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\begin{document}
\begin{center}
{\Large\bf
Metric structures of inviscid flows}
\end{center}
\begin{center}
{\large Rub\'en A.\ Pasmanter}\\
{\em
KNMI, P.O.Box 201, 3730 AE \quad De Bilt\\
e-mail: pasmante@knmi.nl}
\end{center}
\begin{center}
{\Large Abstract}
\end{center}
\begin{center}
{\begin{minipage}{4.5in}
\small
An intrinsic metric tensor, a flat
connexion and the corresponding
distance-like function are constructed in the
configuration space formed by the velocity field
{\bf and} the thermodynamic variables of an inviscid
fluid.
The kinetic-energy norm is obtained as a limiting case;
all physical quantities are Galilean invariant.
Explicit expressions are given for the case of an ideal
gas.
The flat connexion is {\bf not} metric-compatible.
These results are achieved by applying
the formalism of statistical manifolds
\cite{amari,otros} to the statistical mechanics of a
moving fluid.
\end{minipage}}
\end{center}
\begin{center}
Submitted for publication.
\end{center}
\section{Introduction}
\vspace*{-0.1cm}
Typically,
the configuration of a simple fluid
at a certain time
$t$ is described by its density field
$\rho(\vec x,t)$, its temperature field
$T(\vec x,t)$ and its velocity field
$\vec u (\vec x,t)$ where $\vec x$ is the
position vector in a $d$-dimensional space.
This $d$-dimensional space is often the
everyday
three-dimensional Euclidean space or a
two-dimensional surface embedded
in three-dimensional
space, e.g., a plane or a sphere.
In that $d$-dimensional space, it is possible to
calculate the square of the distance
between two nearby positions,
say $\vec x$ and
$\vec x + d\vec x$, in the form of an expression
\be
M_{\lambda\nu}(\vec x) dx^\lambda dx^\nu
\label{rie}
\ee
where $M_{\lambda\nu}(\vec x)$ is the
metric tensor, $dx^\lambda$ is the
$\lambda$-component of $d\vec x$ in some coordinate
system, $1 \le \lambda \mbox{\ and\ } \nu \le d$,
and the convention of
summation over repeated lower and upper
indices holds.
For example, on a two-dimensional
($d=2$) sphere of radius $r$,
in the coordinate system defined by the angles $\vartheta$
and $\phi$ (latitude and longitude),
the non-vanishing components of the metric tensor are
\ba
M_{\vartheta\vartheta} = r^2,\\
M_{\phi\phi} = r^2 \sin^2\vartheta,
\ea
and the corresponding volume element is
$\sqrt{\det\,(M_{\lambda\nu})}= r^2\sin\vartheta$.
Such a metric tensor is also what
is required in order to compute the scalar
quantity we call the norm
of a (tangent) vector, e.g., the norm
of the velocity vector
$d\vec x/dt$ is given by
\be
M_{\lambda\nu}(\vec x) \frac{dx^\lambda}{dt\,}
\frac{dx^\nu}{dt\,} .
\label{velo}
\ee
Similarly, one can compute the angle between two
(tangent) vectors, e.g., two velocities, at
each position in the $\vec x$-space.
The situation in the configuration space of a
fluid with coordinates $(\rho , T, \vec u)$ is
totally different from that in the $\vec x$-spaces
described above as long as it is not known how to
introduce, in a natural and intrinsic way, a geometric
structure, e.g., something similar to the metric tensor
$M_{\lambda\nu}(\vec x)$
in (\ref{rie}) and (\ref{velo}).
Such a lack of geometric
structures imposes many restrictions on the
kind of computations that one can perform:
it is impossible to talk of ``the distance" between
two states of the fluid, i.e.,
between two positions in the space $(\rho , T, \vec u)$;
neither is it possible to talk of ``the norm"
of the vector formed by the rate of
change of the dynamical variables $(d\rho/dt , dT/dt,
d\vec u/dt)$;
it is not possible
to consider the angle between two such vectors;
there is no volume element defined in configuration space,
therefore, it does not make sense to talk about ``the
density" of a distribution of points in that space;
etc.
These limitations are too restrictive since,
to name but a few examples: 1) A norm is needed when
studying the (in)stability of flows, especially when the
linear growth of
the perturbations is only transient \cite{energynorm};\,
2) The angle between two directions is required in order
to determine how strongly a given perturbation projects
along an optimal perturbation \cite{Farrell};\,
3) A volume element is needed in order to determine the
density distribution of ensemble simulations of flows
\cite{Mureau}.
Therefore, it is often opted to
circumvent these limitations by introducing an acceptable,
if somewhat arbitrary, metric
tensor, e.g., the kinetic-energy metric in the case
of incompressible, isentropic flows \cite{energynorm}.
Such a situation is
not unique to fluid mechanics: often
our knowledge of a dynamical system is
limited to the variables that define its
phase space and to the equations that
determine the time evolution of these
variables while any
information on the geometry of the
phase space is lacking.
In this article it is shown how to generate,
in a natural way,
not only the analogue of the spatial metric
tensor $M_{\lambda\nu}(\vec x)$ but also
a rich geometric structure
in the configuration space of
an inviscid, moving fluid starting from the fluid's
probability distribution.
The derivation of these results makes use
of ideas and techniques from statistical mechanics,
see e.g., \cite{balescu,grand} and from statistical
manifolds theory, see e.g., \cite{amari,otros}, a
particular branch of differential geometry, see e.g.,
\cite{geometry,Erwin}.
The presentation is as self-contained as possible.
A complete description of the required background material,
however, was not attempted;
detailed bibliographic references are given for the
benefit of those readers willing to go deeper into
some aspects.\\
The paper is structured as follows:
In Section 2, the probability
density of a moving fluid in local thermal equilibrium
is presented.
In Section 3, some
fundamental concepts of
the theory of statistical manifolds are introduced;
additional information is given in Appendix A.
In Section 4
these ideas are applied to the
probability density ofa moving fluid, the
natural metric tensor in the fluid's configuration
space is derived and an application is
described.
In Section 5, it is shown that the probability
density generates a flat, non-Riemannian,
geometry in the configuration space.
Basic concepts on flatness of affine connexions)
are reviewed in Appendix B.
Based on this flatness, a distance-like function is
defined in Section 6.
The computed metric and distance-like function are
Galilean invariant, as they should.
When the thermodynamic variables can be ignored,
both the metric and the distance-like function
reduce to the above-mentioned
kinetic-energy norm times a conformal factor.
For the purpose of illustration, the expressions
corresponding to an ideal gas are explicitly given in
Sections 4 and 6.
In Section 7, we comment on related
work done by others; in Appendix C,
the differences between the approach
developed in the present paper and a particular
Hamiltonian formulation of fluid dynamics are
pointed out.
Finally, in Section 8, we review the results and
discuss their generalizations, e.g., to systems far
from thermal equilibrium.
%
%
%
\section{Statistical mechanics of a moving fluid}
\vspace*{-0.1cm}
The first step in our construction of an intrinsic
geometric structure in the $(\rho , T, \vec u)$
space consists in associating a probability density
to these variables.
In this Section we
derive the fluid's probability distribution
by applying statistical mechanics to
a moving fluid in local thermal equilibrium.
Consider a simple fluid characterized, on a
macroscopic level,
by its density field $\rho(\vec x,t)$, temperature
field $T(\vec x,t)$ and velocity field
$\vec u(\vec x,t)$;
from here onwards, we do not express the
$(\vec x,t)$-dependence explicitly.
We adopt the standard assumptions that make
possible the derivation of the Euler equations
and, less rigorously, of the
Navier-Stokes equations from local thermal
equilibrium statistical mechanics,
see, e.g., \cite[Chapters 1 to 3 of Part 1]{Spohn} and
\cite[Chapters 1 to 3]{Demasi}, to wit:
One considers systems in which the
above-mentioned fields vary on length scales
that are much larger than the intermolecular
distances and on time scales that are much larger
than the microscopic time scales
needed for local thermal equilibration.
Then the space can be divided
into small volume elements
$\Delta V$ with a typical size much smaller than
the length scales of the macroscopic fields but
much larger than the intermolecular distances so
that the
statistical mechanical description applies to them,
i.e., on the length scales
of $\Delta V$, the fluid is in local thermal equilibrium.
The dynamics of the fluid is assumed to be
such that the following
extensive quantities are conserved:
a)
the number
of (indistinguishable) particles $N$,
b)
the total momentum of these particles
\be
\vec M :=m\sum_k^N\vec v_k,
\label{M}
\ee
where $m$ is the mass of the particles,
$\vec v_ k$ is the velocity of the $k$-th particle,
and
c) their total energy
\be
E:=\sum_k^N{\left( \frac{1}{2}m
|\vec v_k|^2 +
\frac{1}{2}\sum_{l\neq k}^N V_{kl}\right) },
\label{E}
\ee
$V_{kl}$ being the potential interaction between
particles $k$ and $l$. (The symbol to the left
of $:=$ is defined by the expression to the right.)
Let us denote the conserved quantities by
$\{ H_i (\xi_N)\, |i=1,\dots ,s\}$ where
$\xi_N := \{\vec x_1,\dots ,\\
\vec x_N , m\vec v_1 ,\dots
,m\vec v_N\, | N=1, \dots , \infty\}$
and the values of $\{ H_i\}$ in a
thermal equilibrium state by $\{\eta_i \}$.
These two elements, i.e., the expressions defining
the conserved quantities $\{ H_i \}$ as functions of the
microscopic variables $\{ \xi_N \}$ and the values
$\{\eta_i \}$ characterizing the macroscopic state, are the
essential ingredients needed for the statistical mechanical
description of the system.
Statistical mechanics establishes that
the probability density in the phase space
$\xi_N $,
of a system in $\Delta V$, in thermal equilibrium with and
free to exchange particles, momentum and energy with a
surrounding thermal bath is
\be
p(\xi_N; \theta) :=
\exp {\left[ \theta^i H_i(\xi_N) \right] }
/c^N N! {\cal Z}(\theta),
\label{pd}
\ee
see, e.g., \cite[Chapter 4]{balescu},
\cite[Chapters 9, Section B.3]{grand}.
We use the standard short-hand conventions:
repeated lower and upper indices are
summed up; $\theta$ stands for $\{\theta^i|i=1,
\dots , s\}$.
The ${\cal Z}(\theta)$ in the denominator,
called the grand partition function,
is the normalizing factor needed to make
sure that
$
\sum_N^\infty
\int\! d\xi_N \, p(\xi_N; \theta) =1,
$
i.e.,
\be
{\cal Z}(\theta) :=
\sum_N^\infty
\int\! d\xi_N \,
\exp {\left[ \theta^i H_i(\xi_N) \right] }
/c^N N! ,
\label{Z}
\ee
the constant $c$ makes each of the terms
contributing to ${\cal Z}$ dimensionless.
From (\ref{pd}) and (\ref{Z}) we see that the
average values of the conserved quantities
$\{H_i(\xi_N)\}$, which we call $\{\eta_i\}$,
can be written as
\be
\eta_i (\theta) =
\frac{\partial \ln {\cal Z}(\theta)}%
{\partial{\theta^i}},\quad i=1,\dots ,s .
\label{eta}
\ee
% One says then that $\eta_i$ and $\theta^i$ are
% {\em conjugate} to each other.
Since the thermal equilibrium state of a system with
$s$ conserved quantities is completely
characterized by their macroscopic values
$\{\eta_i\}$,
the last expression makes evident that the whole
%macroscopic description, including the
thermodynamics can be obtained from the grand
partition function ${\cal Z}(\theta)$.
In this way, one identifies $\ln {\cal Z}(\theta)$
as the thermodynamic potential which is minimized
at fixed values of the
$\theta$-variables and the $\theta$-variables as the
intensive thermodynamic parameters (like temperature,
pressure, chemical potential, etc).
Since the system in $\Delta V$ is in thermal
equilibrium with its surroundings, the values of
these intensive variables must be
equal to those of its thermal bath,
see, e.g., \cite[Chapter 4]{balescu},
\cite[Chapters 9, Section B.3]{grand}.
In the case of a moving fluid,
the conserved quantities are
$(N,E,\vec M)$, confer (\ref{M}) and (\ref{E}),
so that $s= d + 2$ and
the probability density (\ref{pd}) reads
\be
p(N,E,\vec M;\gamma,-\beta,\vec\kappa) =
\exp [\gamma N -\beta E + \vec\kappa\cdot\vec M ] /
c^N N! \cal Z(\gamma,-\beta,\vec\kappa),
\label{probability}
\ee
i.e., $\theta = (\gamma,-\beta ,\vec \kappa)$.
It follows then that the average, macroscopic
values of $(N,E,\vec M)$ are given by
\ba
\langle N\rangle =
\partd{\ln{\cal Z}(\gamma,-\beta,\vec\kappa)}{\gamma}
\label{relation1}\\
\langle E\rangle =
-\partd{\ln{\cal Z}(\gamma,-\beta,\vec\kappa)}{\beta}
\label{relation2}\\
\langle M_\lambda \rangle=%m\langle N\rangle u^i=
\partd{\ln{\cal Z}(\gamma,-\beta,\vec\kappa)}{\kappa^%
\lambda},\quad\lambda =1,\dots , d,
\label{relation3}
\ea
where the pointed brackets indicate average over
the probability distribution (\ref{probability}).
It is convenient to introduce the macroscopic velocity
of the fluid $\vec u$ by
\be
m \langle N\rangle \vec u := \langle\vec M\rangle
\label{u}
\ee
and the specific internal energy
$\epsilon(\gamma,\beta,\vec\kappa)$ as
\be
\langle N\rangle ( \epsilon + \frac{m}{2} u^2 ) :=
\langle E\rangle .
\label{epsilon}
\ee
Relating these expressions to the thermodynamics of
the system, one identifies $\ln {\cal Z}(\gamma,
-\beta,\vec \kappa)$ as the
thermodynamic grand potential, i.e.,
\be
\ln {\cal Z}(\gamma,-\beta,\vec\kappa)=
\beta \Delta V P,
\ee
where $P= P(\gamma,\beta,\vec\kappa )$
is the pressure,
$\beta$ as the inverse local temperature,
$\beta\equiv (kT)^{-1}$, $k$ is Boltzmann's
constant and $\gamma$ as
$\gamma = \beta\mu$
with $\mu$ the local chemical potential;
see, e.g., \cite[Chapter 4]{balescu},
\cite[Chapters 9, Section B.3]{grand}.
Due to the simple dependence of $\vec M$ and of
$E$ upon the particles' velocities $\vec v_k$,
confer (\ref{M}) and (\ref{E}), the integrals over
the velocities in (\ref{Z}) can be performed;
from (\ref{relation3}) and (\ref{u})
one obtains then that
\be
\vec\kappa= \beta\vec u
\label{kappa}
\ee
and that
\be
{\cal Z}(\gamma,-\beta,\vec\kappa)=
{\cal Z}(\gamma - \frac{m\kappa^2}{2\beta},-\beta,\vec 0).
\label{shift}
\ee
Galilean invariance implies then that
\be
\gamma = \bar{\gamma} + \frac{m\kappa^2}{2\beta},
\label{gamma}
\ee
where $\bar{\gamma}$ is the value of $\gamma$ for the same
system at rest.
Making use of (\ref{kappa}), one sees then that
$\mu = \bar\mu + m u^2/2$, i.e., the
chemical potential of the fluid moving with velocity
$\vec u$ is shifted by an amount $m u^2/2$ with
respect to $\bar\mu$, the chemical potential of the
fluid at rest.
In the theory of statistical manifolds \cite{amari,otros}
it is shown that a probability density like
$p(\xi_N; \theta)$
in (\ref{pd}) induces a geometric structure in the
parameter-space $ \theta$, i.e., in the space
of the intensive thermodynamic variables.
This is worked out in the following Sections.
%
%
%
%
\section{Statistical manifolds}
\vspace*{-0.1cm}
Given two probability densities, say
$p(\xi_N ,\theta_1)$ and $p(\xi_N ,\theta_2)$,
how should one quantify their (dis)similarity? Or
given three distributions corresponding to $\theta_1 ,
\theta_2$ and $\theta_3$ respectively,
is it possible to determine which pair of distributions
is ``closer" than the two other pairs?
These and related questions have been extensively
studied in statistics, see, e.g., \cite[pages
290--296]{enciclo} and \cite{eguchi}.
In this Section we give a fleeting overview of
some concepts and results from the theory of statistical
manifolds that we apply later to a moving fluid.
If $D(\theta_1 ,\theta_2)$ is a scalar quantity measuring the
difference between two probability densities $p(\xi_N,\theta_1)$
and $p(\xi_N,\theta_2)$, then it should satisfy:\\
1) $D(\theta_1 ,\theta_2)\geq 0$,
the equality holding if, and only if,
$p(\xi_N,\theta_1)\equiv p(\xi_N,\theta_2)$,\\
2) $D(\theta_1 ,\theta_2)$ is sufficiently continuous in
$p(\xi_N,\theta_1)$
and $p(\xi_N,\theta_2)$.\\
In the statistical literature such measures are often called
divergences or contrast functionals;
in order to avoid any possible confusion with
the usual mathematical meaning of divergence, we shall keep
to the second name.\\
We shall impose two additional constraints:\\
3) The value of $D(\theta_1 ,\theta_2)$ should be independent
of our
choice of coordinates in $\xi_N$-space, i.e., it should be
invariant under general coordinate transformations
in the space of the random variables.\\
4)
We shall consider only contrast functionals that are local
in the random variables $\{ \xi_N\}$\footnote{C.R.\ Rao has
considered also non-local contrast functionals,
\cite[pp. 226--227]{otros}.}.\\
In order to satisfy these constraints,
it is sufficient that
the $D(\theta_1 ,\theta_2)$s be
of the following form
\cite{amari}, \cite[pp.\ 349--350]{eguchi},
\be
D(\theta_1 ,\theta_2) =
\int\! d\xi_N\, p(\xi_N,\theta_2)
f \left(
\frac{p(\xi_N,\theta_1)}{p(\xi_N,\theta_2)}
\right),
\label{D}
\ee
where
the function $f$ must be sufficiently
smooth and
%%%%%% ATENTI: las Chernof NO tienen derivada 0 !!!!!
should satisfy $f(1)=0$ and
$f(t)- f'(1)(t -1) > 0$ if $t\ne 1$.
Without any loss of generality, one normalizes $f$ such
that $f''(1)=1$.\\
Consider next two distributions that are infinitesimally
close, i.e., $\theta$ and $\theta + d\theta$.
It turns out that, up to third
order in $d\theta$, the corresponding contrast is
given by \cite[theorem 3.10]{amari}
\be
D(\theta,\theta + d\theta) =
\nonumber\\
\frac{1}{2}\,
g_{ij}(\theta)\, d\theta^i\, d\theta^j +
\frac{1}{2}\left\{ {
[i,j;k] + \frac{\alpha}{ 3!} T_{ijk}(\theta)
} \right\}
\, d\theta^i\, d\theta^j\, d\theta^k
+ O(d\theta^4),
\label{taylor}
\ee
where
the components of $g_{ij}(\theta)$,
known as the Fisher tensor \cite{fisher,Rao},
are given by
\be
g_{ij}(\theta) :=
\left\langle {\partd {\ln p}{\theta^i}
\partd{\ln p}{\theta^j} }\right\rangle ,
\label{g}
\ee
the pointed brackets indicate an average taken over the
$p(\xi_N ,\theta)$ distribution.
From this expression it is evident that $g_{ij}$ is
symmetric in the indices $i \mbox{ and } j$ and that,
since it is an addition of products of
covariant vectors, under a general
coordinates transformation
it transforms like the
product of two covariant vectors, i.e.,
it is a covariant tensor\footnote{For a
more detailed meaning of these geometric
concepts, see, e.g., \cite{geometry,Erwin}.}.
Moreover, any sensible choice of the parameters
$\theta$ ensures that it is non-singular.\\
The first term in the coefficient of the third-order
contribution to the expansion (\ref{taylor}),
$[i,j;k]$, is given by
\be
[i,j;k] := \frac{1}{2} \left( \partd{g_{ik}}{\theta^j}
+ \partd{g_{jk}}{\theta^i} - \partd{g_{ij}}{\theta^k}
\right),
\label{christo}
\ee
its meaning is discussed in Appendix B and in Section 5.
The second term in the third-order coefficient is a
totally symmetric covariant tensor,
often called the skewness,
\be
T_{ijk}(\theta)
:= \left\langle {\partd{\ln p}{\theta^i}
\partd{\ln p}{\theta^j}
\partd{\ln p}{\theta^k}}\right\rangle.
\label{third}
\ee
Both symmetric tensors, $g_{ij}$ and $T_{ijk}$,
will play an important role in the sequel.\\
The constant $\alpha$ in (\ref{taylor}) is given by
$\alpha:= 2f'''(1) + 3$.
A number of points are worth noticing:\\
$\bullet$\quad
The symmetric contravariant tensor $g_{ij}(\theta)$ has all
the credentials for being the natural metric tensor in
the $\theta$-space.
Besides the clear meaning ensuing from (\ref{taylor}),
it has an
important statistical significance \cite{Rao,Cramer}
that is described in Appendix A.\\
$\bullet$\quad
Up to second order, the expansion (\ref{taylor}) is
independent of the particular choice of the function $f$
that defines the contrast $D$, confer (\ref{D}).
(Remember that $f$ has been normalized so that $f''(1)=1$.)\\
$\bullet$\quad
Up to third order, the whole dependence upon the function $f$
has been brought back to a single number,
namely, to $\alpha$.\\
%
%
\section{Fisher's metric}
\vspace*{-0.1cm}
In this and the following Sections, we apply the
theory of statistical manifolds to the probability
density $p(\xi_N ,\theta)$ of a moving fluid.\\
As shown in the previous Section,
the Fisher tensor appears naturally as
the metric in $\theta$-space.
From (\ref{probability}) it follows that
in the coordinate system
$(\theta^1,\dots ,\theta^{d+2}) =
(\gamma,-\beta,\vec\kappa)$,
the metric tensor (\ref{g}) reads,
\be
g_{ij}(\theta)=
\frac{\partial^2\ln {\cal Z}(\theta)}%
{\partial \theta^i\partial\theta^j}=
\Delta V\,\frac{\partial^2\beta P}%
{\partial\theta^i\partial\theta^j} ,
\label{coordina}
\ee
see also (\ref{apeng}) in Appendix A.
We see then that
the metric elements are expressed in terms of
the derivatives of the thermodynamic potential.
Moreover, from (\ref{relation1}--\ref{relation3}) or
from (\ref{eta}) one has that,
in the $(\gamma,-\beta,\vec\kappa)$ coordinate
system,
\be
d\eta_i =
g_{ij}(\theta) d\theta^j,
\label{lower}
\ee
where $\eta_i$ are the average values of the extensive
quantities given in (\ref{relation1}--\ref{relation3}).
I.e., in this coordinate system, the lowering
of the indices corresponds to passing from the intensive
variables $\theta$ to the extensive ones $\eta$.
The inverse transformation is achieved by raising the
indices by means of the inverse metric $g^{ij}(\theta)$.
It is instructive to work out the metric tensor for
the case of an ideal gas, i.e., for a vanishing
intermolecular potential $V_{kl}$ in (\ref{E}).
In this case all the integrals over the molecular
velocities $\vec v_k$ and over their positions
$\vec x_k$ in (\ref{Z}) can be performed and one
finds that
\be
\ln{\cal Z}(\gamma,-\beta,\vec\kappa) = \Delta V \,
\beta^{-d/2} \exp (\frac{m\kappa^2}{2\beta} -\gamma).
\ee
It is convenient to express the results in
the coordinate system $(\rho,\beta,\vec u)$, where
$\rho$ is
the mass density, i.e.,
$\rho := m\langle N\rangle/\Delta V$.
The non-vanishing elements of the metric are,
\ba
{g}_{\rho\rho}= \Delta V({m\rho})^{-1},
\nonumber\\
{g}_{\beta\beta} = \Delta V
\frac{d}{2} %\left(\frac{d}{2}+1\right)
\frac{\rho}{m\beta^2},\nonumber \\
{g}_{u_i u_j}
= \Delta V \rho\beta\delta_{ij}.
\label{gideal3}
\ea
Therefore, each volume element $\Delta V$ contributes
a squared distance $(dL)^2$ between two states
$(\rho,\beta,\vec u)$ and
$(\rho +d\rho,\beta +d\beta,\vec u + d\vec u)$
given by
\be
\frac{\Delta V\rho}{m}
\left[ \left(\frac{d\rho}{\rho}\right)^2 +
\frac{d}{2} %\left(\frac{d}{2}+1\right)
\left(\frac{d\beta}{\beta}\right)^2 +
m\beta\, d\vec u\cdot d\vec u \right].
\label{idealm}
\ee
This expression agrees with our expectations:
1) The metric coefficients are independent of $\vec u$, as
demanded by Galilean invariance\footnote{The
Galilean invariance of the metric in the
general case (non-ideal gas) is proven in Section 6,
confer (\ref{Galil}).};
2) Since no prefered or external scales are available
and taking into account the Galilean invariance,
the differentials must always appear
in the form of $d\rho/\rho$ and $d\beta/\beta$; the gas
temperature determines the scale for measuring the kinetic
energy associated with $d\vec u$;
3) Cross-terms like $d\rho\,d\vec u$ and $d\beta\,d\vec u$
cannot appear due to the rotational symmetry of the gas
and
4) The factor multiplying the square brackets is just the
number of particles in $\Delta V$.\\
What could not have been guessed beforehand, is the
absence of the cross-term $d\rho\, d\beta$ and the precise
form of the coefficients.
One of the simplest applications of a metric tensor
is the computation of the norm of a tangent vector,
as in (\ref{velo}).
Now we can do this in the configuration space of the
moving fluid, i.e., we can compute the norm of
the local rate of evolution of our sytem,
that we denote by $F$, as
\be
F ^2 :=
{g}_{ij}
{\dot{\theta}}^i {\dot{\theta}}^j,
\label{F}
\ee
where the dots indicate the
time derivatives of the corresponding quantities
as given by the Euler equations.
In order to distinguish between this generalized velocity $F$
and the standard velocities $\vec v_k$ and $\vec u$,
we shall call this quantity {\em the rapidity}.
The rapidity squared is a scalar with dimension
${\rm [time]}^{-2}$.
From (\ref{lower}) we see that it
can also be written as
\be
F^2 =
{\dot{\eta}}_i {\dot{\theta}}^i ,
\ee
i.e., it is the contraction between the velocities
of the intensive variables with the velocities of
their corresponding conjugate extensive variables.
Let us compute the rapidity associated with the material
derivatives in the Euler equations with no external
forcing, i.e.,
\ba
\frac{d\rho}{dt} = -\rho \mbox{\,div\,} \vec u, \nonumber\\
\rho c_v\frac{dT}{dt} = -P \mbox{\,div\,}\vec u, \nonumber\\
\rho \frac{d\vec u}{dt} =
-\vec\nabla P ,
\label{material}
\ea
where $c_v$ is the specific heat at constant volume.
In the case of an ideal gas, i.e.,
making use of (\ref{gideal3}) and inserting (\ref{material})
into (\ref{F}) leads to
\be
F^2= \frac{\Delta V\rho}{m}
\left[
\left( {1 + \frac{d}{2}\left(\frac{R'}{c_v}\right)^2}\right)
(\mbox{\,div\,}\vec u)^2
+ R'T \left|{\vec \nabla \ln(\rho T)}\right|^2
\right] ,
\ee
where $R':=k/m$ is the specific constant of the gas.
%
%
%
\section{Flatness of the manifold}
\vspace*{-0.1cm}
Another standard application of a metric is the determination
not only of an infinitesimal distance, as in (\ref{idealm}),
but also of finite distances, say $L({\bf 1},{\bf 2})$
between positions
${\bf 1} := (\rho_1,-\beta_1,\vec u_1)$ and
${\bf 2} := (\rho_2,-\beta_2,\vec u_2)$.
This distance
is defined as the length of the shortest path connecting
${\bf 1}$ with ${\bf 2}$,
\be
L({\bf 1},{\bf 2})
= \min
\int_1^2{\! d\tau \,
\sqrt{
{g}_{ij}(\theta)
\,\frac{d\theta^i}{d\tau}
\frac{d\theta^j}{d\tau} }},
\label{L}
\ee
where the minimum is taken over all paths from
${\bf 1}$ to ${\bf 2}$ and $\tau$ is any parametrization
of these paths.\\
This calculation is not a simple one when
there is no coordinate system in which the metric tensor
$g_{ij}(\theta)$ takes a simple form; this happens in the
case of our moving fluid since, as we will see in this
Section, (\ref{idealChrist}) and comments thereafter,
(\ref{gideal3}) is {\em not} flat.
Luckily, it turns out that the geometry of statistical
manifolds offers more interesting and useful possibilities
than (\ref{L});
this is particularly so in the case of a probability
density of the form (\ref{probability}) as is our case.
In order to realize this, we need some concepts from
differential geometry; for the sake of completeness,
these concepts are listed in Appendix B.
Two simple calculations indicate that the connexions
described in Appendix B
may play an important role in the case of our moving fluid:\\
Let us compute first the $[ij;k]$ connexion compatible with
the Fisher metric (\ref{coordina}); from (\ref{LC}) one has
that, in the coordinate system
$(\theta^1,\dots ,\theta^{d+2}) =
(\gamma,-\beta,\vec\kappa)$,
\be
[ij;k] = \frac{1}{2}
\frac{\partial^3\ln {\cal Z}(\theta)}%
{\partial \theta^i\partial\theta^j\partial \theta^k}.
\label{idealChrist}
\ee
One can compute the corresponding curvature tensor and
check that, even for the ideal gas metric
(\ref{gideal3})-(\ref{idealm}),
this connexion is {\em not}
flat.\\
Next, let us compute the skewness tensor $T_{ijk}$ generated by
(\ref{probability}), confer (\ref{third});
one finds that
\be
T_{ijk}(\theta) =
\frac{\partial^3\ln {\cal Z}(\theta)}%
{\partial \theta^i\partial\theta^j\partial \theta^k}.
\label{expoT}
\ee
From these two last results,
we see that by taking $[ij;k] - (1/2)T_{ijk}$ one gets a
connexion that vanishes identically in this coordinate
system, i.e., {\em this connexion is flat}.\\
It should be noted that the existence of a flat connexion
implies that there is a prefered family of coordinate
systems, in our case, the system of the intensive variables
$\theta$ and their linear combinations. The physical
relevance of these variables becomes even more evident
when one considers the phenomenon of phase-coexistence,
e.g., liquid-vapour coexistence: coexisting phases
correspond to flat portions of the thermodynamic potential
surface only in terms of the intensive variables.
Similarly, one can check that also $[ij;k] + (1/2)T_{ijk}$,
notice the
change in sign, vanishes identically in the {\em extensive}
variables coordinate system $\eta$,
i.e., also this connexion is flat.
Connexions of the form $[ij;k] - (\alpha/2)T_{ijk}$ play an
important role in the theory of statistical manifolds;
they were introduced by Chentsov \cite{chentsov},
Efron \cite{Efron} and Amari
\cite[Chapter 3]{amari}.
In the third reference it is shown that if
such a connexion is flat for a certain $\alpha$ then it is also
flat for $-\alpha$.
In fact, the coordinate systems
$\theta =
(\gamma,-\beta,\vec\kappa)$
and
$\eta =
(\langle N\rangle ,\langle H\rangle ,\langle\vec M\rangle )$
play totally symmetric roles:
it can be shown \cite[Theorems 3.4 and 3.5]{amari}
that $g^{ij}$, the inverse
of the metric tensor, is also given by the second-order
derivatives of a function $\Phi(\eta)$, this time with respect to
the extensive variables $\eta$, i.e.,
\be
g^{ij}(\eta) =
\frac{\partial^2\Phi(\eta)}%
{\partial \eta_i\partial\eta_j},
\label{ginverse}
\ee
that the function $\Phi(\eta)$
is nothing else but the Legendre transform
of $\ln{\cal Z}(\theta)$ and that the coordinate systems are
related to each other as in a Legendre transformation, i.e.,
\ba
\theta^i (\eta) = \frac{\partial\Phi}{\partial\eta_i} ,
\label{theta}\\
\Psi(\theta) + \Phi(\eta) - \theta^i\cdot\eta_i =0,
\label{Legendre}
\ea
where we have introduced $\Psi := \ln {\cal Z}$,
confer also (\ref{eta}).
Recall that all functions are assumed to be sufficiently
smooth so that (\ref{Legendre}) is good enough for the
definition of the Legendre transform; for more complicated
situations, see \cite{Fenchel}.
We have shown then that the probability density of the moving
fluid
generates not only a metric tensor (\ref{coordina}) but also two
flat
connexions, namely $[ij;k] \pm (1/2)T_{ijk}$ and that all this is
closely related to a Legendre transformation of thermodynamic
potentials and variables.
Some interesting consequences of these facts are presented in the next
Section.
%
%
%
\section{Distance-like function}
\vspace*{-0.1cm}
While the computation of the distance $L$ in (\ref{L})
implies difficult calculations,
it was pointed out by Amari \cite[Section 3.5]{amari} that,
when the connexions
$[ij;k] \pm (1/2)T_{ijk}$
are flat,
it is possible to define a
distance-like function $D(\bf 1,\bf 2)$
between positions
${\bf 1} := (\gamma_1,-\beta_1,\vec \kappa_1)$ and
${\bf 2} := (\gamma_2,-\beta_2,\vec \kappa_2)$
as follows
\be
D({\bf 1},{\bf 2}) :=
\Psi ({\bf 1}) + \Phi({\bf 2}) -
\theta^i({\bf 1})\cdot\eta_i({\bf 2}),
\label{amariD}
\ee
with
$\Psi := \ln {\cal Z}$ as in
(\ref{Legendre}), confer also (\ref{ginverse}), (\ref{theta})
and (\ref{eta}).
Amari has shown that
this function shares some essential properties with
the usual distance functions, to wit\footnote{%
The first property is a special instance of Fenchel's theorem
\cite{Fenchel}.}:\\
\ba
1)\qquad
D({\bf 1},{\bf 2}) \geq 0, \nonumber\\
{\rm the\ equality\ holds\ when,\
and\ only\ when,}
\quad {\bf 1} ={\bf 2},
\label{p1}\\
2)\qquad
D(\theta,\theta + d\theta) =
\nonumber\\
\frac{1}{2}\,
g_{ij}(\theta)\, d\theta^i\, d\theta^j +
\frac{1}{2}\left\{ {
[i,j;k] + \frac{1}{ 3!} T_{ijk}(\theta)
} \right\}
\, d\theta^i\, d\theta^j\, d\theta^k
+ O(d\theta^4),\label{again} \\
3) \qquad D({\bf 1},{\bf 2}) =
D({\bf 1},Q) + D(Q,{\bf 2}),\nonumber\\
{\rm where\ }Q{\rm \ is\ connected\ to\ }{\bf 1}{\rm\ by\ a\ }
\theta{\rm -geodesic\ and\
to\ }{\bf 2}{\rm\ by\ an\ }
\nonumber\\
\eta{\rm -geodesic\
and\ these\ two\ geodesics\
intersect\ orthogonally\ at\ }Q
\label{path}\\
{\rm and\ } \qquad 4) \qquad {\rm The\ }
\min_{Q \in \Omega} D({\bf 1},Q)\nonumber
\\
{\rm is\ obtained\ at\ a\ point\ on\ the\ boundary\ of\
the\ smooth\ closed\ region\ }\Omega
\nonumber\\
{\rm\ that\ is\ the\
projection\ of\ }{\bf 1}{\rm\ along\ a\ }
\theta{\rm -geodesic\ orthogonal}\nonumber\\
{\rm to\ the\
boundary\ of\ } \Omega.
\ea
The $\theta$-geodesics above is a linear
interpolation from $\theta({\bf 1})$ to $\theta(Q)$
while the $\eta$-geodesic is a linear interpolation
from $\eta (Q)$ to $\eta({\bf 2})$.
%The second property shows that the
%metric (\ref{g}) and the $\alpha_0$-connexion
%(\ref{amari}) are only a small part of the information
%contained in $D$.
The third property is a generalization
of Pythagoras' theorem to a space with two biorthogonal
coordinate bases, related to $\theta$ and $\eta$ in our case.
Analogously, the fourth property
generalizes the notion of projection to such a space.
The reader should
refer to \cite[Section 3.5]{amari} for the
proofs of the properties listed above. \\
There is one important difference between the usual
distance functions and the $D$ defined above:
In general, the $D$ function is
{\em not} symmetric, i.e., $D({\bf 1},{\bf 2})\ne
D({\bf 2},{\bf 1})$.
This asymmetry is associated with the fact that the path
going from ${\bf 1}$ {\em first} along an $\eta$-geodesic
to $Q'$ and {\em then} along an orthogonal
$\theta$-geodesic to ${\bf 2}$ is, in general, different from the
path described in (\ref{path}). If one is set on defining a
symmetric
distance, then taking the minimum of the values
$D({\bf 1},{\bf 2})$ and
$D({\bf 2},{\bf 1})$ seems to be the most satisfactory solution.\\
From properties (\ref{p1}) and (\ref{again}), we recognize that
this $D$ may belong to the family of contrast functionals
discussed in Section 3.
In fact,
using (\ref{eta}), (\ref{coordina}), (\ref{theta}) and
(\ref{Legendre}),
one finds that
$D({\bf 1},{\bf 2})$ in (\ref{amariD})
can be written as
\be
D({\bf 1},{\bf 2}) %= %\left\langle {\ln
%\frac{p(N,E,\vec M;\theta_1}%
%{p(N,E,\vec M;\theta_2} }
%\right\rangle_1 \nonumber\\
=(\beta_2-\beta_1) \langle H\rangle_1 -
(\vec\kappa_2-\vec\kappa_1)\cdot\langle\vec M\rangle_1 -
(\gamma_2-\gamma_1)\langle N\rangle_1 +
\Delta V (\beta_2 P_2 -\beta_1 P_1),
\ee
where $\langle\cdots\rangle_1$ indicates that an average is taken
over $p(N,E,\vec M;\theta_1)$.
One can check then that, in our case, the last expression for
$D({\bf 1}, {\bf 2})$
is identical to (\ref{D}) with
$f(z)= z \ln z$.
The circle is now complete.
The Galilean invariance of $D({\bf 1},{\bf 2})$
follows from introducing
(\ref{epsilon}), (\ref{kappa}) and (\ref{gamma})
into the last expression above; one finds then that
the contrast functional (\ref{D}) or (\ref{amariD})
can be written as
\be
D({\bf 1},{\bf 2}) =
(\beta_2-\beta_1) \langle N\rangle_1 \epsilon_1 +
\frac{m}{2} \beta_2 |\vec u_2 - \vec u_1|^2
\langle N\rangle_1 -
(\bar{\gamma}_2-\bar{\gamma}_1)\langle N\rangle_1 +
\Delta V (\beta_2 P_2 -\beta_1 P_1).
\label{Galil}
\ee
Combining this with (\ref{taylor}) and with (\ref{again}),
one sees that both $g_{ij}(\theta)$ and
$T_{ijk}(\theta)$ are always
independent of $\vec u$, i.e., Galilean invariant.\\
Notice also that when the variations in the
thermodynamic variables can be ignored,
as is the case
in an (effectively) incompressible, isentropic flow,
the contrast reduces to
$ D({\bf 1},{\bf 2}) =
(\Delta V\rho\beta/2) |\vec u_1 - \vec u_2|^2$
with $\rho =\rho_1 =\rho_2$ and $\beta =\beta_1 =
\beta_2$,
i.e., the kinetic-energy norm
times a conformal factor.\\
In the case of an ideal gas, the
contrast function reads:
\be
D({\bf 1},{\bf 2})
=\frac{\rho_1}{m} \Delta V
\left[
\frac{m}{2} \beta_2 |\vec u_2 - \vec u_1|^2 +
\left( { \ln {\frac{\rho_1}{\rho_2}} +
\frac{\rho_2}{\rho_1} -1 } \right) +
\frac{d}{2} \left( { \ln \frac{\beta_1}{\beta_2} +
\frac{\beta_2}{\beta_1} -1 }\right)
\right].
\label{idealD}
\ee
\section{Related work}
Before closing, we review and
comment some papers
that deal with related questions:\\
Weinhold \cite{Weinhold} proposed
to use the matrix of second-order derivatives of the internal
energy as a metric in the space tangent to the
equation-of-state surface at an equilibrium point.
Some researchers tried then to attach a
physical meaning to the curvature tensor derived from the
Weinhold's metric-compatible connexion,
confer Section 5. This led to a lengthy
discussion which has been summarized in \cite{Andresen};
all these researchers ignored the flat connexions
$[ij;k] \pm\frac{1}{2} T_{ijk}$ and the associated
distance-like function $D$ as discussed in Section 6.
In the context of the present article, Weinhold's metric
comes close to the Fisher metric (\ref{coordina}) while
Gilmore's approach, see \cite{Andresen}, is closer to
the distance-like function $D$ (\ref{amariD}).\\
The Fisher metric was
introduced into thermodynamics by Ingarden
\cite{Ingarden1,Ingarden2}.
Janyszek and Mruga{\l}a \cite{Janyszek} studied the
curvature tensors of the corresponding metric-compatible
connexions and
tried to associate this curvature with
some physical properties; they did not consider
the flat connexions
$[ij;k] \pm\frac{1}{2} T_{ijk}$ neither the associated
distance-like function $D$.
On the 18th of December 1995,
I presented this paper at the Technical
University of Berlin. Prof.\ U.\ Simon informed me then
about affine and projective differential geometry
which deals, among other things, with conjugate
connexions like $[ij;k] \pm\frac{1}{2} T_{ijk}$ and their
flatness. Excellent, up-to-date reviews of this
branch of differential geometry are \cite{Simon} and
\cite{Nomizu}.
There exists a rich literature on the
Hamiltonian (symplectic) structure of
hydrodynamics that can also be
seen as one branch of differential geometry;
see, e.g., the review articles
\cite{Salmon,Shepherd}, the references therein
and \cite{Zeitlin,Rouhi} for another interesting
application.
This approach is based on the Poisson bracket
and it does {\em not} require a
metric tensor. By contraposition,
the present article does not use the Poisson bracket
and leads to a metric tensor, flat connexions, etc;
i.e., the two
approaches are not necessarily related and can be seen
as complementary.
In Arnold's analysis of incompressible flows \cite{Arnold},
a metric plays an important role, however,
this metric is just the metric
of the ambient $d$-dimensional space that appears in
the kinetic energy term,
i.e., is not the type of metric developed in the
present article.
There is
one instance of a Hamiltonian structure for
fluid dynamics that apparently
contains a candidate for a metric tensor
in the configuration space of a moving fluid;
this instance is discussed in Appendix C.
%
%
%
%
\section{Summary and discussion}
\vspace*{-0.1cm}
In this article, it has been shown that the grand canonical
partition function (\ref{probability}) which describes the
thermal fluctuations of a moving fluid in local thermal equilibrium
generates a natural metric tensor $g_{ij}(\theta)$, two flat
connexions $[ij;k]\pm \frac{1}{2}T_{ijk}$ and
the corresponding distance-like
contrast function $D$, confer (\ref{coordina}),
(\ref{idealChrist}), (\ref{expoT}) and (\ref{amariD}).
In the case of an ideal gas, the explicit
expressions for
these quantities have been given in (\ref{idealm}) and in
(\ref{idealD}).
It was also shown that these quantities are Galilean invariant,
as they should, and that they reduce to the
kinetic-energy norm times a
conformal factor when one can ignore variations in
variables other than the velocity field, confer (\ref{Galil}).
Noteworthy aspects of the geometric structure are that the flat
connexions are {\em not} metric-compatible and that the
distance-like function $D$ is {\em not} symmetric.
As pointed out by one of the referees, the analysis presented
in this article can be applied to relativistic fluids, to
plasmas, etc.
In fact, some of the results listed above, e.g.,
the particular form (\ref{coordina}) of the metric tensor,
as well as (\ref{ginverse})--(\ref{Legendre})
and the flatness of the connexions
$[ij;k]\pm \frac{1}{2}T_{ijk}$, are valid for all systems
characterized by Gibbs' probability distributions like
(\ref{pd}).
The following considerations
are relevant with regard to further applications of our approach.
We have seen that all the components of the geometric structure
are generated from $\ln {\cal Z}=\Delta V\beta P$ and
since, in the absence of external forces,
the equation of state $P=P(\bar{\gamma},\beta)$
is all that is needed in order to write down the
Euler equations of fluid motion, it is then clear
that the metric, connexions and contrast function
are closely related to the dynamical equations.
When the system is far from thermal equilibrium, however,
a different approach may be more appropriate. For example,
when considering a fluid in a turbulent stationary state,
the corresponding probability density should be used.
Similarly, when studying the predictability properties of
systems in the presence of noise, a time-dependent probability
density should be employed \cite{yo}.
In this sense, the geometric structures we have described
are not universal;
the specific physical conditions
of the system under consideration determine whether a
thermal-equilibrium, a stationary-state or a time-dependent
probability density is the most appropriate starting point.
\bc {\bf Acknowledgements} \ec
I would like to thank Robert Mureau for numerous conversations
and useful comments,
friends and colleagues,
in particular Gregory Falkovich and
Itamar Procaccia,
for their stimulating interest and Uriel Frisch for
some practical advices.
This article is dedicated to Marta and to Siggy.
%
\appendix
%
%
\bc {\bf Appendix A: On Fisher's metric} \ec
\vspace*{-0.1cm}
Besides the clear meaning due to (\ref{taylor}),
the Fisher metric tensor (\ref{g})
plays an important role in statistics
due to the following theorem \cite{Rao,Cramer}:\\
Suppose that we perform $n$ independent measurements of the
random variables $(N,E,\vec M)$ and use them in order
to estimate the value of the parameters $(\gamma,-\beta,\vec
\kappa)$ in (\ref{probability}).
Denote by $\{\hat\theta^i\}$ an unbiased
estimate of these parameters. Then, for large enough $n$,
the covariance of
these estimated values with respect to the exact values
$\{\theta^i\}=(\gamma,-\beta,\vec \kappa)$
has the following lower bound
\be
{\rm cov\ } \left[
(\hat\theta^i -\theta^i) (\hat\theta^j -\theta^j)
\right] \geq \frac{1}{n} g^{ij},
\ee
where $(g^{ij})$ is the inverse of the matrix $(g_{ij})$.
In other words, given a fixed tolerance error,
the larger the distance between two probability distributions
as measured by $g_{ij}$,
the smaller the number of measurements needed in order
to distinguish between them.
Finally, one should notice the following identity,
\be
g_{ij}(\theta)=
\left\langle {\partd {\ln p}{\theta^i}
\partd{\ln p}{\theta^j} }\right\rangle=
-\left\langle {
\frac{\partial^2 \ln p}{\partial\theta^i\partial\theta^j} }
\right\rangle .
\label{apeng}
\ee
This identity is obtained as follows:
$\int\! dx\, p(x,\theta) =1$ implies that
$\langle \partial
\ln p(x,\theta)/\partial\theta^i\rangle =
\int\! dx\,\partial p(x,\theta)/\partial\theta^i = 0$;
taking then the derivative of
$\langle \partial
\ln p(x,\theta)/\partial\theta^i\rangle$ with respect to $\theta^j$
leads to the identity in (\ref{apeng}).
(It is assumed that the order of derivation with respect to
the $\{\theta^i\}$ and integration over $x$ can be
interchanged.)
%
%
%
\appendix
\bc {\bf Appendix B: On affine connexions} \ec
In some cases, the geometry of a manifold is not
completely defined only by its
metric tensor but also by another geometric object,
called the connexion of the manifold\footnote{When
this happens, we speak of a non-Riemannian
manifold.}, see, e.g.,
\cite[Chapter 4, Sections 28 and 29]{geometry}
and \cite[Chapters I--IX]{Erwin}.
Briefly stated, the connexion expresses mathematically
what it means ``to move a vector along a curve in such
a way that the vector remains constant",
i.e., it defines the parallel transport of a vector.\\
We list some important properties of the connexions that we
need in Section 5:\\
$\bullet$\quad
Under a general transformation of the coordinates, a
connexion does {\em not} transform like a tensor, i.e.,
a connexion is {\em not} a tensor. In particular: a
symmetric
connexion may vanish identically in a particular coordinate
system {\em without} vanishing in a different one while a
tensor that vanishes identically in a particular system of
coordinates does so in all coordinate systems.\\
$\bullet$\quad
The result of adding a third-rank tensor to a connexion
is a connexion.\\
$\bullet$\quad
We say that a manifold is {\em flat} with respect to a
connexion when every vector that is
parallel transported along every closed path returns
to its original condition; otherwise, one says that the
manifold is curved.\\
$\bullet$\quad
A manifold is flat with respect to a connexion if,
and only if, it is possible to find a
coordinate system in which this connexion vanishes identically.\\
$\bullet$\quad
Whether a given connexion is flat or not can be determined in a
coordinate-system independent way: a connexion is flat if, and only if,
it is symmetric
in its two first indices, i.e., torsionless, and
its curvature tensor vanishes identically.\\
$\bullet$\quad
The partial derivatives of a tensor are {\em not} tensors,
however, using the connexion it is possible to define the
so-called covariant derivatives which are tensors.
If the connexion vanishes identically in a given coordinate
system then, in that coordinate system, the covariant derivatives
coincide with the partial derivatives.\\
$\bullet$\quad
There is only one connexion symmetric in its two first indices
such that the covariant
derivatives of the metric tensor vanish identically.
If $g_{ij}$
denotes the metric tensor, then this connexion\footnote{%
Actually, this connexion is obtained by raising the $k$-index
of the $[ij;k]$ in (\ref{LC}), i.e., by $g^{lk}[ij;k]$.} is given by
\be
\frac{1}{2} \left( \partd{g_{ik}}{\theta^j}
+ \partd{g_{jk}}{\theta^i} - \partd{g_{ij}}{\theta^k}
\right).
\label{LC}
\ee
This is precisely the $[ij;k]$ appearing in the
third-order term in (\ref{taylor}), confer (\ref{christo}).
This connexion is called
the metric-compatible or Levi-Civita connexion.
For the proofs of these statements, refer to, e.g.,
\cite[Chapter 4, Sections 28 and 29]{geometry}
and \cite[Chapters I--IX]{Erwin}.
%
\bc {\bf Appendix C: On the Dubrovin-Novikov
approach to flows} \ec
\vspace*{-0.1cm}
In an illuminating article \cite{DN1},
Dubrovin and Novikov introduced a novel
Hamiltonian formalism for
one-dimensional systems of hydrodynamic
type described by $s$ dynamical fields.
(An extensive review of related ideas and
developments can be found in \cite{DN3}.)
One notable aspect of this formalism is
the central role played by a symmetric
covariant tensor of type (2,0), call it
$\Lambda^{ij},\, 1\le (i,j)\le s$:
when
$\det(\Lambda^{ij})\ne 0$,
the Jacobi identity
that the Poisson bracket has to satisfy implies that the
Levi-Civita connexion generated by $\Lambda_{ij}:=
(\Lambda^{-1})_{ij}$
must be flat.
Moreover,
when $\det(\Lambda^{ij})\ne 0$
and other,
relatively mild conditions are satisfied,
it is possible
to reconstruct (up to a constant
factor) the tensor $\Lambda^{ij}$
directly
from the hydrodynamical equations of
motion \cite{Tsarev}.
Therefore, in such cases, it is
possible to associate with the
equations of motion, one metric in the
phase space of the fluid: the inverse of
$(\Lambda^{ij})$.
In practice, these symmetric tensors
cannot be used as metrics but in few,
unrealistic cases due to the following reasons:
1) When the fluid exists in a
$d$-dimensional space, $d$ different
symmetric tensors $\Lambda^{ij}$
must be introduced \cite{DN2} leading, in the
best case, to $d$ different metrics in
the fluid phase space;
2) Even in the case of a
one-dimensional physical space,
one often has that
$\det(\Lambda^{ij}) \equiv 0$,
this is the case, e.g., of a
one-dimensional nonbarotropic fluid
\cite{DN1,DN2},\cite[p. 59]{DN3}.
Moreover, it should be noticed that the
$\Lambda^{ij}$ tensor is part
of the Poisson bracket definition, therefore,
different dynamical systems generated
by different Hamiltonians, i.e., fluids with
different equations of state, but by the
same Poisson bracket share the same
$\Lambda^{ij}$;
this property is obviously not
desirable when one is trying to
identify the geometric structures
that characterize and distinguish
between different hydrodynamical
systems.
Similarly, some hydrodynamical
equations have a multi-Hamiltonian
formulation, i.e., the same set of
equations can be generated from
different Hamiltonians by means of
different Poisson brackets \cite{magri,Nutku};
in such cases,
each of these Poisson
brackets would lead to a different
metric tensor for the same
dynamical equations.
\vspace*{-0.1cm}
\begin{thebibliography}{99}
\bibitem{amari}
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\end{document}
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