Coarse-graining and symplectic non-squeezing

NIKOLAOS KALOGEROPOULOS †

Center for Research and Applications
of Nonlinear Systems (CRANS),
University of Patras, Patras 26500, Greece.

Abstract

We address aspects of coarse-graining in classical Statistical Physics from
the viewpoint of the symplectic non-squeezing theorem. We make some comments
regarding the implications of the symplectic non-squeezing theorem for the
BBGKY hierarchy. We also see the cubic cells appearing in coarse-graining as a
direct consequence of the uniqueness of Hofer’s metric on the group of
Hamiltonian diffeomorphisms of the phase space.

Keywords: Entropy, Coarse-graining, Symplectic non-squeezing, BBGKY hierarchy,
Hofer’s metric.

† Electronic mail<EMAIL_ADDRESS>

##  Introduction

The concept of entropy is fundamental in Statistical Physics. Despite its long
history, and its many successes along the way, many of its aspects are still
not very well understood, and are sometimes grossly misunderstood, at both
conceptual and calculational level. This becomes obvious if one even skims the
contemporary literature. Controversies about entropy are always simmering,
periodically resurfacing, but never seem to get settled, or even come close to
reaching a consensus. Such issues pertinent to entropy and its dual intensive
quantity in thermodynamics, namely the temperature, have come up recently in
the context of conjectured negative absolute temperatures and dark matter,
(in-)equivalence of the classical ensembles, as well as due to the continuing
studies of non-equilibrium processes, among many other motivations. For some
relatively recent views on these matters, see for example [1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

In this work, we point out some issues related to the foundational, but still
not well-understood issue of coarse graining, from the viewpoint of the
symplectic non-squeezing theorem. The latter is a fundamental result in
symplectic geometry, whose potential significance for Statistical Mechanics
has not been fully explored yet. We follow a Hamiltonian approach ignoring the
role that any metrics on phase space may play, as much as possible. In Section
2, we present some well-known facts in symplectic geometry needed in the
sequel. In Section 3, we comment on the relation of the symplectic non-
squeezing theorem to the BBGKY hierarchy. In Section 4, we make some general
comments about coarse-graining. In Section 5, we present an argument
purporting to explain why cubes are the appropriate cells in which to divide
the phase space during coarse-graining. Section 6 contains some comments on
the relation between the symplectic and the metric structures which may be of
some interest to Statistical Mechanics and may be the motivation for future
work in the metric approach to determining the statistical properties of
systems of many degrees of freedom.

##  Symplectic Preliminaries

Let us consider a classical particle system having $n$ degrees of freedom in
3-dimensional space. Such a system is usually analyzed either in a Lagrangian
or in a Hamiltonian way. The Hamiltonian approach has the advantage over the
Lagrangian one that it allows the use of additional symmetries existing
between the canonical positions and momenta which may not be manifest in the
Lagrangian description. Such symmetries are symmetries of the underlying
system in phase space and are central in determining aspects of coarse
graining, as will be elaborated upon in the sequel. Hence we proceed in the
present work following the Hamiltonian approach. We use [18] as our basic
reference for symplectic geometry in the sequel. Let the microscopic
Hamiltonian of the system under study be denoted by $\mathcal{H}$ and the
phase space $\mathcal{M}$ be parametrized by canonical coordinates
$(q_{i},p_{i}),\ i=1,\ldots,3n$. The phase space $\mathcal{M}$ is naturally
endowed with a symplectic structure $\omega_{0}$ which is locally expressed,
according to Darboux’s theorem, as

$\omega_{0}=\sum_{i=1}^{3n}dq_{i}\wedge dp_{i}$ (1)

Symplectic manifolds such as $(\mathcal{M},\omega_{0})$ are devoid of any
local geometric structure. Indeed all such symplectic manifolds of the same
dimension are locally diffeomorphic to $(\mathbb{R}^{6n},\omega_{0})$. This
statement is intimately related to the fact that the group of symplectic
diffeomorphisms/canonical transformations is infinite dimensional, in sharp
contrast to the group of isometries of a Riemannian manifold. As a result, all
symplectic manifolds of the same dimension are locally “the same”. As long as
they are of the same dimension, such manifolds only differ from each other in
their global aspects, which are encoded in their topology. For this reason,
the existence of the canonical symplectic structure $\omega_{0}$ alone on
$\mathcal{M}$ may be considered to be too “rigid” for an effective description
of the behavior of the associated physical system. After all, the overwhelming
majority of physical theories do have local degrees of freedom which
distinguish one from another.

If we want to remain within the realm of symplectic manifolds for Hamiltonian
systems, without assuming an additional structure on the phase space
$\mathcal{M}$ such as a metric, or some conditions such as integrability of
the associated almost complex structure, then we can proceed as follows. We
know that Hamiltonian flows preserve the symplectic volume: this is
Liouville’s theorem. Its immediate result is that the Gibbs entropy of an
isolated system remains constant if the system is described by a Hamiltonian
flow. But, symplectic maps turned out to have a more fundamental property
which distinguishes them from volume-preserving maps: the symplectic non-
squeezing property and its generalizations. There are symplectic invariants
called “symplectic capacities” [18, 22] which, incidentally, have nothing to
do with the volume of the underlying manifold, which volume-preserving flows
do not possess. The non-squeezing property essentially quantifies that the
symplectic maps are more “rigid” than the volume-preserving ones, and that the
former are a proper subset of the latter. To be more precise, it establishes
that the set of symplectic diffeomorphisms of $\mathbb{R}^{2n}$ is
$C^{0}$-closed in the set of all diffeomorphisms of $\mathbb{R}^{2n}$, rather
than its closure being the set of volume-preserving diffeomorphisms of
$\mathbb{R}^{2n}$ as one might possibly have expected.

The symplectic non-squeezing theorem was first proved by M. Gromov in [19],
subsequent proofs are given by [20, 21, 22], and since that time it has been
the foundation of an explosive growth in the field of symplectic geometry. To
formulate it, let $\|x\|$ indicate the Euclidean norm of $x\in\mathbb{R}^{2n}$
and let

$B_{2n}(r)=\left\\{x\in\mathbb{R}^{2n}:\ \|x\|\leq r\right\\}$ (2)

denote the closed ball of radius $r$ centered at $0\in\mathbb{R}^{2n}$. Let

$Z_{2n}(r)=B_{2}(r)\times\mathbb{R}^{2n-2}=\left\\{x\in\mathbb{R}^{2n}:q_{i}^{2}+(p_{1})^{2}\leq
r^{2}\right\\}$ (3)

be the cylinder in $\mathbb{R}^{2n}$ whose base lies in the symplectic 2-plane
$(q_{1},p_{1})$ having base radius $r$. If there exists a symplectic embedding
$B_{2n}(r)\hookrightarrow Z_{2n}(R)$, then $r\leq R$. This “non-squeezing
theorem” is not valid if the base of the cylinder lies on an isotropic
2-plane, such as $(q_{1},q_{2})$. The theorem was extended by F. Lalonde and
D. McDuff [23] to any symplectic manifold. During the last two decades,
several more rigidity results have been obtained for symplectic embeddings
[24], of mostly 2- or 4-dimensional symplectic objects having a simple
geometry onto similar classes of objects or their stabilized analogues. From
the viewpoint of Statistical Mechanics, it would be desirable to have similar
results on embeddings of high dimensional symplectic manifolds, which are not
stabilized versions of lower dimensional ones. However, no such results of
potential physical interest have been obtained, so far as the author knows.

The symplectic non-squeezing theorem can be reformulated as follows [25]. Let
$\mathbb{P}$ denote the orthogonal projection, with respect to the Euclidean
metric, of an object in $\mathbb{R}^{2n}$ onto the 2-dimensional symplectic
plane $(q_{1},p_{1})$ and let ${\sf Vol}_{2}$ indicate the symplectic volume
(“area”) of a 2-dimensional set on the $(q_{i},p_{i}),\ i=1,\ldots,3n$ plane.
Consider a symplectic embedding
$\varphi:B_{2n}(r)\hookrightarrow\mathbb{R}^{2n}$. Then

${\sf Vol}_{2}(\mathbb{P}\varphi(B_{2n}(r))\geq\pi r^{2}$ (4)

Moreover, it is known [26] that analogues of (4) exist for intermediate
dimensions, namely that

${\sf Vol}_{2k}(\mathbb{P}\varphi(B_{2n}(r)))\geq\gamma_{2k}r^{2k},\hskip
42.67912pt1\leq k\leq n$ (5)

but only for linear symplectic automorphisms
$\varphi:\mathbb{R}^{2n}\rightarrow\mathbb{R}^{2n}$ onto complex linear
subspaces of $\mathbb{R}^{2n}$ of real dimension $2k$. In (5), $\gamma_{2k}$
stands for the volume of the $2k$-dimensional ball $B_{2k}(1)$ of unit radius.
Such a result does not hold for general, non-linear, symplectic maps [27, 26].
Indeed, for every $\varepsilon>0$ there exists a smooth symplectic embedding
$\varphi:B_{2n}(1)\rightarrow\mathbb{R}^{2n}$ such that

${\sf Vol}_{2k}(\mathbb{P}\varphi(B_{2n}(1)))<\varepsilon$ (6)

Hence, the intermediate dimensional non-squeezing theorem does not hold
anymore when passing from linear to non-linear symplectic maps $\varphi$. The
transition from linear to non-linear symplectic maps resulting in the
transition from the validity to the non-validity of the symplectic non-
squeezing theorem for intermediate dimensions was examined in detail in [26].

The conclusion of the symplectic non-squeezing theorem does not hold, either
for linear or non-linear symplectic maps, if the word “projections” is
replaced by the word “sections” in the above statements. Due to (5), (6), we
only have to formulate the case for sections for $\varphi$ being a linear
symplectic map. Let $\mathcal{V}\subset\mathbb{R}^{2n}$ be a complex linear
subspace of real dimension $2k,\ 1\leq k\leq n$. Then

${\sf Vol}_{2k}(\mathcal{V}\cap\varphi(B_{2n}(r)))\leq\gamma_{2k}r^{2k}$ (7)

and the equality holds if and only if the linear subspace
$\varphi^{-1}(\mathcal{V})$ is complex.

Despite its foundational role in geometry, the symplectic non-squeezing
theorem, further developments stemming from it, and its applications in
Mathematics, have been largely ignored by the Physics community. An exception
to this is the string theory community where the ideas contained in [19],
which gave rise to the various versions of Floer homology and quantum
cohomology, are used in a variety of contexts, not the least of which are the
various statements associated with mirror symmetry [28]. Another exception has
been the work of M. de Gosson who has strongly advocated for the use of the
symplectic non-squeezing theorem in Physics in general, but especially in the
context of the semi-classical limits of quantum systems [29, 30, 31, 32, 33,
34, 35, 36, 37, 38] during the last two decades. Being influenced by his work,
we also presented some views on the possible implications of the non-squeezing
theorem for the macroscopic time irreversibility of Hamiltonian systems in
[39, 40]. These contributions illustrate the viewpoint which is adopted and
has largely motivated the present work.

##  Symplectic non-squeezing and the BBGKY hierarchy

One could argue that for a systems with many degrees of freedom which are
modelled by flows on high-dimensional manifolds like $\mathcal{M}$, statements
like (4) which involve projections onto 2-dimensional symplectic planes cannot
really be of any significance. In other words, for a Hamiltonian evolution in
a high dimensional phase space $\mathcal{M}$ (4) is too weak of a constraint,
or provides too small a “rigidity” due to its dimension alone, to be of any
significance in determining the macroscopic properties of the underlying
system.

One could start uttering some objections this viewpoint, which is very
reasonable, by invoking the following argument. As is well known, the behavior
of a set can be determined either directly, geometrically, or by determining
the behavior of appropriate sets of functions defined on it. The most obvious
such choice is the characteristic function of the set. Micro-canonical
probability distributions of isolated systems are uniform on the sets of
constant energy, therefore they are constant multiples of such characteristic
functions. So, instead of working directly with sets, one could as well work
with the micro-canonical probability densities.

In this spirit, consider dual description of a phase space and its subsets, in
terms of probability distributions defined on them. The Bogoliubov-Born-Green-
Kirkwood-Yvon (BBGKY) hierarchy which is extensively used in the derivation of
kinetic equations for particle systems describes the time evolution of the
marginal distribution functions of such particle systems, starting from
Liouville’s equation for the distribution function of all particles in the
system and gradually descending to marginals of fewer and fewer particles
until it reaches the distribution function of a single particle [41]. To be
more concrete, let $f_{n}(q_{1},p_{1};q_{2},p_{2};\ldots;q_{n},p_{n};t)$
denote the joint probability density function describing the behavior of all
$n$ particles in the system under study. Since the system is not assumed to be
in equilibrium, then $f_{n}$ can explicitly depend on some variable $t$
describing evolution, let us call it “time”. Since $f_{n}$ is a probability
density it is normalized as

$\int_{\mathcal{M}}\ f_{n}(q_{1},p_{1};q_{2},p_{2};\ldots;q_{n},p_{n};t)\
d{\sf Vol}_{6n}$ (8)

where the volume of the phase space $\mathcal{M}$ is

$d{\sf Vol}_{6n}=\prod_{i=1}^{n}\ d^{3}q_{i}\ d^{3}p_{i}$ (9)

For most macroscopic systems $n\sim 10^{23}$. Ideally, we would like to be
able to determine $f_{n}$ since knowing it would allow us to calculate any
quantity of physical interest related to the system, as well as the evolution
of any such quantity. Of course this is not possible practically, nor is
necessarily required for an effective macroscopic description of the system,
so we have to, and we can, settle with less. Liouville’s theorem is the
statement that

$\frac{df_{n}}{dt}=0$ (10)

namely that the co-moving volume of a system on $\mathcal{M}$ remains
invariant under its Hamiltonian evolution. This theorem does not state
anything, however, about the evolution of the exact shape of this volume,
something which is left to be found as a solution of Hamilton’s equations, or
equivalently, it is determined by the evolution generated by the Hamiltonian
vector field $X_{\mathcal{H}}$. Such a vector field $X_{\mathcal{H}}$
associated to the Hamiltonian function $\mathcal{H}$ is defined by requiring
it to obey

$\omega_{0}(X_{\mathcal{H}},\cdot)=d\mathcal{H}$ (11)

The Poisson bracket between two vector fields $X_{\mathcal{A}},\
X_{\mathcal{B}}$ associated to the scalar functions $\mathcal{A},\
\mathcal{B}:\mathcal{M}\rightarrow\mathbb{R}$ respectively, is defined by

$\left\\{\mathcal{A},\mathcal{B}\right\\}=\omega_{0}(X_{\mathcal{A}},X_{\mathcal{B}})$
(12)

The one-particle probability density is defined as

$f_{1}(q_{1},p_{1};t)\ =\ n\int_{\mathcal{V}_{n-1}}\
f_{n}(q_{1},p_{1};q_{2},p_{2};\ldots;q_{n},p_{n};t)\ \prod_{i=2}^{n}\
d^{3}q_{i}\ d^{3}p_{i}$ (13)

which is normalized as

$\int_{\mathcal{V}_{1}}\ f_{1}(q_{1},p_{1};t)\ d^{3}q_{1}\ d^{3}p_{1}\ =\ n$
(14)

with $\mathcal{V}_{n-1}$ being the submanifold of $\mathcal{M}$ which is
parametrized by $(q_{2},p_{2};\ldots;q_{n},p_{n})$ and $\mathcal{V}_{1}$ being
the submanifold of $\mathcal{M}$ which is parametrized by $(q_{1},p_{1})$. In
a similar manner, and for identical particles, one defines the 2-particle
distribution function by

$f_{2}(q_{1},p_{1};q_{2},p_{2};t)\ =\ n(n-1)\int_{\mathcal{V}_{n-2}}\
f_{n}(q_{1},p_{1};q_{2},p_{2};\ldots;q_{n},p_{n};t)\ \prod_{i=3}^{n}\
d^{3}q_{i}\ d^{3}p_{i}$ (15)

and, more generally, the marginal $k$-particle distribution function for
$1\leq k\leq n$ is defined as

$f_{k}(q_{1},p_{1};q_{2},p_{2},\ldots;q_{k},p_{k};t)\ =\
\frac{n!}{(n-k)!}\int_{V_{n-k}}\
f_{n}(q_{1},p_{1};q_{2},p_{2};\ldots;q_{n},p_{n};t)\ \prod_{i=k}^{n}\
d^{3}q_{i}\ d^{3}p_{i}$ (16)

Let us assume that the Hamiltonian of the particle system is

$\mathcal{H}\ =\
\sum_{i=1}^{n}\frac{p_{i}^{2}}{2m_{i}}+\sum_{i<j}U(q_{i}-q_{j})$ (17)

where $U$ stands for the potential energy between pairs of particles. We
ignore contributions to $U$ from triplets and higher numbers of particles.
Then the evolution of the marginal distribution $f_{1}$ is given by

$\frac{\partial f_{1}(q_{1},p_{1};t)}{\partial t}\ =\
\left\\{\mathcal{H}_{1},f_{1}\right\\}+\int\frac{\partial
U(q_{1}-q_{2})}{\partial q_{1}}\cdot\frac{\partial f_{2}}{\partial p_{1}}\
d^{3}q_{2}\ d^{3}p_{2}$ (18)

where

$\mathcal{H}_{1}=\frac{p_{1}^{2}}{2m_{1}}$ (19)

and $\cdot$ indicates the inner product on $\mathcal{M}$. In general

$\frac{\partial f_{k}}{\partial t}\ =\
\left\\{\mathcal{H}_{k},f_{k}\right\\}+\sum_{i=1}^{k}\int\frac{\partial
U(q_{i}-q_{k+1})}{\partial q_{i}}\cdot\frac{\partial f_{k+1}}{\partial p_{i}}\
d^{3}q_{k+1}\ d^{3}p_{k+1}$ (20)

where

$\mathcal{H}_{k}\ =\ \sum_{i=1}^{k}\frac{p_{i}^{2}}{2m_{i}}+\sum_{i<j\leq
k}U(q_{i}-q_{j})$ (21)

As is well-known and can be immediately seen, the BBGKY hierarchy (20), (21)
is of little practical interest unless one can find a plausible set of
arguments allowing them to truncate it at the distribution function $f_{k}$ of
a very small number of particles, such as $k=3$ for instance. As a result, in
the BBGKY hierarchy a central role is played by the $1-$ and the $2-$ particle
distribution functions $f_{1}$ and $f_{2}$ given by (13) and (15)
respectively. Hence the most important distribution functions one is dealing
with in the BBGKY hierarchy have support on $2-$ and $4-$ dimensional subsets
of the phase space $\mathcal{M}$. For this reason, additional constraints
placed upon the $1-$ and $2-$ particle distribution functions which are not
related to Liouville’s theorem may have substantial consequences for the
description of the evolution the whole system. The symplectic non-squeezing
theorem (4) provides such a constraint for the $1-$particle distribution
function, as its validity is independent of the validity of Liouville’s
theorem. This is true since in the probabilistic language, the $1-$particle
distribution functions are marginals, therefore they correspond to the
projections of their support to the $2-$dimensional symplectic planes of the
canonically conjugate variables.

Consider a system with many degrees of freedom whose initial probability
distribution has as support a Euclidean ball of dimension $6n$ and radius $R$.
The physical model that we have in mind is an isolated system of coupled
mechanical harmonic oscillators of equal masses and equal frequencies. The
total energy of the system is the only constraint hence it gives rise to this
ball in $\mathcal{M}$. Granted that this is an integrable system, so it is
very unlike the typical systems one encounters in Statistical Mechanics.
Still, we use it for illustrative purposes. In the evolution of such a system,
the support of the probability distribution will change shape but will keep
its overall volume constant. This is described by stating that $f_{n}$ should
be a constant. However, according to the symplectic non-squeezing theorem this
“micro-canonical” distribution is not the only constraint that $f_{n}$ should
obey. Each marginal distribution $f_{1}$ should also have as support a disk
whose radius is at least $R$. As the system evolves, in order to keep its
overall symplectic volume fixed, it will have to change shape, as is dictated
by the Hamiltonian flow. For a small amount of time though, the shape of such
a support will not change appreciably from that of a ball, if the ball is
relatively small. Its projections to symplectic 2-planes will be almost
spherical but will have a radius that will be at least as big as $R$. Hence
the value of $f_{1}$ will decrease on its support as a function of time.

##  On coarse-graining

The issue of coarse-graining is fundamental in Statistical Physics. It
expresses the fact that in order to determine the macroscopic behavior of a
system, many of its microscopic features have to be ignored. At the purely
classical level, there is no obvious way to implement such a coarse graining.
One usually considers the phase space $\mathcal{M}$ and partitions it in small
cubes. The only requirement is that such cubes should be much bigger in size
than any of the microscopic details of the Hamiltonian system describing the
physical system under study, so one can perform some kind of averaging inside
each such a cube and in essence “gloss over” the unimportant features of each
of the microscopic degrees of freedom of the system.

The view of Boltzmann is to enumerate all such microscopic states, let their
number be denoted by $\mathcal{N}$, giving rise to the same macroscopic state
in determining the entropy of the system, whether it is in equilibrium or not.
Let the coarse-grained phase space of the system be denoted by
$\overline{\mathcal{M}}$. The overwhelming majority of the microscopic states
of $\overline{\mathcal{M}}$ correspond to macroscopic equilibrium, which
justifies the increase of the Boltzmann entropy

$\mathcal{S}_{B}=k_{B}\log\mathcal{N}$ (22)

for an isolated system approaching equilibrium, where $k_{B}$ is Boltzmann’s
constant.

The view of Gibbs, who instead works with probability distributions in his
ensemble theory, is somewhat dual: one replaces the given probability
distribution $\rho$ on phase space $\mathcal{M}$, with its (constant) average
value $\overline{\rho}$ inside each cube/element of $\overline{\mathcal{M}}$
and then calculates the coarse-grained Gibbs entropy

$\overline{\mathcal{S}_{G}}[\overline{\rho}(x)]\ =\
-k_{B}\int_{\overline{\mathcal{M}}}\ \overline{\rho}(x)\
\log{\overline{\rho}(x)}\ d\mathsf{Vol}_{\overline{\mathcal{M}}}$ (23)

In (23) one should be, strictly speaking, using a summation instead of an
integration, but due to the small sizes of the cubes of the partition from a
macroscopic viewpoint, integration works in practice equally well. Then, one
sees that even though the Gibbs entropy expressed through the actual
probability distribution $\rho$ cannot change for an isolated system due to
Liouville’s theorem, its coarse-grained counterpart
$\overline{\mathcal{S}_{G}}$ does change as the system approaches equilibrium.

The existing literature on coarse-graining is vast, because such a central
issue has occupied the attention of practitioners since its earliest days,
when it was introduced by Gibbs, and by P. and T. Ehrenfest [42]. From the
time of its introduction, all kinds of works have appeared covering aspects of
coarse-graining, ranging from the purely philosophical/ontological to the very
formal ones. Some indicative recent references are [1, 2, 17, 43, 44, 45, 46,
47, 48, 49]. In general, most works referring to the foundations of
Statistical Mechanics and the meaning of entropy, have some discussion on
coarse-graining.

A troubling aspect of this process, is that even though there is considerable
arbitrariness in performing a coarse-graining, the final results arising from
calculations of the resulting thermodynamic potentials are physical/measurable
quantities, hence completely independent of any subjective choices leading to
such a coarse-graining. This may be due to statistics, where features of
individual variables are glossed over in favor of statistically significant
quantities. As such, reasonable but subjective choices leading to coarse-
graining of particular systems may be statistically unimportant. Hence
statistics seems to effectively eliminate subjectivity.

The question that we address in this section is whether there is any
preferred, or even optimal, way of performing such a coarse-graining. A
related question is what, if any, constraints should be placed to such a
coarse-graining by the microscopic features of the system. At a first thought,
the answer is negative to both questions. Coarse-graining is, after all, a
process that someone performs “by hand” when making a transition from the
microscopic to a mesoscopic or macroscopic description of a system and there
is a substantial degree of arbitrariness to it. However, this question is
crucial, since the calculation of entropy or any of the associated
thermodynamic potentials relies on coarse-graining.

If one adopts the viewpoint of Boltzmann the description of a macroscopic
system, then there is no obvious way of choosing how to coarse-grain the phase
space $\mathcal{M}$. To get around this difficulty, one uses inherent
uncertainties about a physical system: it is actually impossible to measure
with absolute accuracy the properties of any physical system, even in
classical Physics. Hence uncertainties can creep in two places. There are
uncertainties in the Hamiltonian of the system; indeed, no system can be
absolutely isolated, except the Universe itself. This lack of absolute
isolation can be described either by introducing terms coupling the system
under study to its environment, or by introducing the possibility for
fluctuations of the degrees of freedom in the underlying Hamiltonian. It is
not unreasonable to assume that such fluctuations are small, certainly away
from phase transitions, and therefore treat them as small perturbations of the
underlying Hamiltonian. Incidentally, these two approaches give equivalent
results, heuristically at least, in the perturbative regime of relativistic
quantum field theories, which are systems however that we do not discuss in
the present work.

These small perturbations can either be described either as being stochastic
(“noise”) or as being sustained for particular systems. As a result, the
evolution of the Hamiltonian system is not described by a single trajectory in
$\mathcal{M}$ but by the evolution of a small volume around it, which expands
at some rate depending on the perturbation, as the system evolves. This is the
process of $\varepsilon$-fattening of the trajectories of the underlying
Hamiltonian system. The fact that this is the generic behavior, rather than
the exception for multi-dimensional dynamical systems, was proved in [50].
Given this, someone can coarse-grain $\mathcal{M}$ by making a judicious
choice of scale based on the features of such a perturbation and the
subsequent rate of volume expansion. The process of $\varepsilon$-fattening
can be most effectively described in a Riemannian or metric-measure
theoretical setting, so we will not elaborate on it any further.

A second way to introduce coarse-graining into the system is by considering
not just one, but a whole set of initial conditions. This reflects our lack of
absolute precision in preparing the physical system. Such a set of initial
conditions can have any “shape” in phase space, but has to be “small” in order
to reflect the small macroscopic uncertainties in preparing the system. For
simplicity, which however is compatible with the symplectic structure as will
be explained in the sequel, let us choose such a set of initial conditions to
have the shape of a ball around the desired initial conditions of the system.
Such a ball provides a natural coarse-graining, a natural lower cut-off scale
in $\mathcal{M}$. The fact that its volume remains invariant under the
Hamiltonian flow is guaranteed by Liouville’s theorem. We also know from the
symplectic non-squeezing theorem that its projections onto the symplectic
$2-$planes of the conjugate variables will also have to remain invariant or
increase in size as the system evolves. There is no required minimum size of
such initial conditions in classical physics; in fact it is exactly zero. For
further discussion, see [29, 30, 31, 33, 34, 36] .The scale for such a minimum
size is set by Quantum Physics. This approach, in essence, allows us to speak
about a minimal size, as for the case of “quantum blobs” [33].

However, we know of no constraints on the projections of the initial
conditions on the isotropic $2-$planes. Hence, the symplectic embeddings give
no information on the geometric features of $2-$dimensional isotropic
submanifolds. This is not totally surprising $2-$dimensional symplectic planes
are essentially almost complex curves embedded in a very high dimensional
symplectic manifold. It seems unlikely that the geometry of such curves may
have anything to say about the ambient space. The situation is markedly
different for Lagrangian subspaces though, since the latter are maximal
dimensional isotopic sub-manifolds of the ambient space, which allows them to
have a special significance in symplectic geometry [18]. It is not an accident
that pseudo-holomorphic curves [19], which are objects having real dimension
$2$, have been applied most effectively in symplectic and enumeration problems
on $4-$dimensional manifolds. From our viewpoint, it may be worth pointing
out, that Lagrangian sub-manifolds inside any constant energy hyper-surface of
a symplectic manifold, as we assume in our case, remains invariant under the
Hamiltonian flow. This fact might have considerable consequences if a middle-
dimensional symplectic non-squeezing theorem were valid which, as [26] argued,
does not, except for the case of linear symplectic maps.

##  On the shape of the coarse-graining cells

In Section 2, we discussed the case of symplectic embeddings of balls and its
implication for the marginal ($1-$particle) distributions appearing in the
BBGKY hierarchy. Such balls, even if convenient objects from a metric
viewpoint, appear not have the appropriate shape for providing a coarse-
graining of the phase space $\mathcal{M}$ of the system. As was previously
pointed out, such coarse-graining usually takes place by using
cubes/parallelepipeds whose sides are parallel to the coordinate axes, at
least as long as $\mathcal{M}$ is $\mathbb{R}^{6n}$. From a metric viewpoint,
one could not easily expect that a cube and a ball are close to each other. In
a sense, these are two convex bodies that are as far from each other as
possible, at least when one uses the Banach-Mazur distance [52]. We have used
this maximal distance/extremal mismatch to argue about coarse-graining in
[53].

In a symplectic context things are radically different though. To explain
this, let us consider instead of $\mathcal{M}$ which is usually non-compact, a
compact approximation. This is quite common in mainstream treatments, ones not
aspiring to mathematical rigor of physical systems, at least in position
space: we resort to systems confined in a cubic box, with appropriate boundary
conditions. We then let the width of such a box increase without any upper
bound. Similar things can be done about canonical momenta, as in
hydrodynamics, when one provides an upper bound to the magnitude of the
momenta of the particles of the system by confining our attention to the
behavior of the low-frequency/large wavelength modes. To proceed and to be in
accordance with the fact that cubes are used in typical coarse grainings of
$\mathcal{M}$, let us investigate how close is such a cube to balls of equal
dimension $2d$, from the viewpoint of symplectic maps.

We can rephrase this question as a packing problem: consider the unit cube
$C_{2d}=[0,1]^{2d}$ in $\mathbb{R}^{2d}$. What fraction of the volume of
$C_{2d}$ can someone fill by using symplectic embeddings of $k$ equal disjoint
balls? If $k$ is large, the answer is: $1$. To be more precise, let
$\mathcal{P}_{k}(\mathcal{M},\omega)$ denote the $k$-th packing number of a
compact $2d$-dimensional symplectic manifold $(\mathcal{M},\omega)$. This is
defined as

$\mathcal{P}_{k}(\mathcal{M},\omega)=\frac{\sup_{r}\mathsf{Vol}_{2d}(\sqcup_{k}B_{2d}(r))}{\mathsf{Vol}_{2d}(\mathcal{M},\omega)}$
(24)

where the supremum is taken over all radii $r$ for which there exists a
symplectic embedding of the disjoint union of $k$ balls of radius $r$ into
$(\mathcal{M},\omega)$. Then $(\mathcal{M},\omega)$ has packing stability if
there exists some integer $n_{0}(\mathcal{M},\omega)$ such that
$\mathcal{P}_{l}(\mathcal{M},\omega)=1$ for all $l\geq
n_{0}(\mathcal{M},\omega)$. In [54] it was proved that if
$(\mathcal{M},\omega)$ is rational, namely if $[\omega]\in
H^{2}(\mathcal{M},\mathbb{Q})\subset H^{2}(\mathcal{M},\mathbb{R})$, then it
has packing stability. The condition on the second singular homology group of
$(\mathfrak{M},\omega)$ is obviously satisfied for the cube $C_{2d}$, since
the cube is homologically trivial. The same result applies if one uses small
ellipsoids instead of balls [54]. As a result of this theorem, one can use
sufficiently many balls $(k\rightarrow\infty)$, instead of cubes, for coarse
graining the phase space $\mathcal{M}$ of a physical system of many degrees of
freedom.

A question that naturally arises has to do with the shape of the allowed
partitions of phase space. It was pointed out initially by Boltzmann, and ever
since, that an arbitrary partition of phase space is not appropriate if one
wishes to get correct physical results. Clearly cubes or parallelepipeds are
the first obvious choice based on naturality and simplicity. However, the
experience of integration theory makes us skeptical of adopting such “obvious”
choices without further investigation. We would expect any partition of the
phase space to be acceptable, if we accept that partitions are man-made
devices allowing us to proceed with our coarse-graining arguments and beyond.
However this is obviously not true. Therefore, there may be more to phase
space coarse-graining cells than just being a convenient device arbitrarily
chosen “by hand” to serve our purposes.

It turns out that to get correct results on has to use a “right” or the
“right” partition. It seems that phase space partition through cubes whose
sides are parallel to the local canonical position and momenta axes gives
correct results. Some other partitions do not. But it is unclear why a
partition by such cubes works, or what other partitions might work and how to
distinguish good/acceptable partitions from not good ones (see [44] pp.
27-28).

A brief digression is in order here. The question of cubes as fundamental
cells of coarse-graining in phase space is strongly reminiscent of a similar
question in Quantum Mechanics: the uncertainty principle of Quantum Mechanics
has its simple commutator form only if one uses Cartesian coordinates. For any
other coordinate system, the canonical position and momentum commutator is not
so trivial, or natural, to obtain. However, it is widely accepted in Physics,
at least since the advent of General Relativity that coordinates have no
fundamental meaning in a physical theory, but are just labels associated to a
particular observer. They may be convenient technical devices allowing someone
to perform explicit calculations in a specific context, but the results of
physical measurements should either not depend on them at all, or might depend
on them in a “covariant” way. The question of why the Cartesian coordinates
seem to play this special role in the formulation of the uncertainty principle
has been around since the earliest days of Quantum Mechanics and but has not
been adequately resolved. The advent of geometric quantization did little to
answer such a question, apart from placing it in a manifestly coordinate
independent form. Maybe an answer to this question could be found along the
lines that we propose for finding a resolution to the issue of coarse-graining
of phase spaces in this section. After all, the probabilistic philosophy and
formalism established in Classical Statistical Mechanics pervades Quantum
Mechanics, despite their different domains of applicability, goals and
formalisms. Maybe the analogies that may be drawn, and the lessons learned,
could be transferred from one of these disciplines to the other.

In what follows, we attempt to provide a possible answer to the question
regarding the “right” shape(s) of the partition cells of phase space by
resorting to dynamics. One could argue about using such cubes for the phase
coarse-graining, based on simplicity, or even elegance, as noted above. But
such an approach is not convincing enough. Something stronger is needed.
Aesthetic or simplicity considerations do not really shed any light into the
suitability of such cubes as fundamental cells of the phase space coarse-
graining.

To achieve our goal we will be examining the geometry of the set of
Hamiltonian diffeomorphisms on the given phase space $(\mathcal{M},\omega)$
[55]. We see, to begin with, that a cube with sides parallel to the coordinate
axes in $\mathbb{R}^{n}$ is the unit ball of the $L^{\infty}$, otherwise known
as the $\sup$-norm. This is defined as follows. Let
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a measurable function. Define the
Lebesgue spaces $L^{s}(\mathbb{R}^{n},\mathbb{R})$ as the set of such
functions so that their $s$-norm is finite

$\|f\|_{s}=\left\\{\int_{\mathbb{R}^{n}}|f|^{s}\
d\mathsf{Vol}_{n}\right\\}^{\frac{1}{s}}\ <\ \infty\hskip 56.9055pt1\leq
s<\infty$ (25)

and, in addition, define the $L^{\infty}(\mathbb{R}^{n},\mathbb{R})$ as the
Lebesgue space whose $\sup$-norm is finite

$\|f\|_{\infty}=\sup\left\\{|f(x)|:\ x\in\mathbb{R}^{n}\right\\}<\infty$ (26)

It is not too far-fetched to expect that

$\lim_{s\rightarrow\infty}\|f\|_{s}=\|f\|_{\infty}$ (27)

which turns out to be true. These definitions can be straightforwardly
extended to $n$-dimensional differentiable manifolds $\mathcal{M}$.

Let $(\mathcal{M},\omega)$ be a closed symplectic manifold. Most of the times
in Physics, the phase space is a cotangent bundle, therefore non-compact.
However, we can use compact approximations as regulators, whose diameters and
widths at the end of each calculation are sent to infinity, as we commented
above. This approach can be dually expressed by considering only compactly
supported sets of functions on phase space.

To simplify matters, we will also assume that $\mathcal{M}$ has no boundary.
We could use the set of all smooth functions on $\mathcal{M}$, but since it is
compact, we can equivalently use the set of all mean zero value functions on
it, which we denote by $C_{0}^{\infty}(\mathcal{M})$. The results of this and
the next paragraphs hold for general, non-autonomous Hamiltonians
$\mathcal{H}_{t}$ and their corresponding Hamiltonian vector fields
$X_{\mathcal{H}_{t}}$ even though that level of generality is not needed in
this work. Consider the group of Hamiltonian diffeomorphisms
$\mathsf{Ham}(\mathcal{M},\omega)$ of such a phase space $\mathcal{M}$ which
consists of all Hamiltonian flows whose time parameter has been set to $1$.
This is an infinite dimensional Lie group. The corresponding Lie algebra
$\mathcal{G}$ is $C_{0}^{\infty}(\mathcal{M})$. Let $\|\cdot\|$ be a norm on
$\mathcal{G}$. We can use $\|\cdot\|$ to define the length of the flow
$\phi:[0,1]\rightarrow\mathsf{Ham}(\mathcal{M},\omega)$ by

$l(\phi)\ =\ \int_{0}^{1}\|\frac{d\phi}{dt}\|\ dt\ =\
\int_{0}^{1}\|\mathcal{H}_{t}\|\ dt$ (28)

Here $\mathcal{H}_{t}$ the unique Hamiltonian
$\mathcal{H}:[0,1]\times\mathcal{M}\rightarrow\mathbb{R}$ obeying

$\int_{\mathcal{M}}\mathcal{H}_{t}\ \omega^{n}\ =\ 0$ (29)

for all $t\in[0,1]$. Then, a Finslerian distance function $\mathsf{dis}$
between two elements $\phi_{1},\phi_{2}\ \in\
\mathsf{Ham}(\mathcal{M},\omega)$ associated to the norm $\|\cdot\|$ is
defined as

$\mathsf{dis}(\phi_{1},\phi_{2})\ =\ \inf_{\psi}\ l(\psi)$ (30)

where $\psi$ is a Hamiltonian path connecting $\phi_{1}$ and $\phi_{2}$ and
the infimum is taken over all such paths. It turns out that $\mathsf{dis}$ is
non-negative, it is symmetric, it is non-degenerate and it satisfies the
triangle inequality, therefore it is a genuine metric on
$\mathsf{Ham}(\mathcal{M},\omega)$. The existence of such a metric is non-
trivial since $\mathsf{Ham}(\mathcal{M},\omega)$ is an infinite dimensional
group, hence no such metric is guaranteed to exist.

As in the finite-dimensional case, the group
$\mathsf{Ham}(\mathcal{M},\omega)$ acts on its Lie algebra
$C_{0}^{\infty}(\mathcal{M})$ through the adjoint action

$Ad_{\phi}f\ =f\circ\phi^{-1}$ (31)

for $\phi\in\mathsf{Ham}(\mathcal{M},\omega)$ and for $f\in
C_{0}^{\infty}(\mathcal{M})$. One can see that the distance function (30) is
bi-invariant under the adjoint action, namely

$\mathsf{dist}(\phi_{1},\phi_{2})\ =\
\mathsf{dist}(\phi_{1}\psi,\phi_{2}\psi)\ =\
\mathsf{dist}(\psi\phi_{1},\psi\phi_{2})$ (32)

Hofer [56] considered the $L^{\infty}$ norm on $C_{0}^{\infty}(\mathcal{M})$
which induces bi-invariant metric on $(\mathbb{R}^{2n},\omega_{0})$. His
construction was generalized by [23] for general symplectic manifolds. We may
recall at this point that two distance functions $d_{1},\ d_{2}$ in a general
metric space are equivalent if there is a positive constant $C>0$ such that

$\frac{1}{C}\ d_{1}\ \leq\ d_{2}\ \leq\ C\ d_{1}$ (33)

With these definitions, in a series of works [57, 58] culminating with [59],
it was proved that for a closed symplectic manifold $(\mathcal{M},\omega)$,
any bi-invariant Finslerian metric on the set of its Hamiltonian
diffeomorphisms induced by a norm on $C_{0}^{\infty}(\mathcal{M})$ which is
continuous with respect to the $C^{\infty}$ topology is either zero or
equivalent to Hofer’s metric. All other $L^{s}$ metrics are zero [57]. Hofer’s
metric, which belongs to the class $L^{\infty}$ is the unique non-trivial
metric among all $L^{s}$ bi-invariant metrics under the group of Hamiltonian
diffeomorphisms on a closed symplectic manifold.

Incidentally, the approach of understanding the geometry of some space by
understanding properties of its group of automorphisms/symmetries is what the
“Erlangen programme” proposed by F. Klein (1872) is all about. This is what we
follow in this section to argue for the suitability of cubes for the phase
space coarse-graining.

It might appear in this result that the metric equivalence (33), which is not
an equality, would compromise the uniqueness of Hofer’s metric, hence the
exclusive use of its unit balls which are cubes in phase space coarse-
graining. However this is not the case. To see this, we only have to recall
the forms of both the Boltzmann and the Gibbs expressions for the entropy. Let
us use the latter for concreteness. We have, if we are careful, that the Gibbs
entropy is defined by

$\mathcal{S}_{G}\ =\ -k_{B}\int_{\mathcal{M}}\ \rho(x)\
\log\left\\{\frac{\rho(x)}{\rho_{0}}\right\\}\ d\mathsf{Vol}_{\mathcal{M}}$
(34)

In this expression $\rho_{0}$ is taken to be a constant distribution on
$(\mathcal{M},\omega)$. It is present in order to make the logarithm
dimensionless. The constant distribution $\rho_{0}$ has no physical
significance, since it is only differences/derivatives of the entropy that
have any physical meaning, and not its absolute value.

A redefinition of distances by a constant factor $C$ as indicated in (33), can
be absorbed through a redefinition of $\rho_{0}$, in both the Boltzmann and
the Gibbs entropy formulae. As such, the metric equivalence assumed in the
proof of the uniqueness of Hofer’s metric has no implications for the coarse-
graining process of the classical phase space $(\mathcal{M},\omega)$. The
cubes which are the cells of the phase space coarse-graining process, are
replaced by cubes of a different side length, under such an equivalence.
Consequently, all the points made in this section remain valid even
considering two equivalent (33), but not equal, metrics.

##  Discussion and outlook

We presented in this work some comments on possible implications of the
symplectic non-squeezing theorem for Statistical Physics and for coarse-
graining. It is a fact that very few things are really known about high-
dimensional manifolds which would be the phase spaces of systems of many
degrees of freedom. But, some things are known about infinite dimensional
manifolds [60, 61] being the phase spaces of some nonlinear partial
differential equations. The latter may have some common features with the
former, which may turn out to be useful not only for the thermodynamic limit
of particle systems, but also for their implications to field theories, as
long as relevant model-independent results can be obtained. Given that
symplectic structures are inherently two-dimensional, and that symplectic
manifolds lack any local geometry, it may be of some interest to see what new
results, if any, may come out of such investigations.

On the other hand, more structure is present in a system of many degrees of
freedom, such as a metric. One can probe several features of a system by
following a metric approach. But such a choice of a metric is not unique. And
a symplectic structure always induces sets of almost complex structures on a
symplectic manifold [18]. The problem is that such almost complex structures,
even if compatible or tame, are usually not integrable. If they were, then
high dimensional phase spaces would be Kähler manifolds, which have a lot of
structure and about which several things are known, since Kähler manifolds
have symplectic, Hermitian and algebraic-geometric properties. But most
symplectic manifolds are not Kähler, and this fact begs for a different
approach [19].

A slightly different question, which may allow us to reach some conclusions
pertinent to our goals, is whether a high dimensional phase space of a
physical system, is reasonably close (in the rather strong $C^{\infty}$, or in
the weaker measured Gromov-Hausdorff sense) to a Kähler manifold. If this is
true, it might allow us to draw some conclusions about the underlying physical
system. An immediate/uneducated guess to answering this question would be
negative. If such an answer turns out to be true, one might narrow the scope
of the question and ask which systems have phase spaces for which such a
statement might be true and what its implications might be. We believe that it
may be worth further investigating this matter in the future.

The obvious weak point of the present treatment is the lack of concrete
examples on which explicit calculations can be performed, and have such ideas
checked for their physical relevance and significance. This however has been a
persistent problem plaguing the dynamical foundations of Statistical Mechanics
for decades. Analytical results are scarce, and hard to establish from first
principles, especially if one does not assume classical ensemble equivalence,
is interested in far from equilibrium phenomena, explores the treatment of
systems having long-range interactions, and even going as far as considering
the use of different, from the Boltzmann and Gibbs, entropic functionals.
Coming up with physically relevant but also analytically tractable models to
test some of the above ideas is a challenge that is certainly worth pursuing,
in our opinion.

Acknowledgements: The author is grateful to Prof. Panayotis Benetatos for
numerous discussions on the topics of the present work, and of Statistical
Physics in general, for his continuous encouragement to keep pursuing such
foundational issues and for providing many references on which this work
relies. The author is also grateful to Prof. Anastasios Bountis for his
encouragement, and constant support over many years, which made the present
work possible.

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