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{\Large \bf Bounded Fluctuations and Translation Symmetry} \par \vskip 3
truemm
{\Large \bf Breaking: A Solvable Model} \par \vskip 5 truemm
{\bf B. Jancovici}\footnote{Laboratoire de Physique Th\'eorique,
Universit\'e de Paris-Sud, B\^atiment 210, 91405 Orsay, France (Unit\'e
Mixte de Recherche n$^{\circ}$ 8627 - CNRS).\\
\noi E-mail: Bernard.Jancovici@th.u-psud.fr.}\ {\bf and
J.L. Lebowitz}\footnote{Departments of Mathematics and Physics, Rutgers
University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA.\\
\noi E-mail: lebowitz@math.rutgers.edu.}
\end{center}
\vspace{1 cm}
\begin{abstract}
The variance of the particle number (equivalently the total charge) in a
domain of length ${\cal L}$ of a one-component plasma (OCP) on a cylinder
of circumference $W$ at the reciprocal temperature $\beta=2$, is shown to
remain bounded as ${\cal L} \to \infty$. This exactly solvable system with
average density $\rho$ has a measure which is periodic with period $(\rho
W)^{-1}$ along the axis of the infinitely long cylinder. This illustrates
the connection between bounded variance and periodicity in (quasi)
one-dimensional systems \cite{AGL}. When $W \to \infty$ the system
approaches the two-dimensional OCP and the variance in a domain $\Lambda$
grows like its perimeter $|\partial \Lambda|$\. In this limit, the system
is translation invariant with rapid decay of correlations.
\end{abstract}
\vspace{8 mm}
{\bf KEY WORDS:} Coulomb systems; bounded fluctuations; translation symmetry
breaking.
\vspace{5 mm}
%\noindent LPT Orsay 00-69 \par
%\noindent July 2000 \par
\noi {\bf 1.INTRODUCTION AND SUMMARY OF PREVIOUS WORK}
\noi Aizenman, Goldstein, and Lebowitz\cite{AGL} have shown that bounded
fluctuations in a one-dimensional one-component particle system imply the
existence of a periodic structure. The present note illustrates this
general property in a quasi one-dimensional system: the two-dimensional
one-component plasma on the surface of a cylinder. This system is exactly
solvable at the reciprocal temperature $\beta=2$, where it was
studied by Choquard, Forrester, and Smith\cite {C,CFS}.
The finite volume model is defined as follows. There are $N$ particles of
unit charge on the surface of a cylinder of circumference $W$ and length
$L$. The coordinates of a particle are ${\bf r}=(x,y)$ such that $-L/2\leq
x\leq L/2$, $-W/2\leq y\leq W/2$. It is convenient to use also the complex
coordinate $z=x+iy$.
The interaction energy $\phi ({\bf r}_1,{\bf r}_2)$
between two particles at ${\bf r}_1$ and ${\bf r}_2$ is a
two-dimensional Coulomb potential required to be periodic in $y$ with
period $W$:
$$\phi ({\bf r}_1,{\bf r}_2)=-\ln |2\sinh \frac{\pi (z_2-z_1)}{W}|
\eqno(1)$$
\noi (this is equivalent to eq.(4) of ref.\cite{CFS}). There is also a
neutralizing background and the full Hamiltonian also includes
particle-background and background-background interactions. At small
distances $|{\bf r}_2-{\bf r}_1|\ll W$ the interaction (1) behaves like
the two-dimensional Coulomb interaction $-\ln|{\bf r}_2-{\bf r}_1|$,
while at large distances along the cylinder $|x_2-x_1|\gg W$ it behaves
like the one-dimensional Coulomb interaction $-(\pi/W)|x_2-x_1|$.
At the special value $\beta=2$ of the inverse temperature, the model is
exactly solvable. It can be regarded as a variant of the strictly
one-dimensional one-component plasma \cite{L,K,AM}, with however a more
explicit solution.
Choquard et al.\ calculated the one and two-particle distribution functions
and their limits when $N,L\rightarrow \infty$,\ where $\rho=N/LW$ stays
constant. In this limit the one-particle distribution function, i.e.\ the
density, is a periodic function of $x$ with period $1/\rho W$: it is a sum
of equidistant identical Gaussians. The location of the centers of the
Gaussians depends on the way the $L \to \infty$ limit is taken. If this
limit is defined by the sequence of odd values of $N$, then one of the
Gaussians stays centered at $x=0$. If, on the contrary, the thermodynamic
limit is defined by the sequence of even values of $N$, the whole array of
Gaussians is shifted by half a period and the origin is in the middle
between two Gaussians. (It is also possible to take limits along sequences
which don't keep the density exactly constant or use periodic boundary
conditions along the $x$-axis. The latter would yield a translation
invariant density corresponding to a uniform superposition of periodic
ones).
In the following we shall, without loss of generality, concentrate on the
case of odd values of $N$. Defining reduced values of the coordinates
$\zeta=\rho Wx$, $\lambda=(y_2-y_1)/W$, and putting $\xi=\rho W^2$, the
limit along such an odd sequence yields [3] the density
$$n(x)=\frac{(2\xi)^{1/2}}{W^2}\sum_{l=-\infty}^{\infty}
\exp[-2\pi (\zeta -l)^2/\xi] \eqno(2)$$
\noi and the truncated two-particle distribution function
$$\rho^T({\bf r}_1,{\bf r}_2)=-\frac{2\xi}{W^4}
\exp[-\pi (\zeta_2-\zeta_1)^2/\xi]
\left|\sum_{l=-\infty}^{\infty}\exp[-2\pi
(\frac{\zeta_1+\zeta_2}{2}-l)^2/\xi]+2\pi il\lambda]\right|^2 \eqno(3)$$
\noi In the limit $W\rightarrow
\infty$, the sums on $l$ in (2) and (3) can be replaced by
integrals and it can be easily checked that one recovers the expressions
appropriate for a two-dimensional system in the whole plane, i.e.\
$n(x)=\rho$ and\cite{J} $\rho^T=-\rho^2\exp[-\pi\rho
({\bf r}_2-{\bf r}_1)^2]$.\\
\noi {\bf 2.VARIANCE OF THE CHARGE IN AN INTERVAL}
We now calculate the variance of the net charge, which here is equal to the
variance $-^2$ of the number of particles $N_I$ in some
interval $I$ of the cylinder of circumference $W$ and length
${\cal L}$. We show
that this variance remains uniformly bounded as ${\cal L}$ increases and
has a limit when ${\cal L} \to \infty$ along specified sequences.
After integration upon the $y$ coordinates, one obtains the one-dimensional
density $Wn(x)$ and the truncated two-particle density (when convenient, we
use the reduced variable $\zeta =\rho Wx$ instead of $x$):
$$R^T(\zeta_1,\zeta_2)=\int_{-W/2}^{W/2}dy_1\int_{-W/2}^{W/2}dy_2
\:\rho^T({\bf r}_1,{\bf r}_2)$$
$$=-\frac{2\xi}{W^2}\exp[-\pi (\zeta_2-\zeta_1)^2/\xi]\sum_{l=-\infty}
^{\infty}\exp[-4\pi(\frac{\zeta_1+\zeta_2}{2}-l)^2/\xi] \eqno(4)$$
For simplicity, we choose the extremities of the interval $I$ at the
centers of two of the Gaussians, e.g.\
$I$ is the interval $0\leq \zeta < m$, where $m$ is some positive
integer. The variance in $I$ is then given by
$$
-^2\: = m + \int_0^{m/\rho W}dx_1
\int_0^{m/\rho W}dx_2\: R^T (\zeta_1,\zeta_2)
$$
$$
=m-\frac{4}{\xi}\int_0^m du\exp[-\pi u^2/\xi]
\int_{u/2}^{m-(u/2)}dv\sum_{l=-\infty}^{\infty}
\exp[-4\pi (v-l)^2/\xi] \eqno(5)$$
\noi where $u=\zeta_2-\zeta_1$
and $v=(\zeta_1+\zeta_2)/2$, and the symmetry between $\zeta_1$
and $\zeta_2$ has been taken into account,
The integral on $v$ can be written as the difference
between integrals in the ranges $(-(u/2),m-(u/2))$ and
$(-(u/2),(u/2))$. The first one, when combined with the
sum on $l$, like in (7), gives $m$ times the integral of a Gaussian from
$-\infty$ to $\infty$ which has a simple value. In the second one, one
takes into account that the sum on $l$ is an even function of $v$. One
finally obtains for the variance in $I$,
$$-^2\:=m\left(1-2\xi^{-1/2}\int_0^m du\exp[-\pi u^2/\xi]
\right)$$
$$+\frac{8}{\xi}\int_0^m du\exp[-\pi u^2/\xi]\int_0^{u/2} dv
\sum_{l=-\infty}^{\infty}\exp[-4\pi (v-l)^2/\xi] \eqno(6)$$
The variance (6) is clearly bounded uniformly in $m$ and its limit as
$m \to \infty$ is given by
$$-^2 \longrightarrow_{m \to \infty}
\frac{8}{\xi}\int_0
^{\infty}du\exp[-\pi u^2/\xi]\int_0^{u/2}dv\sum_{l=-\infty}
^{\infty}\exp[-4\pi (v-l)^2/\xi] \eqno(7)$$\\
(the first term in
the r.h.s. of (6) goes to 0 as $m \rightarrow \infty$).
\noi {\bf 3.SMALL AND LARGE CIRCUMFERENCE LIMITING CASES}
The variance (7) depends on the cylinder circumference $W$ through
$\xi=\rho W^2$. Let us investigate limiting cases.\\
\noi {\bf 3.1.Small Circumference}
For $W \to 0$, only the $l=0$ term contributes to
(7). The resulting double integral can be evaluated by rescaling the
variables as $u\xi^{-1/2} \rightarrow u$ and
$2v\xi^{-1/2} \rightarrow v$, and using the symmetry of the integrand
in the new $u,v$ coordinates, with the result
$$\lim_{W\to 0} [-^2]\:=\frac{1}{2} \eqno(8)$$
The result (8) has a simple interpretation. For small $\xi$, each
Gaussian in (2) becomes a narrow peak while $R^T$ is such
that there is one particle in each peak and these particles are
otherwise uncorrelated. The system behaves like a one-dimensional
one-component plasma at zero temperature since the interaction behaves
like $-(\pi/W)|x_2-x_1|$ with $W \rightarrow 0$. The variance
is thus entirely due to the particles located near each end
of the interval $I$. Each of these particles has a probability 1/2 of
being inside $I$ and thus contributes 1/4 to the variance of $N_I$.\\
\noi {\bf 3.2.Large Circumference}
For large $\xi$, the sum on $l$ in (7) can be replaced by an integral
which is just $\xi^{1/2}/2$. One finds a variance which
increases with $W$ as
$$-^2\\:\sim (\rho^{1/2}/\pi)W \eqno(9)$$
This is the variance expected\cite{JLM} for a large two-dimensional
one-component plasma at $\beta=2$ with a boundary length $2W$.\\
\noi {\bf 4.ANOTHER CHOICE OF INTERVAL}
The choice of intervals of Section 3 is likely to generate the largest
possible variance. For illustrating that the variance in a large
interval $I$ is very sensitive to the precise positions of the
extremities, we now choose these extremities in the middle between two
adjacent Gaussians. For instance, $I$ is the range $1/2 \leq \zeta \leq
m+(1/2)$. Calculations similar to the ones in Section 3 now lead to
$$-^2\:=m\left(1-2\xi^{-1/2}\int_0^m du\exp[-\pi u^2/\xi]
\right)$$
$$+\frac{4}{\xi}\int_0^m du\exp[-\pi u^2/\xi]\int_{(-u+1)/2}^{(u+1)/2}
dv \sum_{l=-\infty}^{\infty}\exp[-4\pi (v-l)^2/\xi] \eqno(10)$$
\noi In the limit $m \rightarrow \infty$,
$$-^2\:\to\frac{4}{\xi}\int_0^{\infty} du\exp[-\pi u^2/\xi]
\int_{(-u+1)/2}^{(u+1)/2}dv \sum_{l=-\infty}^{\infty}
\exp[-4\pi (v-l)^2/\xi] \eqno(11)$$
For small $\xi$, (11) is dominated by the terms $l=0$ and $l=1$, and the
variance has a factor $\exp(-\pi/\xi)$ which vanishes exponentially as
$W \rightarrow 0$. This is as expected, since the density vanishes as a
Gaussian tail at the extremities of the interval $I$, through which
particles enter or leave $I$.
For large $W$, the variance is still given by (9). In that limit
the density oscillations disappear and the precise
locations of the extremities of $I$ become irrelevant.
\bigskip
\noi {\bf Acknowledgments}: J. L. thanks M. Aizenman and S. Goldstein for
many valuable discussions. The
work of J.L. was carried out in part during a visit to the I.H.E.S. in
Bures-sur-Yvette and was supported by NSF Grant DMR-9813268, and AFOSR
Grant F49620-98-1-0207.
\bigskip
\noi {\bf REFERENCES}
\begin{references}
\bibitem{AGL}
M.Aizenman, S.Goldstein, and J.L.Lebowitz, {\it J.Stat.Phys.}, previous
paper in this issue.
\bibitem{C}
Ph.Choquard, {\it Helv.Phys.Acta}, {\bf 54}:332(1981).
\bibitem{CFS}
Ph.Choquard, P.J.Forrester and E.R.Smith, {\it J.Stat.Phys.} {\bf
33}:13(1983).
\bibitem{L}
A.Lenard, {\it J.Math.Phys.} {\bf 4}:533(1963).
\bibitem{K}
H.Kunz, {\it Ann.Phys.} {\bf 85}:303(1974).
\bibitem{AM}
M.Aizenman and Ph.A.Martin, {\it Comm.Math.Phys.} {\bf 78}:99(1980).
\bibitem{J}
B.Jancovici, {\it Phys.Rev.Lett.} {\bf 46}:386(1981).
\bibitem{JLM}
B.Jancovici, J.L.Lebowitz, and G.Manificat, {\it J.Stat.Phys.} {\bf
72}:773(1993).
\end{references}
\end{document}