This analysis focuses on the probability distribution function, $P_i(\\mathbf{r}_i-\\mathbf{r}_\\mathrm{cm})$, of the $i$th segment relative to the center of mass in an ideal chain. An ideal chain is modeled as a multidimensional random walk with independent steps. The distribution of each step, with a mean length of $b$, is assumed to be Gaussian: $P(\\mathbf{r})\\sim\\mathcal{N}(0,b^2)$. <\/p>\n\n
Let $\\mathbf{b}_i=\\mathbf{r}_{i+1}-\\mathbf{r}_{i}$ represent the $i$th bond vector. Then, the position of the $i$th segment is given by:<\/p>\n
$\\mathbf{r}_i=\\sum_{j=1}^{i-1} \\mathbf{b}_j$<\/p>\n\n
The center of mass, $\\mathbf{r}_\\mathrm{cm}$, is calculated as:<\/p>\n
$\\mathbf{r}_\\mathrm{cm}=\\frac{1}{N}\\sum_{i=1}^{N} \\mathbf{r}_i = \\frac{1}{N}\\sum_{j=1}^{N-1}(N-j)\\mathbf{b}_j$<\/p>\n\n
Therefore, the displacement of the $i$th segment relative to the center of mass is:<\/p>\n
$\\mathbf{r}_i-\\mathbf{r}_\\mathrm{cm}=\\sum_{j=1}^{i-1}\\frac{j}{N}\\mathbf{b}_j+\\sum_{j=i}^{N-1}\\frac{N-j}{N}\\mathbf{b}_j$<\/p>\n\n
If $X$ is a Gaussian random variable with variance $\\sigma^2$, then $aX$ is also a Gaussian random variable with variance $a^2\\sigma^2$. Using this property, we can write the characteristic function for $P_i(\\mathbf{r}_i-\\mathbf{r}_\\mathrm{cm})$ of a $d$-dimensional ideal chain:<\/p>\n
$\\phi_i(\\mathbf{q})=\\Pi_{j} \\phi_{\\mathbf{b}_j’}(\\mathbf{q})=\\exp\\left(-\\frac{1}{2}\\mathbf{q}^T\\left(\\sum_{j=1}^{N-1}\\Sigma_j\\right)\\mathbf{q}\\right)$<\/p>\n\n
where $\\mathbf{b}’_j=\\frac{j}{N}\\mathbf{b}_j$ for $j\\le i-1$ and $\\frac{N-j}{N}\\mathbf{b}_j$ for $i\\le j \\le N-1$. $\\phi_{\\mathbf{b}_j’}=-\\exp(-0.5\\mathbf{q}^T\\Sigma\\mathbf{q})$ is the characteristic function of the probability distribution of bond $j$. $\\Sigma_j=\\frac{j^2 b^2}{d N^2}\\mathbb{I}_d$ for $j\\le i-1$ and $\\Sigma_j=\\frac{(N-j)^2 b^2}{d N^2}\\mathbb{I}_d$ for $i\\le j \\le N-1$, where $\\mathbb{I}_d$ is the $d$-dimensional identity matrix. Setting $b=1$ for convenience, we obtain:<\/p>\n
$\\phi_i(\\mathbf{q})=$\n$\\exp \\left(-\\frac{\\left(6 i^2-6 i (N+1)+2 N^2+3 N+1\\right) \\left(q_x^2+q_y^2+q_z^2\\right)}{36 N}\\right)$<\/p>\n\n
The corresponding distribution of this characteristic function is still a Gaussian distribution with $\\Sigma=\\frac{b^2}{3} \\mathbb{I}_3$, where the equivalent bond length $b^2=\\frac{\\left(6 i^2-6 i (N+1)+2 N^2+3 N+1\\right)}{6 N}$. The 6th moment is calculated as $\\frac{1}{N}\\sum_{i=1}^N \\langle(\\mathbf{r}_i-\\mathbf{r}_\\mathrm{cm})^6\\rangle=\\frac{58 N^6-273 N^4+462 N^2-247}{1944 N^3}$. For large $N$, only the leading term, $\\frac{29}{972} N^3$, is significant. For $N=20$, the result is $235.886$, which agrees with simulations. Another example is for $N=5$, where the $R_g^2$ is $0.8$ compared to the expected value of $5\/6=0.8\\dot{3}$.<\/p>\n\n
Here is the simulation code:<\/p>\n
ch = np.random.normal(size=(100000,20,3),scale=1\/3**0.5)\nch[:,0,:]=0\nch = ch.cumsum(axis=1)\nch -= ch.mean(axis=1,keepdims=1)\nm6 = np.mean(np.linalg.norm(ch,axis=-1)**6)\n<\/code><\/pre>\n\n","protected":false},"excerpt":{"rendered":"This analysis focuses on the probability distribution function, $P_i(\\mathbf{r}_i-\\mathbf{r}_\\mathrm{cm})$, of the $i$th segment relative to the center of mass in an ideal chain. An ideal chain is modeled as a multidimensional random walk with independent steps. The distribution of each step, with a mean length of $b$, is assumed to be Gaussian: $P(\\mathbf{r})\\sim\\mathcal{N}(0,b^2)$. Let $\\mathbf{b}_i=\\mathbf{r}_{i+1}-\\mathbf{r}_{i}$ […]<\/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-234","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\/234","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=234"}],"version-history":[{"count":1,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts\/234\/revisions"}],"predecessor-version":[{"id":235,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts\/234\/revisions\/235"}],"wp:attachment":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/media?parent=234"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/categories?post=234"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/tags?post=234"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}