End-to-End Distribution Function of Two-Dimensional Stiff Polymers\\ for all Persistence Lengths
B.~Hamprecht$^1$
W.~Janke$^2$ and H.~Kleinert$^1$
We set up and solve a recursion relation for all even moments of a two-dimensional stiff polymer (Porod-Kratky wormlike chain) and determine from these moments a simple analytic expression for the end-to-end distribution at all persistence lengths.
2d.tex
% Sun Jul 20 21:49:44 CEST 2003 (WJ)
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\begin{document}
%
\title{End-to-End
Distribution Function
of Two-Dimensional Stiff Polymers\\ for all Persistence Lengths
}
%
\author{B.~Hamprecht$^1$, W.~Janke$^2$ and H.~Kleinert$^1$}
\affiliation{$^1$ Institut f�r Theoretische Physik, Freie Universit�t Berlin,\\
Arnimallee 14, D-14195 Berlin, Germany\\
$^2$ Institut f�r Theoretische Physik, Universit�t Leipzig,\\
Augustusplatz 10/11, D-04109 Leipzig, Germany\\
{\scriptsize e-mail: bodo.hamprecht@physik.fu-berlin.de}
{\scriptsize e-mail: wolfhard.janke@itp.uni-leipzig.de}
{\scriptsize e-mail: hagen.kleinert@physik.fu-berlin.de}}
%
%
\begin{abstract}
We set up and solve a recursion relation for all even moments
of a two-dimensional stiff polymer
(Porod-Kratky wormlike chain)
and determine from these moments
a simple analytic expression for
the end-to-end distribution
at all persistence lengths.
\end{abstract}
%
\maketitle
%
%
\section{Introduction}
%
In a recent
note \cite{BodoHagen},
two of us
set up a recursion relation
for
the even moments of the end-to-end distribution of stiff polymers
in $D$ dimensions and used the resulting moments
of high order
to
 construct an simple analytic
 distribution function
of the end-to-end distance ${\bf R}$.
For large persistence length $\xi$,
the result agrees
well
 with
perturbative and Monte Carlo results
of Wilhelm and Frey \cite{Frey}, for small
$\xi$ with the Daniels correction \cite{Dan}.

Recently, Dahr and Chaudhuri \cite{Dhar} have pointed out
the existence of
an interesting dip-structure in the distribution function
for two dimensions
appearing
at intermediate $\xi$-values
 in the {\em spatial\/} density function $P_L({\bf R})\equiv P_L(R)/R$
normalized to unity as $\int d^2x P_L({\bf R})=1$.
The often-plotted {\em radial\/} density function $P_L(R)$
with the normalization $\int P_L(R)~dR=1$
makes this dip almost invisible.
If one wants this dip tp
show up in a simple analytic approximation
of the type in Ref.~\cite{BodoHagen},
several more parameters are needed.
The purpose of this note
is to construct
such an analytic expression
which fits excellently
high-precision Monte Carlo data.

The end-to-end
distribution function
of a stiff polymer  in two dimensions
is
given by the path integral
 \cite{Hagen}
\begin{align}
P_L({\bf R})\! \propto  \!\!
\int d\phi_b
\, d\phi_a
 {\cal D}\phi(s) e^{ -{E_{\rm b}}/{k_B T}}\! \delta ^{(2)}\!
        \left( \!{\bf R}\!-\!\!\int_0^L\!\! ds
         {\bf u}(s)\!\right)
\label{15.82nc}
\end{align}
%
with
the bending energy
\begin{align}
E_{\rm b} =\frac{\kappa}2~\int _0^Lds\,[{\bf u}'(s)]^2\,.
\end{align}
%
where ${\bf u}(s)=(\cos\phi(s),\sin \phi(s))$
are the direction vectors of the polymer links,
and
 $\kappa $ is the stiffness
which defines
the persistence length
$\xi\equiv 2{ \kappa }/{k_BT}$.
%
%
%
Due to the presence of the $ \delta^{(2)}$-function
in the integrand, the path integral is not exactly
solvable.
It is, however, easy to find
arbitrarily high even moments
of the radial distribution
of the end-to-end distance
$P_L(R)$:
%
\begin{eqnarray}
 \left\langle { R}^{2n}\right\rangle
 \equiv
 \int dR\,{ R}^{2n}
P_L(R).
\label{@mome}\end{eqnarray}
%
%
%
%
This was explained
for  $D$-dimensions
in Ref.~\cite{BodoHagen}.
In two dimensions,
the
moments (\ref{@mome})
can be obtained
from the coefficient of $\lambda ^{2n} /2 ^{2n} (2n)!$
 in the expansion of
the integral
 %
\begin{eqnarray}
f(\tau ;\lambda )\equiv \int_{0}^\pi d\theta \,
\psi(\theta,\tau ; \lambda ),
\label{@}\end{eqnarray}
%
in powers of $ \lambda $, evaluated at the euclidean time
$\tau = \xi$.
Here we use natural units with $L=1$.
  The wave function $\psi(\theta,\tau ; \lambda )$ is a solution of the
Schr�dinger equation in euclidean time
\begin{eqnarray}
\hat H\psi(\theta,\tau; \lambda )=
-\frac{d}{d\tau}\psi(\theta,\tau; \lambda ),
\label{15.82nc4}
\end{eqnarray}
%
where
%
\begin{eqnarray} \!\!\!\!\!\!\!
\hat H =
 -\frac{1}{2}\Delta+\frac{1}{2}\lambda \,z
\equiv -\frac{1}{2}\frac{d^2}{d\theta^2}+
\frac{1}{2}\lambda \,\cos\theta.
\end{eqnarray}
%
%
\section{Recursive Solution of the Schroedinger Equation.}
%
The function
$f( L ; \lambda )$
has a spectral representation
%
\begin{align}
f( L ; \lambda ) \!
\equiv \!\sum_{l=0}^\infty\frac{\int_{0}^\pi d\theta\,\varphi^{(l)}{}^\dagger(\theta)
 \exp{\left(-{E^{(l)}L}\right)}
~ \varphi^{(l)}(0)}{\int_{0}^\pi d\theta\,
\varphi^{(l)}
{}^\dagger(\theta)~\varphi^{(l)}(\theta)},
\label{II1}
\end{align}
%
where the $\varphi^{(l)}(\theta)$ are
arbitrarily
 normalized eigensolutions of the time-independent interacting Schr�dinger equation
$\hat H\varphi^{(l)}(\theta)=E^{(l)}\varphi^{(l)}(\theta)$.
%
Applying perturbation theory to this problem, we start from
the eigenstates
$|l\rangle $
of the
unperturbed Hamiltonian $\hat H_0=-\Delta /2$,
with eigenvalues $E_0^{(l)}=l^2/2$, and
form the
eigenfunctions $\varphi^{(l)}(\theta)=\left\langle \theta|l\right\rangle $
with the explicit form
$\varphi^{(0)}(\theta)=1/\sqrt{4\pi}$
and  $\varphi^{(l)}(\theta)=\cos(l\theta)/\sqrt{\pi}$.
Note that the ground state wave function is
not normalized to unity
 on purpose,
for later convenience.
Now we set up a recursion scheme for the expansion coefficients
 $\gamma^{(l)}_{l',i}$
and
$\epsilon^{(l)}_j$ of the eigenfunctions and their energies:
%
%
%
%
\begin{eqnarray}
|\varphi^{(l)} \rangle =\sum_{l',i=0}^\infty~\gamma^{(l)}_{l',i}~\lambda ^i~\left|l'\right\rangle,
~~
E^{(l)}
=\sum_{j=0}^\infty ~\epsilon^{(l)}_j~\lambda ^j.
\label{II3}
\end{eqnarray}
%
%%
The procedure is
 described in \cite{BodoHagen,HaPe}.
The properties  of the unperturbed system for $\lambda=0$
determine the initial
conditions for the recursion:
%
%
\begin{equation}
\label{GAMMA}
\gamma_{l,i}^{(l)}=\delta_{i,0}\,, \qquad \gamma_{k,0}^{(l)}=\delta_{l,k}\,,
\qquad \epsilon _0^{(j)}=j^2/2\,.
\end{equation}
%
%
To proceed, we need the the matrix elements of the
perturbing Hamiltonian $\hat H_I$ in the basis of the
unperturbed eigenstates. They are simply
 $\langle n |\hat H_I|n \pm 1 \rangle=\lambda/2$.
Inserting the
expansions (\ref{II3})
into the Schr�dinger equation (\ref{15.82nc4}), projecting the result onto some base vector $\langle k|$,
 and extracting the coefficient of $\lambda^i$, we obtain the following recursion
relations:
\begin{align}
\label{EQ1_1}
\epsilon_i^{(0)} = \gamma_{1,i-1}^{(0)},\qquad
\epsilon_i^{(l)} = (\gamma_{l+1,i-1}^{(l)}+\gamma_{l-1,i-1}^{(l)})/2 ,
\end{align}
and
%
\begin{align}
\label{EQ1_2}
\gamma_{0,i}^{(l)} = \frac{2}{l^2}\left(\gamma_{1,i-1}^{(l)}-\sum\limits_{j=1}^{i-1}
\epsilon_j^{(l)}\gamma_{0,i-j}^{(l)}\right),
\end{align}
%
%
%
%
%
\begin{align}
\label{EQ1_3}
\gamma_{k,i}^{(l)} = \frac{1}{l^2-k^2} \left(\gamma_{k+1,i-1}^{(l)}+\gamma_{k-1,i-1}^{(l)}-2\sum\limits_{j=1}^{i-1}
\epsilon_j^{(l)}\gamma_{k,i-j}^{(l)}\right)\,.
\end{align}
%
%
%
Starting from the initial values (\ref{GAMMA}), these recursion
relations determine
successively the
 higher-order expansion coefficients in (\ref{II3}).
Inserting the resulting
expansions (\ref{II3}) into Eq.~(\ref{II1}),
only the constant parts in $\varphi^{(l)}(\theta)$ which are independent of $\theta$
will survive the integration in the numerators. Therefore $\varphi^{(l)}(\theta)$ in the numerators of (\ref{II1}) may be replaced by the constants:
%
\begin{align}
%\label{@15@II4}
\varphi^{(l)}_{\rm symm}
\equiv
\int _0^{2\pi}d\theta\,
\varphi^{(0)}{}^\dagger(\theta)
\varphi^{(l)}(\theta)
=\frac{1}{2}\sum_{i=0}
^\infty\gamma^{(l)}_{0,i}\, \lambda ^i\,,
\end{align}
%
the factor $1/2$ reflecting the
special normalization of $\varphi^{(0)}(\theta)
$.
The denominators of (\ref{II1}) become explicitly
%
\begin{align}
\int\limits_{0}^\pi d\theta\,
\varphi^{(l)}
{}^\dagger(\theta)~\varphi^{(l)}(\theta)=\sum_i
\left(|\gamma_{0,i}^{(l)}|^2/2 + \sum_{l'}|\gamma_{l',i}^{(l)}|^2\right)
 \lambda^{2i},
\end{align}
%
where the sum over $i$ is limited by the
power of $\lambda^2$ up to which we want to carry the perturbation series;
also $l'$ is restricted to a finite number of terms,
because of a band-diagonal structure of the $\gamma_{l',i}^{(l)}$
(see \cite{BodoHagen}).
\noindent
Extracting the coefficients of the power expansion in $ \lambda $ from
(\ref{II1}) we
obtain all desired  moments
of the end-to-end distribution,
the
lowest two being:
%
%
\begin{equation}
\!\!\!\!\!\!
\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!
\langle R^2\rangle = {2} \left\{
\xi L-{\xi ^2} \left[ 1-e^{-L/\xi }\right] \right\},
\label{II5}
\end{equation}
%
\begin{align} \label{R4}
\langle R^4\rangle
=&\,8L^2 \xi ^2 - L\xi ^3\left(30+\frac{40}{3}e^{-L/\xi}\right)
~~~~~~~~~~\nonumber \\
+&\xi ^4
\left(
\frac{87}{2}
-\frac{392}{9}e^{-L/\xi}+\frac{1}{18}e^{-4L/\xi}
\right).
\end{align}
%
The calculation of higher moments
is straightforward
with
a Mathematica
program, which we have placed
on the internet
in notebook form
\cite{NBK}.
The above lowest moments agree with those in
Ref.~\cite{Dhar}.
%
%
\section{End-to-End Distribution}
%
As in the previous paper,
we shall now
set up an
analytic  distribution function
for $p_L(r)\equiv
P_L({\bf R})$, where $r=R/L$:
\begin{align} \!\!\!\!
p_L(r)=(a_0\!+\!a_2x^2\!+\!a_4x^4\!+\!a_6x^6)x^k(1-x^\beta)^m\,.
\label{modell}
\end{align}
%
The
parameters $a_0,\dots,a_6,\,k,\,\beta,\,m$
are functions of $\xi/L$ and are
determined by forcing the moments of
(\ref{modell}),
\begin{align}
\displaystyle\left\langle { R}^{2n}\right\rangle=L^{2n}
\int_0^1 r^{2n+1}\,p_L(r)\,dr\,,
\end{align}
%
 to fit the exact moments in the
range $0 \leq n \leq {\rm Max}(6,10\,{\xi}/L)$.
For floppy chains with $\xi<1$ we set $k=0$;
for very floppy chains with $\xi<1/20$ the choice $a2=a4=a6=0$
guarantees excellent accuracy.
 A comparison of $p_L(r)$ with Monte-Carlo data is shown in Fig.~\ref{D2},
and with higher resolution near ${\bf R}=0$ in Fig.~\ref{D2L}. The
associated coefficients are listed in Table~\ref{T2}.
%
%
\begin{table*}
\begin{tabular}{||r|r|r|r|r|r|r|r||}
\hline \hline
$\hat{\xi}$ & $a_0$ & $a_2$  & $a_4$  & $a_6$ & $k$  & $m$ & $\beta$ \\
\hline \hline
$1/400$ & $400.0$  &$0$ & $0$ & $0$ & $0$ & $196.784$   & $1.99496$   \\
\hline
$1/100$ & $100.0$  &$0$ & $0$ & $0$ & $0$ & $47.5378$   & $1.98197$   \\
\hline
$1/50$ & $50.0$  &$0$ & $0$ & $0$ & $0$ & $22.9930$   & $1.97564$   \\
\hline
$1/30$ & $29.5302$  &$-58.9195$ & $77.9373$ & $-87.3526$ & $0$ & $12.0224$   & $2.00505$   \\
\hline
$1/15$ & $14.0952$  &$-29.8504$ & $66.8842$ & $-68.1985$ & $0$ & $5.62896$   & $2.21169$   \\
\hline
$1/10$ & $9.20629$  & $-34.7515$ & $50.6223$ & $-26.2289$ & $0$ & $13.2737$  & $10.5486$   \\
\hline
$1/5$  & $4.18239$  &$-7.45808$ & $11.616$ & $-7.30855$ & $0$  & $10.2031$  & $16.6444$    \\
\hline
$1/4$  & $3.12655$  &$-4.9930$ & $13.1086$ & $-10.0222$ & $0$  & $9.42195$   & $20.0750$   \\
\hline
$3/10$ & $2.38054$  &$-3.38168$ & $12.8823$ & $-9.51483$ & $0$  & $9.16782$   & $23.0164$   \\
\hline
$7/20$ & $1.82132$  &$-2.062292$ & $11.2343$ & $-7.24306$ & $0$  & $8.84230$   & $25.5206$   \\
\hline
$2/5$  & $1.39171$  &$-0.952158$ & $8.94986$ & $-4.33545$ & $0$  & $8.49552$   & $27.9120$   \\
\hline
$1/2$  & $0.800939$  &$0.647524$ & $4.36711$ & $1.36933$ & $0$  & $8.06681$   & $33.2814$   \\
\hline
$1$    & $42.8376$  &$-173.308$ & $263.327$ & $-123.515$ & $4.9880$ & $11.4933$   & $85.8428$  \\
\hline
$2$   & $504.624$  &$-1832.52$ & $2271.51$ & $-925.829$ & $13.4792$  & $30.4949$  & $244.143$  \\
\hline
\hline
\end{tabular}
%
\caption{Coefficients of the analytic
distribution
function for $p_L(r)$ in Eq.~(\ref{modell}) for
various  values of the persistence length
$\hat{\xi}$. They
are obtained by making six or seven even  moments of  $p_L(r)$
 agree with the exact ones. }
\label{T2}
\end{table*}
%
%
\begin{figure}[htp!]
\begin{center}
\setlength{\unitlength}{1cm}
\begin{picture}(8.5,5.5)
\put(0,0){\scalebox{.8}[.8]{\includegraphics*{grafd2.eps}}}
\put(2.1,0.1){0.2}
\put(3.5,0.1){0.4}
\put(4.9,0.1){0.6}
\put(6.3,0.1){0.8}
\put(7.88,0.1){1}
%
\put(0.3,1.1){ 2}
\put(0.3,2.4){ 6}
\put(0.3,3.7){10}
\put(3.88,-0.3){$r=R/L$}
\put(4.8,4.1){$p_L(r)$}
\put(2.65,3.7){$\xi/L=1/15$}
\put(6.35,3.7){$\xi/L=2$}
%
\end{picture}
~\\
~\\[-1em]
\caption[D2]{End-to-end distribution $p_L(r)$ in $D=2$
dimensions as a function of $r=R/L$ for various values
of the stiffness ${\xi}/L=1/15,\,1/10,\,1/5,\,1/4,\,3/10,\,7/20,\,2/5,\,1/2,\,1,\,2$. The solid curves show the model functions (\ref{modell}) with parameters from Table~\ref{T2}. The dots
show the Monte Carlo data.
 }
\label{D2}
\end{center}
\end{figure}
\begin{figure}[htp!]
%
\begin{center}
\setlength{\unitlength}{1cm}
\begin{picture}(8.5,6)
\put(0,0){\scalebox{.8}[.8]{\includegraphics*{grafd2l.eps}}}
%\put(0.38,0.1){r}
\put(3.25,0.1){0.2}
\put(6.1,0.1){0.4}
%
\put(0.2,1.5){0}
\put(0.2,2.55){2}
\put(0.2,3.6){4}
%
\put(.8,2.3){$\xi/L=1/400$}
\put(6.15,2.3){$\xi/L=1/30$}
\put(4.2,-0.1){$r=R/L$}
\put(4.8,4.1){$\log p_L(r)$}
%
\end{picture}
\caption[D2L]{
End-to-end distribution near ${\bf R}=0$
in $D=2$ dimensions plotted logarithmically
as a function of $r=R/L$
for various values of the stiffness for
floppy polymers with
${\xi}/L=1/400,\,1/100,\,1/50,\,1/30$. The solid curves show the model functions (\ref{modell}) with parameters from Table~\ref{T2}. The dots represent Monte Carlo data.
 }
\label{D2L}
\end{center}
\end{figure}
%
%
%
The calculation of
the coefficients in (\ref{modell})
requires some care
 to
% avoid
guarantee sensitivity to
possible local minima,
and to avoid
running  into unphysical oscillations.
The latter may arise from the
 existence of polynomials
in which
all moments lower than some $n$ vanish.
Such oscillations are avoided by controlling
the high moments and using only
low polynomial
coefficients in (\ref{modell}).
 For $\xi>1/20$, a more involved strategy is necessary to avoid
low-quality local solutions. We proceed as follows:
\begin{itemize}
\item
In a first step we set $a_2=a_4=a_6=0$, $\beta=2$, and determine
preliminary values for $k$ and $m$ by fitting
 two higher moments of $n$ near $10\,{\xi}/L$. The first
coefficient $a_0$ is  fixed by normalization.
This gives a reasonable starting value for $m$.
\item
In a second step,
we introduce one more of the higher moments and improve
the solution for $k$, $m$, and $\beta$.
\item
Next we solve for the coefficients $a_j$ by bringing yet more moments into play.
If ${\xi}/L < 1$, we take $k=0$ and solve for the $a_j$, keeping $\beta$ and $m$ fixed, based on four properly chosen moments.
Then we solve for  $\beta$ and $m$ keeping  the $a_j$ fixed,
based on a choice of two moments.
This alternating procedure is repeated three times.
Finally, we solve for the $a_j$, $\beta$, and $m$ simultaneously,
based on six properly chosen moments.
\item
For ${\xi}/L \ge 1$, we proceed similarly, but allow for
 $k\ne 0$. The search for good
coefficients
 $a_j$ alternating with a search for good $k$, $\beta$,
 and $m$ is repeated until it converges, with no further attempt to
solve once more for
all seven parameters simultaneously.
\end{itemize}


~\\
\begin{table}[h]
\begin{tabular}{||c|c|c|c||}
\hline \hline
${\xi}/L$& $\Sigma$ & $\Delta_{\rm abs}$ & $\Delta_{\rm rel}(N_{\rm max})$ \\
\hline \hline
$1/400$ & $3\times 10^{-12}$ & $0.000\,126$   & $0.9\%(16)$ \\
\hline
$1/100$ & $2\times 10^{-13}$ & $0.000\,033$   & $8\%(16)$  \\
\hline
$1/50$ & $1\times 10^{-10}$ & $0.000\,071$   & $1\%(16)$  \\
\hline
$1/30$ & $5\times 10^{-9}$ & $0.000\,717$   & $8\%(16)$ \\
\hline
$1/15$ & $2\times 10^{-6}$ & $0.000\,057$   & $0.9\%(16)$                         \\
\hline
$1/10$ & $5\times 10^{-5}$  & $0.000\,038$   & $0.8\%(24)$                         \\
\hline
$1/5$  & $4\times 10^{-5}$  & $0.000\,048$    & $0.5\%(24)$                         \\
\hline
$1/4$  & $9\times 10^{-5}$  & $0.000\,048$    & $0.4\%(24)$                         \\
\hline
$3/10$ & $13\times 10^{-5}$ & $0.000\,047$   & $0.3\%(24)$                          \\
\hline
$7/20$ & $2\times 10^{-4}$  & $0.000\,087$   & $0.8\%(36)$                          \\
\hline
$2/5$  & $2\times 10^{-4}$  & $0.000\,100$    & $0.8\%(36)$                         \\
\hline
$1/2$  & $2\times 10^{-4}$  & $0.000\,139$    & $0.8\%(36)$                         \\
\hline
$1$    & $2\times 10^{-4}$  & $0.000\,217$      & $0.4\%(48)$                       \\
\hline
$2$    & $8\times 10^{-5}$  & $0.004\,705$      & $1.6\%(48)$                       \\
\hline
\hline
\end{tabular}
%
\caption{To illustrate
the accuracy of our
analytic approximation
(\ref{modell}) of the end-to-end
distribution
 we list the quantity $\Sigma$ of Eq.~(\ref{sigma})
which measures the
deviation
of the even moments
 from the exact ones.
The other two columns
illustrate the accuracy of the Monte Carlo data
for the end-to-end distribution
by listing
the maximal deviation $\Delta_{\rm abs}$ of its moments
and the relative
 deviation $\Delta_{\rm rel}(N_{\rm max})$ up to the moment $N_{\rm max}$.
}
\label{T1}
\end{table}
%

We check
the quality of
our simple distribution function (\ref{modell})
with the parameters
of Table~\ref{T2}
by calculating its moments and comparing them with the exact ones.
The comparison is shown
in Table~\ref{T1}
for a large range of the persistence length $\xi$.
%
As a measure of the quality of approximation
we use
the quantity
 $\Sigma$,
listed in the second column of Table~\ref{T1},
 which sums up all squared deviations of the
moments of the model from the
exact ones in a relevant range of $\xi$:
%
\begin{align}
\label{sigma}
\Sigma(\xi)=\sqrt{\sum_{n=0}^N \left(
 \langle R^{2n}\rangle_{p_L}-
\langle R^{2n}\rangle_{P_L} \rangle
\right)^2}\,,
\end{align}
%
where we have extended
the sum up to the  moments
of order $N=12$ for ${\xi}/L<1/5$, and  $N=24$ for ${\xi}/L \geq 1/5$.
% We also have listed a check of the accuracy of the Monte Carlo data




Let us also convince ourselves
of the high accuracy
of our Monte Carlo data
for the end-to-end distribution
in Figs.~\ref{D2}
and \ref{D2L}
by listing
the maximal deviation
%
\begin{align}
\Delta_{\rm abs}=\sup_{n=0}^\infty \left|
 \langle R^{2n}\rangle_{MC}-
\langle R^{2n}\rangle_{P_L} \rangle
\right|
\end{align}
%
of its moments
and the relative
 deviation
%
\begin{align}
\Delta_{\rm rel}(N_{\rm max})=\sup_{n=0}^{N_{\rm max}} \left|
 \langle R^{2n}\rangle_{MC}/
\langle R^{2n}\rangle_{P_L} \rangle-1
\right|
\end{align}
%
 up to the moment $N_{\rm max}$.

Remarkably,
in spite of the simplicity
of the model,
it is a nontrivial task
to obtain accurate Monte Carlo results
for $P_L({\bf R})$ near ${\bf R}=0$ which
are sensitively displayed in the plots of Fig.~\ref{D2}
but which are almost ignored by the moments.
The reason for the difficulty is the small configuration space for
the small-${\bf R}$
data since  the binning of the data is done in $R$
to estimate the density $P_L({\bf R})$.
One is caught in the
competition between large systematic errors resulting from
a too large bin size
$\Delta R$, and statistical errors
from a too small $\Delta R$.
 As a compromise
we employed in our simulations a uniform
bin size $\Delta R/L = 0.01$ which in combination with
a single-cluster update procedure and a statistics
of $10^8$ sampled chains yielded satisfactory accuracy
near
${\bf R}=0$, as shown in the high-resolution plot in Fig.~\ref{D2L}.



\vspace{.7cm}
~\\Acknowledgment\\
%[-.9em]
This work was partially supported by
ESF COSLAB Program and by the Deutsche Forschungsgemeinschaft
under Grant Kl-256. % (DFG).
WJ acknowledges partial support by the German-Israel-Foundation (GIF) under
contract No.~I-653-181.14/1999.
%\pagebreak
%\vspace{.5cm}
%
%
%
\begin{thebibliography}{99}
%
\bibitem{BodoHagen}
\aut{B.~Hamprecht and H.~Kleinert,}
{\em End-To-End Distribution Function of Stiff Polymers for all Persistence Lengths\/}, cond-mat/0305226.
%

\bibitem{Frey}

          \aut{J. Wilhelm} and \aut{E. Frey},
 Phys. Rev. Lett. {\bf 77}, 2581 (1996).
%
%
\bibitem{Dan}
{H.E. Daniels}, Proc.~Roy.~Soc.~Edinburgh {\bf 63A},  29  (1952).
 For details see also Chapter 15 of
the textbook \cite{Hagen}.
\bibitem{Dhar}
A. Dhar and D. Chaudhuri, Phys. Rev. Lett {\bf 89},
 065502 (2002).
%
%
\bibitem{Hagen}
\aut{H.Kleinert}, {\it Path Integrals
 in Quantum Mechanics, Statistics and Polymer Physics} (World Scientific, Singapore, 1995)
{\tt (http://www. physik.fu-berlin.\linebreak de/\~{}{}kleinert/b5)}.
%
%
%\bibitem{Hagena}

%See Subsection 15.7.2 in the textbook \cite{Hagen}.

%\bibitem{Verena}
%{ H.Kleinert, V.Schulte-Frohlinde
%{\it Critical Properties of $\Phi^4$-Theories}, World Scientific, Singapore, 2001}
%{\tt (http://www.physik.fu-berlin.de/{}\~{}{}kleinert/b8)}.

%
\bibitem{HaPe}

See Appendices of Chapter 3 in
the textbook  \cite{Hagen},
and the article
by B. Hamprecht and A. Pelster in:
{\em Fluctuating Paths and Fields\/}
 - Festschrift Dedicated to Hagen Kleinert on the
 Occasion of his 60th Birthday,
 Eds. \aut{W. Janke}, \aut{A. Pelster}, \aut{H.-J. Schmidt},
 and \aut{M. Bachmann}
 (World Scientific, Singapore, 2001), p.~347.
%
%\bibitem{Abramow}
%I.S.~Gradshteyn and {I.M.~Ryzhik}, {\em Table of Integrals,
%Series, and Products\/}, Academic Press, New York, 1980,
% Formulas  7.313.
%   or
%{M.~Abramowitz and I.A.~Stegun, Handbook of Mathematical Functions, Dover publications, Chapter 22: Orthogonal Polynomials.}
%
%\bibitem{RR}
%ibid., %{I.S.~Gradshteyn} and {I.M.~Ryzhik}, op. cit.,
% Formula  8.933.1.
\bibitem{NBK}
The Mathematica notebook can be obtained
from {\tt http://www.physik.fu-berlin.de/\~{}kleinert/b5/pgm15}.

%\bibitem{schutz}
%S. Stepanow and M. Sch\"utz, Europhys. Lett. (in press)
%(cond-mat/0206338).
%%
%\bibitem{3D}
%J. Samuel and S. Sinha, Phys. Rev. E {\bf 66} 050801(R) (2002),
%
\end{thebibliography}
%


%


\end{document}


cond-mat0307530.extracted_bib
\begin{thebibliography}{99}
%
\bibitem{BodoHagen}
\aut{B.~Hamprecht and H.~Kleinert,}
{\em End-To-End Distribution Function of Stiff Polymers for all Persistence Lengths\/}, cond-mat/0305226.
%

\bibitem{Frey}

          \aut{J. Wilhelm} and \aut{E. Frey},
 Phys. Rev. Lett. {\bf 77}, 2581 (1996).
%
%
\bibitem{Dan}
{H.E. Daniels}, Proc.~Roy.~Soc.~Edinburgh {\bf 63A},  29  (1952).
 For details see also Chapter 15 of
the textbook \cite{Hagen}.
\bibitem{Dhar}
A. Dhar and D. Chaudhuri, Phys. Rev. Lett {\bf 89},
 065502 (2002).
%
%
\bibitem{Hagen}
\aut{H.Kleinert}, {\it Path Integrals
 in Quantum Mechanics, Statistics and Polymer Physics} (World Scientific, Singapore, 1995)
{\tt (http://www. physik.fu-berlin.\linebreak de/\~{}{}kleinert/b5)}.
%
%
%\bibitem{Hagena}

%See Subsection 15.7.2 in the textbook \cite{Hagen}.

%\bibitem{Verena}
%{ H.Kleinert, V.Schulte-Frohlinde
%{\it Critical Properties of $\Phi^4$-Theories}, World Scientific, Singapore, 2001}
%{\tt (http://www.physik.fu-berlin.de/{}\~{}{}kleinert/b8)}.

%
\bibitem{HaPe}

See Appendices of Chapter 3 in
the textbook  \cite{Hagen},
and the article
by B. Hamprecht and A. Pelster in:
{\em Fluctuating Paths and Fields\/}
 - Festschrift Dedicated to Hagen Kleinert on the
 Occasion of his 60th Birthday,
 Eds. \aut{W. Janke}, \aut{A. Pelster}, \aut{H.-J. Schmidt},
 and \aut{M. Bachmann}
 (World Scientific, Singapore, 2001), p.~347.
%
%\bibitem{Abramow}
%I.S.~Gradshteyn and {I.M.~Ryzhik}, {\em Table of Integrals,
%Series, and Products\/}, Academic Press, New York, 1980,
% Formulas  7.313.
%   or
%{M.~Abramowitz and I.A.~Stegun, Handbook of Mathematical Functions, Dover publications, Chapter 22: Orthogonal Polynomials.}
%
%\bibitem{RR}
%ibid., %{I.S.~Gradshteyn} and {I.M.~Ryzhik}, op. cit.,
% Formula  8.933.1.
\bibitem{NBK}
The Mathematica notebook can be obtained
from {\tt http://www.physik.fu-berlin.de/\~{}kleinert/b5/pgm15}.

%\bibitem{schutz}
%S. Stepanow and M. Sch\"utz, Europhys. Lett. (in press)
%(cond-mat/0206338).
%%
%\bibitem{3D}
%J. Samuel and S. Sinha, Phys. Rev. E {\bf 66} 050801(R) (2002),
%
\end{thebibliography}