#  Roles of Energy and Entropy in Multiscale Dynamics and Thermodynamics

Miroslav Grmela 111 e-mail<EMAIL_ADDRESS>
École Polytechnique de Montréal, C.P.6079 suc. Centre-ville,
Montréal, H3C 3A7, Québec, Canada

###### Abstract

Multiscale thermodynamics is a theory of relations among levels of
description. Energy and entropy are its two main ingredients. Their roles in
the time evolution describing approach of a level (starting level) to another
level involving less details (target level) is examined on several examples,
including the level on which macroscopic systems are seen as composed of
microscopic particles, mesoscopic levels as kinetic theory of ideal and van
der Waals gases, fluid mechanics, the level of chemical kinetics, and the
level of equilibrium thermodynamics. The entropy enters the emergence of the
target level in two roles. It expresses internal energy, that is the part of
the energy that cannot be expressed in terms of the state variables used on
the starting level, and it reveals emerging features characterizing the target
level by sweeping away unimportant details. In the case when the target level
is a mesoscopic level involving time evolution the roles of the energy and the
entropy is taken by two different potentials that are related to their rates.

###### Contents

  1. 1 Introduction
  2. 2 starting level$\longrightarrow$ level of equilibrium thermodynamics
    1. 2.1 Static theory (MaxEnt)
    2. 2.2 Dynamic theory
    3. 2.3 Roles of Energy and Entropy
  3. 3 Particular Realizations
    1. 3.1 Boltzmann’s kinetic theory
      1. 3.1.1 Roles of Energy and Entropy
    2. 3.2 Gibbs equilibrium statistical mechanics
      1. 3.2.1 Roles of Energy and Entropy
    3. 3.3 Euler-Navier-Stokes-Fourier fluid dynamics
      1. 3.3.1 Extended fluid mechanics
      2. 3.3.2 Roles of Energy and Entropy
    4. 3.4 van der Waals gas
      1. 3.4.1 Roles of Energy and Entropy
  4. 4 starting level$\longrightarrow$ target level with dynamics
    1. 4.1 Target level: Level of chemical kinetics (ChemKin)
    2. 4.2 Starting level: Level of extended chemical kinetics (ExtChemKin)
    3. 4.3 (ExtChemKin)$\longrightarrow$ (ChemKin)
  5. 5 Concluding Remarks

##  1 Introduction

Our point of departure is a pair of autonomous levels of description of
macroscopic systems. Both have arisen from certain type of experimental
observations. The types of observations are different different levels. One
level, that we call starting level, is based on more detailed observations
than the second level called target level. We emphasize that both levels are
autonomous in the sense that both levels separately provide a good description
of the experimentally observed behavior without a need of other levels. In
particular, we investigate the levels of particle mechanics, kinetic theory
and fluid mechanics as starting levels. All these three levels are paired with
the level of the classical equilibrium thermodynamics as the target level. In
Section 4 we also investigate pairs of levels in which both the starting and
the target levels are mesoscopic levels involving time evolution.

Since we assume that both the starting and the target levels are autonomous
and well established (i.e. predictions of the theory agree with results of
observations), one observation (among all observations made on the starting
level) has to be an observation of the approach to the target level. In other
words, the autonomous existence of both levels guarantees the possibility to
prepare macroscopic systems for the target level and the visibility of the
preparation process in the time evolution observed on the starting level. We
emphasize that in passing from the starting level to a target level that
involves less details we loose details but gain emerging overall features.

We use the following terminology. Solutions of the governing equations on the
starting level are trajectories, their collection is called a phase portrait.
We distinguish three types of energy. Sources of the external energy are
outside the macroscopic system under investigation. The potential energy in
gravitational field is an example of the external energy. The inner energy is
the energy that can be expressed in terms of the state variables used on the
chosen level of description. The total energy in the Gibbs theory or the
kinetic energy of a fluid in fluid mechanics are examples of the inner energy.
The former is expressed in terms of the $n$-particle distribution function
that serves as the state variable in the Gibbs theory and the latter in terms
of the velocity and mass fields that serve as state variables in fluid
mechanics. The internal energy is the energy that cannot be expressed in terms
of the state variables that are used on the chosen level of description.
Microscopic details that are not seen on the chosen level of description are
needed to express it.

The objective of this paper is to investigate the roles that the energy and
that entropy play in the passage from the starting level to the target level.

Energy provides the force generating the phase portrait.

Entropy plays two roles: (Ent I) it expresses the internal energy,  (Ent II)
it makes patterns, emerging in the phase portrait during the time evolution,
manifestly visible by sweeping away unimportant details. The patterns
represent the target levels inside the starting level.

We explore this viewpoint of energy and entropy in the setting of GENERIC time
evolution (a combination of Hamiltonian and gradient dynamics) describing the
passage from a starting level to a target level. The ”universal competition”
between energy and entropy is also discussed, but with a different
perspective, in [1].

Before starting our discussion we briefly recall history of GENERIC dynamics.
The first step in its formulations, made by Vladimir Arnold [2], was casting
the Euler fluid mechanics into the form of noncanonical Hamiltonian dynamics.
Vladimir Arnold has also realized connection with older results obtained in
investigations of Lie groups. Such connection then allowed to formulate
nondissipative parts of kinetic [3] and other mesoscopic time evolution
equations as noncanonical Hamilton’s equations. In proceedings of the
conference devoted to this subject (organized by Jerrold Marsden in the summer
of 1983 in Boulder, Colorado) the formulation of the complete Boltzmann
kinetic equation (that includes the dissipative collision term) as a
combination of Hamiltonian and generalized gradient dynamics has appeared [4].
Many other mesoscopic time evolution equations have been put into such form in
[5], [6], [7], [8], [9], [10]. In [11], [12] the combination of Hamiltonian
and generalized gradient dynamics has been called GENERIC (an acronym for
General Equation for Non Equilibrium Reversible Irreversible Coupling). The
natural geometrical setting for the Hamilton-gradient dynamics is contact
geometry [13], [14].

##  2 starting level$\longrightarrow$ level of equilibrium thermodynamics

The target level in this section is the level of the classical thermodynamics,
the starting levels vary. Historically, the passages of this type were
investigated by Boltzmann [15], Gibbs [16], and Prigogine [17]. Their
comparison leads to the abstract formulation presented below. Its four
particular realizations (Boltzmann’s kinetic theory, Gibbs equilibrium
statistical mechanics, Navier-Stokes-Fourier fluid mechanics, van der Waals
theory) provide an insight into the roles of energy and entropy.

###  2.1 Static theory (MaxEnt)

The state variable chosen on the starting level is denoted by the symbol $x$,
the energy is $E(x)$, the entropy $S(x)$, and the number of moles $N(x)$. In
the Gibbs theory $x$ is the $n$-particle distribution function ($n\sim
10^{23}$), in the Boltzmann theory, discussed in Section 3.1, the state
variable $x$ is the one particle distribution function, in fluid mechanics
theories discussed in Section 3.3 the state variable $x$ is a collection of
hydrodynamic fields, and in the classical equilibrium thermodynamics the state
variable $x=(E,N,V)$, where $E$ is the energy, $N$ number of moles, and $V$ is
the volume.

The input into the passage starting level$\longrightarrow$ level of
equilibrium thermodynamics is the fundamental thermodynamic relation on the
starting level consisting of three real valued functions

$S=S(x);\,\,y=y(x)$ (1)

where $y=\left(\begin{array}[]{cc}E\\\ N\end{array}\right)$ are state
variables on the target level that is in this section always the level of the
classical equilibrium thermodynamics, $E$ is the equilibrium thermodynamic
energy and $N$ is the equilibrium thermodynamic number of moles. By
$y(x)=\left(\begin{array}[]{cc}E(x)\\\ N(x)\end{array}\right)$ we denote the
energy and the number of moles given in the fundamental thermodynamic relation
(1) on the starting level. Both functions $S(x)$ and $y(x)$ are assumed to be
sufficiently regular and the entropy $S(x)$ is moreover assumed to be concave.

With (1) we construct thermodynamic potential

$\Phi(x,y^{*})=-S(x)+<y^{*},y(x)>$ (2)

The covector $y^{*}=(E^{*},N^{*})$ denotes the Lagrange multipliers. We
require

$E^{*}>0$ (3)

In the standard equilibrium thermodynamic notation $E^{*}=\frac{1}{T}$ and
$N^{*}=-\frac{\mu}{T}$, where $T$ is the absolute equilibrium thermodynamic
temperature in energy units and $\mu$ is the equilibrium thermodynamic
chemical potential. The requirement (3) is thus the requirement that the
absolute temperature $T$ is positive. By $<,>$ we denote pairing in the
equilibrium thermodynamic state space; $<y^{*},y>=E^{*}E+N^{*}N$.

With the thermodynamic potential $\Phi(x,y^{*})$ we are now in position to
make the passage $staring\,\,level\,\,\rightarrow\,\,target\,\,level$. The
passage is made by

$minimizing\,\,the\,\,thermodynamic\,\,potential\,\,\Phi(x,y^{*})$ (4)

Solutions of

$\Phi_{x}(x,y^{*})=0$ (5)

is denoted $x_{eq}(y^{*})$ and called equilibrium states. We use a shorthand
notation $\Phi_{x}=\frac{\partial\Phi}{\partial x}$. If $x$ is an element of
an infinite dimensional space (e.g. a distribution function) then the
derivative is an appropriate functional derivative.

The Legendre transformation $S^{*}(y^{*})$ (that belongs to the target level)
of $S(x)$ (that belongs to the starting level) is given by

$S^{*}(y^{*})=\Phi(x,y^{*})|_{x=x_{eq}(y^{*})}$ (6)

The function $S^{*}(y^{*})$ is a Legendre transformation of $S(y)$ that is the
entropy on the target level implied by the entropy $S(x)$ on the starting
level. Explicitly,

$S(y)=\Phi^{*}(y^{*},y)_{y^{*}=y^{*}_{ex}(y)}$ (7)

where $\Phi^{*}(y^{*},y)=-S^{*}(y^{*})+<y^{*},y>$ and $y^{*}_{ex}(y)$ is a
solution of $\Phi^{*}_{y^{*}}(y^{*},y)=0$.

So far we have not addressed yet the region $\Omega\subset\mathbb{R}^{3}$ in
which the macroscopic system under investigation is confined. We do it now but
only on the target level. We characterize $\Omega$ only by its volume $V$. We
take it into account by extending $S(y)$ introduced in (7) into $S=S(y,V)$. We
require that both $S$ and $y$ are extensive state variables in the sense that
$S(\lambda E,\lambda N,\lambda V)=\lambda S$, where $\lambda\in\mathbb{R}$.
From the Euler relation we have $S=<\frac{\partial S}{\partial
y},y>+\frac{\partial S}{\partial V}V$. From (7) we have $\frac{\partial
S}{\partial y}=y^{*}$. The Euler relation takes the form $S=<y^{*},y>+V^{*}V$,
where $V^{*}=\frac{\partial S}{\partial V}$. In the standard notation of the
classical thermodynamics $V^{*}=\frac{P}{T}$, where $P$ is the equilibrium
pressure. Consequently,

$S^{*}(y^{*})=-V^{*}$ (8)

Summing up, we have passed from the starting level with state variables $x$
and the fundamental thermodynamic relation (1) to the target level with state
variables $y$. The input is the fundamental thermodynamic relation (1), the
output is the equilibrium state $f_{eq}(y^{*})$ and the fundamental
thermodynamic relation

$S=S(y);\,\,y=y$ (9)

The equilibrium state $f_{eq}(y^{*})$ places the level of equilibrium
thermodynamics inside the starting level and $S(y)$ is the fundamental
thermodynamic relation on the level of equilibrium thermodynamics that is
inherited from the fundamental thermodynamics relation on the starting level.
We note that the passage $starting\,\,level\,\,\rightarrow\,\,target\,\,level$
made by MaxEnt (4) is in fact a reducing Legendre transformation.

The conjugate variables $y^{*}$ introduced in the thermodynamic potential (2)
belong to the target level (that is in this section the level of equilibrium
thermodynamics). One of them, namely $e^{*}=\frac{1}{T}$, relates energy to
entropy and is therefore of particular interest in this paper. We shall
discuss it in more detail in Section 2.3 below.

###  2.2 Dynamic theory

The postulated maximization of the entropy (4) is replaced in the dynamic
formulation by the time evolution equation

$\frac{\partial x}{\partial
t}=L(x)\mathfrak{E}_{x}(x)-\Xi_{x^{*}}(x,x^{*})|_{x^{*}=\mathfrak{E}_{x}}$
(10)

where

$\mathfrak{E}(x,y^{*})=\frac{1}{e^{*}}\Phi(x,y^{*})=E(x)-\frac{1}{e^{*}}S(x)+\frac{n^{*}}{e^{*}}N(x)$
(11)

is called, in accordance with the standard terminology (see ([1]), an
available free energy. The maximization of the entropy $S(x)$ subjected to
constraints $E(x),N(x)$ is made by following the time evolution governed by
(10). The asymptotic solution $t\rightarrow\infty$ of (10) is the equilibrium
state $x_{eq}(y^{*})$ that is also a solution to (5).

We now explain the meaning of the symbols introduced in the GENERIC equation
(10) and show that its solutions indeed approach $x_{eq}(y^{*})$. The first
term on the right hand side of (10) is the Hamiltonian vector field. It is the
covector $\mathfrak{E}_{x}$ transformed into vector by the Poisson bivector
$L(x)$. This bivector is defined by the bracket $\\{A,B\\}=<A_{x},LB_{x}>$
that is required to be a Poisson bracket satisfying the following properties:

$\displaystyle antisymmetry\,\,\\{A,B\\}=-\\{B,A\\}$ $\displaystyle
Jacobi\,\,identity\,\,\\{A,\\{B,C\\}\\}+\\{B,\\{C,A\\}\\}+\\{C,\\{A,B\\}\\}=0$
(12)

$A(x)$ and $B(x)$ are real valued and sufficiently regular function of $x$.
From the physical point of view, the Poisson bracket expresses the kinematics
of the state variable $x$.

Next, we turn to the second term on the right hand side of (10). The symbol
$\Xi$ stands for a real valued function of $(x,x^{*})$, called a dissipation
potential, satisfying the following properties

$\displaystyle\Xi(x,x^{*})|_{x^{*}=0}=0$
$\displaystyle\Xi(x,x^{*})\,\,reaches\,\,its\,\,minimum\,\,at\,\,x^{*}=0$
$\displaystyle\Xi(x,x^{*})\,\,is\,\,a\,\,convex\,\,function\,\,of\,\,x^{*}\,\,in\,\,a\,\,neighborhood\,\,of\,\,x^{*}=0$

We note that for small $x^{*}$ (i.e. in a small neighborhood of equilibrium
states) all dissipation potentials are quadratic functions of $x^{*}$.

With the requirements (2.2), (2.2), (15) Eq.(10) implies

$\frac{\partial\Phi}{\partial t}\leq 0$ (14)

This inequality makes the thermodynamic potential (2) the Lyapunov function
(provided (3) is taken into account and provided not only $-S(x)$ but also
$\Phi(x)$ is a convex function) indicating the approach to $x_{eq}(y^{*})$. A
rigorous proof of the approach requires an additional analysis. We shall
comment about it at the end of Section 3.1.

If we equip (10) with an extra structure

$\displaystyle LS_{x}=0;\,\,LN_{x}=0$ (15)
$\displaystyle\Xi\,\,depends\,\,on\,\,x^{*}\,\,only\,\,through\,\,the\,\,dependence\,\,on$
$\displaystyle
the\,\,thermodynamic\,\,force\,\,X=Kx^{*}\,\,where\,\,K\,\,is\,\,a\,\,linear\,\,operator$
$\displaystyle satisfying\,\,KE_{x}=0;\,\,KN_{x}=0$

then (10) turns into the familiar GENERIC equation ( see e.g. [18])

$\frac{\partial x}{\partial
t}=L(x)E_{x}(x)+\widehat{\Xi}_{x^{*}}(x,x^{*})|_{x^{*}=S_{x}}$ (16)

where $\widehat{\Xi}=\frac{1}{e^{*}}\Xi$. Equation (16) with the structure (15
then implies

$\displaystyle\frac{dE}{dt}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\frac{dN}{dt}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\frac{dS}{dt}$ $\displaystyle\geq$ $\displaystyle 0$ (17)

is a stronger property than (14). The result (2.2) implies (14) but gives a
more information about solutions to (16).

###  2.3 Roles of Energy and Entropy

The roles that the energy and the entropy play on various starting levels
depend on the levels. They will be discussed below. Here we recall the roles
that they play on the target level that is in this section the level of the
classical equilibrium thermodynamics. On this level the entropy plays the role
of one of the state variables. There is no inner energy, the only energy is
the internal energy. Moreover, the internal energy is in one-to-one relation
with the entropy. The entropy thus plays only the role (Ent I).

There is no time evolution on the level of equilibrium thermodynamics. The
entropy therefore does not play the role (Ent II). However, equilibrium
thermodynamics considers processes in which states change. The changes are due
to interactions through walls surrounding the macroscopic systems. The walls
can pass or prevent passing the internal energy, can expand or shrink the
system, and can enlarge or diminish the number of moles. An investigation of
the time evolution involved in such changes would require to step outside the
level of equilibrium thermodynamics on a level involving more details. When
remaining inside the equilibrium level, the processes are considered as
sequences of equilibrium states. Their final outcome is determined by MaxEnt.
In this sense the entropy plays on the level of equilibrium thermodynamics
also the role (Ent II). The unimportant details that are swept away in this
pattern recognition process are details of initial arrangements of the
subsystems.

Since the relation between internal energy and entropy is invertible their
roles as state variables can be exchanged. The fundamental thermodynamic
relation $E=E(S,N,V)$ can be replaced by

$S=S(E,N,V)$ (18)

In addition, the set of state variables $(S,N,V)$ can be enlarged by adopting
variables characterizing overall states of macroscopic systems. For example it
can be the overall velocity or coordinates characterizing placement of the
macroscopic system in an imposed force field (e.g. gravitational field). In
such case the total energy is a sum of the inner energy (the energy that can
be expressed in terms of the newly adopted state variables, for example the
overall kinetic and/or potential energy) and the remaining internal energy
expressed in terms of $(S,N,V)$.

Because our objective in this paper is to investigate relations between energy
and entropy, a particularly important concept is the temperature. On the level
of equilibrium thermodynamics the temperature is defined by
$\frac{1}{T}=S_{E}$ (see (7)). MaxEnt principle implies that two systems, that
are connected by a wall that freely passes the internal energy $E$ and both
are surrounded by a wall that prevents such passing approach an equilibrium
state at which both systems, have the same temperature. This is the way the
temperature on the level of equilibrium thermodynamics is measured. One of the
two systems is a thermometer for which the fundamental thermodynamic relation
is known and thus its temperature can be read in other quantities (e.g.
volume). A general temperature can be defined [19] as a measure of internal
energy, its measurements involve a process of equilibration. Different
meanings that can be given to ”internal energy” and ”equilibration” lead to
different meanings of the temperature.

##  3 Particular Realizations

The multiscale formulation of the passage starting level$\rightarrow$ level of
equilibrium thermodynamics (Section 2)) is a common structure extracted from
investigations of many such passages with specific choices of the starting
level. Some of these passages are now presented as particular realizations of
the multiscale formulation. In all realizations we always begin with the
static theory, continue with the dynamic theory and end with a discussion of
the roles of energy and entropy.

###  3.1 Boltzmann’s kinetic theory

Boltzmann’s investigation [15] of dynamics of ideal gases was the first step
towards understanding the approach of macroscopic systems to equilibrium. The
insight allowing to recognize the equilibrium pattern in the phase portrait is
the realization that collisions of gas particles are the source of unimportant
details that have to be swept away in order that the equilibrium pattern is
revealed. We present below the Boltzmann theory as a particular realization of
the multiscale formulation of starting level$\rightarrow$ level of equilibrium
thermodynamics.

Before placing ourselves on the level of kinetic theory, we recall that on the
level of equilibrium thermodynamics the individual nature of an ideal gas is
expressed in the equilibrium-thermodynamics fundamental thermodynamic relation

$S=\frac{5}{2}k_{B}N+NR\ln\left[\left(\frac{E}{V}\right)^{3/2}\left(\frac{V}{N}\right)\right]$
(19)

where $k_{B}$ is the Boltzmann constant and $R$ the universal gas constant. On
the level of equilibrium thermodynamics, this relation is obtained from
experimental observations of the behavior of ideal gases. Our objective is to
get (19) from investigating the passage level of kinetic
theory$\rightarrow$level of equilibrium thermodynamics. Our objective is to
introduce a particular realization of (10) which implies (19).

We begin with the static theory. The state variable on the level of kinetic
theory is the one particle distribution function

$x=f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ (20)

We put the mass of one particle equal to one, $r$ is the position coordinate
and $v$ momentum of one particle. The ideal gas under investigation is assumed
to be confined in the region $\Omega\subset\mathbb{R}^{3}$ with periodic
boundary conditions (i.e. all integrals over the boundary that arise in by
parts integrations equal zero).

The fundamental thermodynamic relation of an ideal gas on the level of kinetic
theory is

$\displaystyle S(f)$ $\displaystyle=$ $\displaystyle-\int
d{\boldmath\mbox{$r$}}\int
d{\boldmath\mbox{$v$}}f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})\ln
f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ $\displaystyle E(f)$
$\displaystyle=$ $\displaystyle\int d{\boldmath\mbox{$r$}}\int
d{\boldmath\mbox{$v$}}f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})\frac{{\boldmath\mbox{$v$}}^{2}}{2}$
$\displaystyle N(f)$ $\displaystyle=$ $\displaystyle\int
d{\boldmath\mbox{$r$}}\int
d{\boldmath\mbox{$v$}}f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ (21)

The energy $E(f)$ is the kinetic energy. The physical interpretation of
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ as a distribution function
leads directly to the number of moles $N(f)$, and the entropy $S(f)$ is
postulated.

The MaxEnt principle (4) leads (see (5)) to the equilibrium state

$f_{eq}({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})=\left(\frac{1}{2\pi
T}\right)^{3/2}\exp\left(\frac{\mu}{T}-1\right)\exp\left(-\frac{{\boldmath\mbox{$v$}}^{2}}{2T}\right)$
(22)

and the ideal gas fundamental thermodynamic relation (19) on the level of
equilibrium thermodynamics.

Next, we turn to the dynamic theory. The postulated MaxEnt principle (4) is
replaced by the Boltzmann kinetic equation

$\frac{\partial f}{\partial
t}=-\frac{\partial}{\partial{\boldmath\mbox{$r$}}}\left(f\frac{\partial
E_{f}}{\partial{\boldmath\mbox{$v$}}}\right)-\Xi_{f^{*}}|_{f^{*}=S_{f}}$ (23)

that is a particular realization of (16) with the Poisson bracket

$\\{A,B\\}=\int d{\boldmath\mbox{$r$}}\int
d{\boldmath\mbox{$v$}}f\left(\frac{\partial
A_{f}}{\partial{\boldmath\mbox{$r$}}}\frac{\partial
B_{f}}{\partial{\boldmath\mbox{$v$}}}-\frac{\partial
B_{f}}{\partial{\boldmath\mbox{$r$}}}\frac{\partial
A_{f}}{\partial{\boldmath\mbox{$v$}}}\right)$ (24)

(where $A(f),B(f)$ are real valued sufficiently regular functions of
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$) and the dissipation
potential

$\Xi(f,X)=\int d{\boldmath\mbox{$r$}}\int d{\boldmath\mbox{$v$}}\int
d{\boldmath\mbox{$v$}}_{1}\int d{\boldmath\mbox{$v$}}^{\prime}\int
d{\boldmath\mbox{$v$}}^{\prime}_{1}W(f,{\boldmath\mbox{$v$}},{\boldmath\mbox{$v$}}_{1},{\boldmath\mbox{$v$}}^{\prime},{\boldmath\mbox{$v$}}^{\prime}_{1})\left(e^{X}+e^{-X}-2\right)$
(25)

where $W\neq 0$ only if
${\boldmath\mbox{$v$}}+{\boldmath\mbox{$v$}}_{1}={\boldmath\mbox{$v$}}^{\prime}+{\boldmath\mbox{$v$}}^{\prime}_{1}$
and
${\boldmath\mbox{$v$}}^{2}+({\boldmath\mbox{$v$}}_{1})^{2}=({\boldmath\mbox{$v$}}^{\prime})^{2}+({\boldmath\mbox{$v$}}^{\prime}_{1})^{2}$.
In addition, $W$ is symmetric with respect to the interchange of
${\boldmath\mbox{$v$}}\rightarrow{\boldmath\mbox{$v$}}_{1}$ and
${\boldmath\mbox{$v$}}^{\prime}\rightarrow{\boldmath\mbox{$v$}}^{\prime}_{1}$
and with respect to the interchange of
$({\boldmath\mbox{$v$}},{\boldmath\mbox{$v$}}_{1})$ and
$({\boldmath\mbox{$v$}}^{\prime},{\boldmath\mbox{$v$}}^{\prime}_{1})$. The
thermodynamic force $X$ is given by

$X(f,f^{*})=\mathbb{K}f^{*}=f^{*}({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}}^{\prime})+f^{*}({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}}^{\prime}_{1})-(f^{*}({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})+f^{*}({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}}_{1})).$
(26)

The Poisson bracket (24) expresses kinematics of the one particle distribution
function $f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ (see e.g. [18]).
Regarding the dissipation potential (25), a direct verification proves that it
satisfies the required properties (2.2). The degeneracy requirements (15) can
also be proved by a direct verification. All functions $C(f)=\int
d{\boldmath\mbox{$r$}}\int
d{\boldmath\mbox{$v$}}\zeta(f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}}))$
where $\zeta:\mathbb{R}\rightarrow\mathbb{R}$ satisfy $\\{A,C\\}=0\,\,\forall
A$. Such functions are called Casimirs of the bracket $\\{A,B\\}$.
Consequently, the Poisson bivector $L$ defined by (24) satisfies thus the
degeneracy requirement (15). The required degeneracy of the dissipation
potential (25) can also be directly verified.

With the above specification of the building blocks of (10), the thermodynamic
potential (2) plays the role of the Lyapunov function for the time evolution
governed by Eq.(23). Maximization of the entropy subjected to constraints of
energy and number of moles is thus made by following the time evolution
governed by (23) to its conclusion.

There are still missing pieces in a rigorous proof of the approach of
solutions of (23) to the equilibrium state (22). In particular it is the
existence of solutions of the Boltzmann equation and an additional analysis
(in addition to identifying the Lyapunov function) needed to prove the
approach to the equilibrium state. The former, provided in [20], demonstrates
agreement with experimental observations. The time evolution of ideal gases is
seen in experimental observations to exist. Solutions to the Boltzmann kinetic
equation describing the time evolution of ideal gases in their mathematical
representation is proven in [20] to exist. Results proven in [21], [22], [23]
are even more physically significant. Solutions to the Boltzmann equation with
the Hamiltonian term missing approach the local equilibrium (that is the
equilibrium state (22) in which $n$ and $T$ are unspecified functions of the
position coordinate $r$). It is the coupling with the Hamiltonian term that
brings solutions to the total equilibrium (22). The Hamiltonian term by itself
does not produce any dissipation. The enhancement of dissipation, that is due
to the coupling of gradient dynamics with Hamiltonian dynamics, is called
Grad-Villani-Desvillettes enhancement of dissipation. Very likely this is the
principal mechanism making the transformation of time reversible and
nondissipative Hamiltonian time evolution of $\sim 10^{23}$ particles to time
irreversible and dissipative GENERIC time evolution. A very small instability
on the microscopic level may be enhanced by the Grad-Villani-Desvillettes
enhancement to macroscopic dissipation bringing macroscopic systems to
equilibrium states. So far only the damping that brings the local equilibrium
to the global equilibrium in the Boltzmann dynamics [21], [22] and the Landau
damping occurring in the Vlasov dynamics [23], [24] have been proven
rigorously. An argument supporting the general importance of the Grad-Villani-
Desvillettes enhancement of dissipation is presented in Section 3.3.2.

####  3.1.1 Roles of Energy and Entropy

Ideal gas particles do not interact among themselves. Their kinetic energy,
that is expressed in terms of $f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$
and is thus an inner energy, is the only energy. The entropy plays only the
role (Ent II). In the process of its maximization (or equivalently in the
process of following solutions of the Boltzmann kinetic equation to
$t\rightarrow\infty$) unimportant details in the phase portrait are swept away
and the equilibrium pattern in the phase portrait emerges. The entropy is not
postulated in the Boltzmann theory. It arises from investigating solutions of
the Boltzmann kinetic equation. The individual nature of ideal gases is
expressed only in the energy (that is the inner energy). The entropy is
universal. We also note that on the starting level there is no direct relation
between the energy and the entropy. Both are functions of
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ but
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ cannot be eliminated between
them. On the other hand, on the target level the energy and the entropy are
directly related, there is a one-to-one relation between them.

There is no temperature on the level of kinetic theory because there is no
internal energy. Only after the inner energy has been transformed by MaxEnt
(or equivalently by following the time evolution generated by the Boltzmann
equation), to the internal energy on the level of equilibrium thermodynamics,
the temperature can be defined on the submanifold composed of the equilibrium
states $f_{eq}(y)$ $\left(\frac{3}{2}T=(\int d{\boldmath\mbox{$r$}}\int
d{\boldmath\mbox{$v$}}f)^{-1}\int d{\boldmath\mbox{$r$}}\int
d{\boldmath\mbox{$v$}}\frac{{\boldmath\mbox{$v$}}^{2}}{2}\right)$. We can also
introduce local temperature that arises on the manifold composed of the local
equilibrium states. .

Before leaving the Boltzmann theory, we note that the abstract multiscale
thermodynamics in Section 2 is in fact an extraction of the mathematical
structure that is present in the Boltzmann equation. By transforming the
original Boltzmann theory into the form presented in Section 2 we have in fact
extended its applicability to general macroscopic systems.

###  3.2 Gibbs equilibrium statistical mechanics

.

Gibbs’ investigation [16] is limited to static situations but is applicable to
all macroscopic systems. The state variable is chosen to be $n$-particle
distribution function $f(z)$, where $n\sim 10^{23}$ is a fixed number of
microscopic particles composing macroscopic systems,
$z=({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})=(z_{1},...,z_{n})=({\boldmath\mbox{$r$}}_{1},...,{\boldmath\mbox{$r$}}_{n}{\boldmath\mbox{$v$}}_{1},...,{\boldmath\mbox{$v$}}_{n})$,
where $z_{i}=({\boldmath\mbox{$r$}}_{i},{\boldmath\mbox{$v$}}_{i})$, is the
position coordinate and the momentum if $i-th$ particle, $i=1,2,...,n$, $n\sim
10^{23}$. The Gibbs equilibrium statistical mechanics is traditionally
introduced (see e.g. [25]) in two steps. The equilibrium distribution function
$f_{eq}(z)$ in the first step, the entropy in the second step. Gibbs assumes
that $f_{eq}(z)$ depends only on constants of motion (i.e. on the energy and
the number of moles). Other details of their trajectories do not enter
$f_{eq}(z)$ due to the ergodic hypothesis according to which the particle
trajectories are uniformly spread. The exponential dependence of $f_{eq}(z)$
on the energy and the number of moles then follows from noting that the energy
and the number of moles of two independent subsystems is a sum of their
energies and the numbers of moles while $f(z)$ is a multiplication of their
two distribution functions. The Gibbs entropy appears by requiring that
$f_{eq}(z)$ arises in MaxEnt.

Below, we present the Gibbs theory as a particular realization of the static
multiscale formulation of level of particle mechanics$\longrightarrow$ level
of equilibrium thermodynamics in Section 2.1. We shall also supplement the
static theory with a corresponding to it dynamic theory that is a particular
realization of the multiscale dynamic theory in Section 2.2.

The state variable on the starting level of the Gibbs theory is the
$n$-particle distribution function

$x=f(z)$ (27)

The fundamental thermodynamic relation is

$\displaystyle S(f)$ $\displaystyle=$ $\displaystyle-\int dzf(z)\ln f(z)$
$\displaystyle E(f)$ $\displaystyle=$ $\displaystyle\int dzf(z)h(z)$
$\displaystyle N(f)$ $\displaystyle=$ $\displaystyle\int dzf(z)$ (28)

where $h(z)$ is the particle Hamiltonian and $k_{B}$ is the Boltzmann
constant. With these specifications the MaxEnt principle leads to the
equilibrium state

$f_{eq}(z)=\exp\left(\frac{\mu}{T}-1\right)\exp\left(-\frac{h(z)}{T}\right)$
(29)

and the fundamental thermodynamic relation.

$S^{*}(\mu,T)=\exp\left(\frac{\mu}{T}-1\right)\int
dz\exp\left(-\frac{h(z)}{T}\right)$ (30)

Now we turn to the dynamic theory. The Gibbs theory addresses the time
evolution only in the energy conservation and in the ergodic hypothesis.
Inspired by Boltzmann, we suggest a particular realization of (10) replacing
the ergodic hypothesis by the time evolution.

The time evolution of $f(z)$ (a lift of the Hamiltonian and reversible time
evolution of $z$ to the time evolution of real valued functions of $z$ [18])
is governed by the Liouville equation

$\frac{\partial f(z)}{\partial
t}=-\sum_{i=1}^{n}\sum_{\alpha=1}^{3}\frac{\partial}{\partial
r_{i\alpha}}\left(f(z)\frac{\partial h(z)}{\partial
v_{i\alpha}}\right)+\frac{\partial}{\partial
v_{i\alpha}}\left(f(z)\frac{\partial h(z)}{\partial r_{i\alpha}}\right)$ (31)

We note that this equation is a Hamilton equation $\frac{\partial
f(z)}{\partial t}=L(f)E(f)$ with the energy $E(f)$ given in (3.2) and the
Poisson bivector $L(z)$ given by the Poisson bracket [18]

$\\{A,B\\}=<A_{f},L(f)B_{f}>=\int
dzf(z)\sum_{i=1}^{n}\sum_{\alpha=1}^{3}\left(\frac{\partial A_{f}}{\partial
r_{i\alpha}}\frac{\partial B_{f}}{\partial v_{i\alpha}}-\frac{\partial
B_{f}}{\partial r_{i\alpha}}\frac{\partial A_{f}}{\partial
v_{i\alpha}}\right)$ (32)

This means that the Liouville equation is a particular realization of (10)
with the second term on its right hand side missing. We also note that the
lift from the time evolution of $z$ to the time evolution of $f(z)$ is a
linearization. Typically very nonlinear equations governing the time evolution
of $z$ become linear Liouville equations (31) governing the time evolution of
$f(z)$.

Can we modify the Liouville equation (31) in such a way that: (i) it becomes a
particular realization of (10), and (ii) the time evolution that it generates
maximizes the Gibbs entropy subjected to constraints of the energy and the
number of moles (specified in (3.2)). Our goal is to make the emergence of the
equilibrium pattern in the Liouville phase portrait (i.e. collection of
solutions to (31)) manifestly visible in the phase of appropriately modified
Liouville equation. The Gibbs equilibrium phase portrait will appear as an
attractive fixed point in its phase portrait.

The modification of Eq.(31) that we search should ideally result from a
thorough analysis if its solutions, in particular then from an analysis of
overall features of its solutions. We do not follow this path. Instead, we
suggest a formal modification of (31) by adding to its right hand side an
appropriate (inspired by the Boltzmann collision term) particular realization
of the second term on the right hand side of (16).

When the macroscopic system is an ideal gas then, following Boltzmann, a
separate treatment (separate from the free flow of particles) of binary
collisions leads to such modification. For general macroscopic systems we
replace binary collisions with transformations $z\leftrightarrow z^{\prime}$
that preserve (i.e. $h(z)=h(z^{\prime})$). We introduce thermodynamic force

$X(f^{*})=f^{*}(z^{\prime})-f^{*}(z)$ (33)

where $f^{*}(z)=S_{f(z)}(f)$ with S(f) given in (3.2). With the dissipation
potential

$\Xi(f;z,z^{\prime})=\int dz\int
dz^{\prime}W(f;z,z^{\prime})\left(e^{X}+e^{-X}-2\right)$ (34)

the modified Liouville equation becomes

$\displaystyle\frac{\partial f}{\partial t}$ $\displaystyle=$ $\displaystyle
L(f)E_{f}+\Xi_{f^{*}}|_{f^{*}=S_{f}}$ (35) $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial{\boldmath\mbox{$r$}}}\left(f\frac{\partial
h}{\partial{\boldmath\mbox{$v$}}}\right)+\frac{\partial}{\partial{\boldmath\mbox{$v$}}}\left(f\frac{\partial
h}{\partial{\boldmath\mbox{$r$}}}\right)+\int
dz^{\prime}W(f;z,z^{\prime})(f(z^{\prime})-f(z))$

The function $W$ in (34) is required to satisfy the following three
properties: (i) $W=0$ if $h(z)\neq h(z^{\prime})$; (ii) $W>0$ if
$h(z)=h(z^{\prime})$; (iii) $W$ is symmetric with respect to $z\leftrightarrow
z^{\prime}$. Equation (35) (already suggested in [26]) is indeed a particular
realization of (16). Also the degeneracy requirement (15) is clearly
satisfied. Its asymptotic solution is the equilibrium distribution function
(29) of the Gibbs theory. We note that the linearity of the Liouville equation
has not been erased in its modification. The modified Liouville equation (35)
remains a linear equation.

The dynamic formulation (35) of the static Gibbs theory remains formal. Both
the physical basis of the transformation $z\rightarrow z^{\prime}$ and a
detailed analysis of solutions to (35) needed to complete the proof of the
approach of its solutions to $f_{eq}(z)$ remain open. This type of
investigation reaches beyond the scope of this paper.

####  3.2.1 Roles of Energy and Entropy

The roles that the energy and the entropy play in the Gibbs theory are the
same as in the static Boltzmann theory. The total energy is an inner energy
characterizing completely the individual nature of macroscopic systems. The
entropy is universal and serves only to sweep away unimportant details (i.e.
it plays only the role (Ent II)). There is no internal energy. As in the
Boltzmann theory, there is no direct relation between the energy and the
entropy on the starting level. Both are functions of $f(z)$ but $f(z)$ cannot
be eliminated between them. The Boltzmann theory and the Gibbs theory differ
in the domain of applicability and in their dynamic formulations. The static
Boltzmann theory is applicable only to ideal gases, the static Gibbs theory to
all macroscopic systems. In their dynamical formulations, binary collisions
that drive gases to equilibrium in the Boltzmann theory, have no obvious
parallel in the Gibbs theory. The transformations $z\rightarrow z^{\prime}$
playing the role of binary collisions in the dynamic extension of the Gibbs
theory (35) remain formal. The entropy in the Boltzmann theory is not
postulated as in the Gibbs theory but emerges in the analysis of the time
evolution. Independent arguments supporting the Gibbs entropy come for
instance from the standard introduction of the Gibbs theory that we have
recalled in the first paragraph of this section or from connections with the
information theory [27].

As in the Boltzmann theory, there is no temperature on the starting level of
the Gibbs theory because there is no internal energy. The temperature arises
in the submanifold of equilibrium states $f_{eq}(z)$ in the same way as in the
Boltzmann theory.

###  3.3 Euler-Navier-Stokes-Fourier fluid dynamics

Starting levels in the two previous realizations of the mesoscopic
thermodynamics had no internal energy. Only after the passage to the target
level their initial starting-level inner energy turned to the target-level
internal energy on the level of equilibrium thermodynamics. In this section we
choose the level of fluid mechanics as the starting level. This level has both
the inner energy and the internal energy. The level of fluid mechanics is an
extension of the level of equilibrium thermodynamics to spatially
inhomogeneous fluids. Fluids are composed of fluid particles. A fluid particle
of unit volume at ${\boldmath\mbox{$r$}}\in\mathbb{R}^{3}$ has the momentum
${\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}})$, an internal structure
characterized by the internal energy $\epsilon({\boldmath\mbox{$r$}})$, and
the mass $\rho({\boldmath\mbox{$r$}})$. The energy $E$ that serves as the
state variable on the level of equilibrium thermodynamics turns in the
extension to the local internal energy $\epsilon({\boldmath\mbox{$r$}})$, the
number of moles $N$ to $\frac{1}{M^{(mol)}}\rho({\boldmath\mbox{$r$}})$, where
$M^{(mol)}$ is the molar mass. The volume $V$ on the level of equilibrium
thermodynamics enters the extension in assigning unit volume to fluid
particles and by requiring that the energy, the mass and the entropy are
extensive variables. The most important new feature in the extension is the
motion of the fluid particles characterized by the new state variable
${\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}})$. The motion of the fluid
particles then brings new contribution
$\frac{{\boldmath\mbox{$u$}}^{2}({\boldmath\mbox{$r$}})}{\rho({\boldmath\mbox{$r$}})}$
to the energy (the kinetic energy) that is the inner energy since it is
expressed in terms of the state variables. The total energy is
$E=E^{(in)}+E^{(int)}$, where $E^{(in)}=\int
d{\boldmath\mbox{$r$}}\frac{{\boldmath\mbox{$u$}}^{2}}{2\rho}$ is the total
kinetic energy and $E^{(int)}=\int
d{\boldmath\mbox{$r$}}\epsilon({\boldmath\mbox{$r$}})$ the total internal
energy.

Global conservation $\frac{dA}{dt}=0$ of a quantity $A$ becomes in the
extension to the local field $a({\boldmath\mbox{$r$}})$ (that is related to
$A$ by $A=\int d{\boldmath\mbox{$r$}}a({\boldmath\mbox{$r$}})$) the local
conservation $\frac{\partial a({\boldmath\mbox{$r$}})}{\partial
t}=-\frac{\partial{\boldmath\mbox{$J$}}^{(a)}}{\partial{\boldmath\mbox{$r$}}}$,
where ${\boldmath\mbox{$J$}}^{(a)}(a({\boldmath\mbox{$r$}}))$ is a flux of the
field $a({\boldmath\mbox{$r$}})$. The local conservation of $A$ requires thus
to specify the flux ${\boldmath\mbox{$J$}}^{(a)}$ and boundary conditions. As
to whether the local conservation implies the global one is decided by the
boundary conditions. In this paper we always assume periodic boundary
conditions that make integrals over boundary equal zero. The local
conservation thus always implies in this paper the global conservation.

Fluids that we investigate in this section on the level of fluid mechanics are
characterized on the level of equilibrium thermodynamics by fundamental
thermodynamic relations (18). Our objective is to make the passage level of
fluid mechanics$\rightarrow$level of equilibrium thermodynamics which ends up
with a placement of the equilibrium thermodynamic state space inside the fluid
mechanics state space (i.e. with the fluid-mechanics equilibrium state) and
the equilibrium fundamental thermodynamic relation (18) with which the
extension started. In other words, we begin with fluids that are characterized
on the level of equilibrium thermodynamics by (18), extend their investigation
to fluid mechanics and then by following the time evolution end up with the
fundamental thermodynamic relation (18) with which the extension started. Our
focus is put again on the roles that energy and entropy play in such passage.

The variables

$x=(s({\boldmath\mbox{$r$}}),\rho({\boldmath\mbox{$r$}}),{\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}))$
(36)

playing the role of state variables on the level of fluid mechanics are
physically interpreted as local entropy per unit volume, local mass per unit
volume, and local momentum respectively.

The fundamental thermodynamic relation is

$\displaystyle S(s,\rho,{\boldmath\mbox{$u$}})$ $\displaystyle=$
$\displaystyle\int d{\boldmath\mbox{$r$}}s({\boldmath\mbox{$r$}})$
$\displaystyle E(s,\rho,{\boldmath\mbox{$u$}})$ $\displaystyle=$
$\displaystyle\int
d{\boldmath\mbox{$r$}}e(s,\rho,{\boldmath\mbox{$u$}};{\boldmath\mbox{$r$}})=E^{(in)}(\rho,{\boldmath\mbox{$u$}})+E^{(int)}(s,\rho,{\boldmath\mbox{$u$}})$
$\displaystyle E^{(in)}(\rho,{\boldmath\mbox{$u$}})$ $\displaystyle=$
$\displaystyle\int
d{\boldmath\mbox{$r$}}\frac{{\boldmath\mbox{$u$}}^{2}}{2\rho}$ $\displaystyle
E^{(int)}(s,\rho)$ $\displaystyle=$ $\displaystyle\int
d{\boldmath\mbox{$r$}}\epsilon(s,\rho;{\boldmath\mbox{$r$}})$ $\displaystyle
N(\rho)$ $\displaystyle=$ $\displaystyle\frac{1}{M_{mol}}\int
d{\boldmath\mbox{$r$}}\rho({\boldmath\mbox{$r$}})$ (37)

The energy $E^{(in)}(\rho,{\boldmath\mbox{$u$}})$ is the kinetic energy that
is in the classical hydrodynamics the inner energy, i.e. the part of the
energy that can be expressed in terms of state variables excluding the
entropy. Moreover,
$s_{\epsilon}({\boldmath\mbox{$r$}})=\frac{1}{\tau({\boldmath\mbox{$r$}})}$ is
physically interpreted as a local temperature and is assumed to be positive
(an extension of (3) to local fields). If the function
$\epsilon(s,\rho;{\boldmath\mbox{$r$}})$ is assumed to be the same as
$E(S,N,V)|_{V=1,N=\frac{\rho}{M^{(mol)}}}$ on the level of equilibrium
thermodynamics then such assumption is called an assumption of local
equilibrium. Identification of $\tau({\boldmath\mbox{$r$}})$ with the local
temperature is a weaker form of the local equilibrium assumption. The
positivity of the local temperature implies that there is a one-to-one
relation between the energy field $e({\boldmath\mbox{$r$}})$ and the entropy
field $s({\boldmath\mbox{$r$}})$. For later use we recall the relations

$s_{e}=\frac{1}{e_{s}};\,\,s_{{\boldmath\mbox{$u$}}}=-\frac{e_{{\boldmath\mbox{$u$}}}}{e_{s}};\,\,s_{\rho}=\frac{e_{\rho}}{e_{s}}$
(38)

With the fundamental thermodynamic relation (3.3) and the local equilibrium
assumption, the MaxEnt principle leads to the equilibrium state

${\boldmath\mbox{$u$}}_{eq}({\boldmath\mbox{$r$}})=0;\,\,\left(s_{e}({\boldmath\mbox{$r$}})\right)_{eq}=\frac{1}{\tau_{eq}({\boldmath\mbox{$r$}})}=\frac{1}{T};\,\,\left(s_{\rho}({\boldmath\mbox{$r$}})\right)_{eq}=-\frac{\mu}{T}$
(39)

and the fundamental thermodynamic relation (9). With this result we end the
static theory.

Now we turn to the dynamical theory. We look for a particular realization of
(16) that replaces the postulated maximization in the MaxEnt principle.. We
know already the potentials (3.3) entering (16), we only need to identify the
Poisson bracket expressing kinematics of (36) and a dissipation potential.

Motion of continuum are transformations
$\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$. These transformation form a Lie
group. The momentum field $u$ is an element of the dual of the algebra
corresponding to the group. The structure of the group manifests itself in the
dual of the Lie algebra that corresponds to the group as the Poisson bracket
$\\{A,B\\}=\int d{\boldmath\mbox{$r$}}u_{i}\left(\frac{\partial
A_{u_{i}}}{\partial r_{j}}B_{u_{j}}-\frac{\partial B_{u_{i}}}{\partial
r_{j}}A_{u_{j}}\right)$ [28],[18] (the convention of summation over repeated
indices is used). The remaining two fields $\rho({\boldmath\mbox{$r$}})$ and
$s({\boldmath\mbox{$r$}})$ are let to be passively advected by
${\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}})$. The Poisson bracket that
expresses kinematics of the state variables (36) is [18]

$\displaystyle\\{A,B\\}$ $\displaystyle=$ $\displaystyle\int
d{\boldmath\mbox{$r$}}\left[u_{i}\left(\frac{\partial A_{u_{i}}}{\partial
r_{j}}B_{u_{j}}-\frac{\partial B_{u_{i}}}{\partial r_{j}}\right)\right.$ (40)
$\displaystyle\left.+\rho\left(\frac{\partial A_{\rho}}{\partial
r_{i}}B_{u_{i}}-\frac{\partial B_{\rho}}{\partial
r_{i}}A_{u_{i}}\right)\right.$ $\displaystyle\left.+s\left(\frac{\partial
A_{s}}{\partial r_{i}}B_{u_{i}}-\frac{\partial B_{s}}{\partial
r_{i}}A_{u_{i}}\right)\right]$

The Hamilton time evolution equation $\frac{\partial x}{\partial
t}=L\mathfrak{E}_{x}$ with $x$ given in (36) and $L$ in (40) are the Euler
equations

$\displaystyle\frac{\partial u_{i}}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\partial(u_{i}\mathfrak{E}_{u_{j}})}{\partial
r_{j}}-\frac{\partial p}{\partial r_{i}}$ $\displaystyle\frac{\partial
s}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\partial(s\mathfrak{E}_{u_{i}})}{\partial r_{i}}$
$\displaystyle\frac{\partial\rho}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\partial(\rho\mathfrak{E}_{u_{i}})}{\partial r_{i}}$ (41)

where

$p({\boldmath\mbox{$r$}})=-e+\rho\mathfrak{E}_{\rho}+s\mathfrak{E}_{s}+u_{i}\mathfrak{E}_{u_{i}}$
(42)

We note in particular that the degeneracy requirements (15) hold and that the
entropy contributes, through its presence in the local pressure
$p({\boldmath\mbox{$r$}})$, to drive the time reversible Hamiltonian time
evolution. The gradient of the local pressure is partially an entropic force.
Entropy takes this new role because of the presence of the internal energy. In
the absence of such energy, as it is in the case of the Boltzmann and the
Gibbs theories, the entropy does not participate in the Hamiltonian part of
the time evolution.

The forces that drive fluids to thermodynamic equilibrium are the Fourier
force $X^{F}$ and the Navier Stokes forces $X^{NS}$ and $X^{NSvol}$

$\displaystyle X_{i}^{F}({\boldmath\mbox{$r$}})$ $\displaystyle=$
$\displaystyle\frac{\partial s_{e}}{\partial r_{i}}$ $\displaystyle
X_{ij}^{NS}({\boldmath\mbox{$r$}})$ $\displaystyle=$
$\displaystyle\left(\frac{\partial e_{u_{i}}}{\partial r_{j}}+\frac{\partial
e_{u_{j}}}{\partial r_{i}}\right)$ $\displaystyle X^{NSvol}$ $\displaystyle=$
$\displaystyle div(e_{{\boldmath\mbox{$u$}}})$ (43)

Dissipation potential

$\Xi^{flm}=\frac{1}{2}\int d{\boldmath\mbox{$r$}}\lambda
X_{i}^{F}({\boldmath\mbox{$r$}})X_{i}^{F}({\boldmath\mbox{$r$}})+\frac{1}{2}\int
d{\boldmath\mbox{$r$}}\eta
X_{ij}^{NS}({\boldmath\mbox{$r$}})X_{ij}^{NS}({\boldmath\mbox{$r$}})+\frac{1}{2}\int
d{\boldmath\mbox{$r$}}\eta^{(vol)}(X^{NSvol})^{2}$ (44)

generates the Navier Stokes Fourier gradient terms supplementing the Euler
equations (3.3)

$\frac{\partial}{\partial t}\left(\begin{array}[]{ccc}{\boldmath\mbox{$u$}}\\\
e\\\ \rho\end{array}\right)=\left(\begin{array}[]{ccc}Euler\\\ equations\\\
\end{array}\right)+\left(\begin{array}[]{ccc}\Xi_{s_{{\boldmath\mbox{$u$}}}}\\\
\Xi_{s_{e}}\\\ 0\end{array}\right)$ (45)

The coefficients $\lambda>0,\eta>0,\eta^{(vol)}>0$ introduced in the
dissipation potential (44) are functions of (36). The calculations that are
needed to turn (45) into the familiar Navier Stokes Fourier equations use the
relations (38) (see [18] for details of the calculations) .

####  3.3.1 Extended fluid mechanics

The level of kinetic theory playing the role of the starting level in Section
3.1 involves more details than the level of fluid mechanics that plays the
role of the starting level in this section. Indeed, the physical
interpretation of the one particle distribution function
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ and of the hydrodynamic
fields
$({\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}),\rho({\boldmath\mbox{$r$}}))$
implies that the hydrodynamic fields can be expressed in terms of
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ as its moments
$\rho({\boldmath\mbox{$r$}})=\int
d{\boldmath\mbox{$v$}}f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ and
${\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}})=\int
d{\boldmath\mbox{$v$}}\,{\boldmath\mbox{$v$}}f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$.
Are there autonomous levels that lie between the level of kinetic theory and
fluid mechanics? Such levels, if they exist, are then expected to provide a
theoretical framework for complex fluids (as for example viscoelastic
polymeric fluids or suspensions) that are found to be outside the domain of
applicability of the classical fluid mechanics.

Intermediate levels can be constructed in two ways: bottom up and top down.
The former is an extension of the level of fluid mechanics by adopting extra
fields (e.g. higher order moments of
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ or fields characterizing an
internal structure) as extra state variables. The latter as a reduction from
the level of the kinetic theory or other theories involving more details than
fluid mechanics. We first discuss the former and then the latter
constructions.

There are two kinds of the bottom up extensions. The state variables in the
first are
$x=({\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}),\rho({\boldmath\mbox{$r$}}),{\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}}))=({\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}),\rho({\boldmath\mbox{$r$}}),a_{1}({\boldmath\mbox{$r$}}),...,a_{k}({\boldmath\mbox{$r$}}))$
and in the second
$x=({\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}),\rho({\boldmath\mbox{$r$}}),s({\boldmath\mbox{$r$}}),{\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}}))$.
In the first extension we proceed as in the Boltzmann and the Gibbs theories
but with the fields
$({\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}),\rho({\boldmath\mbox{$r$}}),{\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}}))$
replacing the distribution function
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ (or $f(z)$ in the Gibbs
theory). We need to specify the fundamental thermodynamic relation (i.e. we
need to specify $S=\int
d{\boldmath\mbox{$r$}}s({\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}),\rho({\boldmath\mbox{$r$}}),{\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})),E=\int
d{\boldmath\mbox{$r$}}e({\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}),\rho({\boldmath\mbox{$r$}}),{\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})),N=\int
d{\boldmath\mbox{$r$}}n({\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}),\rho({\boldmath\mbox{$r$}}),{\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}}))$,
Poisson bracket expressing kinematics of the fields
$({\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}),\rho({\boldmath\mbox{$r$}}),{\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}}))$,
thermodynamic forces, and dissipation potential. As in the Boltzmann and the
Gibbs theory, there is no internal energy. In the second extension we proceed
as in fluid mechanics but with the kinetic energy replaced by an enlarged
inner energy involving the extra fields
${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})$. The total energy involves an
internal energy that is expressed in terms of the entropy field that serves as
one of the state variables.

The most familiar example of the top down extension is the reformulation of
the Boltzmann kinetic equation (23) into Grad’s hierarchy [29]. The one
particle distribution function
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ is replaced by infinite
number of fields
${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})^{(\infty)}({\boldmath\mbox{$r$}})=(a_{1}({\boldmath\mbox{$r$}}),...)$
that are moments of $f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ in the
momentum $v$. There are two ways to rewrite (23) into an equation governing
the time evolution of
${\boldmath\mbox{$a$}}^{(\infty)}({\boldmath\mbox{$r$}})$. First, it is a
direct method [29] consisting of multiplying (23) by $v_{i}$ and integrating
over $v$, then by $v_{i}v_{j}$ and integrating over $v$, and repeating this
process for all moments. In the second method we regard (23) as a particular
realization of the GENERIC equation (10) and rewrite into moments separately
all its building blocks (for example the Poisson bracket (24) is rewritten
into Grad’s moments in [30]). The two methods lead to two different infinite
hierarchies [30].

If our objective is to obtain an extended fluid mechanics that involves a
finite number of fields ${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})$, that
addresses physics of complex fluids, and that is compatible with equilibrium
thermodynamics in the sense of Section 2 then, in both bottom up and top down
approaches, we need to identify the finite number of fields
${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})=(a_{1}({\boldmath\mbox{$r$}}),...,a_{k}({\boldmath\mbox{$r$}}))$
joining the hydrodynamic fields
$(\rho({\boldmath\mbox{$r$}}),{\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}))$
and corresponding to them building blocks of GENERIC. An inspiration can come
from an insight into the physics of the internal structure (e.g. various
models of macromolecules composing polymeric fluids [31], [32], [33]), from an
insight into the geometrical structure of continuum dynamics ([34], [35], and
from hierarchy reformulations of kinetic equations [29],[36], [37], [38].
Poisson brackets expressing kinematics of some of extended sets of
hydrodynamic fields
$({\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}),\rho({\boldmath\mbox{$r$}}),{\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}}))$
are identified in [13],[39].

In the top down approach to extensions the passage from an infinite number of
fields ${\boldmath\mbox{$a$}}^{(\infty)}({\boldmath\mbox{$r$}})$ to a finite
number
${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})=(a_{1}({\boldmath\mbox{$r$}}),...,a_{k}({\boldmath\mbox{$r$}}))$
is called a closure of the infinite hierarchy. Difficulties encountered in
finding appropriate closures are well illustrated in the problem of finding
Poisson brackets expressing kinematics of a finite number of moments
${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})=(a_{1}({\boldmath\mbox{$r$}}),...,a_{k}({\boldmath\mbox{$r$}}))$
in the Grad hierarchy. Two arguments developed in [40] suggest that there are
only three autonomous fluid mechanics theories that can be based on the Grad
hierarchy: $\mathbb{F}\mathbb{M}^{\infty}$ that is fluid mechanics with all
infinite number of moments playing the role of state variables (such fluid
mechanics is equivalent to kinetic theory), $\mathbb{F}\mathbb{M}^{5}$ that is
the classical fluid mechanics with the classical hydrodynamic fields playing
the role of state variables, and $\mathbb{F}\mathbb{M}^{\infty-5}$ that is
fluid mechanics in which all Grad moments except the hydrodynamic fields play
the role of state variables. The first argument is based on the analysis of
the Lie algebra of moments, the second argument is physical. We recall the
latter argument. After the onset of turbulence the originally simple fluid
whose behavior is well described by the classical hydrodynamic fields becomes
a complex fluid with an internal structure that needs higher Grad moments to
describe its behavior. From the Kolmogorov cascade we know that the complexity
of the flow passes gradually to smaller and smaller scales until the molecular
scale is reached. At the molecular scale the inner energy (i.e. the energy
expressed in terms of ${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})$) of the
complex turbulent flow turns into an internal energy. If there were an
autonomous fluid mechanics with a finite number $k$ of Grad moments then the
Kolmogorov cascade would have a plateau. The turbulent decay would stop on the
$k$-scale (i.e. the scale associated with $k$-th Grad moments), the flow would
become $k$-laminar (i.e. laminar in the setting of the extended fluid
mechanics in which $k$ Grad moments play the role of the state variables) and
then a $k$-onset of turbulence would be needed to continue the turbulent decay
to the molecular scale. No such plateau in the Kolmogorov cascade is, to the
best of our knowledge, observed.

####  3.3.2 Roles of Energy and Entropy

Fluids are composed of fluid particles that have an internal structure and
associated with it an internal energy. Its presence distinguishes the level of
fluid mechanics from two levels discussed in the two previous illustrations of
mesoscopic thermodynamics. The total energy $e({\boldmath\mbox{$r$}})$ of the
fluid particle at the position $r$ is a sum of the inner energy (that is the
kinetic energy
$\frac{{\boldmath\mbox{$u$}}^{2}({\boldmath\mbox{$r$}})}{2\rho({\boldmath\mbox{$r$}})}$)
and the internal energy $\epsilon(s,\rho;{\boldmath\mbox{$r$}})$. The total
energy $E=\int d{\boldmath\mbox{$r$}}e({\boldmath\mbox{$r$}})$ of the fluid is
then the sum of the energy of all fluid particles. The energy that generates
the Hamiltonian part of the time evolution involves entropy and thus the
entropy participates in the Hamiltonian time evolution. Specifically, the
gradient of the local pressure $p({\boldmath\mbox{$r$}})$ is the entropic
force. The entropy generates in addition also the dissipative forces driving
the fluids to total equilibrium at which the fluid particles do not move and
their internal structures are identical. The entropy plays thus on the
starting level both roles (Ent I) and (Ent II). A weak local equilibrium
assumption is carrying the one-to-one relation between internal energy and
entropy from the level of equilibrium thermodynamics to the level of fluid
mechanics, a complete local equilibrium assumption is carrying the complete
equilibrium thermodynamic relation to individual fluid particles.

An interesting insight into the dependence of the passages starting
level$\rightarrow$ level of equilibrium thermodynamics on the starting level
can be gained by comparing the passages with kinetic theory and fluid
mechanics playing the role of starting levels. For all starting levels the
time evolution making the passages are governed by (10). The differences are
in the strength of the dissipation in the second term on the right hand side
of (10). In the context of fluid mechanics the passage from local equilibrium
to the total thermodynamics equilibrium is made by the Navier-Stokes-Fourier
dissipative forces (3.3) and the dissipation potential (44). In the context of
the Boltzmann kinetic theory the passage from local Maxwellian distribution
functions (that we can see as states corresponding to local equilibrium states
in fluid mechanics) to the total Maxwellian distribution functions (22) does
not require an extra explicit dissipation. It is made simply by coupling the
dissipation driving to the local Maxwellian distribution functions with the
Hamiltonian completely nondissipative part of the time evolution (by the Grad-
Villani-Desvillettes enhancement of dissipation [22]). More precisely,
solutions of the Boltzmann equation get in the course of the time evolution
only close to the local equilibrium and reach it only in the total equilibrium
(22). This comparison supports the conjecture that a very weak dissipation
(instability) on the microscopic level is sufficient to grow by the Grad-
Villani-Desvillettes enhancement of dissipation to macroscopic dissipation
towards the level equilibrium thermodynamics.

Temperature on the level of fluid mechanics has the same meaning as the
temperature on the level of the equilibrium thermodynamics. The situation is
different if the fluid mechanics is extended by adopting fields
${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})$ as extra state variables. If
the state variables of the extended fluid mechanics do not include the entropy
field $s({\boldmath\mbox{$r$}})$ then such setting is essentially the same as
the setting of the Boltzmann and the Gibbs theory. There is no internal energy
and no temperature on the starting level. If the entropy field
$s({\boldmath\mbox{$r$}})$ is included in the set of state variables then the
energy that cannot be expressed in terms of
$({\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}),\rho({\boldmath\mbox{$r$}}),s({\boldmath\mbox{$r$}}),{\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}}))$
is the internal energy. But such internal energy is different from the
internal energy in fluid mechanics in which the extra fields
${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})$ are missing in the set of state
variables. The part of the energy
$\widehat{\epsilon}({\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}}))$ that is
expressed in terms of ${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})$ is
excluded from the internal energy in the extended fluid mechanics. The meaning
of temperature depends on what kind of walls we use in its measurement and on
what kind of control we have over the fields
${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})$. The walls can either pass or
prevent passing the energy
$\epsilon({\boldmath\mbox{$r$}})-\widehat{\epsilon}({\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}}))$
($Walls^{(extfm)}$) or they are the same as the ones used in the classical
fluid mechanics in which the fields
${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})$ are not included in the set of
state variables ($Walls^{(fm)}$). With $Walls^{(fm)}$ and no control over
${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})$ the measured temperature is the
same as in the classical fluid mechanics. With $Walls^{(extfm)}$ and with
control over ${\boldmath\mbox{$a$}}({\boldmath\mbox{$r$}})$ the measured
temperature is a different temperature.

###  3.4 van der Waals gas

In this section we extend the kinetic theory discussed in Section 3.1. The
extension is not made by adopting extra distribution functions as state
variables (the one particle distribution function
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ remains the only state
variable) but by introducing an internal energy. From the microscopic point of
view, the van der Waals (vdW) gas particles become interacting particles. Two
types of interactions are considered: a long range attraction and a short
range hard-core repulsion. The former is considered as an inner energy and the
latter as an internal energy. The inner energy is the mean-field type Vlasov
energy, the internal energy is expressed in a modification of the Boltzmann
entropy. The hard core repulsion is replaced by an excluded-volume type
constraint. From the point of view of hydrodynamic and equilibrium-
thermodynamic type observations the vdW gas differs from the ideal gas by
experiencing transition from gas to liquid. The Vlasov gas shows the approach
to spatially homogeneous distribution (Landau damping). How do the roles of
the energy and the entropy change when ideal gas becomes vdW or Vlasov gas?.

Before discussing the vdW gas on the level of kinetic theory, we recall the
vdW fundamental thermodynamic relation

$S=\frac{5}{2}N+NR\ln\left[\left(\frac{E}{V}+a\frac{N}{V}\right)^{3/2}\left(\frac{V}{N}-b\right)\right]$
(46)

on the level of equilibrium thermodynamics We note that (46) is a two
parameter $(a,b)$ deformation of the ideal gas fundamental thermodynamic
relation (19) on the level of equilibrium thermodynamics. If $a=0$ and $b=0$
then (46) reduces to (19). On the level of equilibrium thermodynamics, the
relation (46) is based on heuristic arguments and experimental observations.

As in the previous sections we begin with the static theory. The state
variable is the same one particle distribution function (20) as in the
Boltzmann theory. The fundamental thermodynamic relation extending (3.1) is
[42]

$\displaystyle S(f)$ $\displaystyle=$ $\displaystyle-\int
d{\boldmath\mbox{$r$}}\int
d{\boldmath\mbox{$v$}}(f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})\ln
f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})+k_{B}f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})\ln(1-bn({\boldmath\mbox{$r$}})))$
$\displaystyle E(f)$ $\displaystyle=$ $\displaystyle\int
d{\boldmath\mbox{$r$}}\int
d{\boldmath\mbox{$v$}}\frac{{\boldmath\mbox{$v$}}^{2}}{2}f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$
$\displaystyle+\frac{1}{2}\int d{\boldmath\mbox{$r$}}\int
d{\boldmath\mbox{$r$}}_{1}\int d{\boldmath\mbox{$v$}}\int
d{\boldmath\mbox{$v$}}_{1}f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})V_{pot}(|{\boldmath\mbox{$r$}}-{\boldmath\mbox{$r$}}_{1}|)f({\boldmath\mbox{$r$}}_{1},{\boldmath\mbox{$v$}}_{1})$
$\displaystyle N(f)$ $\displaystyle=$ $\displaystyle\int
d{\boldmath\mbox{$r$}}\int
d{\boldmath\mbox{$v$}}f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ (47)

where $n({\boldmath\mbox{$r$}})=\int
d{\boldmath\mbox{$v$}}f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$. MaxEnt
principle implies the equilibrium state

$f_{eq}({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})=\left(\frac{1}{2\pi
T}\right)^{3/2}n_{eq}({\boldmath\mbox{$r$}})\exp\left(-\frac{{\boldmath\mbox{$v$}}^{2}}{2T}\right)$
(48)

where $n_{eq}({\boldmath\mbox{$r$}})$ is a solution to

$\ln
n({\boldmath\mbox{$r$}})+\frac{b}{1-bn({\boldmath\mbox{$r$}})}+\frac{1}{T}\int
d{\boldmath\mbox{$r$}}^{\prime}V_{pot}(|{\boldmath\mbox{$r$}}-{\boldmath\mbox{$r$}}^{\prime}|)n({\boldmath\mbox{$r$}}^{\prime})-\frac{3}{2}\ln(2\pi
T)-\frac{\mu}{T}$ (49)

and the fundamental thermodynamic relation (46) on the level of equilibrium
thermodynamics.

Gas-liquid phase transition manifests itself in (48) (as the familiar P-V-T
curves - see e.g [41]) and also in the manifold of equilibrium states (48) (as
a bifurcation in solutions to (49) - see [42], [43], [18]).

Now we turn to the dynamic theory. Our objective again is to replace the
maximization of the entropy postulated in the static theory with the time
evolution describing the experimentally observed approach to equilibrium. How
does the kinetic equation (23) change when the ideal gas change into the vdW
gas? There are at least two answers to this question. First it is a particular
realization of (10) with the potentials (3.4) and the second is the Boltzmann
kinetic equation with two changes: the Vlasov term expressing the influence of
the attractive force is added to its right hand side and the Boltzmann
collisions of point particles are replaced by Enskog collisions of particles
having a finite size. The modified kinetic equation (called Enskog Vlasov
equation [44]) seems to be very natural from the physical point of view but
its compatibility with the static theory presented above as well as with the
van der Waals fundamental thermodynamic relation (46) requires additional
assumptions that do not appear to be natural from the physical point of view
[45], [43]. Below, we follow the path (already suggested in [46]) on which the
kinetic equation emerges as a particular realization of (10) with potentials
(3.4).

The state variable is the same as in the Boltzmann kinetic theory, its
physical interpretation is the same, and thus also its kinematics expressed in
the Poisson bracket (24) is the same. We leave also the same thermodynamic
forces (26) as well as the thermodynamic potential (25). The binary collisions
in which the entropy plays the role ((ENT II) are thus left in the vdW gas the
same as in the ideal gas. The finite size of the particles is taken into
account in the Hamiltonian part of the time evolution by changing the
available free energy $\mathfrak{E}(f)$ (3.4). With the building blocks
specified above, Eq.(10) becomes

$\frac{\partial f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})}{\partial
t}=-\frac{\partial}{\partial{\boldmath\mbox{$r$}}}\left(Tf\frac{\frac{\partial\Phi}{\partial
f}}{\partial{\boldmath\mbox{$v$}}}\right)+\frac{\partial}{\partial{\boldmath\mbox{$v$}}}\left(Tf\frac{\frac{\partial\Phi}{\partial
f}}{\partial{\boldmath\mbox{$r$}}}\right)-\frac{\partial\Xi}{\partial
f^{*}({\boldmath\mbox{$r$}})}|_{f^{*}({\boldmath\mbox{$r$}})=\frac{\partial\Phi}{\partial
f({\boldmath\mbox{$r$}})}}$ (50)

Using the thermodynamic potential (3.4), the kinetic equation (50) gets the
form

$\displaystyle\frac{\partial
f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial{\boldmath\mbox{$r$}}}\left(\frac{{\boldmath\mbox{$v$}}}{m}f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})\right)$
(51) $\displaystyle+T\frac{\partial
f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})}{\partial{\boldmath\mbox{$v$}}}\frac{\partial}{\partial{\boldmath\mbox{$r$}}}\left(\ln(1-bn({\boldmath\mbox{$r$}}))-\frac{bn({\boldmath\mbox{$r$}})}{1-bn({\boldmath\mbox{$r$}}))}\right)$
$\displaystyle+\frac{\partial
f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})}{\partial{\boldmath\mbox{$v$}}}\frac{\partial}{\partial{\boldmath\mbox{$r$}}}\int
d{\boldmath\mbox{$r$}}_{1}V_{pot}(|{\boldmath\mbox{$r$}}-{\boldmath\mbox{$r$}}_{1}|)n({\boldmath\mbox{$r$}}_{1})$
$\displaystyle-\frac{\partial\Xi(f,f^{*})}{\partial
f^{*}({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})}|_{f^{*}=\frac{\partial\Phi}{\partial
f}}$

Solutions to this kinetic equation describe approach to the equilibrium vdW
theory. Equation (51) is a particular realization of the GENERIC equation
(10). It shares the GENERIC structure with all well established equations
describing approach to the level of equilibrium thermodynamics. The
particularly interesting and new feature of (51) is that vdW equilibrium
theory addresses gas-liquid phase transitions and consequently (51) addresses
its dynamical aspects. In order to see them we need detailed solution to (51).
We hope to present them in a future paper.

####  3.4.1 Roles of Energy and Entropy

When comparing the static kinetic theory of ideal gases (Section 3.1) with the
static kinetic theory of the vdW gas, the most important new feature in the
vdW theory is the presence of internal energy that is expressed as a
modification (3.4) of the Bolktzmann entropy (3.1). From the physical point of
view, the hard-core repulsive potential cannot be expressed in terms of only
one particle distribution function (i.e. it is not an inner energy). At least
two particle distribution function is needed to express it. In the setting of
the kinetic theory in which only one particle distribution function serves as
the state variable the hard core repulsive potential has to be considered as
an internal energy. Its expression in terms of entropy then follows from
replacing the hard-core potential with the excluded-volume type constraint
[42]. We note that if we do not modify the entropy in (3.4) (i.e. if we keep
only the Boltzmann entropy (3.1)) then there is no modification of the inner
energy $E(f)$ in (3.4) that would imply the vdW fundamental thermodynamic
relation (46) on the level of equilibrium thermodynamics. This result is just
another manifestation of a well known result from equilibrium statistical
mechanics, namely that the Gibbs equilibrium theory does not provide a setting
for phase transitions unless it is somehow extended (e.g. by carrying it to
thermodynamic limit
$N\rightarrow\infty;\,\,V\rightarrow\infty;\,\,\frac{N}{V}=const.$) [47]. An
internal energy that modifies the Gibbs entropy (in the setting of the Gibbs
theory) or the Boltzmann entropy (in the setting of the Boltzmann theory) is
needed to make phase transitions visible in geometrical features of the
manifold of equilibrium states and geometrical features of the fundamental
equilibrium thermodynamics relation, both obtained by MaxEnt.

In the dynamical view of the kinetic theory of the vdW gas we keep the setting
of the static theory. As in fluid mechanics (Section 3.3), the internal energy
(expressed in terms of the entropy in kinetic theory in (3.4)) generates one
of the forces that drive the reversible Hamiltonian mechanics. Unlike fluid
mechanics, the total entropy is not conserved. Only the available free energy
(11) is. We note that if the extra term
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})\ln(1-bn({\boldmath\mbox{$r$}})))$
in the entropy in (3.4) were replaced by
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})\ln(1-bf({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})))$
then the modified entropy would remain a Casimir of the Poisson bracket (24)
and the total entropy would be conserved in the Hamiltonian part of the time
evolution. With such modification the equilibrium states
$f_{eq}({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ as well as the implied
fundamental thermodynamic relation on the level equilibrium thermodynamics
would be however modified. In particular, the dependence of
$f_{eq}({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ on $v$ would be a
modified Maxwell distribution. From the physical point of view (i.e. in order
to guarantee the experimentally observed approach to equilibrium) the
conservation of $\mathfrak{E}(f)$ and not the conservation of $S(f)$ is
essential in the Hamiltonian part of the time evolution.

##  4 starting level$\longrightarrow$ target level with dynamics

In this section we change the target level. The level of equilibrium
thermodynamics is replaced by a level involving less details than the starting
level but still involving the time evolution. Not only the roles of the energy
and the entropy change but also the energy and the entropy themselves are
replaced by different potentials. The well known and thoroughly investigated
passage from kinetic theory to fluid mechanics will serve us as a guide. There
are essentially two points of view of the passage  a mesoscopic level
involving time evolution$\longrightarrow$ another mesoscopic level that
involves less details but still involves time evolution. In the first we
regard it as an intermediate step in the passage  a mesoscopic level involving
time evolution$\longrightarrow$ level of equilibrium thermodynamics. In the
second we regard it in the space of vector fields rather than in the state
space.

The first viewpoint is, roughly speaking, the viewpoint of Chapman and Enskog
[48], [49] in their investigation of level of kinetic theory$\longrightarrow$
level of fluid mechanics. The idea is to look in the state space for an
invariant (or approximately invariant) attractive manifold. The time evolution
describing approach to the invariant manifold is then the time evolution
describing the passage  a mesoscopic level involving time
evolution$\longrightarrow$ another mesoscopic level that involves less details
but still involves time evolution. This viewpoint has two disadvantages. The
first is that passages to mesoscopic levels with time evolution exist also
when the level of equilibrium thermodynamics is not well established (due to
the presence of external forces) and consequently the passage  a mesoscopic
level involving time evolution$\longrightarrow$ level of equilibrium
thermodynamics does not exist. The second disadvantage is that the
thermodynamics based on the approach to fixed points (Section 2) does not
directly extend to thermodynamics based on the approach to invariant (or
approximately invariant) submanifolds.

On the other hand, when we lift the time evolution in the state space to the
time evolution in the tangent (or alternative cotangent) space, the approach
involved in the passage  a mesoscopic level involving time
evolution$\longrightarrow$ another mesoscopic level that involves less details
but still involves time evolution becomes the approach to fixed point
(approach to vector fields governing the time evolution on the target level).
The viewpoint of thermodynamics presented in Section 2 becomes directly
applicable. Moreover, this second viewpoint does not require that the approach
to equilibrium states exists and thus is directly applicable to externally
forced systems. In the context of the extensive literature devoted to
investigations of the passage level of kinetic theory$\longrightarrow$ level
of fluid mechanics the second viewpoint of the approach to mesoscopic levels
with the time evolution is essentially the one used in the Grad type
investigations [29].

We follow below the second viewpoint. Our main objective is to show that the
passage  a mesoscopic level involving time evolution$\rightarrow$ another
mesoscopic level that involves less details but still involves time evolution
leads to thermodynamics but with entropy and energy replaced by, roughly
speaking, their rates. We shall restrict our investigation in this section to
working out one particular example taken from chemical kinetics.

###  4.1 Target level: Level of chemical kinetics (ChemKin)

Chemical components $A_{1},...,A_{p}$ undergo one chemical reaction

$\alpha_{1}A_{1}+...+\alpha_{p}A_{p}\leftrightarrows\beta_{1}A_{1}+...+\beta_{p}A_{p}$
(52)

where $\alpha_{1},...,\alpha_{p}$ and $\beta_{1},...,\beta_{p}$ are
stoichiometric coefficients. The state variables

$x={\boldmath\mbox{$n$}}=\left(\begin{array}[]{ccc}n_{1}\\\ \vdots\\\
n_{p}\end{array}\right)$ (53)

are number of moles of the components. Experimentally observed approach to
chemical equilibrium
${\boldmath\mbox{$n$}}_{eq}=\left(\begin{array}[]{ccc}(n_{1})_{eq}\\\
\vdots\\\ (n_{p})_{eq}\end{array}\right)$ is governed by a particular
realization of (10)

$\frac{dn_{i}}{dt}=-\Xi_{n_{i}^{*}}|_{{\boldmath\mbox{$n$}}^{*}=\Phi_{{\boldmath\mbox{$n$}}}}$
(54)

The potential $\Phi({\boldmath\mbox{$n$}})$ is a thermodynamic potential and
$\Xi$ a dissipation potential.

We make a few observations about Eq.(54).

(i) If the dissipation potential $\Xi$ depends on ${\boldmath\mbox{$n$}}^{*}$
only through its dependence on the thermodynamic force $X=\gamma_{i}n^{*}_{i}$
(called in chemical kinetics chemical affinity) then (54) takes the form

$\frac{dn_{i}}{dt}=\gamma_{i}J$ (55)

where

$\gamma_{i}=\alpha_{i}-\beta_{i};\,\,i=1,...,p$ (56)

and

$J=\Xi_{X}$ (57)

(ii) Particular forms of the potentials $\Phi$ and $\Xi$ for which (54)
becomes the familiar Guldberg-Waage mass action law can be found in [50]. In
our analysis we can keep $\Phi$ and $\Xi$ unspecified.

(iii) Since (54) is a particular realization of (10) its solutions approach
(as $t\rightarrow\infty$) equilibrium ${\boldmath\mbox{$n$}}_{eq}$ that is a
solution to $\Phi_{{\boldmath\mbox{$n$}}}=0$.

###  4.2 Starting level: Level of extended chemical kinetics (ExtChemKin)

Chemical reactions, seen on the scale of molecules, are complex processes
described by quantum mechanics. Is there an intermediate level between the
level on which the mass action law is formulated and the level on which the
chemical reactions are seen on the scale of molecules? We follow here [50]
where the extension of (54) is made by including chemical inertia. The
chemical flux $J$ introduced in (57) is adopted as an extra state variable. It
has been argued in [51] that this extension indeed carries the mass action law
(54) towards the microscopic level. We assume that the extended mass action
law described below represents a well established level.

With the state variables

$x=\left(\begin{array}[]{cc}{\boldmath\mbox{$n$}}\\\
J\end{array}\right)=\left(\begin{array}[]{cccc}n_{1}\\\ \vdots\\\ n_{p}\\\
J\end{array}\right)$ (58)

the particular realization of (10) governing the time evolution takes the form

$\left(\begin{array}[]{cc}\frac{d{\boldmath\mbox{$n$}}}{dt}\\\
\frac{dJ}{dt}\end{array}\right)=\left(\begin{array}[]{cc}0&\Gamma\\\
-\Gamma^{T}&0\end{array}\right)\left(\begin{array}[]{cc}{\boldmath\mbox{$n$}}^{*}\\\
J^{*}\end{array}\right)-\left(\begin{array}[]{cc}0\\\
\Theta_{J^{*}}\end{array}\right)$ (59)

where $L=\left(\begin{array}[]{cc}0&\Gamma\\\ -\Gamma^{T}&0\end{array}\right)$
is a Poisson bivector (see more in [Ogul]),

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\gamma_{1}\\\ \vdots\\\
\gamma_{p}\end{array}\right)$ (63) $\displaystyle\Gamma^{T}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\gamma_{1}&\cdots&\gamma_{p}\end{array}\right)$
(65)

are the stoichiometric matrices, $\Theta$ is a dissipation potentiasl (its
relation to the dissipation potential $\Xi$ appering in (54) is given later in
(70)), and ${\boldmath\mbox{$n$}}^{*}=\Phi^{(ext)}_{{\boldmath\mbox{$n$}}}$,
$J^{*}=\Phi^{(ext)}_{J}$, where $\Phi^{(ext)}({\boldmath\mbox{$n$}},J)$ is a
thermodynamic potential extending the thermodynamic potential
$\Phi({\boldmath\mbox{$n$}})$ introduced in (54).

Regarding solutions to (59), we note that this equation is a particular
realization of (10) and thus the thermodynamic potential
$\Phi^{(ext)}({\boldmath\mbox{$n$}},J)$ plays the role of the Lyapunov
function indicating approach to equilibrium state
$\left(\begin{array}[]{cc}{\boldmath\mbox{$n$}}\\\ J\end{array}\right)_{eq}$
that is a solution to
$\Phi^{(ext)}_{{\boldmath\mbox{$n$}}}=0,\,\,\Phi^{(ext)}_{J}=0$. Static view
of (59) is the MaxEnt passage to the equilibrium state
$\left(\begin{array}[]{cc}{\boldmath\mbox{$n$}}\\\ J\end{array}\right)_{eq}$.
Other properties of solutions to Eq.(59) that address its relation to (54) are
discussed in the next section.

###  4.3 (ExtChemKin)$\longrightarrow$ (ChemKin)

Now we are in position to investigate in the setting of chemical kinetics
described above the passage  a mesoscopic level involving time
evolution$\longrightarrow$ another mesoscopic level that involves less details
but still involves time evolution. The target level is the level represented
by (59) and the starting level is represented by (54).

If the term $\Theta_{J^{*}}$ is dominant in the second equation (59) then $J$
evolves faster that $n$. As a first approximation, we can solve (59) by
following first the time evolution governed by the second equation in (59) to
its conclusion and then follow the time evolution governed by the first
equation in (59) in which $J^{*}$ is replaced by
$J^{*}_{stat}({\boldmath\mbox{$n$}}^{*})$ that is a solution to

$\Psi_{J^{*}}(J^{*},(J^{*})^{{\dagger}})|_{(J^{*})^{{\dagger}}=-<\Gamma^{T},{\boldmath\mbox{$n$}}^{*}>}=0$
(66)

where

$\Psi({\boldmath\mbox{$n$}},J^{*};(J^{*})^{{\dagger}})=-\Theta({\boldmath\mbox{$n$}},J^{*})+J^{*}(J^{*})^{{\dagger}}$
(67)

Solution of (66) is

$J^{*}_{stat}({\boldmath\mbox{$n$}}^{*})=\Theta^{{\dagger}}_{(J^{*})^{{\dagger}}}((J^{*})^{{\dagger}})|_{(J^{*})^{{\dagger}}=-<\Gamma^{T},{\boldmath\mbox{$n$}}^{*}>}$
(68)

where $\Theta^{{\dagger}}((J^{*})^{{\dagger}})$ is the Legendre transformation
of $\Theta(J^{*})$.

After $J$ reached the state at which
$J^{*}=J_{stat}({\boldmath\mbox{$n$}}^{*})$ the time evolution continues as
the time evolution of $n$ governed by the first equation in (59) in which
$J^{*}$ is replaced by $J^{*}_{stat}({\boldmath\mbox{$n$}}^{*})$ given in
(68). The time evolution of $n$ is thus governed by

$\frac{dn_{i}}{dt}=-\Theta^{{\dagger}}_{{\boldmath\mbox{$n$}}^{*}}((J^{*})^{{\dagger}})|_{(J^{*})^{{\dagger}}=-<\Gamma^{T},{\boldmath\mbox{$n$}}^{*}>}$
(69)

By comparing this equation with Eq.(54) we see that the two dissipation
potentials $\Xi$ and $\Theta$ are related by

$\Xi({\boldmath\mbox{$n$}},{\boldmath\mbox{$n$}}^{*})=\Theta^{{\dagger}}((J^{*})^{{\dagger}})|_{(J^{*})^{{\dagger}}=-<\Gamma^{T},{\boldmath\mbox{$n$}}^{*}>}$
(70)

Summing up, we have reformulated (59) into two equations: Eq.(54) and

$\frac{\partial J^{*}}{\partial
t}=\mathbb{G}\Psi_{(J^{*})^{{\dagger}}}|_{(J^{*})^{{\dagger}}=-<\Gamma^{T},{\boldmath\mbox{$n$}}^{*}>}$
(71)

In (71) we have used $J^{*}=\mathbb{G}J$, where
$\mathbb{G}=\Phi^{(ext)}_{JJ}$, and (70) in (54). The three equations (59),
(54) and (71) describe three reductions. We emphasize again that the existence
of all three reductions is guaranteed by our assumption that all three levels
(i.e. the level of equilibrium thermodynamics, the level of the Guldberg-Waage
chemical kinetics, and the level of the extended Guldberg-Waage chemical
kinetics) are autonomous well established levels. All tree equations (59),
(54), and (71) are particular realization of (10). The first two describe
approach to the level of equilibrium thermodynamics and the third approach to
the level of the Guldberg-Waage chemical kinetics. The first two equations are
thus two additional examples of the time evolution equations investigated in
Section 2, the third is new. We shall now discuss it in a more detail.

First, we establish terminology. In order to make a clear distinction between
passages starting level$\rightarrow$level of equilibrium thermodynamics and
starting level$\rightarrow$target level with target levels that are different
from the level of equilibrium thermodynamics we use the adjective ”rate” in
investigations of the latter. We call the thermodynamic potential $\Psi$ in
(67) a rate thermodynamic potential, $\Theta$ in (71) a rate entropy, the time
evolution governed by (71) a rate time evolution. The adjective ”rate” points
to the fact that the space in which the time evolution takes place is the
tangent (or alternatively cotangent) space.

Both the entropy $S^{(ext)}({\boldmath\mbox{$n$}}.J)$ and the rate entropy
$\Theta({\boldmath\mbox{$n$}},J^{*})$ appear in both equations (59) and (71).
Their roles are however very different. In Eq.(59) (governing the reduction to
the level of equilibrium thermodynamics) the entropy
$S^{(ext)}({\boldmath\mbox{$n$}}.J)$ drives the reduction and the rate entropy
$\Theta({\boldmath\mbox{$n$}},J^{*})$ plays the role of the geometrical
structure transforming gradient of the entropy into a vector. On the contrary,
in Eq.(71) (governing the reduction to the level of the Guldberg-Waage
chemical kinetics) the rate entropy $\Theta({\boldmath\mbox{$n$}},J^{*})$
drives the reduction and the entropy $S^{(ext)}({\boldmath\mbox{$n$}}.J)$ (in
the form of the Hessian $\mathbb{G}=\Phi^{(ext)}_{JJ}$) is the geometrical
structure in the vector field.

In a small vicinity of thermodynamic equilibrium states the rate entropy
$\Theta$ can be seen, roughly speaking, as entropy production in the Guldberg-
Waage time evolution. This follows from: (i) the relation (70) between the
dissipation potential $\Xi$ and the rate entropy, (ii) from
$\frac{d\Phi}{dt}=-<n^{*},\Xi_{n^{*}}>|_{{\boldmath\mbox{$n$}}^{*}=\Phi_{{\boldmath\mbox{$n$}}}}$
implied by (54), from the fact that close to equilibrium all dissipation
potentials are quadratic (see the text following (2.2)), (iii) and from the
fact that Legendre transformations of quadratic potentials as well as the
scalar product of the variable with gradient of the potential remain quadratic
potentials..

Skeptical views of the relevance of the entropy production in mesoscopic
dynamics, occasionally appearing in the literature, originate most likely from
unsuccessful attempts to see the entropy production as Lyapunov-like function
in the time evolution equations describing approach to thermodynamic
equilibrium states. The entropy production (more precisely the rate entropy)
does play such role but in different time evolution equations. Equations
describing the approach to equilibrium stats are replaced by equations
describing the approach to lower level (i.e. involving less details than the
dynamics on the starting level) mesoscopic dynamics. The rate entropy has
moreover a more general applicability than the entropy since the approach to a
lower level dynamics exists also when the approach to thermodynamic
equilibrium states does not exist (e.g. due to presence of external forces).

Finally, we make an observation about the static version of (71). We recall
that the static version of (54) is the MaxEnt principle. The passage from the
starting level to the level of equilibrium thermodynamics, made by following
the time evolution governed by (54), is made in the static version of (71)
simply by maximizing the entropy subjected to constraints
$(\Phi_{{\boldmath\mbox{$n$}}}=0)$. Analogically, the static version of (71)
is minimization of the rate entropy subjected to constraints (i.e. by solving
$\Psi_{J^{*}}=0)$. This view of (71) is in fact the Onsager variational
principle [52], [53], [54], [55] applied to chemical kinetics. The time
evolution equation (71) can be seen thus as an extension of the Onsager
variational principle to chemical kinetics similarly as (54) is an extension
of MaxEnt to dynamics. When seeing (71) in the context of the Onsager
variational principle, the rate thermodynamic potential $\Psi$ is called
Rayleighian. We note that the rate entropy $\Theta$ decreases to its minimum
in the time evolution governed by (71) . We can therefore call the
minimization of the rate entropy MinRent similarly as we call the maximization
of the entropy MaxEnt.

##  5 Concluding Remarks

The experimentally observed approach to equilibrium in externally unforced
macroscopic systems is driven by gradients of energy and entropy. If the
complete energy can be expressed in terms of the chosen state variables then
the gradient of energy generates the reversible Hamiltonian time evolution and
the gradient of entropy the irreversible time evolution during which
unimportant details are swept away and the equilibrium pattern in the phase
portrait emerges. This situation is illustrated in the Boltzmann and the Gibbs
theories. If there is a part of energy that cannot be expressed in terms of
the chosen state variables (called an internal energy) then the entropy is
used to express it and the gradient of entropy becomes also one of driving
forces of the reversible Hamiltonian time evolution. This situation is
illustrated in fluid mechanics and kinetic theory of the van der Waals gas.
The latter illustration also shows that the gas-liquid phase transition
appears as a geometrical feature of the manifold of equilibrium states and of
the equilibrium fundamental thermodynamic relation implied by the kinetic
theory only if the energy in the kinetic theory formulation involves an
internal energy (in the van der Waals gas it is the energy generating hard-
core repulsive forces) that cannot be expressed in terms of one particle
distribution function and is expressed in terms of entropy.

The experimentally observed approach to mesoscopic dynamical theories
involving less details is driven by gradients of potentials that are related
to rates of the energy and the entropy. This situations is illustrated in
chemical kinetics. The static version of such reduction becomes the Onsager
variational principle in the setting of chemical kinetics.

Acknowledgement

I would like to thank Oǧul Esen, Václav Klika, and Michal Pavelka, for
stimulating discussions.

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