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\DOI{10.5802/crphys.234}
\datereceived{2024-07-04}
\daterevised{2025-01-10}
\dateaccepted{2025-01-17}
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\dateposted{2025-03-07}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{On the ``mosaic'' picture of liquids and glasses}

\alttitle{Sur l'image en \og mosa{\"i}que \fg{} des liquides et des verres}

\dedicatory{Dedicated to the memory of G\'erard Toulouse}

\author{\firstname{Jorge} \lastname{Kurchan}\CDRorcid{0000-0001-5367-3688}}
\address{Laboratoire de Physique de l'\'Ecole Normale Sup\'erieure,
ENS, Universit\'e PSL, CNRS, Sorbonne Universit\'e, Universit\'e de
Paris, F-75005 Paris, France}
\email{kurchan.jorge@gmail.com}

\keywords{\kwd{Order}\kwd{Solidity}\kwd{Configurational entropy}\kwd{Glasses}}

\altkeywords{\kwd{Ordre}\kwd{Solidit\'e}\kwd{Entropie configurationnelle}\kwd{Verres}}

\begin{abstract}
Supercooled liquids are sometimes described as being composed of a
mosaic of patches that may be listed in a ``library'', each one having
some form of non-periodic order. Looking closer, one finds this
\mbox{construction} elusive. In attempting to give the notion of mosaic
a precise sense, we find that we are inevitably led to the construction
of a procedure for compressing the information in the particle
configuration, essentially the same as that used for texts. The amount
of stored information needed by an optimal \mbox{encoding} directly
defines the configurational entropy. A solid, in this view, is a
particle arrangement described by a low amount of information, that can
only flow by breaking into uncorrelated pieces, thus increasing
its{\break} complexity.
\end{abstract}

\begin{altabstract} 
Les liquides surfondus sont parfois d\'ecrits comme \'etant compos\'es
d'une mosa\"ique de fragments pouvant \^etre r\'epertori\'es dans une
\og biblioth\`eque \fg, chacun ayant une
forme d'ordre non p\'eriodique. En examinant de plus pr\`es, cette
construction se r\'ev\`ele insaisissable. En tentant de donner un sens
pr\'ecis \`a la notion de mosa\"ique, nous sommes in\'evitablement
conduits \`a la construction d'une proc\'edure pour compresser
l'information dans la configuration des particules, essentiellement la
m\^eme que celle utilis\'ee pour les textes. La quantit\'e
d'information n\'ecessaire pour un encodage optimal d\'efinit
directement l'entropie configurationnelle. Un solide, dans cette
perspective, est une disposition de particules d\'ecrite par une faible
quantit\'e d'information, qui ne peut circuler qu'en se fragmentant en
pi\`eces non corr\'el\'ees, augmentant ainsi sa complexit\'e.
\end{altabstract}

\maketitle
{\vspace*{-.6pc}}

\end{noXML}

\section{Introduction}

As we cool a liquid to lower and lower temperatures, the viscosity goes
up dramatically, and some two-time correlations develop a very slow
relaxation. Perhaps there is a divergence at some temperature---this
would be the ``ideal glass transition'': an interesting question that
will not be our main concern here, and whose answer we are far from
knowing. ``Ideal'' then means that the timescales are ideally infinite,
and the system is a true solid, rather than being one for all practical
purposes.

The development of solidity, even an imperfect one as in a supercooled
liquid, is a highly nontrivial phenomenon~\cite{anderson1972more}. To
convince oneself of this, it is better to consider particles that have
no hard core and are thus interpenetrable. Now, these are known to form
both crystalline solids and glasses, showing that the image of a solid
being made of particles ``sitting upon each other''{\break} is
misleading, and that something more subtle is at work. What is
happening is an order that develops, involving more and more particles
(see Figures~\ref{shear} and~\ref{budda}).

\begin{figure}
\includegraphics{fig01}
\caption{\label{shear}Thanks to its order, a sheared crystal made of
soft particles develops infinite energy barriers in the thermodynamic
limit. Here the sketch shows one such barrier, though certainly not the
lowest nor the one the system takes.}
{\vspace*{.8pc}}
\end{figure}

\begin{figure}
\includegraphics{fig02}
\caption{\label{budda}A solid sustains shear thanks to (some for of)
order.}
\end{figure}

A way to formalize this idea is to consider the point-to-set
construction~\cite{biroli2004diverging, bouchaud2004adam,
montanari2006rigorous}, as in Figure~\ref{pts}. We consider a
supercooled liquid. At some time we freeze (in the computer) all
particles around a sphere of radius $R$ and we reshuffle the particles
inside. If, upon letting them relax, they come back to a configuration
close to the original one, this tells us that the boundary determines
the interior. We repeat for larger $R$ up to the point in which this no
longer happens: the maximal length obtained this way is the
``point-to-set length'' $R_{\mathrm{ps}}$. Now, it is clear that if the
particles are soft, the barrier heights the system can construct at
finite $R_{\mathrm{ps}}$ are bounded by some function of
$R_{\mathrm{ps}}$~\cite{bouchaud2004adam, biroli2004diverging,
montanari2006rigorous}. Our question is then to what extent does
$R_{\mathrm{ps}}$ grow, and, a more academic one, if there is a true
divergence.

\begin{figure}
\includegraphics{fig03}
{\vspace*{-.2pc}}
\caption{\label{pts}The point-to-set construction: the outside is
frozen, the cavity is rethermalized.}
{\vspace*{.5pc}}
\end{figure}

This kind of ideas has naturally led to the picture in which a
supercooled liquid---and its out of equilibrium version, a glass---are
a mosaic~\cite{lubchenko2007theory, xia2000fragilities,
xia2001microscopic} of patches of ordered particles, ``ordered'' in the
sense, for example, that they are the interior of a point-to-set cavity
constructed above. These are not all equal up to rotations, as we would
have in an assembly of microcrystals, and therein lies the problem.

I will first describe how one makes a mosaic model meaningful. For
simplicity, on a first approach (Section~\ref{mosaic}) I will neglect
the important question of when do we consider two patches to be the
same---what precision we require---and how do we factor out the rapid
thermal fluctuations (see Figure~\ref{vibrations}). These issues will
be dealt with in Section~\ref{temp}.

\begin{figure}
\includegraphics{fig04}
{\vspace*{-.2pc}}
\caption{\label{vibrations}Low energy excitations around an ordered
system involve vibrations as well as small topological excitations that
do not destroy order.}
\end{figure}

The approach described here, and the inspiration coming from the Random
First Order scenario~\cite{kirkpatrick1987p, kirkpatrick1989scaling,
wolynes1997entropy, xia2000fragilities, xia2001microscopic,
lubchenko2007theory} takes us directly to a spatial characterization of
effective temperatures~\cite{cugliandolo1997energy}
(\mbox{Section}~\ref{renyi}) and the Parisi--Monasson $X$
parameter~\cite{monasson1995structural} in terms of the Renyi
entropies that measure which ordered patches are typical and which are
rare. It turns out that a supercooled liquid has the whole repertoire
of ordered patches, some more rare and some more common, as
temperature (or time, in an aging system) is changed, rare patches
become typical and typical rare.\looseness=-1

This overview is based on work with Dov Levine and Ittai
Fraenkel~\cite{kurchan2009correlation, kurchan2010order,
fraenkel2024information}.

\section{Mosaics} \label{mosaic}
\subsection{A misleading picture} \label{sec2.1}

There is the temptation to think of mosaics as follows. Consider a
$p$-state Potts model, and let us call its $p$ possible ordered phases
red, blue, green, etc. A mosaic then looks as 
Figures~\ref{microcrystal},~\ref{Potts}.
There is a large correlation inside each region, but their boundaries
are disordered in the sense that the fracture lines are randomly
placed.

\begin{figure}
\includegraphics{fig05}
\caption{\label{microcrystal}Top left: a crystal, right: three
microcystallites. Below: a tile, two tiles make a new tile, two
fragments at an angle do not.}
{\vspace*{.5pc}}
\end{figure}

\begin{figure}
\includegraphics{fig06}
{\vspace*{-.15pc}}
\caption{\label{Potts}An ordered Potts model, its states and a mosaic
of them.}
\end{figure}

We may construct a ``library'' with exactly $p$ states and thus
classify each ``tile'' of the mosaic according to its color. Here we
need to pause and think: what we are storing in the library is an
infinite---e.g.\ all-green---configuration, and we are recognizing a
tile by it being a fragment of these infinite---idealized---states.

\subsection{A better one} \label{ff}

The pitfalls of the previous picture become evident as soon as we
consider any form of non-periodic order. The example of a
quasicrystal~\cite{shechtman1984metallic, levine1984quasicrystals} is
enough for our purposes here.

A quasicrystal is a nonperiodic arrangement; we give here the example
of a Penrose tiling~\cite{grunbaum1987tilings}---see
Figures~\ref{penrose00} and~\ref{boundary}. We may make one such
structure infinitely large, without incurring empty spaces, i.e.\ 
``defects''. Consistent with the ``point to set'' idea, appropriate
boundaries uniquely determine the interior.

\begin{figure} 
{\vspace*{2pt}}
\includegraphics{fig07}
{\vspace*{2pt}}
\caption{\label{penrose00}A Penrose tiling.}
{\vspace*{2pt}}
\end{figure}

\begin{figure}
\includegraphics{fig08}
{\vspace*{2pt}}
\caption{\label{boundary}Boundary determines interior in a crystal and
a quasicrystal.}
{\vspace*{2pt}}
\end{figure}

It is well known that in a large quasiperiodic tiling, patches of every
size repeat often~\cite{grunbaum1987tilings}---see Figure~\ref{star},
although in a manner that does not make the system periodic. This
rather surprising property is, in fact, a very general feature of any
nonperiodic arrangement: energetically ``convenient'' patches reappear
often.

\begin{figure}
\includegraphics{fig09}
\caption{\label{star}Frequently recurring motifs.}
\end{figure}

Now consider a two-tile system as in Figure~\ref{mismatched}, obtained
by sticking together incorrectly two random pieces. We immediately see
that our image of a ``library'' in the naive version of the previous
section breaks down: different fragments have many elements in common.
What are the two ``different colors''? What shall we store in our
library? 

A possible answer could be to spot the lines of defects. The problem is
that defects are not obviously lines (see Figure~\ref{defect}), and
furthermore, a patch near a defect may seem superficially ``right'',
but it inevitably leads to a defect: these are called ``deceptions'' in
the context of quasiperiodic tilings. Are deceptions ordered in spite
of not belonging to any perfect quasicrystal? Are they then a defect,
even if the obvious defect is not within them, and the energetic cost
is not yet paid?

\subsection{The structure of the library} \label{sec2.3}

What size are the patches in our library? Infinite, as in the case of
the Potts model above? Or finite, for example obtained by the
point-to-set procedure. Both possibilities seem problematic for 
constructing mosaics. 

Looking at Figure~\ref{composite}, we begin to see a way out. The
crucial idea is to give up attempting to store infinite configurations
in your library and to start constructing it bottom-up and
recursively\footnote{One can insist on discussing full amorphous order
in the large size limit, this requires doing it with care, introducing
the concept of ``metastate''. See below.}. In Figure~\ref{composite} we
see how a star shape is often repeated, although sometimes it is
continued into a ``gem'' and sometime into a ``spiky'' figure. The
situation is analogous to a dictionary where composite words are
allowed: from ``man'' we can go to ``manpower'' or ``manslaughter''.
Our dictionary should label man ${=}$ 1, slaughter ${=}$ 2, power ${=}$ 3,
full ${=}$ 4 and then manslaugher ${\equiv}$ 5 ${=}$ 1 ${+}$ 2, manpower
${\equiv}$ 6 ${=}$ 1 ${+}$ 3, powerful ${=}$ 7 ${=}$ 3 ${+}$ 4. If this article
were in German, we could go to the next steps and make composites of
composites.

\begin{figure}
{\vspace*{-2pt}}
\includegraphics{fig10}
{\vspace*{-2pt}}
\caption{\label{composite}The ``star'' may be continued into a ``gem''
or into a ``big star'', just as composing words.}
{\vspace*{-2pt}}
\end{figure}

\begin{figure}
\includegraphics{fig11}
\caption{\label{mismatched}Two mismatched fragments of a Penrose
tiling.}
\end{figure}

\begin{figure}
\includegraphics{fig12}
\caption{\label{defect}An defect (low, left) that can be only cured
with large rearrangements. The surroundings are what is called a
``deception'', despite their apparent lack of disorder, they don't
belong to an infinite perfect quasicrystal.}
\end{figure}

In this way, our library' contains building blocks, building blocks
composed of smaller building blocks, and so on. If we ever hit against
a defect, because of its random nature, it will be \mbox{seldom}
repeated in the same form; the analogy would be a word with a spelling
mistake, necessarily a rare occurrence. This will be more so the
larger the number of defects, as with two spelling mistakes in the same
word. 

There is nothing new in our construction: we are just following the
steps to devise a compression program. Many theorems have been proven
about the optimality of such a procedure{\break}
\cite{lempel1986compression, ziv1977universal, ziv1978compression,
ziv1993measure}, i.e.\ that our dictionary has all it needs and nothing
in excess. For example, for a perfect quasicrystal, we know that the
amount of information is polynomial in the number of tiles, and hence
the information content (its logarithm) is subextensive in the
large-size limit. Clearly, breaking this quasicrystal randomly into
many pieces, there are exponentially many ways of reassembling them,
and this can only be coded with an extensive information, scaling with
the exponential of the number of pieces.

What we have described here is perfectly applicable to liquids and
glasses, but only once we deal correctly with fast relaxations such as
vibrations and small defects.

Before closing this section, let us remark that there is an extensive
literature on the classification of the relevant mosaics for melts of
particles interacting via a spherically symmetric potential, see for
example the ``Topological Cluster Classification'' of
Ref.~\cite{malins2013identification}.

\section{Treating fast degrees of freedom correctly} \label{temp}

Let us now face the problem of ``factoring out'' fast events, including
vibrations. 

\subsection{Total entropy} \label{sec3.1}

We start from the standard construction for the entropy.\ We consider a
very large sample, and select many patches of size~$L$.\ We classify the
configurations according to a precision~$\epsilon$,{\break} as~in
Figure~\ref{entropy}. Our aim is to compute~$S$, or, more generally,
the Renyi entropies (for a detailed perspective, see
Ref.~\cite{ozawa2024perspective}). We consider a large sample
containing $M$ different patches $\mathbf{a}_{\mathbf{1}}, \ldots,
\mathbf{a}_{\mathbf{M}}$ each of volume~$V$. Denoting by
$n_a^{(V)}(\epsilon)$ the number of patches that coincide in this way
with the patch~$\mathbf{a}$, we have:
{\begin{equation} \label{GP3}
K_q \sim - \lim_{\epsilon\rightarrow 0} \lim_{V\rightarrow\infty}
\frac{1}{(q-1)V} \ln\left\{\frac{1}{M} \sum_a [n_a^{(V)}
(\epsilon)]^{(q-1)}\right\}
\end{equation}}\unskip
and in particular $S = \lim_{q \rightarrow 1} K_q$ is the usual
entropy. As usual with the definition of entropy, the coarse-graining
parameter $\epsilon$ affects the number of patches that do
coincide---and is zero when $\epsilon \rightarrow 0$---but one is saved
by the fact that this happens only in a subdominant way with respect to
the volume $S \sim Vs + a \ln \epsilon$ so the appropriate limit is
well-defined. 

\begin{figure}
\includegraphics{fig13}
{\vspace*{.4pc}}
\caption{\label{entropy}On a single large system, or on many
realizations of a small one---we compute the number of configurations,
equality being defined as up to a grid of precision~$\epsilon$.}
{\vspace*{-.5pc}}
\end{figure}

A choice of parameters in these equations is clearly important,
especially given that one works with a finite system. A practical
computational way to optimize this procedure and obtain a good limit
was devised by Grassberger and
Procaccia~\cite{grassberger1983measuring}, and is extensively used. It
is based on graphically detecting a scaling region, which makes
extrapolation possible. For the moment we do not have an application of
this procedure to an actual glass, the best we have to compute glass
entropies is Ref.~\cite{fraenkel2024information}, but the model used is
very schematic.

A repeating patch or subpatch will naturally make this measure smaller,
but aside from this it may not seem obvious how this quantity is
actually the one needed by a hierarchical library. This is indeed the
object of mathematical proofs~\cite{ziv1977universal,
ziv1978compression, ziv1993measure}.

\subsection{Configurational entropy} \label{sec3.2}

Solids in general, supercooled liquids, and glasses exhibit
timescale-separation: there is a fast dynamics involving vibrations or
nearby particle swaps, and then there are the rearrangements whose
timescale scales with the viscosity. For a crystal, this looks as in
Figure~\ref{vibrations}. The correlation functions, on the other hand, 
clearly show an extensive plateau that separate fast from~``$\alpha$''
scales, as shown in Figure~\ref{14}.

\begin{figure}
\includegraphics{fig14}
\caption{\label{14}Computing the ``vibrational'' and ``structural''
entropies only relies on timescale-separation.}
\end{figure}

The configurational entropy is a measure of entropy of all
configurations, but treats as identical all the class that are a fast
rearrangement away from each other. To compute and define complexity
properly, a strategy akin as the one used to detect common authorship
between books~\cite{benedetto2002language} can be employed: pick
configurations at $N$ different times within the plateau
(Figure~\ref{14}) and concatenate them into a a single figure. Then,
compute the system's entropy $S_N$ obtained this way. Comparing the
entropies of concatenated versus non-concatenated configurations we
have:
{\begin{equation}
S_N = NS_{\mathrm{fast}} + \Sigma; \quad S = S_{\mathrm{fast}}+\Sigma
\end{equation}}\unskip
which allows us to define the complexity $\Sigma$ as the mutual entropy
{\begin{equation}
\Sigma = \frac{NS - S_N}{(N-1)}.
\end{equation}}\unskip

The same definition may be used for the Renyi entropies. Also very
important: {\it we have never assumed equilibrium here}, so the
definition holds even for an aging glass.

\section{Physical situations and definitions} \label{definitions}

The mathematical literature discusses the definition of pure states in
equilibrium and sending the system size to infinity, a construction
that need be done with care. The procedure is the ``metastate
construction''~\cite{newman2022metastates, read2022complexity}. 

From a pragmatic standpoint, experiments involve quenching systems 
from high temperatures to a low (possible zero) temperature bath. At
all finite times the boundary conditions are irrelevant, provided they
are sufficiently (i.e.\ a few correlation lengths) far away, a condition
that can be checked at each step---the statistics are ideally obtained
by repeating the experiment. The complexity, and from it the typical
length, may be evaluated also in this out of equilibrium situation. We
will see the former decrease slowly, and possibly saturate at some
point in time. The question of the existence of an ideal glass thus
becomes straightforward: would this length continue to grow
indefinitely with more time, or does it always eventually reach a
saturation point? The reason for the stop of the growth could stem from
the system equilibrating at finite time, or the nucleation of a crystal
making the glass irrelevant, or yet that another form of transition
intervenes.

We have no answer to this question. However, from the physical reality
point of view one must realize that the lengths involved in real
glasses are exceedingly short; only a few molecules are correlated. The
estimates made in the point-to-set framework are done in polydisperse
systems by refilling the cavity with particles of the same sizes as the
original ones, rather than randomly sampling them from the
distribution. This is legitimate, but it clearly leads to longer PTS
lengths. On the other hand, freezing an out of equilibrium
configuration, the PTS length may be dramatically longer than the
entropy-related one for the same situation. One can easily understand
this in the case of hard spheres crunched to high density or in general
at very low temperatures: introducing the same particles, the chances
that another variant of the configuration contributing to equilibrium
inside the cavity are vanishing.

\section{The growth of correlations, Renyi complexities and effective
temperatures} \label{renyi}

A picture that has emerged of a glass, based on the mean field
solution~\cite{cugliandolo1993analytical, cugliandolo1997energy} but
also with important precedents at the phenomenological
level~\cite{tool1946viscosity}, is as follows. The system is taken at
high temperature and put in contact with a low temperature bath. The
landscape is composed of a large number of metastable states,
distributed in terms of their internal free energy as in
Figure~\ref{microcrystal}. As the liquid relaxes, it does so exploring the
states at a given level {\it democratically}: this remarkable fact can
be rationalized by realizing that every new relaxation is very much
slower than the preceding ones. The probability distribution of a state
``$a$'' is then approximated by
{\begin{equation} \label{monasson}
p_a \propto \mathrm{e}^{-\frac{1}{T_{\mathrm{eff}}} f_a}
\end{equation}}\unskip
where $T_{\mathrm{eff}}$ slowly decreases with time 
(see Figure~\ref{config}), until finally
reaching $T_{\mathrm{eff}}=T$. This is the construction of
Monasson~\cite{monasson1995structural}. Because fast relaxations occur
at the bath temperature, but slow ones are essentially thermalized at a
slowly decreasing effective temperature, this explains why, at low
frequencies the fluctuations and dissipations are
related~\cite{cugliandolo1997energy} by $T_{\mathrm{eff}}$ precisely 
(see the construction in Figure~\ref{microcrystal}).
Clearly, this is only
literally true for mean field systems, but there is abundant evidence
that the scenario holds, at least qualitatively, in realistic glasses.

\begin{figure}
\includegraphics{fig15}
\caption{\label{config}Effective temperature or replica parameter $X$
in an aging situation.}
\end{figure}

Now, comparing Equation~(\ref{monasson}) with the definition of Renyi
entropies~(\ref{GP3})---the latter being applicable to realistic
glasses---we see that the configurations that dominate the Renyi
entropies $K_q$ are those that dominate also at times such that:
{\begin{equation}
T_{\mathrm{eff}} = \frac{T}{q}.
\end{equation}}\unskip
This establishes a connection between the configurations that dominate
the Renyi entropies calculated in equilibrium, and those explored by
the dynamics on their way to equilibrium. ``Higher'' states dominate
for small~$q$, but these are those that have higher configurational
entropies, and hence smaller mosaics.

\section{Conclusion}

In conclusion, thinking in terms of information and information storage
is not an elegant curiosity, but rather a necessity for the description
of liquids and glasses. Part of the interest of this approach is that
it exempts us from defining ``boundaries'' between patches that are
themselves disordered, a rather perplexing endeavor.

These ideas have yet to be applied fully for the case of true glasses.
It remains to be seen if they will remain a matter of principle or
develop into a practical tool.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Acknowledgments}
An overview based on work in collaboration with Ittai Fraenkel and Dov
Levine.

\back{}

\printbibliography
\refinput{crphys20240547-reference.tex}

\end{document}
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