{"id":230,"date":"2024-08-25T21:16:43","date_gmt":"2024-08-25T13:16:43","guid":{"rendered":"https:\/\/www.shirui.me\/blog\/?p=230"},"modified":"2024-08-25T21:16:43","modified_gmt":"2024-08-25T13:16:43","slug":"saddle-point-method-and-end-to-end-distribution-of-polymer-models","status":"publish","type":"post","link":"https:\/\/www.shirui.me\/blog\/2024\/08\/25\/saddle-point-method-and-end-to-end-distribution-of-polymer-models\/","title":{"rendered":"Saddle point method and end-to-end distribution of polymer models"},"content":{"rendered":"\n<p><b>Introduction<\/b><\/p>\n<p>Saddle point methods are widely used to estimate integrals of the form:<\/p>\n<p>$I = \\int \\exp\\left(-f(x)\\right)\\mathrm{d}x$<\/p>\n<p>where the function $f(x)$ can be approximated by the first two terms of its Taylor series expansion around a point $x_0$:<\/p>\n<p>$f(x)\\approx f(x_0) + f^\\prime (x_0)(x-x_0) + \\frac{1}{2}f^{\\prime\\prime}(x_0)(x-x_0)^2$<\/p>\n<p>The integral is then approximated by its value at the saddle point, where $f^\\prime (x_0)=0$ and $f^{\\prime\\prime}(x_0)&gt;0$:<\/p>\n<p>$\\begin{align} I&amp;\\approx \\int \\exp\\left(-f(x_0) &#8211; \\frac{1}{2}f^{\\prime\\prime}(x_0)(x-x_0)^2\\right) \\mathrm{d}x\\\\ &amp;=\\exp(-f(x_0))\\sqrt{\\frac{2\\pi}{f^{\\prime\\prime}(x_0)}} \\end{align}$<\/p>\n\n<p><b>Examples<\/b><\/p>\n<ul>\n<li><b>Stirling&#8217;s Formula:<\/b><\/li>\n<\/ul>\n<p>Using the definition of the Gamma function, we have:<\/p>\n<p>$N!=\\int_0^\\infty \\exp(-x)x^N\\mathrm{d}x$<\/p>\n<p>Let $f(x):=x-N\\ln(x)$. For large $N$, the negative term is negligible. Solving $f^\\prime (x) = 0$, we obtain:<\/p>\n<p>$N!\\approx\\exp(-N+N\\ln(N))\\sqrt{2\\pi{}N}=\\sqrt{2\\pi{}N}\\left(\\frac{N}{e}\\right)^N$<\/p>\n\n<ul>\n<li><b>Partition Function:<\/b><\/li>\n<\/ul>\n<p>The partition function is given by:<\/p>\n<p>$Z = \\int \\exp(-\\beta U(\\mathbf{x})) \\mathrm{d}\\mathbf{x}$<\/p>\n<p>where $U(\\mathbf{x})$ can be approximated by:<\/p>\n<p>$U(\\mathbf{x})\\approx U(\\mathbf{x}_0) + \\frac{1}{2}(\\mathbf{x}-\\mathbf{x}_0)^T H[U](\\mathbf{x}-\\mathbf{x}_0)$<\/p>\n<p>where $H$ represents the Hessian matrix.<\/p>\n\n<p><b>End-to-End Distribution Function of Random Walk Model of Polymer Chains<\/b><\/p>\n<p>For an $N$-step random walk model, the exact end-to-end vector distribution is:<\/p>\n<p>$\\begin{align} P(\\mathbf{Y})&amp;=\\frac{1}{(2\\pi)^3}\\int \\mathrm{d}\\mathbf{k} \\exp(-i\\mathbf{k}\\cdot\\mathbf{Y})\\tilde{\\phi}^N\\\\ &amp;=\\int_{0}^{\\infty} k\\sin(kY) \\left(\\frac{\\sin(kb)}{kb}\\right)^N \\mathrm{d}k \\end{align}$<\/p>\n<p>where $\\phi(\\mathbf{x})=\\frac{1}{4\\pi b^2}\\delta(|\\mathbf{x}|-b)$ is the distribution of a single step vector (length=$b$) and $\\tilde{\\phi}$ is its characteristic function; $\\mathbf{Y}:=\\sum_{i=1}^N \\mathbf{x}_i$ is the end-to-end vector. Let $s=kb$ and $f(s):=i\\frac{Y}{Nb}s -\\ln\\frac{\\sin(s)}{s}$. Then we have:<\/p>\n<p>$\\begin{align} P&amp;=\\frac{i}{4\\pi^2 b^2 Y}\\int_{-\\infty}^{+\\infty} s \\exp\\left(-is\\frac{Y}{b}\\right)\\left(\\frac{\\sin(s)}{s}\\right)^N\\mathrm{d}s\\\\ &amp;=\\frac{i}{4\\pi^2 b^2 Y} \\int s \\exp(-Nf(s)) \\mathrm{d} s\\end{align}$<\/p>\n<p>The integral is extended to $(-\\infty, +\\infty)$ due to the symmetry of the sine and cosine functions. The first sine function is replaced with an exponential form using Euler&#8217;s formula: $\\exp(ix)=\\cos(x)+i\\sin(x)$.<\/p>\n\n<p>Solving $f^\\prime(s)=0$, we find that $is$ satisfies:<\/p>\n<p>$\\coth(is)-\\frac{1}{is}=\\frac{Y}{Nb}$<\/p>\n<p>This is the Langevin function, $is_0=L^{-1}(\\frac{Y}{Nb})$. Therefore, we have:<\/p>\n<p>$\\begin{align} P &amp;\\approx\\frac{s_0}{4\\pi^2b^2Y}\\sqrt{\\frac{2\\pi}{Nf^{\\prime\\prime}(s_0)}}e^{-Nf(s_0)}\\\\ &amp;=\\frac{1}{(2\\pi{}Nb^2)^{3\/2}}\\frac{L^{-1}(x)^2}{x(1-(L^{-1}(x)\\csc\\left(L^{-1}(x)\\right))^2)^{1\/2}}\\\\&amp;\\times\\left(\\frac{\\sinh\\left(L^{-1}(x)\\right)}{L^{-1}(x)\\exp\\left(xL^{-1}(x)\\right)}\\right)^N\\end{align}$<\/p>\n<p>where $x:=\\frac{Y}{Nb}$.<\/p>\n\n","protected":false},"excerpt":{"rendered":"<p>Introduction Saddle point methods are widely used to estimate integrals of the form: $I = \\int \\exp\\left(-f(x)\\right)\\mathrm{d}x$ where the function $f(x)$ can be approximated by the first two terms of its Taylor series expansion around a point $x_0$: $f(x)\\approx f(x_0) + f^\\prime (x_0)(x-x_0) + \\frac{1}{2}f^{\\prime\\prime}(x_0)(x-x_0)^2$ The integral is then approximated by its value at the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[29],"class_list":["post-230","post","type-post","status-publish","format-standard","hentry","category-notes","tag-polymer-physics"],"_links":{"self":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts\/230","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/comments?post=230"}],"version-history":[{"count":1,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts\/230\/revisions"}],"predecessor-version":[{"id":231,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts\/230\/revisions\/231"}],"wp:attachment":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/media?parent=230"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/categories?post=230"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/tags?post=230"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}