\documentstyle[graphicx,multicol,epsf,aps]{revtex}
\begin{document}
\title{Entropic Tightening of Vibrated Chains}
\author{M. B. Hastings$^{1,2}$,
  Z. A. Daya$^{2,3}$, E. Ben-Naim$^{1,2}$, and R. E. Ecke$^{2,3}$}
\address{$^1$Theoretical Division, $^2$Center for Nonlinear Studies,
$^3$Condensed Matter \& Thermal Physics Group\\
Los Alamos National Laboratory, Los Alamos, NM 87545}
\maketitle
\begin{abstract}
We investigate experimentally the distribution of configurations of a
ring with an elementary topological constraint, a ``figure-8'' twist.
Using a system far from thermal equilibrium, a vibrated granular
chain, we show that configurations where one loop is small and the
second is large are strongly preferred.  Despite the highly
non-equilibrium nature of the system, our results are consistent with
recent predictions for equilibrium properties of
topologically-constrained polymers. The dynamics of the tightening
process weakly violates a (coarse-grained) detailed balance,
indicating that the unexpected correspondence with an equilibrium
entropic approach is not exact.  
\vskip1mm {PACS:} 05.40.-a, 81.05.Rm,83.10Nn \vskip1mm
%05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
%81.05.Rm Porous materials; granular materials
%83.10.Nn Polymer dynamics
\end{abstract}

\begin{multicols}{2}

Extending the concept of entropy to nonequilibrium systems remains an
important challenge with only a few scattered examples where an
effective statistical mechanics can be constructed
\cite{gallavotti,egolf00}. Recently, vertically-vibrated granular
media consisting of spherical particles have been studied from the
perspective of kinetic theory and statistical mechanics
\cite{ou,menon}.  Granular chains composed of spherical beads
connected by rods were suggested as a model for polymers driven far
from equilibrium\cite{bdve,bsew}.  In this study, we use this system
to test the relevance of statistical measures, particularly entropy,
to nonequilibrium systems, by analyzing steady state conformations of
topologically constrained chains.

In equilibrium polymers and biomolecules such as DNA, topological
constraints constantly form and
relax\cite{n,fw,sw,sw1,q,bgwb}. Whereas the role entanglements play in
chain dynamics is well appreciated\cite{dg,de}, their effect in
systems far from equilibrium received much less attention. Moreover,
direct dynamical experiments are lacking.  Theoretical studies,
numerical simulations and scaling analysis predict that in
equilibrium, a knotted polymer will generally favor configurations
where the knot is ``tight", {\it i.e.}, localized to a small region of
the chain\cite{sommer,kovds,g,mhdkk}.

We experimentally examine the applicability of this interesting
prediction to vertically-vibrated granular chains.  A vibrating plate
supplies the system with energy, balancing the energy dissipation due
to inelastic collisions experienced by beads\cite{ou,k,jnb,mus}. This
system is well suited for studying topological constraints as
demonstrated by experiments on diffusive relaxation \cite{bdve} and
spontaneous formation \cite{bsew} of knots. Indeed, it allows control
of the chain size and the constraint type, as well as direct
observations of the chain conformation.

We considered the simplest possible topology, a ``figure-8'': a
once-twisted ring consisting of two loops, separated by a single
crossing point, which functions as the topological constraint (see
Fig.~1).  Under appropriate vibration amplitude, the system is
effectively two-dimensional and the crossing point hops along the
chain without flipping open the figure-8.  Surprisingly, despite the
highly nonequilibrium drive applied to the system, we observed strong
entropic tightening.  The microscopic degrees of freedom, the beads,
experience periodic drive and dissipative collisions with the plate,
rods, and other beads, as well as frictional forces.  Remarkably the
macroscopic observable, the loop size, obeys an effective statistical
mechanics.  Detailed balance is only weakly violated and the empirical
loop-size distribution is close to that conjectured on entropic
grounds.

\begin{figure}
\centerline{\includegraphics[width=5.5cm,angle=90]{fig1}}
%\centerline{\epsfxsize=7.6cm\epsfbox{fig1.eps}}
\caption{A vibrated ring with a figure-8 twist.}
\end{figure}

Tightening of knots in polymers can be understood by considering the
simplest case\cite{sommer}, in which electrostatic interactions are
perfectly screened.  Let a knot be considered as located at a point,
with several loops of lengths $N_1,N_2,...$ projecting from the knot.
Ignoring hard-core interactions, the total number of configurations is
then proportional to $e^{cN} \prod_i N_i^{-d/2}$, where $d=2$ is the
spatial dimension, and $c$ is a constant.  Therefore, assuming all
microscopic configurations are equally likely, the probability of
having a given $N_1,N_2,\cdots$ is maximized when $N_1 \approx N$ and
$N_2,\cdots$ are all as short as possible. That is, the knot
spontaneously tightens due to entropic effects.

A ring with a figure-8 constraint is natural for studying this effect
as it simply contains two loops of sizes $N_1$, $N_2$ with the overall
size $N=N_1+N_2$ fixed. The argument above predicts a power law
divergence in the loop-size probability distribution
\begin{equation}
\label{powerlaw}
\rho(n)\propto n^{-\alpha}
\end{equation}
with $n\equiv N_1,N_2$ for $1 \ll n\ll N$. The exponent $\alpha=43/16
\approx 2.7$ can be obtained even when excluded volume interactions
are taken into account \cite{mhdkk,ds}.  Although this is obtained
from large $n,N$ asymptotics, an enumeration for smaller sizes only
leads to small corrections.

Our apparatus consists of an anodized aluminum plate, driven
sinusoidally by an electromechanical vibrator. The bead and rod chains
consist of hollow nickel-plated stainless steel spheres of radius
$1.18 \pm 0.01$~mm connected by thin rigid rods of radius $0.26 \pm
0.01$~mm. The rods constrain both the bending and stretching of the
chain. In particular, the rods must lie within a cone of a half-angle
of roughly $45^\circ$ about the axis of either of the two adjacent
rods, and the separation $b$ between two adjacent beads lies in the
range $0 \leq b \leq 0.94$~mm. The chains were connected end-to-end to
form rings, and then twisted with a single crossing point thereby
forming a figure-8. For the experiments reported here, the number of
beads in the figure-8 ring was between $69$ and $219$, much larger
than the tightest possible $8$-bead loop. The plate was oscillated
harmonically at a frequency of $16$~Hz.

The dynamics of the crossing depend on the rms acceleration of the
plate, $\gamma$ (this dimensionless quantity is in units of the
gravitational acceleration $g$).  For $\gamma\lesssim 1.35$, the
crossing does not move along the chain.  For $\gamma\gtrsim 1.55$, the
vertical motion of the chain is large enough that the number of
crossings in the ring is not fixed: a loop of the figure-8 can easily
flip, untwisting the ring, or creating additional crossings.  We chose
$\gamma=1.5$, for which flipping events remained rare, occurring every
roughly $10^4$ oscillation cycles, while the crossing remained mobile,
with 50\% chance of the loop size changing in a $1/16$-th second
cycle.  The probabilities of the loop size changing by $1,2,3$ in a
single cycle were $37\%,9\%,2\%$ respectively, with larger jumps rare.
The acceleration was constant and uniform across the plate to better
than $1\%$. The $27.2$~cm plate diameter, corresponding to
approximately $115$ beads, was large enough so that collisions with
the sloped acrylic wall were rare.

Digital images of the chain were obtained to determine the loop size
distribution. Image analysis requires a two step procedure involving:
(i) monomer recognition, and (ii) chain reconstruction.  To obtain the
monomer positions, images of resolution $1000 \times1016$ pixels were
acquired.  At this resolution, the reflected light from a bead appears
in the images as a bright spot of about $5 \times 5$ pixels, even
though the bead has a diameter of just over $8$ pixels. The positions
of the beads were determined by fitting the intensity pattern
generated by each bead to a Gaussian, with the peak position taken as
the bead position. We estimate the positional accuracy obtained using
this procedure as $0.05$ bead diameters.

Given these positions, the order of the monomers along the chain was
determined using an efficient greedy algorithm which requires only
$N^2$ operations for an $N$-bead ring. This algorithm utilizes the
aforementioned geometrical restrictions on stretching and bending
imposed by the rods (given two connected beads, the third was searched
only in a properly restricted neighborhood).  Once the ring was
reconstructed, the crossing point was identified, thereby determining
the two loop sizes.

\begin{figure}
\centerline{\includegraphics[width=8.5cm]{fig2}}
%\centerline{\epsfxsize=8.5cm\epsffile{fig2.eps}}
\caption{The observed loop size distribution for chains of length $N=99$
(solid line), $N=149$ (dashed line).  To compare the two distributions,
the range of possible loop sizes, $8\leq i\leq N-8$, is scaled to unity.}
\end{figure}


We first sampled the loop size at a much slower rate of $0.25$ Hz.
Starting with two equal-size loops at each run we obtained the
loop-size distributions shown in Fig.~2 for chains of size $N=99,149$.
The distributions both have sharp peaks located at the smallest
possible loop size.  We repeated the experiments using larger beads,
changing the driving frequency to $13$ Hz, and using chains of length
49, 69, and 219, as well as using a plate with different
roughness. Further, many sets of images at arbitrary phase with
respect to the driving were analyzed. In all these cases, the same
qualitative loop-size distribution emerged.  Hence, the loop is tight.

There were, however, some quantitative differences with the peak
height varying by about 20\%.  Since the number of hopping events
before flipping ($\sim 10^4$) happens to be of the same order as the
number required for the loop size to reach its minimum ($\sim N^2$),
the observed distribution depends on the initial loop size.  This
effect is more pronounced for larger chains as seen in in Fig.~2,
where the $N=149$ chain exhibits a small maximum at the symmetric
configuration, a remnant of the initial conditions.  Although the
Rouse time of an ideal chain also scales as $N^2$, the chain
conformation relaxes on a much faster time scale than the loop size,
so the dependence on the initial chain conformation is negligible.

To find the true peak height, {\it with flipping events removed}, we
measured the loop size at a much faster frame rate of $16$ Hz, and
experimentally determined the transition probability $t_{i,j}$ from a
loop of size $i$ to a loop of size $j$. To sample $t_{i,j}$ for all
$i$, the twisted ring was started manually at various equally-spaced
loop sizes, and then allowed to run for 200 cycles to let the chain
equilibrate, after which 200 frames were taken to measure $t_{i,j}$.
By taking 20,000 total frames over 100 separate runs, we obtained an
accuracy of $10$\% on individual $t_{i,j}$.  After equilibration,
correlations between successive transitions were negligible, implying
a Markov process.

Given Markovian dynamics, it is possible to calculate the steady state
probability, $\rho_i$, of the loop having size $i$, from the $t_{i,j}$
by performing a Monte Carlo simulation of the transition process.  For
a chain of length 149, we found the distribution shown as the solid
line in Fig.~3.  Compared to that measured directly, this histogram is
characterized by a sharper peak and considerable curvature at the
center because flipping events and dependence on initial conditions
were eliminated, respectively.  The dashed line shows the theoretical
curve $\rho_i\propto \left[ i^{-\alpha} (N-i)^{-\alpha}\right]$, with
$\alpha=43/16$.  The two curves are consistent, although the peak is
more sharply defined in the solid line.  Making quantitative
statements about the relation between the curves would require much
larger chain lengths $N$ to obtain a sufficient scaling region.
Further, the statistical error in $\rho_i$ is most pronounced in the
center of the histogram where $\rho_i$ is small.  Other chain lengths
have similar histograms, in this case using $0.25$ Hz transition
rates.
\begin{figure}
\centerline{\includegraphics[width=9cm]{fig3}}
%\centerline{\epsfxsize=9cm\epsffile{fig3.eps}}
\caption{Loop-size distribution obtained from transition rates (solid line),
and equilibrium result (dashed line).}
\end{figure}

The transition rates enable us to check detailed balance, a sharp test
of the nonequilibrium nature of the system.  For a system in thermal
equilibrium, detailed balance implies a vanishing net flux between any
two microscopic states, namely $\rho_i t_{i,j}=\rho_j t_{j,i}$, or
\hbox{$f_j(i)\equiv \ln{\left[(\rho_i t_{i,i+j})/ (\rho_{i+j}
t_{i+j,i})\right]}=0$}.  Interestingly, we have found that $f_1(i)>0$
and $f_2(i)<0$, namely, short jumps tend to tighten the loop more than
long jumps.  This is shown in Fig.~4, where we plot a moving average
of $f_1(i)$ as the solid line, and a moving average of $f_2(i)$ as the
dashed line.  The average of $f_1(i)$ is $0.1\pm 0.02$ for $i>100$.


As this is an important point, we have made several additional checks.
We also considered
$f_1(i)+f_1(i+1)-f_2(i+2)$, which measures the flux around a three
point neighborhood.  This $\rho$-independent quantity was positive,
demonstrating violation of detailed balance directly from the
$t_{i,j}$.  Furthermore, we tested detailed balance
using $0.25$ Hz transition rates for other chain lengths, and again
found that short jumps tighten the loop more than long jumps.  As a
final check, we have tested this procedure using surrogate data from
a simulated process which satisfies detailed balance to verify that
the violation is not an artifact of the reconstruction of $\rho$ from
the $t_{i,j}$.

\begin{figure}[!t]
\centerline{\includegraphics[width=9cm]{fig4}}
%\centerline{\epsfxsize=9cm\epsffile{fig4.eps}}
\caption{Plot of smoothed $f_1(i)$ (solid line), and smoothed
$f_2(i)$ (dashed line).}
\end{figure}

Since it is surprising that an argument based on counting of states
should be relevant in a nonequilibrium system, we now consider a
simplified model.  We show that, even out of equilibrium, there are
entropic tightening forces, despite violation of detailed balance and
possible quantitative changes in the size distribution.  We consider a
chain with linear elastic interactions between neighboring beads and
ignore self-avoidance.  We will first consider the equilibrium case,
with the chain subject to thermal forcing and damping, and then
generalize to athermal drive.  Number the beads from $1$ to $N$, and
label the position of bead $i$ by $\vec x(i)$.  Let the crossing occur
at beads $n_1,n_1+1$ and $n_2,n_2+1$.  We will compute the forces on
the crossing, and from this derive an effective dynamics.

Assume $n_1<n_2$, and consider the loop formed by beads 
$n_1+2,...,n_2-2$.
This loop exerts a force on bead $n_1+1$ proportional to $\vec x(n_1+2)-
\vec x(n_1+1)$.  This force tries to tighten the loop.
Summing over normal modes of the loop, the
average force of a loop of length $n$ is found to be
$k T(c-1/n)$, with $c>0$.  The fluctuations in this force introduce
an effective noise into the dynamics of $n_1$.  There is also
an effective dissipation:
if the crossing moves to tighten a given loop, the force exerted by that loop
temporarily increases (both these contributions are determined by
short distance effects).
We then make an approximation that $n_1,n_2$ can be treated as
particles of mass $M$ subject to the above forces.
The resulting dynamics for $n=n_2-n_1$ when $n<<N$ is,
\begin{equation}
\label{cdyn}
M\partial_{t}^2n=-\frac{kT}{n}-\nu \partial_t n+\eta(t),
\end{equation}
where $\nu$ is the dissipation parameter and $\eta$ is the noise.  By
the fluctuation-dissipation theorem, noise and dissipation in
(\ref{cdyn}) are such that $n$ also behaves as a thermal particle at
temperature $T$, giving an equilibrium distribution $\rho(n)\propto
n^{-d/2}$.

Suppose instead that the chain is subject to athermal forcing with
large, non-Gaussian fluctuations at short distances, a reasonable
assumption given the collisions with the plate.  The average force
will remain proportional to $-1/n$, but the noise in (\ref{cdyn}) also
becomes athermal.  The effect of this is most easily understood in the
overdamped limit of (\ref{cdyn}).  Then, $n$ executes a biased random
walk with a drift of order $1/n$, with additional random jumps of
varying size due to noise.  As a result, large jumps are less biased
than small jumps, and detailed balance is violated as in the
experiment.  At large $n$ this gives biased diffusion, with the
diffusivity determined by the mean-square step size.  This yields
$\rho\propto n^{-\alpha}$, $\alpha \neq d/2$.  For smaller $n$, the
probability distribution is determined only by the smaller, more
biased jumps, sharpening the peak in the distribution at a width of
order the largest jump size, consistent with observations.  This
behavior is independent of the precise form of the noise, as confirmed
by our numerical simulations of (\ref{cdyn}).

Finally, we consider the dynamics of constraints in linear chains
instead of rings.  In this case, knots can open at the ends of the
chain.  Consider a linear chain which crosses itself at one point.
Let the crossing occur at links $n_1,n_2$, with \hbox{$0<n_1<n_2<N$}.
The points $n_1,n_2$ describe a random walk, with boundary conditions
$n_1<n_2$, and a bias proportional to $1/(n_2-n_1)$.  For $\alpha>1$,
the two walkers form a bound state, and the time for the knot to open
will behave for large $N$ as for a single random walker.  This
contrasts with the behavior found in a larger acceleration regime, for
which experimental measurements of knot opening times showed a purely
diffusive behavior\cite{bdve}, with negligible entropic interaction
between walkers.  We speculate that the reason for the reduced
interaction in the larger acceleration regime is that the increased
drive takes the system further out of equilibrium.

In conclusion, we have observed spontaneous tightening of topological
constraints in vertically-vibrated granular chains so that the
presence of the constraint merely reduces the chain length by a fixed
amount, rather then leading to an extensive size reduction.  For
equilibrium polymers the tightening arises from entropy. Here, because
of the strongly nonequilibrium drive applied to the system, the bead
dynamics are athermal and the crossing dynamics breaks detailed
balance.  Nevertheless, the loop-size distribution remains close to
equilibrium.  This system provides further possibilities for
experimental examination of the role of entropic forces in
nonequilibrium statistical mechanics.  For example, it can be used to
probe fluctuation-dissipation relations by a quantitative comparison
between the forces on the crossing point and the velocity fluctuations
in the beads, namely the granular temperature. Furthermore, monitoring
steady state chain conformations may illuminate the ergodic properties
of the system.

We thank Charles Reichhardt for useful discussions. This work was
supported by US DOE(W-7405-ENG-36) and by the Canadian NSERC.

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\end{multicols}

\end{document}
