Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Segment Distribution around the Center of Gravity of Branched Polymers   Mathematical expressions for mass distributions around the center of gravity
are derived for branched polymers with the help of the Isihara formula. We
introduce the Gaussian approximation for the end-to-end vector,
$\vec{r}_{G\nu_{i}}$, from the center of gravity to the $i$th mass point on the
$\nu$th arm. Then, for star polymers, the result is \begin{equation}
\varphi_{star}(s)=\frac{1}{N}\sum_{\nu=1}^{f}\sum_{i=1}^{N_{\nu}}\left(\frac{d}{2\pi\left\langle
r_{G\nu_{i}}^{2}\right\rangle}\right)^{d/2}\exp\left(-\frac{d}{2\left\langle
r_{G\nu_{i}}^{2}\right\rangle}s^{2}\right)\notag \end{equation} for a
sufficiently large $N$, where $f$ denotes the number of arms. It is found that
the resultant $\varphi_{star}(s)$ is, unfortunately, not Gaussian. For
dendrimers \begin{equation}
\varphi_{dend}(s)=\sum_{h=1}^{g}\omega_{h}\left(\frac{d}{2pi\left\langle
r_{G_{h}}^{2}\right\rangle}\right)^{d/2}\exp\left(-\frac{d}{2\left\langle
r_{G_{h}}^{2}\right\rangle}s^{2}\right)\notag \end{equation} where $\omega_{h}$
denotes the weight fraction of masses in the $h$th generation on a dendrimer
constructed from $g$ generations, so that $\sum_{h=1}^{g}\omega_{h}=1$. To be
specific, $\omega_{1}=1/N$ and $\omega_{h}=(f-1)^{h-2}/N$ for $h\ge 2$. These
distributions can be described by the same grand sum of each Gaussian function
for the end-to-end distance from the center of gravity to each mass point. Note
that for a large $f$ and $g$, the statistical weight of younger generations
becomes dominant. As a consequence, the mass distribution of unperturbed
dendrimers approaches the Gaussian form in the limit of a large $f$ and $g$. It
is shown that the radii of gyration of dendrimers increase logarithmically with
$N$, which leading to the exponent, $\nu_{0}=0$. An example of randomly
branched polymers is also discussed.

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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: physics
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The categories of this paper are: patt-sol