]\n\\end{equation}\n\n\u53ef\u89c1\uff0c\u65e0\u8bba$\\mu_r$\u591a\u5927\uff0c$p_r\\left(r\\right)$\u5176\u5b9e\u53ea\u8ddf$r$\u4e0e$\\mu_r$\u4e4b\u5dee\u6709\u5173\u3002\u5f0f(\\ref{eq:3})\u53ea\u9700\u8981\u8ba1\u7b97$\\exp\\left[-\\left(r-\\mu_r\\right)^2\\right]$\u3002\u7531\u4e8eRice\u5206\u5e03\u7684$r$\u8303\u56f4\u4e3b\u8981\u5728$\\mu_r$\u9644\u8fd1\uff0c\u56e0\u6b64$r$\u4e0e$\\mu_r$\u4e4b\u5dee\u662f\u5f88\u5c0f\u7684\uff0c\u56e0\u6b64\u8fd9\u4e00\u8ba1\u7b97\u4e0d\u4f1a\u6ea2\u51fa\u3002\n\n\u5b9e\u9645\u8ba1\u7b97\u65f6\uff0c\u53ef\u5148\u5224\u65ad$\\mu_r$\u548c$\\sigma^2$\u7684\u53d6\u503c\u662f\u5426\u4f7f\u5f0f(\\ref{eq:1})\u7684\u8ba1\u7b97\u6ea2\u51fa\u4e86\uff0c\u82e5\u6ea2\u51fa\uff0c\u6309\u5f0f(\\ref{eq:3})\u8ba1\u7b97\u65f6\uff0c\u53ef\u6839\u636e$\\mu_r$\u548c$\\sigma^2$\u7684\u53d6\u503c\u5927\u5c0f\u51b3\u5b9a\u820d\u53bb\u7ea7\u6570\u7684\u54ea\u4e9b\u5c3e\u9879\u3002\u6309\u7167MATLAB\u7684\u8ba1\u7b97\u6781\u9650\uff0c\u80fd\u8ba9\u5f0f(\\ref{eq:1})\u6ea2\u51fa\u7684\u60c5\u51b5\uff0c\u4e5f\u5c31\u5fc5\u7136\u80fd\u4f7f\u5f0f(\\ref{eq:3})\u7684\u6240\u6709\u5c3e\u9879\u90fd\u820d\u53bb\u4e86\u3002\n<\/p>\n<!-- kcite active, but no citations found -->\n<\/div> <!-- kcite-section 7088 -->","protected":false},"excerpt":{"rendered":"<p>[latexpage] \u6211\u5728\u6781\u5750\u6807\u4e0b\u7684\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u4e2d\u8ba8\u8bba\u5230Rice\u5206\u5e03\uff1a \\begin{equation}\\label{eq:1} p_R\\left(r\\right)=\\frac{r}{\\sigma^2}I_0\\left(\\frac{r\\mu_r}{\\sigma^2}\\right)\\exp\\left(-\\frac{r^2+\\mu_r^2}{2\\sigma^2}\\right) \\end{equation} \u5f53\u65e0\u89c4\u884c\u8d70\u53d1\u751f\u7684\u201c\u4f4d\u7f6e\u201d\uff08$\\mu_r$\uff09\u79bb\u539f\u70b9\u5f88\u8fdc\u65f6\uff0c$\\mu_r$\u503c\u5f88\u5927\uff0c\u800c\u5206\u5e03\u7684\u6781\u503c\u4e5f\u5927\u81f3\u5904\u4e8e$r=\\mu_r$\u5904\uff0c\u56e0\u6b64\uff0c\u80fd\u770b\u5230\u5206\u5e03\u4e3b\u8981\u7279\u5f81\u7684$r$\u8303\u56f4\u4e5f\u5728$\\mu_r$\u9644\u8fd1\u3002\u8fd9\u65f6\uff0c\u5f0f(\\ref{eq:1})\u4e2d\u7684\u7b2c\u4e00\u7c7b\u4fee\u6b63\u8d1d\u585e\u5c14\u51fd\u6570$I_0\\left(\\frac{r\\mu_r}{\\sigma^2}\\right)$\u7684\u503c\u4f1a\u5f88\u5927\uff0c\u800c\u81ea\u7136\u6307\u6570\u9879$\\exp\\left(-\\frac{r^2+\\mu_r^2}{2\\sigma^2}\\right)$\u4f1a\u5f88\u5c0f\uff0c\u4f46\u6574\u4e2a\u5f0f(\\ref{eq:1})\u7684\u503c\u4ecd\u7136\u662f\u9002\u4e2d\u7684\u3002\u5728MATLAB\u8f93\u5165\u4e0a\u5f0f\u8ba1\u7b97\u65f6\uff0c\u4f1a\u56e0\u4e3a\u8ba1\u7b97\u8fc7\u7a0b\u4e2d\u6d89\u53ca\u5230\u5f88\u5927\u7684\u548c\u5f88\u5c0f\u7684\u503c\uff0c\u6240\u4ee5\u5c3d\u7ba1\u6700\u7ec8\u7ed3\u679c\u7684\u503c\u662f\u9002\u4e2d\u7684\uff0c\u8ba1\u7b97\u4e5f\u4f1a\u6ea2\u51fa\u3002\u8fd9\u65f6\u53ef\u4ee5\u5229\u7528\u7b2c\u4e00\u7c7b\u4fee\u6b63\u8d1d\u585e\u5c14\u51fd\u6570\u7684\u6e10\u8fd1\u5c55\u5f00\u3002\u5f53$\\alpha$\u4e3a\u5b9a\u503c\u3001$\\left|z\\right|$\u5f88\u5927\u4e14$\\left|\\mathrm{arg}z\\right|<\/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":[212,303],"tags":[300,287],"class_list":["post-7088","post","type-post","status-publish","format-standard","hentry","category-tagged","category-rheo","tag-brownian-motion","tag-particle-tracking"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p1aPvF-1Qk","jetpack-related-posts":[{"id":7025,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/7025","url_meta":{"origin":7088,"position":0},"title":"\u6781\u5750\u6807\u4e0b\u7684\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70","author":"Andrew","date":"2016\u5e746\u670811\u65e5","format":false,"excerpt":"[latexpage] \u8fd9\u51e0\u5929\u6211\u5728\u674e\u4fca\u6770\u3001Mathematica\u7684\u5e2e\u52a9\u4e0b\u63a8\u5bfc\u4e86\u6781\u5750\u6807\u4e0b\u7684\u4e8c\u7ef4\u5e03\u6717\u8fd0\u52a8\u7684\u7edf\u8ba1\u3002\u4e00\u5f00\u59cb\u6211\u5c31\u89c9\u5f97\uff0c\u8fd9\u662f\u5f88\u57fa\u672c\u7684\u95ee\u9898\uff0c\u5e94\u8be5\u4f1a\u6709\u76f8\u5e94\u7684\u4f8b\u9898\u6216\u8bfe\u4ef6\u76f4\u63a5\u7ed9\u51fa\u7b54\u6848\u3002\u4f46\u662f\u4e00\u5f00\u59cb\u641c\u90fd\u641c\u4e0d\u5230\u3002\u6211\u7528\u4f53\u80b2\u8001\u5e08\u6559\u7684\u6570\u5b66\u63a8\u4e86\u534a\u5929\uff0c\u51fa\u7ed3\u679c\u4e4b\u540e\uff0c\u5c31\u7acb\u9a6c\u641c\u5230\u8d44\u6599\u4e86\u3002\u53ea\u5f53\u4f5c\u4e0a\u5929\u903c\u6211\u7ec3\u4e60\u4e00\u4e0b\u5927\u5b66\u7684\u6570\u5b66\u5427\u3002 \u6211\u8981\u8003\u8651\u7684\u95ee\u9898\u662f\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u7684\u6781\u5750\u6807\u7684\u7edf\u8ba1\uff0c\u5373\u6781\u5f84\u548c\u6781\u89d2\u7684\u5206\u5e03\u3002\u4e4b\u524d\u6211\u603b\u7ed3\u8fc7\u4e86\u4e8c\u7ef4\u548c\u4e09\u7ef4\u65e0\u89c4\u884c\u8d70\u7684\u6781\u5f84\u5206\u5e03\u5206\u522b\u662fRayleigh\u5206\u5e03\u548cMaxwell\u5206\u5e03\uff0c\u8fd9\u90fd\u662f\u76f4\u89d2\u5750\u6807\u7684\u671f\u671b\u503c\u4e3a\u96f6\u7684\u60c5\u51b5\u3002\u672c\u6587\u4e00\u662f\u8981\u8003\u8651\u671f\u671b\u503c\u4e0d\u4e3a\u96f6\u7684\u60c5\u51b5\uff0c\u4e8c\u662f\u4e0d\u5149\u8003\u8651\u6781\u5f84\u7684\u5206\u5e03\uff0c\u8fd8\u8981\u8003\u8651\u6781\u89d2\u7684\u3002 \u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u7684\u201c\u8f68\u8ff9\u4e2d\u5fc3\u201d \u4e00\u4e2a\u8d70\u4e86N\u6b65\u7684\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u8f68\u8ff9\u5e94\u8be5\u662f\u6709\u5176\u4e2d\u5fc3\u7684\u3002\u4f8b\u5982\uff0c\u53ef\u4ee5\u5b9a\u4e49\u8fd9\u4e2a\u4e2d\u5fc3\u4e3a\u65e0\u89c4\u884c\u8d70\u7684x\u5750\u6807\u548cy\u5750\u6807\u7684\u671f\u671b\u503c$\\mu_X$\u548c$\\mu_Y$\u3002\u8fd9\u6837\u7684\u5b9a\u4e49\u4e25\u683c\u6765\u8bf4\u5e76\u4e0d\u5b9e\u7528\uff0c\u56e0\u4e3a\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u6709\u9650\u6b65\u6570\u7684\u8f68\u8ff9\uff0c\u6211\u4eec\u65e0\u6cd5\u77e5\u9053\u671f\u671b\u503c\uff0c\u53ea\u80fd\u8ba1\u7b97\u5747\u503c\u3002\u540e\u8005\u53ea\u6709$N\\rightarrow\\infty$\u65f6\u624d\u662f\u671f\u671b\u503c\u3002\u4e3a\u4e86\u5b9e\u9a8c\u65b9\u4fbf\uff0c\u6211\u5b9a\u4e49\u4e8c\u7ef4\u65e0\u89c4\u884c\u8d70\u7684\u8f68\u8ff9\u201c\u4e2d\u5fc3\u201d\u662f\u5176\u8d77\u59cb\u5750\u6807$\\left(x_0,y_0\\right)$\u3002\u8fd9\u6837\uff0c\u4e0d\u4ec5\u5bf9\u4efb\u610f\u4e00\u4e2a\u7ed9\u5b9a\u7684\u8f68\u8ff9\u90fd\u80fd\u76f4\u63a5\u8bf4\u51fa\u5176\u201c\u4e2d\u5fc3\u201d\uff0c\u8fd8\u80fd\u4e3b\u52a8\u5b9e\u73b0\u4e00\u4e2a\u7ed9\u5b9a\u4e2d\u5fc3\u7684\u201c\u65e0\u89c4\u884c\u8d70\u201d\u2014\u2014\u53ea\u8981\u4ece\u4f60\u7684\u884c\u8d70\u662f\u4ece\u90a3\u4e2a\u4e2d\u5fc3\u5f00\u59cb\u7684\u5c31\u884c\u3002\u51ed\u76f4\u89c2\u60f3\u8c61\uff0c\u4e00\u4e2a\u4ece\u70b9$\\left(x_0,y_0\\right)$\u51fa\u53d1\u7684\u65e0\u89c4\u884c\u8d70\uff0c\u53ea\u8981\u6b65\u6570$N$\u8db3\u591f\u5927\uff0c\u5c31\u6709$\\mu_X=x_0$\u548c$\\mu_Y=y_0$\u3002\u5f53\u7136\uff0c\u8fd9\u662f\u4e3a\u4e86\u5b9e\u9a8c\u7684\u601d\u8003\u3002\u63a5\u4e0b\u6765\u7684\u63a8\u5bfc\u90fd\u662f\u8c08\u671f\u671b\u503c \u671f\u671b\u503c\u4e3a\u96f6\u7684\u60c5\u51b5 \u9996\u5148\uff0c\u8003\u8651\u4e00\u4e2a\u201c\u5728\u539f\u70b9\u5904\u201d\u7684\u65e0\u89c4\u884c\u8d70\uff0c\u5373\u5176x\u5750\u6807\u548cy\u5750\u6807\u7684\u671f\u671b\u503c$\\mu_X=\\mu_Y=0$\uff0c\u65b9\u5dee\u90fd\u662f$\\sigma^2$\uff08\u5404\u5411\u540c\u6027\uff09\u3002\u8fd9\u65f6\u5176\u6781\u5750\u6807\u7684\u6781\u5f84\u548c\u6781\u89d2$r$\u548c$\\theta$\u7684\u5206\u5e03\u5206\u522b\u4e3a\u4e00\u4e2aRayleigh\u5206\u5e03\u548c\u4e00\u4e2a\u5747\u5300\u5206\u5e03\uff1a \\begin{equation}\\label{eq:1} p_R\\left(r\\right)=\\frac{r}{\\sigma^2}\\exp\\left[-\\frac{r^2}{2\\sigma^2}\\right] \\end{equation} \\begin{equation}\\label{eq:2} p_\\Theta\\left(\\theta\\right)=\\frac{1}{2\\pi} \\end{equation} \u671f\u671b\u503c\u4e0d\u4e3a\u96f6\u7684\u60c5\u51b5 \u8003\u8651\u201c\u4f4d\u4e8e\u6781\u5750\u6807\u4e2d\u70b9$\\left(\\mu_r,\\mu_\\theta\\right)$\u5904\u7684\u65e0\u89c4\u884c\u8d70\uff0c\u5373\u5176\u76f4\u89d2\u5750\u6807x\u548cy\u7684\u671f\u671b\u503c\u4e3a$\\mu_X\u548c\\mu_Y$\uff0c\u65b9\u5dee\u90fd\u662f$\\sigma^2$\uff08\u5404\u5411\u540c\u6027\uff09\u3002\u4e8e\u662f \\begin{equation}\\label{eq:3} \\begin{aligned} \\mu_r&=\\sqrt{\\mu_X^2+\\mu_Y^2} \\\\ \\mu_\\theta&=\\left\\{\\begin{matrix} \\arctan\\frac{\\mu_Y}{\\mu_X}, & x>0\\\\ \\arctan\\frac{\\mu_Y}{\\mu_X}+\\pi, &x<0 \\end{matrix}\\right. \\end{aligned}\\end{equation} \u5219\u6781\u5f84$r$\u548c\u6781\u89d2$\\theta$\u7684\u8054\u5408\u5206\u5e03\u4e3a \\begin{equation}\\label{eq:4} p_{R\\Theta}\\left(r,\\theta\\right)=\\frac{r}{2\\pi\\sigma^2}\\exp\\left[-\\frac{r^2+\\mu_r^2-2r\\mu_r\\cos\\left(\\theta-\\mu_\\theta\\right)}{2\\sigma^2}\\right] \\end{equation} \u7531\u4e8e\u65e0\u89c4\u884c\u8d70\u7684\u6781\u5f84\u548c\u6781\u89d2\u662f\u76f8\u4e92\u72ec\u7acb\u7684\u968f\u673a\u91cf\uff0c\u6240\u4ee5\u6c42\u4e24\u4e2a\u8fb9\u7f18\u5206\u5e03\u5c31\u53ef\u4ee5\u4e86\u3002 \\begin{equation} \\begin{aligned} p_R\\left(r\\right)&=\\int_0^{2\\pi}p_{R\\Theta}d\\theta\\\\ p_\\Theta\\left(\\theta\\right)&=\\int_0^\\infty p_{R\\Theta}dr \\end{aligned} \\end{equation} \u8fd9\u4e24\u4e2a\u79ef\u5206\u90fd\u65e0\u6cd5\u7528\u521d\u7b49\u51fd\u6570\u8868\u793a\u3002\u6211\u662f\u7528Mathematica\u6765\u7b97\u7684\uff0c\u7ed3\u679c\u5982\u4e0b\uff1a \\begin{equation}\\label{eq:6} p_R\\left(r\\right)=\\frac{r}{\\sigma^2}I_0\\left(\\frac{r\\mu_r}{\\sigma^2}\\right)\\exp\\left(-\\frac{r^2+\\mu_r^2}{2\\sigma^2}\\right) \\end{equation} \\begin{equation}\\label{eq:7} \\begin{aligned} p_\\Theta\\left(\\theta\\right)=&\\frac{1}{\\sqrt{8\\pi\\sigma^2}}\\mu_r\\cos\\left(\\theta-\\mu_\\theta\\right)\\exp\\left[-\\frac{\\mu_r^2\\sin^2\\left(\\theta-\\mu_\\theta\\right)}{2\\sigma^2}\\right]\\times \\\\ &\\left[\\mathrm{erf}}\\left(\\frac{\\mu_r\\cos\\left(\\theta-\\mu_\\theta\\right)}{\\sqrt{2\\sigma^2}}\\right)+1\\right]+\\frac{1}{2\\pi}\\exp\\left(-\\frac{\\mu_r^2}{2\\sigma^2}\\right) \\end{aligned} \\end{equation}\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":"Rice distributions of varying expetation","src":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2016\/06\/untitled-480x360.png?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":6836,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/6836","url_meta":{"origin":7088,"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 &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":6981,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/6981","url_meta":{"origin":7088,"position":2},"title":"\u975e\u9ad8\u65af\u7cfb\u6570\u7b26\u53f7\u4e3a\u4ec0\u4e48\u6709\u4e2a\u4e0b\u68072\uff1f","author":"Andrew","date":"2016\u5e743\u670831\u65e5","format":"aside","excerpt":"[latexpage] \u5728\u7528\u7c92\u5b50\u793a\u8e2a\uff08particle tracking)\u65b9\u6cd5\u7814\u7a76\u8f6f\u7269\u8d28\u7684\u8bba\u6587\u4e2d\u7ecf\u5e38\u7528van Hove\u51fd\u6570\u7684\u975e\u9ad8\u65af\u6027\u6765\u8868\u5f81\u4f53\u7cfb\u7684\u52a8\u6001\u4e0d\u5747\u5300\u6027\u3002\u5f80\u5f80\u8fd8\u6709\u4e00\u4e2a\u201c\u975e\u9ad8\u65af\u53c2\u6570\u201d\u6765\u8868\u5f81\u5b9e\u9a8c\u7ed3\u679c\u504f\u79bb\u9ad8\u65af\u5206\u5e03\u7684\u7a0b\u5ea6\uff0c\u5e38\u5e38\u7528$\\alpha_2$\u6765\u8868\u793a\u3002\u5b83\u5176\u5b9e\u662f\u7531\u5cf0\u5ea6\uff08kurtosis\uff09\u5b9a\u4e49\u7684\uff0c\u628a\u5b9e\u9a8c\u5206\u5e03\u7684\u5cf0\u5ea6\u62ff\u9ad8\u65af\u5206\u5e03\u7684\u5cf0\u5ea6\u4e00\u76f8\u9664\uff0c\u518d\u51cf1\uff0c\u4f7f\u5f97\u4e00\u4e2a\u9ad8\u65af\u5206\u5e03\u7684$\\alpha_2$\u7684\u503c\u4e3a0\uff0c\u975e\u96f6\u7684$\\alpha_2$\u5c31\u6807\u5fd7\u7740\u975e\u9ad8\u65af\u7684\u5206\u5e03\u3002 \\begin{equation}\\label{eq:1} \\alpha_2=\\frac{3\\langle\\Delta x^4\\rangle}{5\\langle\\Delta x^2\\rangle^2}-1 \\end{equation} \u65e2\u7136\u662f\u8fd9\u6837\u7684\u8bdd\uff0c\u4e3a\u4ec0\u4e48\u4e0d\u76f4\u63a5\u7528\u4e00\u4e2a\u7b80\u5355\u7684\u5b57\u6bcd\u6765\u8868\u793a\uff0c\u800c\u662f\u8981\u7528\u4e00\u4e2a\u5e26\u6709\u6570\u5b57\u4e0b\u68072\u7684\u7b26\u53f7\u5462\uff1f\u96be\u9053\u8fd8\u6709$\\alpha_1$\u3001$\\alpha_3$\uff1f\u6211\u4e00\u76f4\u6709\u8fd9\u4e2a\u7591\u95ee\uff0c\u4eca\u5929\u6211\u8003\u7a76\u4e86\u4e00\u4e0b\u3002\u5176\u5b9e\uff0c\u4f7f\u7528$\\alpha_2$\u6765\u8868\u793a\u8fd9\u4e2a\u91cf\uff0c\u6700\u65e9\u662f\u51fa\u4e8eA. Rahman\u7684\u5de5\u4f5c\u3002\u4f5c\u8005\u63a8\u5bfc\u4e86\u6db2\u4f53\u7684\u65f6\u95f4\u4f9d\u8d56\u76f8\u5173\u51fd\u6570\u7684\u5c55\u5f00\u5f0f\u3002\u7b2c\u4e00\u9879\u5f53\u7136\u662f\u7eaf\u5e03\u6717\u8fd0\u52a8\u7684\u9ad8\u65af\u9879\uff0c\u7b2c\u4e8c\u9879\u5f00\u59cb\u7684\u9ad8\u9636\u9879\u90fd\u662f\u975e\u9ad8\u65af\u9879\uff0c\u5176\u4e2d\u5404\u7684\u7cfb\u6570\u53ef\u4ee5\u7528$\\alpha_n$\u6765\u8868\u793a\uff0c\u5176\u901a\u5f0f\u662f \\begin{equation}\\label{eq:2} \\alpha_n=\\frac{\\langle r^{2n}\\rangle}{C_n\\langle r^2\\rangle^n}-1 \\end{equation} \u5176\u4e2d$C_n=1\\cdot3\\cdot5\\cdots\\left(2n+1\\right)\/3^n$\u3002\u56e0\u6b64$\\alpha_2$\u6070\u597d\u5c31\u662f\u5f0f(\\ref{eq:1})\u3002\u6216\u8005\u8bf4\uff0c\u901a\u8fc7\u5c55\u5f00\u5f97\u5230\u7684\u7b2c\u4e8c\u9879\u7cfb\u6570\uff0c\u6070\u597d\u5c31\u662f\u4e00\u4e2a\u8ddf\u5cf0\u5ea6\u6709\u5173\u7684\u91cf\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c$\\alpha_2$\u5176\u5b9e\u662f\u65f6\u95f4\u76f8\u5173\u51fd\u6570\u5c55\u5f00\u5f0f\u7684\u7b2c2\u9879\uff0c\u6240\u4ee5\u4e0b\u6807\u6709\u4e2a2\u3002\u81ea\u6587\u732e\u540e\uff0c$\\alpha_2$\u5f88\u5feb\u88ab\u91c7\u7528\u4e3a\u201c\u975e\u9ad8\u65af\u7cfb\u6570\u201d\u7528\u6765\u63cf\u8ff0\u6db2\u4f53\u6027\u8d28\uff0c\u6210\u4e3a\u6db2\u4f53\u7406\u8bba\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u53c2\u6570\u3002\u540e\u6765\u80f6\u4f53\u7269\u7406\u7684\u5174\u8d77\uff0c\u7531\u4e8e\u5176\u7406\u8bba\u57fa\u7840\u57fa\u672c\u4e0a\u662f\u6db2\u4f53\u7269\u7406\uff0c\u6240\u4ee5\u5f88\u81ea\u7136\u5c31\u7528\u4e86\u76f8\u4f3c\u7684\u8ba8\u8bba\u8303\u5f0f\uff0c\u4f7f\u7528$\\alpha_2$\u8fd9\u4e2a\u7b26\u53f7\u3002\u518d\u4e00\u6b21\u8bf4\u660e\uff0c\u8981\u60f3\u505a\u597d\u80f6\u4f53\u7269\u7406\u7684\u7814\u7a76\uff0c\u5f88\u5e94\u8be5\u5e94\u8be5\u4ece\u6db2\u4f53\u7269\u7406\u5f00\u59cb\u8865\u4e60\u3002\u4e8b\u5b9e\u4e0a\uff0c\u6587\u732e\u662f\u6700\u65e9\u7528\u8ba1\u7b97\u673a\u6765\u8ba1\u7b97\u6db2\u4f53\u7684\u76f8\u5173\u51fd\u6570\u7684\u5de5\u4f5c\u3002\u4f5c\u8005Aneesur Rahman\u662f\u201c\u5206\u5b50\u52a8\u529b\u5b66\u4e4b\u7236\u201d\u3002\u5728\u6709\u8ba1\u7b97\u673a\u4e4b\u524d\uff0c\u76f8\u5173\u51fd\u6570\u53ea\u6709\u7406\u60f3\u6c14\u4f53\u548c\u683c\u5b50\u6676\u4f53\u624d\u80fd\u8ba1\u7b97\uff0c\u6db2\u4f53\u7684\u76f8\u5173\u51fd\u6570\u7531\u4e8e\u591a\u4f53\u6548\u5e94\u800c\u96be\u4ee5\u8ba1\u7b97\u3002Aneesur Rahman\u5728\u5f53\u65f6\u7684CDC 3600\u8d85\u7ea7\u8ba1\u7b97\u673a\u4e0a\u8ba1\u7b97\u4e86864\u4e2aLennard-Jones\u4f5c\u7528\u52bf\u7684\u539f\u5b50\uff0c\u6a21\u62df\u6db2\u6c29\u3002","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":9108,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/9108","url_meta":{"origin":7088,"position":3},"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$\u4e3a\u6807\u51c6\u5dee\u7684\u9ad8\u65af\u51fd\u6570 \\[P\\left(x\\middle|\\xi\\right)=\\frac{1}{\\sqrt{2\\pi\\xi^2}}\\exp\\left[-\\frac{\\left(x-\\mu\\right)^2}{2\\xi^2}\\right]\\] \u5728\u56fe\u50cf\u53bb\u566a\u7684\u4e0a\u4e0b\u6587\u4e2d\uff0c\u4ee5deconvlucy\u4e3a\u4f8b\uff0c\u6838\u51fd\u6570$P\\left(x\\right)$\u662f\u4e00\u4e2a\u56fa\u5b9a\u53c2\u6570$\\xi$\u7684\u51fd\u6570\uff0c\u4e14$\\xi$\u7684\u53d6\u503c\u8303\u56f4\uff08\u5373$P\\left(x\\right)$\u7684\u201c\u5bbd\u5ea6\u201d\uff09\u8fdc\u7a84\u4e8e$x$\u7684\u8303\u56f4\uff08\u5728\u56fe\u50cf\u8bed\u5883\u4e2d\u662f\u56fe\u50cf\u7684\u5927\u5c0f\uff09\u3002\u4f46\u5728Lucy\u539f\u6587\u7684\u8bed\u5883\u4e2d\uff0c$\\phi\\left(x\\right)$\u3001$P\\left(x\\middle|\\xi\\right)$\u548c$\\psi\\left(\\xi\\right)$\u90fd\u662f\u652f\u6491\u4e3a\u6574\u4e2a\u5b9e\u6570\u7684\u5206\u5e03\u5bc6\u5ea6\u51fd\u6570\u3002\u53ef\u4ee5\u8bf4\uff0c\u56fe\u50cf\u53bb\u566a\u5e94\u7528\uff0c\u53ea\u662fLucy\u539f\u6587\u7b97\u6cd5\u7684\u5176\u4e2d\u4e00\u4e2a\u5f88\u7279\u6b8a\u7684\u4f8b\u5b50\u3002 \u6570\u503c\u8ba1\u7b97\u7684\u8bbe\u8ba1\u827a\u672f \u5728\u6211\u7684\u7814\u7a76\u4e2d\uff0c\u6070\u597d\u9700\u8981\u89e3\u51b3Lucy\u539f\u6587\u610f\u4e49\u7684\u95ee\u9898\uff0c\u56e0\u6b64\u6211\u662f\u8ba4\u771f\u9605\u8bfb\u4e86Lucy\u7684\u539f\u6587\u591a\u6b21\u7684\uff0c\u6709\u4e9b\u5176\u4ed6\u65b9\u9762\u7684\u611f\u53d7\u3002 \u539f\u6587\u7684\u6587\u5b57\u6781\u5176\u6e05\u6670\u548c\u6d41\u7545\uff0c\u903b\u8f91\u5341\u5206\u4e25\u5bc6\uff0c\u8bb0\u53f7\u4ed4\u7ec6\uff08\u65e2\u4e0d\u6ee5\u7528\u53c8\u4e0d\u6df7\u7528\uff09\u3002\u6211\u89c9\u5f97\u8fd9\u662f\u6709\u6210\u5c31\u7684\u4f5c\u8005\u7684\u5171\u6027\u3002\u8bfb\u5230\u8fd9\u6837\u7684\u6587\u5b57\u5c31\u80fd\u8bf4\u660e\u4f5c\u8005\u662f\u4e8b\u5b9e\u4e0a\u7684\u5927\u5e08\uff08\u5c3d\u7ba1\u4e16\u4fd7\u540d\u8a89\u4e0a\u672a\u5fc5\uff09\u3002 \u5728\u8bba\u6587\u53d1\u8868\u76841970\u5e74\u4ee3\uff0c\u7535\u5b50\u8ba1\u7b97\u673a\u5728\u79d1\u5b66\u8ba1\u7b97\u4e2d\u7684\u5e94\u7528\u5df2\u7ecf\u6bd4\u8f83\u666e\u53ca\u3002\u539f\u6587\u6ca1\u6709\u63d0\u53ca\u6240\u62a5\u9053\u7684\u9a8c\u8bc1\u5b9e\u9a8c\u662f\u5728\u4ec0\u4e48\u8ba1\u7b97\u673a\u4e0a\u8fdb\u884c\u7684\uff0c\u53ea\u5728\u81f4\u8c22\u4e2d\u8bf4\u5230\u4e86NASA\u7684Goddard Institute for Space Studies (GISS)\u63d0\u4f9b\u4e86\u673a\u65f6\u3002\u6211\u76f8\u4fe1\uff0c\u8fd9\u5e94\u8be5\u662f\u4e00\u79cd\u9700\u8981\u7533\u8bf7\u8282\u70b9\u7684\u5927\u578b\u8ba1\u7b97\u673a\uff0c\u673a\u65f6\u8d44\u6e90\u5e94\u8be5\u662f\u6bd4\u8f83\u6602\u8d35\u7684\u3002 \u5728\u4eca\u5929\uff0c\u50cf\u6211\u8fd9\u79cd\u6570\u503c\u8ba1\u7b97\u7684\u5916\u884c\uff0c\u53ef\u4ee5\u5728MATLAB\u5f00\u53d1\u73af\u5883\u4e2d\u91cd\u590d\u8fd0\u884c\u591a\u6b21\u6765\u5b66\u4e60\u4e00\u4e2a\u6ca1\u6709\u4ece\u539f\u7406\u4e0a\u5403\u900f\u7684\u7b97\u6cd5\u7684\u884c\u4e3a\uff0c\u56e0\u4e3a\u5f88\u591a\u8ba1\u7b97\u5728\u4eca\u5929\u7684\u666e\u901a\u7b14\u8bb0\u672c\u7535\u8111\u4e0a\u8fd0\u884c\u90fd\u6beb\u65e0\u538b\u529b\u3002\u4f46\u662f\u5728\u5f53\u5e74\uff0c\u8fd9\u79cd\u201c\u4f5c\u5f0a\u201d\u7684\u505a\u6cd5\u662f\u4e0d\u63d0\u5021\u7684\u3002\u7ed9\u5b9a\u4e00\u4e2a\u7b97\u6cd5\uff0c\u4f60\u7406\u5e94\u52aa\u529b\u5730\u5728\u8349\u7a3f\u7eb8\u4e0a\u5206\u6790\u5b83\u7684\u597d\u5904\u548c\u574f\u5904\u3002\u8fd9\u79cd\u529f\u592b\u6211\u6ca1\u6709\uff0c\u4f46\u4ece\u8fd9\u7bc7\u8bba\u6587\u4e2d\u8fd8\u662f\u9886\u6559\u4e86\u5b83\u7684\u4f18\u96c5\u3002 \u4ece\u7b97\u6cd5\u7684\u539f\u7406\uff0c\u5c31\u80fd\u770b\u51fa\u5b83\u7684\u7ed3\u679c\u53ea\u5bf9\u957f\u6ce2\u957f\u566a\u97f3\u654f\u611f\uff0c\u800c\u5bf9\u77ed\u6ce2\uff08\u9ad8\u9891\uff09\u566a\u97f3\u8fdf\u949d\uff0c\u5b83\u5728\u5934\u51e0\u4e2a\u8fed\u4ee3\u5c31\u80fd\u5feb\u901f\u6536\u655b\uff0c\u5b83\u5728\u6837\u672c\u6570$N$\u592a\u5c11\u65f6\u4f1a\u6709\u4ec0\u4e48\u5f02\u5e38\u2026\u2026\u7b49\u7b49\uff0c\u90fd\u901a\u8fc7\u5206\u6790\uff0c\u5728\u4e0d\u653e\u5230\u8ba1\u7b97\u673a\u4e2d\u778e\u8bd5\u4e4b\u524d\uff0c\u5c31\u90fd\u6e05\u695a\u4e86\u3002\u800c\u6211\u7684\u505a\u6cd5\uff0c\u5e38\u5e38\u662f\u8fb9\u8bd5\u8fb9\u6539\u7b97\u6cd5\u3002\u6781\u5ea6\u6d6a\u8d39\u8ba1\u7b97\u8d44\u6e90\u6765\u8fc1\u5c31\u6211\u5728\u5206\u6790\u4e0a\u7684\u61d2\u60f0\u3002\u8fd9\u5728\u4eca\u5929\u4e5f\u8bb8\u4e0d\u662f\u4ec0\u4e48\u7f6a\u8fc7\uff0c\u751a\u81f3\u4f5c\u4e3a\u4e00\u4e2a\u6570\u503c\u8ba1\u7b97\u7684\u5916\u884c\uff0c\u8fd9\u53ef\u4ee5\u8bf4\u662f\u5728\u806a\u660e\u5730\u8282\u7701\u65f6\u95f4\uff0c\u4f46\u6211\u4ecd\u7136\u5bf9\u539f\u6587\u8fd9\u79cd\u201c\u6570\u503c\u8ba1\u7b97\u7684\u8bbe\u8ba1\u827a\u672f\u201d\u611f\u5230\u656c\u4f69\u3002 \u6709\u90a3\u4e48\u4e00\u4e9b\u8bba\u6587\uff0c\u6211\u662f\u79f0\u4e4b\u4e3a\u201c\u6559\u5b66\u8bba\u6587\u201d\u7684\uff0c\u5c31\u662f\u5b83\u597d\u5230\u53ef\u4ee5\u62ff\u6765\u4f5c\u4e3a\u5178\u8303\uff0c\u7ed9\u7814\u7a76\u751f\u4f5c\u4e3a\u8303\u6587\uff0c\u53bb\u5b66\u4e60\u5f88\u591a\u8d85\u51fa\u8bba\u6587\u5177\u4f53\u5185\u5bb9\u7684\u4e1c\u897f\uff0c\u6bd4\u5982\u8bba\u6587\u7ed3\u6784\u3001\u5b66\u672f\u82f1\u8bed\u5199\u4f5c\u3001\u6279\u5224\u6027\u903b\u8f91\u3001\u4e43\u81f3\u79d1\u5b66\u7cbe\u795e\u3002Lucy\u7684\u8fd9\u7bc7\u8bba\u6587\u5c31\u53ef\u4ee5\u79f0\u4e4b\u4e3a\u4e00\u7bc7\u201c\u6559\u5b66\u8bba\u6587\u201d\u3002 \u8be5\u4f5c\u8005\u7684\u5168\u540d\u662fLeon Brian Lucy\u3002\u5173\u4e8e\u4ed6\u7684\u8be6\u7ec6\u4fe1\u606f\uff0c\u53ef\u89c1\u5176\u4e00\u7bc7\u8ba3\u544a\u548c\u7eaa\u5ff5\u6587\u7ae0\u3002","rel":"","context":"In &quot;\u4e2a\u4eba\u6587\u96c6&quot;","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":4296,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/4296","url_meta":{"origin":7088,"position":4},"title":"\u91cd\u65b0\u8003\u8651Laponite\u7c92\u5b50\u95f4\u4f5c\u7528\u52bf\u80fd","author":"Andrew","date":"2011\u5e7411\u67086\u65e5","format":false,"excerpt":"\u9000\u56de\u7684\u5ba1\u7a3f\u610f\u89c1\u53cd\u6620\u7a3f\u5b50\u7684\u6700\u5927\u95ee\u9898\u5728\u4e8e\u5bf9Laponite\u7c92\u5b50\u76f8\u4e92\u4f5c\u7528\u7684\u8003\u8651\u4e0d\u5468\u3002\u9700\u8981\u5bf9\u539f\u672c\u7684model\u8fdb\u884c\u6bd4\u8f83\u5927\u7684\u4fee\u6539\u3002 Laponite\u7c92\u5b50\u5f62\u72b6\u4e0e\u5e26\u7535\u7684\u5404\u5411\u5f02\u6027 \u5ba1\u7a3f\u4eba\u53d1\u96be\u7684\u6700\u4e3b\u8981\u4e00\u70b9\u5c31\u662f\u6211\u6ca1\u6709\u8003\u8651Laponite\u7c92\u5b50\u5728\u5f62\u72b6\u4e0a\u548c\u8868\u9762\u5e26\u7535\u5206\u5e03\u4e0a\u7684\u5404\u5411\u5f02\u6027\u3002Laponite\u662f\u7247\u72b6\u7c92\u5b50\uff0c\u4e24\u4e2a\u9762\uff08face\uff09\u5e26\u8d1f\u7535\uff0c\u8fb9\u7f18\uff08edge\uff09\u5e26\u6b63\u7535\u3002\u56e0\u6b64\uff0c\u4e24\u4e2aLaponite\u7c92\u5b50\u7684\u76f8\u4e92\u4f5c\u7528\u5c31\u8981\u5206\u4e3aface-face\u3001face-edge\u548cedge-edge\u6765\u8ba8\u8bba\u3002\u800c\u4e14\uff0c\u8fd9\u4e09\u79cd\u76f8\u4e92\u4f5c\u7528\u53d1\u751f\u7684\u673a\u7387\u5728\u5728\u91cfLaponite\u7c92\u5b50\u7cfb\u7edf\u4e2d\u5982\u4f55\u5206\u5e03\u4e5f\u9700\u8981\u4f30\u8ba1\u3002\u540e\u8005\u6d89\u53ca\u5230\u53d6\u5411\uff0c\u5f53\u521d\u5c31\u4f30\u8ba1\u7cbe\u786e\u8003\u8651\u53d6\u5411\u673a\u7387\u53ef\u80fd\u4f1a\u5f88\u96be\uff0c\u539f\u672c\u5c31\u6253\u7b97\u7cbe\u786e\u6210\u7403\u578b\u7684\u3002\u73b0\u5728\u65e2\u7136\u5ba1\u7a3f\u4eba\u8bf4\u201c\u771f\u7a7a\u7403\u5f62\u9e21\u201d\u4e0d\u9760\u8c31\u4e86\uff0c\u6211\u53c8\u5728\u6709\u9650\u7684\u65f6\u95f4\u5185\u67e5\u9605\u4e86\u4e00\u4e0b\u6587\u732e\uff0c\u8fd8\u662f\u5f88\u96be\u627e\u5230\u6709\u6587\u732e\u76f4\u63a5\u4ece\u7406\u8bba\u4e0a\u8003\u8651Laponite\u7684\u53d6\u5411\u3002 Secor\u548cRadke\u7ed9\u51fa\u4e86Poisson-Boltzmann\u65b9\u7a0b\u5728\u5706\u67f1\u8584\u7247\u72b6\u80f6\u4f53\u7c92\u5b50\uff0c\u8868\u9762\u5e26\u8d1f\u7535\u3001\u8fb9\u7f18\u5e26\u6b63\u7535\u60c5\u51b5\u4e0b\u7684\u89e3\u6790\u89e3\uff0c\u672c\u6765\u5e94\u8be5\u95ee\u9898\u5c31\u7ed3\u4e86\u3002\u4f46\u6839\u636e\u6587\u7ae0\u7684\u6458\u8981\uff08\u5168\u6587\u6211\u4e0b\u4e0d\u4e86\uff09\uff0c\u8ba1\u7b97\u7ed3\u679c\u662f\u9762\u4e0a\u5e26\u7684\u8d1f\u7535\u7535\u52bf\u5206\u5e03\u6ea2\u51fa\u5230\u4e86\u8fb9\u7f18\u4e0a\uff0c\u6240\u4ee5\u5c31\u7b97\u8fb9\u7f18\u4e0a\u5e26\u6709\u6b63\u7535\u8377\uff0c\u4e5f\u662f\u8d1f\u7535\u52bf\uff0c\u5373edge-face\u662f\u540c\u79cd\u7535\u8377\uff0c\u4e0d\u5b58\u5728\u5f15\u529b\u3002\u8fd9\u8ddfLaponite\u7684\u76ee\u524d\u516c\u8ba4\u7684\u7269\u7406\u56fe\u50cf\u4e0d\u7b26\u3002\u4e0d\u8fc7\u5149\u4ece\u6458\u8981\u8fd8\u770b\u4e0d\u51fa\u539f\u6587\u5728\u8ba1\u7b97\u65f6\u8bbe\u5b9a\u7684\u8fb9\u7f18\u6b63\u7535\u8377\u5927\u5c0f\u662f\u591a\u5c11\uff0c\u4e5f\u8bb8\u662f\u53d6\u503c\u592a\u5c0f\u7684\u7f18\u6545\u3002\u9664\u4e86Poisson-Boltzmann\u65b9\u7a0b\u7684\u7cbe\u786e\u89e3\uff0c\u8fd8\u6709\u5404\u79cd\u8fd1\u4f3c\u65b9\u6cd5\u5efa\u7acb\u4e0d\u540c\u53d6\u5411\u65b9\u5f0f\u7684\u4f5c\u7528\u52bf\u80fd\u51fd\u6570\u3002\u4f8b\u5982\uff0cDjikstra\u7b49\u628aLaponite\u7c92\u5b50\u8003\u8651\u6210\u65e0\u9650\u8584\u7684\u5706\u7247\uff0c\u5176\u5e26\u7535\u60c5\u51b5\u8fd1\u4f3c\u4e3a\u7535\u56db\u6781\u77e9\uff08quadrupole\uff09\uff0cKutter\u7b49\u628aLaponite\u7c92\u5b50\u7684\u5e26\u7535\u60c5\u51b5\u8003\u8651\u6210\u82e5\u5e72\u4e2a\u70b9\u7535\u8377\u7684\u5e73\u9762\u9635\u5217\u7684\u603b\u6548\u679c\u3002\u4f46\u65e0\u8bba\u5982\u4f55\u8003\u8651\u52bf\u80fd\u51fd\u6570\uff0c\u90fd\u8981\u518d\u8fdb\u884cMonte Carlo\u6a21\u62df\uff0c\u624d\u5f97\u5230\u53d6\u5411\u76f8\u5173\u51fd\u6570gor(r)\u3002\u6211\u79c1\u4e0b\u7528\u201c\u542c\u5929\u7531\u547d\u201d\u8fd9\u4e2a\u8bcd\u6765\u5f62\u5bb9\u62ff\u8ba1\u7b97\u673a\u6a21\u62df\u6765\u5c1d\u8bd5\u4e8b\u5148\u9884\u6d4b\u4e0d\u6765\u7684\u4e8b\u60c5\uff0c\u56e0\u89c9\u5f97\u8fd9\u4e0d\u4f26\u4e0d\u7c7b\u3002\u672c\u6765\u4f60\u7684\u6a21\u578b\u5c31\u5f85\u5b9e\u9a8c\u9a8c\u8bc1\uff0c\u7ed3\u679c\u4f60\u53c8\u7528\u8fd9\u4e2a\u6a21\u578b\u6765\u505a\u8ba1\u7b97\u673a\u6a21\u62df\uff08\u5373\u5b9e\u9a8c\uff09\u3002 Dur\u00e1n\u7b49\u628aedge\u548cface\u5f53\u6210\u4e24\u76f8\uff0c\u52a0\u4e0a\u6eb6\u5242\u5c31\u4e00\u5171\u4e09\u76f8\u3002\u7136\u540eLaponite\u7c92\u5b50\u95f4\u7684\u53cc\u7535\u5c42\u76f8\u4e92\u4f5c\u7528\u53ef\u901a\u8fc7HHF\u6a21\u578b\u6765\u5206\u522b\u8ba1\u7b97edge-face\u3001face-face\u548cedge-edge\u4e09\u79cd\u60c5\u51b5\u3002\u4f46\u662f\uff0c\u8fd9\u7bc7\u6587\u7ae0\u7684\u903b\u8f91\u662f\u901a\u8fc7\u7535\u6cf3\u6cd5\u5b9e\u6d4b\u6837\u54c1\u7684\u03b6-\u7535\u4f4d\uff0c\u7136\u540e\u6309\u7167\u7c98\u571f\u7684\u7ed3\u6784\u6765\u5047\u8bbe\u6d4b\u5f97\u7684\u03b6-\u7535\u4f4d\u4e2d\u767e\u5206\u51e0\u662fedge\u7684\uff0c\u767e\u5206\u4e4b\u51e0\u662fface\u7684\uff0c\u4ee3\u5165HHF\u6a21\u578b\u5206\u522b\u7b97\u51fa\u4e09\u79cd\u4f5c\u7528\u52bf\u80fd\u3002\u6574\u7bc7\u6587\u7ae0\u7684\u8ba8\u8bba\u4e2d\uff0c\u4e09\u79cd\u4f5c\u7528\u52bf\u80fd\u90fd\u662f\u5206\u5f00\u7684\u8ba8\u8bba\u7684\uff0c\u6ca1\u6709\u7efc\u5408\u8d77\u6765\u8fc7\uff0c\u56e0\u6b64\u4e5f\u7528\u4e0d\u7740\u8003\u8651\u8fd9\u4e09\u79cd\u60c5\u51b5\u8c01\u591a\u8c01\u5c11\u7684\u95ee\u9898\u2014\u2014\u5373Laponite\u53d6\u5411\u5206\u5e03\u7684\u95ee\u9898\u4e86\u3002 \u7531\u4e8e\u771f\u5b9e\u4e16\u754c\uff08\u975e\u8ba1\u7b97\u673a\u6a21\u62df\uff09\u7684\u5b9e\u9a8c\u624b\u6bb5\u5927\u591a\u65e0\u6cd5\u5206\u8fa8Laponite\u7c92\u5b50\u53d6\u5411\u5206\u5e03\u3002\u5e38\u7528\u7684\u52a8\u6001\u5149\u6563\u5c04\u6cd5\u6d4b\u7c92\u5f84\u548c\u7535\u6cf3\u6cd5\u6d4b\u03b6-\u7535\u4f4d\u90fd\u662f\u5047\u8bbe\u7c92\u5b50\u4e3a\u7403\u5f62\u5f97\u5230\u5b9e\u9a8c\u7ed3\u679c\u7684\u3002\u6240\u4ee5\u5c31\u7b97\u4f60\u80fd\u4ece\u7406\u8bba\u4e0a\u7ed9\u51fa\u53d6\u5411\u5206\u5e03\u7684\u5b9a\u91cf\u5173\u7cfb\uff0c\u9664\u8ba1\u7b97\u673a\u6a21\u62df\u4ee5\u5916\u4e5f\u7f3a\u4e4f\u5b9e\u9a8c\u9a8c\u8bc1\u624b\u6bb5\uff0c\u8003\u8651\u4e86\u4e5f\u767d\u8003\u8651\u3002\u56e0\u6b64\u6211\u4e0d\u592a\u7406\u89e3\u5ba1\u7a3f\u4eba\u662f\u5426\u975e\u8981\u6211\u5b9e\u7259\u5b9e\u9f7f\u5730\u8003\u8651face-edge\u76f8\u4e92\u4f5c\u7528\u4e0d\u53ef\u3002\u6211\u770b\u5230University of Washington\u7684J. Berg\u7ec4\u6709\u4e00\u7bc7\u7814\u7a76Laponite\u6d41\u53d8\u5b66\u7684\u6587\u7ae0\u76f4\u63a5\u628a\u7c92\u5b50\u5f53\u7403\u5f62\u8003\u8651\u4e86\uff0c\u4f46\u6709\u4e00\u6bb5\u5173\u4e8eedge\u548cface\u7684\u89e3\u91ca\u6587\u5b57\u5982\u4e0b\uff08\u91cd\u70b9\u662f\u6211\u52a0\u7684\uff09\uff1a The electrical surface potential must now account for the face and edge potential separately. This gives the following estimate [math]\\mathit{\\Phi}_\\textup{e}\\approx 2\\pi\\epsilon\\epsilon_0 a\\psi _\\textup{face}\\psi_\\textup{edge}\\ln\\left[\\frac{1}{1-\\exp\\left(-\\kappa D \\right )} \\right ].[\/math] The Larson anlysis went from this point to replace the surface potential\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":1031,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/1031","url_meta":{"origin":7088,"position":5},"title":"\u4e3a\u4ec0\u4e48G&#039;&#039;\u662f\u8d1f\u503c\uff1f","author":"Andrew","date":"2010\u5e745\u670825\u65e5","format":false,"excerpt":"\u4e3b\u9898\u56de\u5f52\u5230\u6d41\u53d8\u5b66\u4e00\u4e0b\u3002\u4f17\u6240\u5468\u77e5\uff08\u7ebf\u6027\uff09\u7c98\u5f39\u6027\u662f\u5904\u4e8eHookian\u5f39\u6027\u548cNewtonian\u7c98\u6027\u8fd9\u4e24\u4e2a\u6781\u7aef\u4e4b\u95f4\u7684\u6027\u8d28\u3002\u5bf9Hookian\u5f39\u6027\u4f53\uff08Hookian\u5f39\u7c27\u6a21\u578b\uff09\u65bd\u52a0\u6b63\u5f26\u5f62\u53d8\uff0c\u6d4b\u5230\u7684\u662f\u540c\u76f8\u7684\u6b63\u5f26\u5e94\u529b\uff1b\u5411Newtonian\u7c98\u6d41\u4f53\uff08Newtonian\u7c98\u58f6\uff09\u65bd\u52a0\u6b63\u5f26\u5f62\u53d8\u901f\u7387\uff0c\u6d4b\u5230\u7684\u662f\u540c\u76f8\u7684\u6b63\u5f26\u5e94\u529b\u3002\u5411\u4e00\u4e2a\u7ebf\u6027\u7c98\u5f39\u6027\u6837\u54c1\u65bd\u52a0\u6b63\u5f26\u5f62\u53d8\uff0c\u6d4b\u5230\u7684\u662f\u76f8\u4f4d\u89d2\u4e3a[eq]\\delta[\/eq]\u7684\u6b63\u5f26\u5e94\u529b\uff1a[eq]\\gamma_0 \\sin \\left ( \\omega t+\\delta \\right )[\/eq]\u3002 \u6700\u4e3a\u4eba\u719f\u6089\u7684\u6d41\u53d8\u5b66\u6d4b\u8bd5\uff08\u4e5f\u662f\u6700\u591a\u4e0d\u61c2\u6d41\u53d8\u7684\u4eba\u77e5\u9053\u8981\u505a\u7684\u6d4b\u8bd5\uff09\u5c31\u662f[eq]G'[\/eq]\u3001[eq]G''[\/eq]~[eq]\\omega[\/eq]\u66f2\u7ebf\uff0c\u8fd9\u4e2a\u4e5f\u53eb\u505a\u201c\u7c98\u5f39\u8c31\u201d\u3002\u6211\u66fe\u7ecf\u8be6\u7ec6\u89e3\u91ca\u8fc7[eq]G'[\/eq]\u548c[eq]G''[\/eq]\u5230\u5e95\u662f\u600e\u4e48\u6765\u7684\u3002\u5178\u578b\u7ebf\u578b\u805a\u5408\u7269\u7c98\u5f39\u8c31\u5982\u4e0b\u56fe\uff1a \u4e0b\u56fe\u662f\u6211\u914d\u5236\u7684\u5408\u6210\u9502\u85fb\u571fLaponite\u51dd\u80f6\u7684\u7c98\u5f39\u8c31\uff0c\u53d1\u73b0[eq]G''[\/eq]\u5728\u9ad8\u9891\u5904\u51fa\u73b0\u8d1f\u503c\u3002\u4e0b\u9762\u662f\u5bf9\u8fd9\u4e2a\u4e0d\u6b63\u5e38\u73b0\u8c61\u7684\u89e3\u91ca\u3002 \u4e00\u53f0\u5e94\u53d8\u63a7\u5236\u578b\u6d41\u53d8\u4eea\u82e5\u8981\u7ed9\u51fa\u8fd9\u6837\u7684\u56fe\uff0c\u5c31\u8981\u7ed9\u6837\u54c1\u65bd\u52a0\u4e00\u5b9a\u9891\u7387[eq]\\omega[\/eq]\u548c\u5e45\u5ea6[eq]\\gamma[\/eq]\u7684\u6b63\u5f26\u5e94\u53d8[eq]\\gamma \\left ( t \\right ) = \\gamma_0 \\sin \\left ( \\omega t \\right)[\/eq]\uff0c\u7136\u540e\u5939\u5177\u6240\u8fde\u63a5\u7684\u529b\u5b66\u4f20\u611f\u5668\u8bb0\u5f55\u6837\u54c1\u5411\u5939\u5177\u65bd\u52a0\u7684\u8f6c\u77e9\uff0c\u5e76\u6839\u636e\u5939\u5177\u5f62\u72b6\u548cGap\u503c\u6362\u7b97\u6210\u5e94\u53d8[eq]\\sigma \\left ( t \\right )[\/eq]\uff0c\u5728\u7ebf\u6027\u7c98\u5f39\u6027\u6761\u4ef6\u4e0b[eq]\\sigma \\left ( t \\right )=\\sigma_0 \\sin \\left ( \\omega t + \\delta \\right )[\/eq]\u3002\u4eea\u54c1\u4ec5\u9700\u4e14\u5fc5\u987b\u51c6\u786e\u6d4b\u91cf\u51fa\u4e24\u4e2a\u503c\uff1a[eq]\\sigma_0[\/eq]\uff08\u4ece\u5728\u8f6c\u77e9\u6570\u636e\u4e2d\u627e\u5230\u7684\u6700\u503c\u6362\u7b97\uff09\u548c[eq]\\delta[\/eq]\uff08\u901a\u8fc7\u6bd4\u8f83\u8f6c\u77e9\u6570\u636e\u548c\u5e94\u53d8\u6570\u636e\u7684\u76f8\u4f4d\u5dee[eq]\\Delta t[\/eq]\u6362\u7b97\uff0c\u89c1\u7b2c1\u5e45\u56fe\uff09\u3002\u5176\u4e2d\uff0c[eq]\\Delta t=\\frac{\\delta}{\\omega}[\/eq]\u3002\u5176\u4ed6\u8bf8\u5982[eq]G'[\/eq]\u3001[eq]\\eta'[\/eq]\u4e4b\u7c7b\u7684\uff0c\u90fd\u662f\u4ece\u8fd9\u4e24\u4e2a\u6570\u636e\u7b97\u51fa\u6765\u7684\u3002\u4f8b\u5982[eq]G''=\\frac {\\sigma_0}{\\gamma_0} \\sin \\left( \\omega \\right\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":"\u5c0f\u5e45\u632f\u8361\u529b\u5b66\u54cd\u5e94","src":"https:\/\/i0.wp.com\/www.andrewsun.net\/xiangge\/wp-content\/uploads\/2010\/05\/11-300x188.png?resize=350%2C200","width":350,"height":200},"classes":[]}],"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/posts\/7088","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/comments?post=7088"}],"version-history":[{"count":27,"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/posts\/7088\/revisions"}],"predecessor-version":[{"id":7115,"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/posts\/7088\/revisions\/7115"}],"wp:attachment":[{"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/media?parent=7088"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/categories?post=7088"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.andrewsun.net\/panta_rhei\/wp-json\/wp\/v2\/tags?post=7088"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}