Equation-free dynamic renormalization in a glassy compaction model
L. Chen$^1$
I.G. Kevrekidis$^{1,2}$ and P.G. Kevrekidis$^3$
Combining dynamic renormalization with equation-free computational tools, we study the apparently asymptotically self-similar evolution of void distribution dynamics in the diffusion-deposition problem proposed by Stinchcombe and Depken [Phys. Rev. Lett. {88}, 125701 (2002)]. We illustrate fixed point and dynamic approaches, forward as well as backward in time.
glass2.tex
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\begin{document}

\title{Equation-free dynamic renormalization in a glassy compaction model}

\author{L. Chen$^1$, I.G. Kevrekidis$^{1,2}$ and P.G. Kevrekidis$^3$}
\affiliation{
$^1$Department of Chemical Engineering and $^2$PACM, 
Princeton University, 
6 Olden Str.\ Princeton, NJ 08544 USA, \\
$^3$Department of Mathematics and
Statistics, University of Massachusetts,
Amherst MA 01003-4515, USA}

\begin{abstract}
Combining dynamic renormalization with equation-free computational
tools, we study the apparently asymptotically self-similar evolution
of void distribution dynamics in the diffusion-deposition problem
proposed by Stinchcombe and Depken [Phys. Rev. Lett. {\bf 88}, 125701
(2002)].  We illustrate fixed point and dynamic approaches, forward as
well as backward in time.
\end{abstract}

\pacs{05.10.-a, 05.10.Cc, 81.05.Kf, 05.70.Fh}
\maketitle

The study of glasses is an important topic under
intense investigation for several decades now (see e.g., the review
of \cite{pablo}). 
%
Experimental studies in granular compaction (such as the
ones in \cite{chicago}) have offered considerable
insights on the behavior and temporal dynamics of glass forming systems. 
%
From the theoretical point of view, a variety of
modeling/computational approaches have been used to understand
better the key observables and their evolution for
this slow relaxational type of dynamics. 
%
While direct (molecular dynamics) simulations are widely used, 
perhaps more popular are
kinetic approaches including e.g., the Fredrickson-Anderson model
\cite{fa} and more recent variants thereof \cite{sollich}.
%
Another common approach involves the use of mode coupling theory
\cite{g67} examining the time evolution of the decay of density
fluctuations to separate liquid from glassy (non-ergodic due to
structural arrest) dynamics. 
%
More recently, the energy landscape of glass forming systems and the 
role of its ``topography'' has become the focus of numerous works
\cite{sciortino}.

In a previous  paper \cite{PLA} we proposed a simple compaction 
``thought experiment'' for hard spheres: the insertion of
a hard sphere in a gas of hard spheres (accepted when the sphere
does not overlap with previously existing ones).
%
Combining  simple thermodynamic arguments with equilibrium distribution results
we argued that the evolution of the hard sphere density 
should be logarithmic in time (as the maximal density is approached).
%
%{\bf Based on the same arguments, combined with the 
%considerations of \cite{torquato}, the conditional pair distribution 
%function $G$ (a relevant ``statistical measure'' of the system) was found 
%to decay with distance self-similarly as $1/r$ and
%its amplitude to grow in time logarithmically.
%PK: I thought that this could be easily cut but feel free to bring it back.}
% I also commented out Torquato's reference with it.

In a recent paper \cite{Stinchcombe} Stinchcombe and Depken presented 
an interesting 
diffusion-deposition model exhibiting ``glassy'' compaction kinetics
(see also \cite{physd} for a similar model example).
%
The model provides a useful paradigm for testing the hypothesis
of self-similar evolution of the system statistics --in
particular, of the density and the void distribution function--
based on direct simulations.

The techniques that we will use to test this hypothesis 
combine template-based dynamic renormalization \cite{template} with the
so-called ``equation-free'' approach to complex/multi-scale system modeling
\cite{Manifesto,JNNFM}.
%
Dynamic renormalization has been developed in the context of
partial differential equations with self-similar solutions
(e.g. blowups in finite time) \cite{Papanicolaou}.
%{\bf Panagioti, choose a couple}.
%
The equation-free approach targets systems for which a fine scale,
atomistic/stochastic simulator is available, but for which no
closed form coarse-grained, macroscopic evolution equation has
been derived.
%
The main idea is to substitute function and derivative evaluations
of the unavailable coarse-grained evolution equation with
short bursts of appropriately initialized fine scale computation.
%
The quantities required for traditional continuum numerical
analysis are thus estimated {\it on demand} from brief computational 
experimentation with the fine scale solver.
%
We will demonstrate equation-free fixed point approaches to computing
the self-similar shape and dynamics of the void cumulative
distribution function.

%
We start with a brief description of the  model and 
key (for our purposes) results of \cite{Stinchcombe}. 
%
We then perform
equation-free dynamic renormalization computations for this 
problem. 
%using both time evolution and fixed point methods.
%
The coarse-grained description of the system evolution backward in
time is examined through reverse coarse projective integration.
%
We conclude with a brief discussion of potential extensions
of the approach.
%

The model considered by Stinchcombe and Depken
\cite{Stinchcombe} consists of unit-size grains interacting 
through hard-core potentials and performing a Monte Carlo random walk.
%
While some of their results apply in any dimension,
in this paper we will work in one spatial dimension;
this is for convenience--our approach is not limited to 1d.
%
While the grains diffuse on the line, as soon as a 
sufficiently large void is formed, it is instantaneously
filled by an additional grain.
%
We will work with a system of finite size and 
periodic boundary conditions; the system size L (here of
the order of $10^5$) is
large compared to the grain size, which is taken to be 1.
%
If  $\rho$ is the system density,
$\delta = (1-\rho)/\rho$, is the average void size.
%
One of the main findings in \cite{Stinchcombe} is that
the void density $\epsilon = 1 - \rho$ 
goes to zero as an inverse logarithm 
\begin{equation}
\langle \epsilon(t) \rangle \sim \frac{1}{\ln(t)},
\label{eveq0}
\end{equation}
asymptotically for large $t$. 
%
A similar asymptotic behavior follows for $\delta$.

\begin{figure}[htbp]
\centering
\leavevmode
\includegraphics[height = 2.5in]{fig1.eps}
\caption{Dynamic evolution of the model for different initializations.
%
Dotted (resp. dash-dotted) lines: starting with the self-similar
(resp. uniform) void CDF.
%
Solid (resp. dashed) lines: evolution replotted in terms of an
appropriate ``initial time" $t_0$; notice the collapse of the curves
on a straight line, see text.
%
The inset shows void CDFs at different time/density instances
(color-coded as in the main Figure); they all collapse upon rescaling
to a single CDF (smaller inset).
%
}
\label{fig1}
\end{figure}

\begin{figure}[htbp]
\centering
\leavevmode
\includegraphics[height = 1in]{fig2.eps}
\caption{Schematic of coarse dynamic renormalization.}
\label{fig2}
\end{figure}

Density provides a cumulative measure of the system evolution
(it is the zeroth moment of the particle distribution function).
%
We set out to study in detail the evolution of the
void cumulative distribution function, $CDF(x,t)$.
%
This function is defined between $0$ and $1$ in the void size $x$, since at 
no time do we get
voids larger than $1$ (the moment they appear, they get filled).
%
The dotted lines (and the dash-dot line) in Fig. \ref{fig1} show
the evolution of $\delta(t)$ for various initial void distributions (the
choice of initial void distributions for these transients will be
discussed below).
%
The stochastic simulation uses 100,000 particles;
sequential update is used, with a maximum spatial step size of $0.5$.
%
The semilog plot clearly indicates the asymptotic logarithmic
regime in time. 
%
The inset shows the shape of the void $CDF$ for various
simulations and various instances in time, 
%more or less
within the logarithmic regime; all shapes
when appropriately rescaled with the average void size
collapse on a single curve (smaller inset).
%
These two observations (logarithmic time dependence, and
collapse of the distributions upon rescaling) clearly 
suggest the possibility of self-similar evolution dynamics.
%

In particular, the graph of the time evolution of the densities 
suggests that $\delta$ becomes eventually proportional 
%up to an amplitude rescaling, 
to
\begin{equation}
\delta \equiv \frac{1-\rho}{\rho} \sim \frac{1}{A {\rm ln}(t+t_0) + B}.
\label{delta}
\end{equation}
%
Replotting the data in view of the above relation collapses the
Fig. \ref{fig1} curves onto a single line; the role of $t_0$ will be
discussed below.
%
This implies that the dynamical evolution of $\delta$ eventually follows
the differential equation:
%
\begin{equation}
\frac{d \delta}{dt} \sim -\delta^2 \exp \left(-\frac{1}{A \delta}\right).
\label{eveq1}
\end{equation}
%
(and hence a similar equation can easily be obtained for 
$\epsilon$). 
%while the asymptotic considerations of
%\cite{Stinchcombe} yield 
%\begin{eqnarray}
%\frac{d \langle \epsilon \rangle}{dt} \sim 
%\frac{1}{\langle \epsilon \rangle^3} \exp \left(-
%\frac{1}{\langle \epsilon \rangle}\right).
%\label{eveq2}
%\end{eqnarray}
%
The dynamics of (\ref{eveq1}) share the asymptotic
behavior of Eq. (\ref{eveq0}).
%
%The actual exact equation is:
%\begin{eqnarray}
%\frac{d \delta}{dt} = -A \exp\left(\frac{B}{A}\right) 
%\delta^2 \exp \left(-\frac{1}{A \delta}\right)
%\label{eveq1a}
%\end{eqnarray}
%(which of course reduces to the equations we have used when $A=1$ and
%$B=0$).

We now test the self-similarity hypothesis for the $CDF$ evolution.
%
If the dynamics of the $CDF$ are self-similar and stable,
all initial distributions will asymptotically approach
the same self-similar shape (modulo rescaling)
while they densify and slow down.
%
Our goal is to find the self-similar shape at a {\it convenient}
scale -- a relatively low density, when the evolution dynamics
are still fast.
%
Our main assumption is that the void $CDF$ is a good observable 
for the system dynamics -- i.e., that a macroscopic evolution equation 
(possibly averaged over several experiments) conceptually exists for
its dynamics, even though it is not available in closed form.
%
Starting with a given void $CDF_0$
with $\delta = \delta_0$ we {\it lift} to an ensemble
of microscopic realizations of it -- that is, 
grain configurations possessing the given void $CDF$.
%
This ``lifting'' is accomplished here
by randomly selecting the voids from $CDF_0$ as grains
are ``put down'' on the line; $\delta_0$ indirectly selects the scale 
at which we will perform our computation.
%
We then evolve each of these realizations based on {\it true}
system stochastic dynamics for a time horizon $\tau$.
%
Finally, we obtain
the ensemble averaged void $CDF(x,\tau)$ and its $\delta(\tau)=\delta_1$.
%
This is the {\it restriction} step of equation-free computation: 
evaluating the macroscopic observables of 
detailed, fine scale computations.
%
We now {\it rescale} our macroscopic observable (the $CDF(x,\tau)$), using
%as a template 
the ensemble average $\delta_1$, to the original $\delta_0$:
$\widehat{CDF}(\delta_0 x/ \delta_1, \tau) =  CDF(x, \tau)$.
(Clearly this requires the largest void in the
rescaled $\widehat{CDF}$ to be less than $1$ -- a condition that
one may expect to prevail at high enough densities).
%
We then {\it discard} the simulations we performed, and start
a new ensemble of simulations at the original density, but with
the more ``mature'' $\widehat{CDF}$.
%
The map from current void $CDF$ to future void $CDF$ is the
``coarse timestepper'' of the unavailable equation for the
macroscopic observable evolution; 
the composition of this map with the rescaling step constitutes
the ``renormalized coarse timestepper'', the main tool
of our dynamic renormalization procedure, schematically 
summarized in Fig. \ref{fig2}.
%
Given this map, several approaches to the computation
of the long-term coarse self-similar dynamics exist.
%
The simplest is successive substitution -- we repeat coarse
renormalized time-stepping again and again, and observe the
approach of the void $CDF$ to its self-similar shape {\it at
the scale we chose} (parametrized  by $\delta_0$).
%
The fixed point of this procedure lies on the group orbit of
scale invariance for the $CDF$ dynamics; it was selected through
a ``pinning'' or ``template'' condition (our choice of $\delta_0$).
%
This dynamic renormalization iteration, with $\tau = 4$ is shown in Fig. \ref{fig3}a, 
starting from a uniform void distribution with $\delta_0 = 0.2$; the inset shows the third
iteration before (dotted line) and after (solid line) rescaling.
\begin{figure}[htbp]
\centering
\leavevmode
\includegraphics[height = 2in]{fig3.eps}
\caption{a. Successive evolution of an initially uniform CDF profile with
renormalization: the inset shows the profile
at the n=3 iteration before (dashed) and after (solid) rescaling. 
b. Newton-Krylov-GMRES fixed point renormalization calculation, starting from
uniform void CDF; the inset shows the decrease of the norm of the residual vector
at each Newton step.}
\label{fig3}
\end{figure}
%

Successive substitution will converge to {\it stable} self-similar 
solutions.
% 
It is also, however, possible, to use {\it fixed point algorithms}
to converge to the self-similar void $CDF$ {\it at} $\delta_0$.
%
In effect, we wish to solve a functional equation for the fixed point
of the renormalized timestepper:
\begin{equation}
\widehat{CDF}(x,\tau) - CDF(x,0) = 0.
\label{fixp}
\end{equation}
%
Fixed point algorithms (like Newton's method) require the repeated
solution of systems of linear equations involving the Jacobian matrix of
some discretization of Eq. (\ref{fixp}).
%
Since we have no closed form equations for the void $CDF$ evolution,
this Jacobian is unavailable;
%not explicitly available; 
yet equation-free methods
of iterative linear algebra (like GMRES \cite{Kelley_Saad}) allow the
solution of the problem through a sequence of matrix-vector product
estimations.
%
In our case these estimations are obtained through the
``lift-run-restrict-rescale'' protocol performed at appropriately
selected nearby initial distributions (for details, see 
\cite{Manifesto}).
%
Short
bursts of stochastic simulation from nearby initial void distributions allows
us to estimate the action of the Jacobian 
%of the unavailable
%macroscopic Eq. (\ref{fixp}) 
on selected perturbations, and hence the
solution of linear equations and ultimately, of the nonlinear fixed point problem.
%
%
Fig. \ref{fig3}b shows the iterates of such a matrix-free fixed point
computation for Eq. (\ref{fixp}) through a Newton-Krylov GMRES algorithm applied to a 
100-point uniform finite difference discretization of the void $CDF$ in $x$;
our initial guess was a uniform void $CDF$ ($\delta_0 = 0.2$), and we converge 
to the self-similar shape within 
%6 
a few iterations; the inset shows the evolution of the norm
of the residual vector with iteration number.
%
{\it Lifting} from this coarse description (100 numbers) to realizations
of the $CDF$ (200,000 particles) involved linear interpolation of the $CDF$.
%
Relatively short runs ($\tau =1$) were used to construct the coarse
renormalized timestepper in Fig. \ref{fig3}b; its fixed
point is independent of the reporting horizon $\tau$.
%; for problems with power-law behavior in time, the
%critical exponents can be evaluated 
%upon convergence to the self-similar shape.

%
%It shoudl therefore be possible to construct 
%bifuration detection algorithms (wrapped around the coarse
%renormalized timesepper) that -given a good initial guess-
%will converge to the parameter boundary between
%``normal'' and self-similar dynamics.

Wrapping traditional 
numerical algorithms around coarse timesteppers
enables several computational tasks, of which the fixed
point computation above is only one example.
%
Another interesting example is the so-called coarse projective
integration; here, short runs of the stochastic
simulation are used to estimate the local time derivatives
of the macroscopic observables (e.g. of
the void $CDF$); these estimates are used to ``project'' the
void $CDF$ ``far" into the future.
%
%Saving of computational time 
%results for problems characterized with separation
%of time scales between fast and slow dynamics.
%
The simplest such algorithm is projective forward Euler;
see 
\cite{Manifesto} for multi-step and implicit coarse projective integration. 
%
%Once again, the important feature is that the
%microscopic simulations used to estimate the coarse time
%derivatives are {\it discarded}, and new simulations are
%started at the {\it projected} macroscopic state in the future.
%
One does not evolve the microscopic problem; one evolves the
coarse-grained closure of the microscopic problem, which is
postulated to exist, but is not explicitly available.
%
Here we apply coarse projective integration {\it backward in time} 
(as discussed in \cite{GearReverse} this
is appropriate for systems with large time scale separation
between fast stable dynamics and slow dynamics).
%
%In our case, we can use reverse coarse projective integration
%to enable our {\it forward in time} kMC to study the void $CDF$ 
%evolution {\it backward} in time.
%
Starting with the self-similar $CDF$ at some
(relatively large) density (see Fig. \ref{fig4} and its insets) we
lift to microscopic realizations, evolve briefly {\it forward} in time, 
and estimate the {\it rate} of void $CDF$ evolution.
%
\begin{figure}[htbp]
\centering
\leavevmode
\includegraphics[height = 2.6in]{fig4.eps}
\caption{Reverse coarse projective integration: the dotted pink line 
is the forward estimation of the rate; the solid blue line is the projection backward in time.
% 
The upper inset shows (in the same color code as the large dots in the figure)
the CDF profile at the beginning and the end of the overall run.}
\label{fig4}
\end{figure}

This rate (estimated from the short pink run)
is used to project the void $CDF$ for a (longer) interval
{\it backward} in time (in blue).
%
The forward simulations are then discarded, and
new simulations are initialized at the ``earlier'' void $CDF$.
%
Repeating the procedure clearly shows an acceleration in backward 
time as the density becomes lower; the probability of large voids
becomes increasingly larger, the backward dynamics
evolves faster and faster and their fluctuations intensify,
making rate estimation from the short forward runs difficult.
%
Fig. \ref{fig4} seems to suggest a 
``reverse finite time" rate singularity. 
%
This is only a visual suggestion, however; as the density becomes
lower, 
%our {\it lifting} procedure (i.e. the construction of distributions 
%with prescribed density) will give rise to voids larger than 1 with
%increasing frequency. At these conditions, 
we do not expect an evolution
equation for the void CDF to provide a good model.
%
More detailed physical modeling of the void-filling process will
be required for the study of this backward dynamics at lower densities
to become meaningful.
%


It is this ``visual suggestion" of a backward explosion of the density
evolution rate that suggested fitting the data as a function of $t+t_0$;
we find that, if we initialize with the self-similar $CDF$ (at whatever
density), evolution data can be fitted almost perfectly with a straight
line {\it for the appropriate $t_0$} in Fig. \ref{fig1}.
%
Based on Eq. (\ref{delta}), the value of the term $B$ from all
of our trajectories suggests that at time $t+t_0 = 1$ the self-similar
$CDF$ density is $\sim 0.60$; that is, $t_0 - 1$ 
is the time that it
takes for a simulation {\it initialized with the self-similar void $CDF$
at density 0.60}  to evolve to the initial density of our computational 
experiments (our $t=0$).
%
When the initial condition is {\it different} than the self-similar 
$CDF$ (e.g. if it is a uniform distribution, see the dash-dot line in
Fig. \ref{fig1})  finding a good $t_0$
{\it does not} collapse the entire transient on a straight line -- we see,
however, that
the solution asymptotically approaches the self-similar one (dashed line).
%

In summary, in this paper we implemented equation-free, coarse-grained dynamic
renormalization (simulation forward and backward in time, as well
as fixed point computations) to study the evolution of the void $CDF$
for a model glassy compaction problem; we found this evolution to
be governed by apparent asymptotic self-similarity over the range of densities
we could reliably compute.
%
The procedure is, in principle,
capable of converging to both stable {\it and unstable} self
similar solutions, find similarity exponents when they exist,
as well as to quantify the fixed point stability
(by using matrix-free iterative
eigensolvers like Arnoldi).
%
%Upon convergence of the fixed point iteration, matrix-free iterative
%eigensolvers (like Arnoldi) can approximate the {\it coarse}
%eigenvalues and eigenvectors of the dynamic renormalization
%flow, quantifying fixed point stability.
%
In a continuation  / bifurcation context the algorithms can
be used to track 
self-similar solutions in parameter space, detect their
losses of stability and bifurcations; 
%and follow 
%new-born branches (see, for example, \cite{...}); 
of particular interest is the {\it onset}, in parameter
space, of self-similar dynamics, which will appear in our
formulation as a fixed point bifurcation \cite{template}.
%
Finally, coarse projective
integration (forward or reverse) can be performed in physical space 
(void $CDF$ evolution in time) or 
%-more appropriately for self-similar 
%problems, 
in renormalized space (renormalized
void $CDF$ evolution in logarithmic time).
%
We believe that the bridging of continuum numerical techniques with microscopic
simulations we illustrated here may be helpful in the coarse grained study  of glassy dynamics
through atomistic/stochastic models, especially when the dynamics depend
on {\it parameters} of the microscopic rules.


{\it Acknowledgments} IGK acknowledges the support of a NSF-ITR grant.
PGK acknowledges the support of NSF-DMS-0204585, NSF-CAREER and the 
Eppley Foundation for Research. 
 

%\bibliography{glass}
\begin{thebibliography}{99}

\bibitem{pablo} P.G. Debenedetti,
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(Princeton, 1996); P.G. Debenedetti and F.H. Stillinger,
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\bibitem{chicago} J.B. Knight {\it et al.}, Phys. Rev. E {\bf 51}, 3957 
(1995); E.R. Nowak {\it et al.}, Phys. Rev. E {\bf 57}, 1971 (1998);
E.R. Nowak {\it et al.}, Powder Technol. {\bf 94}, 79 (1997); see also
the review by H.M. Jaeger {\it et al.}, 
%S.R. Nagel and R.P. Behringer, 
Rev. Mod.
Phys. {\bf 68}, 1259 (1996);
%\bibitem{bideau} 
P. Philippe and D. Bideau,
Europhys. Lett. {\bf 60}, 677 (2002).

\bibitem{fa} G.H. Fredrickson and H.C. Andersen, 
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\bibitem{sollich} See e.g., P. Sollich and M.R. Evans, Phys. Rev. Lett.
{\bf 83}, 3238 (1999).

\bibitem{g67} W. G{\"o}tze and L. Sj{\"o}gren,
\newblock Z. Phys. B {\bf 65}, 415 (1987);
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\newblock Phys. Rev. A {\bf 29}, 2765 (1984); 
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%W. G{\"o}tze and A. Sj{\"o}lander,
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\bibitem{sciortino} E. La Nave {\it et al.}
%S. Mossa, and F. Sciortino
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%P.G. Debenedetti, and A.Z. Panagiotopoulos
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\bibitem{PLA} P. G. Kevrekidis {\it et al.}, 
%S. Kumar and I.G.Kevrekidis,
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%; also cond-mat/0302102.


%\bibitem{torquato} S. Torquato, 
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\bibitem{template}
D. G. Aronson {\it et al.}, 
%S. Betelu and IGK, November 2001, 
nlin.AO/0111055; 
C. Siettos {\it et al.}, 
%I.G.K. and P. G. Kevrekidis, 
Nonlinearity, {\bf 16} 497 (2003);
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%I.G.K., J. E. Marsden and K. Lust, 
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\bibitem{Manifesto}
I. G. Kevrekidis {\it et al.}, 
%C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and K. Theodoropoulos,
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%, also physics/0209043.
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\bibitem{JNNFM}
L. Chen {\it et al.}, 
%P. G. Debenedetti, C. W. Gear and I. G. Kevrekidis, 
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\bibitem{Papanicolaou} see e.g. M.J. Landman {\it et al.},
Phys. Rev. A {\bf 38}, 3837 (1988); B.J. LeMesurier {\it et al.},
Phys. {\bf 31D}, 78 (1986); {\it ibid.} {\bf 32D}, 210
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\bibitem{Kelley_Saad}
C. T. Kelley, {\it Iterative Methods for Linear and Nonlinear Equations (Frontiers in Applied Mathematics, Vol. 16)}, SIAM, Philadelphia, 1995;
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\bibitem{GearReverse}
C. W. Gear and I.G. Kevrekidis, Phys. Lett. A {\bf 321}, 335 (2004).
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%, also physics/0212108.

\end{thebibliography}

\end{document}
cond-mat0412773.extracted_bib
\begin{thebibliography}{99}

\bibitem{pablo} P.G. Debenedetti,
{\it Metastable Liquids, Concepts and Principles}, Princeton Univ. Press
(Princeton, 1996); P.G. Debenedetti and F.H. Stillinger,
\newblock Nature {\bf 410}, 259 (2001). 

\bibitem{chicago} J.B. Knight {\it et al.}, Phys. Rev. E {\bf 51}, 3957 
(1995); E.R. Nowak {\it et al.}, Phys. Rev. E {\bf 57}, 1971 (1998);
E.R. Nowak {\it et al.}, Powder Technol. {\bf 94}, 79 (1997); see also
the review by H.M. Jaeger {\it et al.}, 
%S.R. Nagel and R.P. Behringer, 
Rev. Mod.
Phys. {\bf 68}, 1259 (1996);
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\end{thebibliography}