#  Multiscale Theory

Miroslav Grmela
École Polytechnique de Montréal, C.P.6079 suc. Centre-ville,
Montréal, H3C 3A7, Québec, Canada

###### Abstract

Boltzmann kinetic equation is put into the form of an abstract time evolution
equation representing links connecting autonomous mesoscopic dynamical
theories involving varying amount of details. In the chronological order we
present results that led to the abstract time equation evolution in both state
space and the space of vector fields. In the final section we list some open
problems.

##  1 History

Sensations of warm and cold connect us with the scale of atoms and molecules.
Our everyday activities, taking place on the macroscopic scale, are directly
and indirectly influenced by activities taking place on the microscopic scale.
Attempts to understand the heat and to use it as a source of energy on the
macroscopic scale gave birth to thermodynamics. From thermodynamics then
unfolds multiscale theory that is a general theory of relations among scales.
In this Note we begin by listing in the chronological order results that led
to its formulation. In Section 2 we formulate the multiscale theory and in
Section 3 we invite the readers to participate in its further development. We
focus in this Note on the physical aspects. Mathematical aspects as well as
specific illustrations are discussed in detail in the cited literature.

In the historical introduction we follow only one line, namely the line
leading to the multiscale theory in which dynamical theories on a larger scale
are seen as patterns in the phase portraits corresponding to the dynamical
theories on the smaller scale. The time evolution of particles composing
macroscopic systems under investigation is governed by classical mechanics.
Collection of all their trajectories is called a microscopic phase portrait.
Macroscopic observers see in it only an overall pattern representing results
of observations on the larger scale. Unimportant microscopic details are
ignored. In this Note we call a description involving more details an upper
description and the one involving less details a lower description. We
emphasize that the passage to a larger scale (in other words, reduction to a
lower theory) is not only a loss (loss of details) but also a gain of new
emerging overall features that are not seen in the upper theory. What is the
pattern in the phase portrait and how it is recognized?

We begin to answer this question with an allegorical image in which the
microscopic phase portrait is seen as an abstract painting. With our innate
ability we recognize in it a pattern that is then an allegorical image of the
phase portrait of the lower theory. The process of pattern recognition can be
seen as chipping off unimportant details (sweeping away the sand in an
archeological excavation) and revealing in this way important overall
features. We now present, in the chronological order, a sequence of results
that provides a more specific guidance on this path to lower theories.

Ludwig Boltzmann, kinetic theory

The first attempt to formulate mathematically the emergence of an autonomous
macroscopic theory in a microscopic phase portrait was made by Ludwig
Boltzmann [1]. The macroscopic theory was the classical equilibrium
thermodynamics and the microscopic theory was classical mechanics of particles
composing a dilute ideal gas. Boltzmann begins the pattern recognition process
with the following two steps. First, he sees the microscopic phase portrait in
the setting in which one particle distribution function
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ plays the role of the state
variable ($r$ is the position vector and $v$ momentum of one particle).
Second, he realizes that particle trajectories are composed of straight lines
(free flow) followed by very large but very localized changes due to binary
collisions. Details of the changes can be ignored in the overall view. The
process of chipping them off is achieved by considering the binary collisions
as ”chemical reactions” in which a pair of two species labeled by two incoming
momenta transforms into another pair of two species labeled by other two
outgoing momenta. The mechanical origin of the trajectories is retained only
in the conservation of the total momentum and the total kinetic energy before
and after collisions. The Boltzmann vector field generating the time evolution
of $f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ is a sum of a term
generating the free flow and a term, called a Boltzmann collision term, coming
from the Guldberg-Waage (mass action law) chemical kinetics. By analysing
solutions to the Boltzmann equation we indeed find that, by following the time
evolution, the pattern (composed of asymptotically reached fixed points)
expressing equilibrium thermodynamics is reached. The process of chipping off
the unimportant details is achieved by increasing a concave potential, called
an entropy (H function function by Boltzmann) that becomes at the final
destination the entropy of an ideal gas. Boltzmann has demonstrated for the
first time that the new concept of entropy, that emerged previously in the
investigation of the transformation of heat to a macroscopic mechanical
energy, actually arises in the mathematical formulation of the observed
approach to equilibrium states at which their behavior can be well described
by an autonomous theory called equilibrium thermodynamics. In other words,
experimental observations show that externally unforced an internally
unconstrained macroscopic systems can be prepared for using equilibrium
thermodynamics to describe and predict their behavior. The entropy is the
potential driving the preparation process. The existence of equilibrium states
constitutes a basis, called 0-th law of thermodynamics [2], of equilibrium
thermodynamics. If we focus our interest only on the equilibrium
thermodynamics that is the final outcome of the preparation process then we
can replace the Boltzmann equation with the maximum entropy principle (MaxEnt
principle). In this principle the entropy (a real valued concave function of
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$, its tendency to increase, as
well as the constraints (the energy and the number of moles) in the
maximization are postulated.

Willard Gibbs, equilibrium statistical mechanics

In Gibbs’ analysis [3] the microscopic phase portrait as well as the
macroscopic theory searched in it are the same as in Boltzmann’s analysis. The
difference is in the enlargement of the class of systems under investigation
(not only ideal gases but all physical systems are considered) and in the
limitation to the static analysis (i.e. to postulating a single entropy that
is assumed to be universally applicable to all microscopic physical systems
and to postulating the MaxEnt principle). The time evolution involved in the
preparation process (that is in Boltzmann’s analysis described by the
Boltzmann equation) is in the Gibbs analysis replaced by requiring only the
energy and the number of particles conservations. The pattern in the
microscopic phase portrait that corresponds to the Gibbs analysis is assumed
to be a pattern with maximal disorder. The universal Gibbs entropy is
interpreted as a measure of disorder.

The outcome of Gibbs’ analysis is a mapping from microscopic Hamiltonians
(which is one of the constraints in the entropy maximization) to fundamental
thermodynamic relations. The individual nature of macroscopic systems is
expressed in the microscopic theory in the Hamiltonians and in the equilibrium
thermodynamics in the fundamental thermodynamic relations.

Ilya Prigogine, nonequilibrium thermodynamics

How can Boltzmann’s analysis be adapted to fluids? The upper theory has to be
replaced by the classical hydrodynamics, the lower theory remains the
classical thermodynamics. In the classical hydrodynamics the state variables
are the same as in the classical thermodynamics except that they are fields
(i.e. functions of the position vector $r$) and in addition there is another
field representing the local momentum. It has been shown [4], [5] that the
entropy that drives the classical hydrodynamic time evolution to the classical
equilibrium thermodynamics is the entropy $S=\int
d{\boldmath\mbox{$r$}}s(\epsilon({\boldmath\mbox{$r$}}),n({\boldmath\mbox{$r$}}))$,
where $\epsilon({\boldmath\mbox{$r$}})$ is the local internal energy (i.e. the
total local energy minus the local kinetic energy), and the dependence of the
local entropy $s({\boldmath\mbox{$r$}})$ on $\epsilon({\boldmath\mbox{$r$}})$
and $n({\boldmath\mbox{$r$}})$ is the same as the dependence of the
equilibrium entropy on the equilibrium energy and the equilibrium number of
moles (assumption of local equilibrium). At the conclusion of the preparation
process the local equilibrium entropy $s({\boldmath\mbox{$r$}})$ (that drives
the process) as well as $\epsilon({\boldmath\mbox{$r$}})$ and
$n({\boldmath\mbox{$r$}})$ become the equilibrium entropy, the equilibrium
energy, and the equilibrium number of moles.

A very important additional result has arisen in investigations of the passage
from hydrodynamics to the equilibrium thermodynamics. The physical regularity
of hydrodynamic equations (i.e. their compatibility with the 0-th law of
thermodynamics) has been proven to imply a mathematical regularity (Dirichlet
problem is well posed) [6], [7].

Alfred Clebsch, Vladimir Arnold, complex fluids

Is it possible to see the Boltzmann and the Prigogine formulations of the 0-th
law of thermodynamics as two particular realizations of a single abstract
formulation that could also be used for other mesoscopic dynamic theories? The
answer to this question is affirmative.

The right hand side of the Boltzmann equation is a sum of a term expressing
the free flow and the Boltzmann collision term. The right hand side of the
hydrodynamic equations is a sum of the Euler term expressing the flow of
continuum and the Navier-Stokes-Fourier term expressing the friction. Clebsch
[8] has put the Euler term into the Hamiltonian form. In Arnold’s formulation
[9] both the Boltzmann free flow term in the kinetic equation and the Euler
term in hydrodynamic equations appear to be particular realizations of an
abstract noncanonical Hamiltonian dynamics. As for the second term on the
right hand side, the concept of the dissipation potential (see (2) below)
allows to formulate the Boltzmann collision term and the Navier-Stokes-Fourier
term as two particular realizations of a single abstract formulation. In other
words, the dissipation arising in chemical kinetics, in binary collisions, and
in fluid flows are different but all can be expressed mathematically as
particular realizations of a single concept of dissipation potential (see more
in Section 2).

The unification of formulations of the 0-th law of thermodynamics was also
motivated by the emergence of plastic materials and associated with it
interest in hydrodynamics of complex fluids. The large polymer macromolecules
inside the polymeric fluids change their conformations on the same time scale
as the hydrodynamic fields do. The necessity to include an internal structure
into hydrodynamics makes the standard framework of hydrodynamics (consisting
of local conservation laws, also called balance laws) unusable. The extra
fields characterizing the internal structure are not, at least in general,
conserved. A new framework for expressing the physics involved in flows of
complex fluids (in rheology) is needed. It has been suggested [10], [11] that
the abstract Boltzmann equation can provide such framework. The state variable
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ in the Boltzmann equation is
replaced in the abstract Boltzmann equation by an unspecified state variable.
In the case of complex fluids, the state variable consists of hydrodynamic
fields supplemented with other fields or distribution functions expressing the
internal structure. The right hand side of the abstract Boltzmann equation is
a sum of the Hamiltonian term and the term generated by a dissipation
potential. Such abstract Boltzmann equation has been then called in [12], [13]
Generic equation (an acronym for General Equation for Non Equilibrium
Reversible-Irreversible Coupling). The first IWNET meeting (International
Workshop of Non Equilibrium Thermodynamics) held in Montreal in the summer of
1996 assembled researches who began to use the Generic framework in
formulations of rheological models. Many examples of rheological applications
of the Generic structure can be found in [14] \- [17]. An equation similar to
the Generic equation (called a metriplectic equation) has also been introduced
in [18] \- [20] in the context of problems arising in plasma physics. The
Generic equation with a particular choice of the dissipation potential
(quadratic dissipation potential) becomes the metriplectic equation.

Sydney Chapman, David Enskog, Harold Grad, Lars Onsager, rate time evolution

So far, we have looked in the upper phase portrait only for a pattern
representing a lower theory on which no time evolution takes place (the
classical equilibrium thermodynamics). Now, we turn our attention to the
passage from an upper theory involving the time evolution to a lower theory
that takes into account less details but still involves the time evolution.
For example, we can think of the passage from the Boltzmann kinetic theory to
hydrodynamics. This enlargement of view makes it also possible to include into
our investigations physical system subjected to external forces (open
systems). Indeed, while only externally unforced and internally unconstrained
macroscopic system approach equilibrium states at which the equilibrium
thermodynamics can be applied, the observed behavior of externally driven
systems can often be faithfully described by mesoscopic theories. For
instance, the time evolution of the Rayleigh-Bénard system (a horizontal thin
layer of fluid heated from below) has been found to be well described by
hydrodynamic equations. This means that any description involving more details
(e.g. the completely microscopic description in which the Rayleigh-Bénard
system is seen as composed of $10^{23}$ particles) has to show approach to the
hydrodynamic description.

The archetype example of the passage from an upper mesoscopic theory to a
lower mesoscopic theory is the passage from the Boltzmann kinetic theory to
hydrodynamics. Two essentially different approaches to this reduction have
been introduced, one by Chapman and Enskog [21] (we shall call it Chapman-
Enskog method) and the other by Grad [22] (we shall call it Grad’s method).
Our strategy in the investigation of the passage to equilibrium thermodynamics
was to extract from the example of the Boltzmann theory an abstract structure
that can be then carried to a larger class of macroscopic systems. We shall
follow the same strategy also in the case of the Chapman-Enskog and Grad’s
methods.

The reduction upper theory $\rightarrow$ lower theory is seen in the Chapman-
Enskog method as a split of the time evolution describing the passage from the
upper theory to equilibrium thermodynamics into a ”fast” time evolution
describing the passage from the upper theory to the lower theory and the
”slow” time evolution describing the passage from the lower theory to the
equilibrium thermodynamics. The problem is to identify a manifold
$\mathcal{M}$ inside the state space $M^{(up)}$ of the upper theory which is
invariant (or almost invariant) in the time evolution governing the passage
from the upper theory to the equilibrium thermodynamics. The time evolution in
the reduced lower theory is then governed by the time evolution in $M^{(up)}$
restricted to $\mathcal{M}$. In the original investigation of Chapman and
Enskog the manifold $\mathcal{M}$ is searched by, first, identifying its
initial approximation (it is the manifold composed of local Maxwellian
distribution functions) and second, deforming it in order to improve its
invariance. Among many investigations that developed and extended this
approach to reduction we mention [23] \- [27].

Taking Grad’s viewpoint of the reduction of an upper to a lower theory, we ask
the following question. Is there a way to see the passage to lower theories
with the time evolution in the same way as the passage to equilibrium
thermodynamics (i.e. to lower theories without the time evolution)? In other
words, can the pattern that we search be composed of fixed points instead of
invariant manifolds? The key to answering this question is the realization
that if we lift the upper time evolution from the upper state space to the
time evolution in the space of vector fields on the upper state space then
indeed the asymptotically approached fixed point will be the vector field
generating the lower time evolution. In our terminology and notation, we shall
use the adjective ”rate” when addressing the concepts and quantities after
making the lift. We thus talk about rate state space (the space of vector
fields on the state space), rate time evolution (the upper time evolution
lifted to the rate state space), rate phase portrait (collection of all
trajectories in the rate phase space), etc.

In Grad’s method, the attention is put in the first step on the process in
which the macroscopic systems under investigation are prepared for
investigating them in the lower theory. The reduced time evolution in the
lower theory arises then in the second step as a solution to the equation
describing the process. Similarly as in the process of preparation for
equilibrium thermodynamics, solutions to equations describing it can also be
found by extremizing a potential (rate entropy). Its value in the extremum is
then the fundamental rate thermodynamic relation in the lower theory that is
inherited from the preparation process. The static version of the passage to a
lower mesoscopic theory turns out to be the Onsager variational principle
[28], [29], [30], that has originally arisen from attempts to express the
qualitative observation that macroscopic systems subjected to external forces
reach states at which their resistance to the forces is minimal.

As the process of preparation for equilibrium thermodynamics (the 0-th law of
thermodynamics) provides a dynamical foundation of MaxEnt principle and the
equilibrium fundamental thermodynamic relation, the process of preparation for
the lower theory (the 0-th law of rate thermodynamics) provides a dynamical
foundation for the Onsager variational principle and the fundamental
thermodynamic relation in the lower theory. Since the Grad method does not
have to be put into the context of the passage to the equilibrium
thermodynamics and is thus readily applicable to externally driven systems,
the extension of thermodynamics to rate thermodynamics that it offers is the
extension of the classical equilibrium thermodynamics to thermodynamics of
systems out of equilibrium that are driven by external forces.

##  2 Multiscale Graph

Autonomous mesoscopic theories are represented as vertices $\mathbb{V}$ in an
oriented graph. The vertex $\mathbb{V}^{(bottom)}$ represents equilibrium
thermodynamics. No time evolution takes place in $\mathbb{V}^{(bottom)}$. The
vertex $\mathbb{V}^{(top)}$ represents the most microscopic theory in which
macroscopic systems are seen as composed on $n\sim 10^{23}$ particles. The
time evolution in $\mathbb{V}^{(top)}$ is governed by classical mechanics. The
vertex $\mathbb{V}^{(up)}$ represents an autonomous mesoscopic theory
involving more details than another autonomous mesoscopic theory represented
by the vertex $\mathbb{V}^{(down)}$. The time evolution takes place in both
$\mathbb{V}^{(up)}$ and $\mathbb{V}^{(down)}$. Reduction of
$\mathbb{V}^{(up)}$ to $\mathbb{V}^{(down)}$ is represented as an oriented
link $\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$.

In this section we present an outline of the mathematical formulation of the
link $\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$. We concentrate on
the physical aspects. The mathematical aspects are discussed in more detail in
the cited references.

The state variable in the vertex $\mathbb{V}^{(up)}$ is denoted by $x\in
M^{(up)}$, where $M^{(up)}$ denotes the state spaces used in the vertex
$\mathbb{V}^{(up)}$.

Pre-Generic equation

Similarly as hydrodynamic equations unfold from the local conservation laws
(balance laws) by specifying constitutive relations (expressing fluxes in
terms of hydrodynamic fields), the equations governing the time evolution of
$x\in M^{(up)}$ that describes the link
$\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$, unfold from the pre-
Generic equation

$\dot{x}=LE_{x}(x)+\Xi_{x^{*}}(x,X(x^{*}))$ (1)

by specifying the state variable $x$ (giving it a physical interpretation),
and specifying constitutive for the quantities

$(x^{*},X,E,L,\Xi)$ (2)

entering (1). The quantity $x^{*}$ is a conjugate state variable. The
constitutive relation relates it to the state variable $x\in M^{(up)}$. The
quantity $E:M^{(up)}\rightarrow\mathbb{R}$ is the energy. By $E_{x}$ we denote
$\frac{\partial E}{\partial x}$. If $x$ is an element of an infinite
dimensional space then the partial derivative is meant to be an appropriate
functional derivative. The first term on the right hand side of (1) represents
the Hamiltonian dynamics. From the physical point of view it is a remnant of
$\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)\left(\begin{array}[]{cc}E^{(part)}_{{\boldmath\mbox{$r$}}}\\\
E^{(part)}_{{\boldmath\mbox{$v$}}}\end{array}\right)$ that drives the time
evolution of particles composing the macroscopic systems; $E^{(part)}$ is the
particle energy, $({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ are position
coordinates and momenta of the particles. In Eq.(1) the Poisson bivector
$\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)$ is replaced with a
general Poisson bivector $L(x)$. We recall that $L(x)$ is a Poisson bivector
if $<A_{x},L(x)B_{x}>=\\{A,B\\}$ is a Poisson bracket, $A$ and $B$ are real
valued functions of $x$ and $<,>$ is the pairing. The bracket $\\{A,B\\}$ is a
Poisson bracket if $\\{A,B\\}=-\\{B,A\\}$ and the Jacobi identity
$\\{A,\\{B,C\\}\\}+\\{B,\\{C,A\\}\\}+\\{C,\\{A,B\\}\\}=0$ holds. The particle
energy $E^{(part)}$ is replaced in (1) with the $\mathbb{V}^{(up)}$-energy
$E(x)$.

The potential $\Xi:M^{(up)}\times\mathbb{M}^{(up)}\rightarrow\mathbb{R}$
appearing in the second term on the right hand side of (1) is called a
dissipation potential; $x\in M^{(up)}$ and $X\in\mathbb{M}^{(up)}$. The
dissipation potential $\Xi(x,X)$ is required to satisfy the following
properties:

$\displaystyle\Xi(x,0)=0\,\,\forall x$
$\displaystyle\Xi\,\,reaches\,\,its\,\,minimum\,\,at\,\,X=0\,\,\forall x$
$\displaystyle\Xi(x,X)\,\,is\,\,a\,\,convex\,\,function\,\,of\,X\,\,in\,\,a\,\,neighborhood\,\,of\,\,X=0\,\,\forall
x$ (3)

Consequently, in a small neighborhood of $X=0$, the dissipation potential
becomes the quadratic potential $\Xi(x,X)=\frac{1}{2}<X,\Lambda X>$, where
$\Lambda(x)$ is a symmetric and positive definite operator. With such
quadratic dissipation potential, Eq.(1) becomes
$\dot{x}=LE_{x}+\left(X_{x^{*}}\right)^{T}\Lambda X$. The quantities
$X\in\mathbb{M}^{(up)}$, called thermodynamic forces, are functions of
$x^{*}\in M^{*(up)}$ that are conjugate state variables. If $M^{(up)}$ is a
linear space then $M^{*(up)}$ is its conjugate.

Summing up, the requirements placed so far on the constitutive relations
specifying (2) are: $L(x)$ is a Poisson bivector and $\Xi(x,X)$ is a
dissipation potential satisfying (2). The quantity $x^{*}$ is so far an
unspecified function of $x$, $X$ is an unspecified function of $x^{*}$, and
$E(x)$ is an unspecified function of $x$ having the physical interpretation of
energy.

In the rest of this section we present the constitutive relations
(specifications of (2)) that make (1) the Generic equation describing the
links $\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(bottom)}$. In particular,
we show the constitutive relations making (1) the Boltzmann kinetic equation.
Subsequently, we lift (1) into an equation governing the time evolution of the
thermodynamic forces $X$. The lifted equation, (we call it a rate Generic
equation) governs the time evolution describing the link
$\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$. Specifically, we
formulate the rate Boltzmann equation.

Generic equation describing
$\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(bottom)}$

The state space $M^{(bottom)}$ used in the equilibrium thermodynamics
$\mathbb{V}^{(bottom)}$ is a two dimensional space with elements
$(\mathbf{E},\mathbf{N})$ denoting the equilibrium energy and the equilibrium
number of moles per unit volume. The individual nature of macroscopic systems
is expressed in equilibrium thermodynamics in the fundamental thermodynamic
relation $\mathbf{S}=\mathbf{S}(\mathbf{E},\mathbf{N})$, where $\mathbf{S}$ is
the equilibrium entropy per unit volume.

The constitutive relations for which (1) becomes an equation describing the
process of preparation for $\mathbb{V}^{(bottom)}$ begin with

$E=E(x);\,\,\,\,N=N(x);\,\,\,\,S=S(x)$ (4)

where $S:M^{(up)}\rightarrow\mathbb{R}$ is the $\mathbb{V}^{(up)}$-entropy.
The function $S:M^{(up)}\rightarrow\mathbb{R}$ is assumed to be sufficiently
regular and concave. Contrary to the energy $E(x)$ and the number of moles
$N(x)$, the entropy is a potential that does not exist in mechanics. It does
arise in mechanics (upper mechanics) only in the process of attempting to
extract overall features (patterns in the upper phase portrait). Allegorically
speaking, we can say that the entropy plays the role of an ambassador of the
reduced lower dynamics emerging in the upper dynamics as an overall pattern in
its phase portrait. We combine the potentials (4) into a new potential

$\Phi(x,\mathbf{E}^{*},\mathbf{N}^{*})=-S(x)+\mathbf{E}^{*}E(x)+\mathbf{N}^{*}N(x)$
(5)

called $\mathbb{V}^{(up)}$-thermodynamic potential. The role that the
quantities $\mathbf{E}^{*}\in\mathbb{R}$, $\mathbf{N}^{*}\in\mathbb{R}$ play
in $\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$ will be clarified
later (see the text after Eq.(10)).

Constitutive relations continue with the specification of the conjugate state
variable

$x^{*}=S_{x}$ (6)

and with the thermodynamic force

$X(x,x^{*})=\mathbb{K}(x)x^{*}$ (7)

where $\mathbb{K}$ is a linear operator.

The two geometrical structures transforming gradients $E_{x}(x)$ and
$S_{x}(x)$ of the potentials $E(x)$ and $S(x)$ (i.e. covectors) into vectors
are: the Poisson bivector $L(x)$ and the dissipation potential
$\Xi(x,X(x^{*}))$. These two structures are required to be appropriately
degenerate in order that (1) and (7) imply

$\dot{N}=0;\,\,\,\,\dot{E}=0;\,\,\,\,\dot{S}=<x^{*},\Xi_{x^{*}}>|_{x^{*}=S_{x}}\geq
0$ (8)

This means that in addition to requiring that $L(x)$ is a Poisson bivector we
also require that $L(x)$ is a degenerate Poisson bivector in the sense that
the entropy $S(x)$ and the number of moles $N(x)$ are its Casimirs (i.e.
$\\{A,S\\}=0$ and $\\{A,N\\}=0$ for all real valued functions $A(x)$).
Moreover, we require that $\Xi(x,X)$ is a dissipation potential satisfying (2)
and in addition $<x^{*},\Xi_{x^{*}}>|_{x^{*}=E_{x}}=0$ and
$<x^{*},\Xi_{x^{*}}>|_{x^{*}=N_{x}}=0$.

The thermodynamic potential (5) plays the role of the Lyapunov function for
the approach, as $t\rightarrow\infty$, of solutions to (1) to the equilibrium
states $\widehat{x}(\mathbf{E}^{*},\mathbf{N}^{*})$ that are solutions to

$\Phi_{x}(x,\mathbf{E}^{*},\mathbf{N}^{*})=0$ (9)

Indeed, $\Phi$ is a convex function of $x$ and $\dot{\Phi}\leq 0$. We refer to
[17] for details of the proof. The quantities $\mathbf{E}^{*}$ and
$\mathbf{N}^{*}$ play the role of the Lagrange multipliers in the maximization
of the entropy $S(x)$ subjected to constraints $E(x)$ and $N(x)$. In the
standard thermodynamic notation
$\mathbf{E}^{*}=\frac{1}{T};\mathbf{N}^{*}=-\frac{\mu}{T}$, where $T$ is the
absolute temperature and $\mu$ the chemical potential.

We make two additional observation about solutions to the Generic equation.
First, we note that if we are interested only in the final destination of the
preparation process (i.e. only in the equilibrium states
$\widehat{x}(\mathbf{E}^{*},\mathbf{N}^{*})$ ) then we can completely avoid
the Generic equation and only use the thermodynamic potential (5) and solve
(7). This static method of making the reduction is called Maximum Entropy
principle (MaxEnt principle).

The second observation is about the inheritance that remains in
$\mathbb{V}^{(bottom)}$ from the preparation process. At the final destination
the thermodynamic potential

$\mathbf{S}^{*}(\mathbf{E}^{*},\mathbf{N}^{*})=\Phi(\widehat{x}(\mathbf{E}^{*},\mathbf{N}^{*}),\mathbf{E}^{*},\mathbf{N}^{*})$
(10)

becomes the Legendre transformation of the fundamental thermodynamic relation
$\mathbf{S}=\mathbf{S}(\mathbf{E},\mathbf{N})$ implied in
$\mathbb{V}^{(bottom)}$ by the passage from $\mathbb{V}^{(up)}$. If, in
addition, we extend $\mathbf{S}=\mathbf{S}(\mathbf{E},\mathbf{N})$ into
$\mathbf{S}=\mathbf{S}(\mathbf{E},\mathbf{N},\mathbf{V})$, where $\mathbf{V}$
is the volume and require that all three quantities
$\mathbf{S},\mathbf{E},\mathbf{N}$ are extensive (i.e. they become
$\lambda\mathbf{S},\lambda\mathbf{E},\lambda\mathbf{N}$ if $\mathbf{V}$
changes to $\lambda\mathbf{V}$) then (if we use the standard thermodynamic
notation)
$S^{*}(\frac{1}{T},-\frac{\mu}{T})=-S_{\mathbf{V}}(\mathbf{E},\mathbf{N},\mathbf{V})=-\frac{P}{T}$,
where $P$ is the pressure.

Finally, we present the specific constitutive relations for which (1) becomes
the Boltzmann kinetic equation. The state variable $x$ is the one particle
distribution function $f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$. Its
kinematics is expressed in 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$}}}\right)$, where $A(f),B(f)$ are real
valued sufficiently regular functions of
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$. The energy $E(f)=\int
d{\boldmath\mbox{$r$}}\int
d{\boldmath\mbox{$v$}}f\frac{{\boldmath\mbox{$v$}}^{2}}{2m}$, the number of
moles is $N(f)=\int d{\boldmath\mbox{$r$}}\int d{\boldmath\mbox{$v$}}f$, by
$m$ we denote mass of one particle. The entropy $S(f)$ is the Boltzmann
entropy $S(f)=-k_{B}\int d{\boldmath\mbox{$r$}}\int d{\boldmath\mbox{$v$}}f\ln
f$, where $k_{B}$ is the Boltzmann constant. The thermodynamic force
$X(f^{*},{\boldmath\mbox{$v$}},{\boldmath\mbox{$v$}}_{1},{\boldmath\mbox{$v$}}^{\prime},{\boldmath\mbox{$v$}}^{\prime}_{1})=\mathbb{K}f^{*}=f^{*}({\boldmath\mbox{$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$}}_{1})$.
The thermodynamic 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$}}^{\prime}_{1})\left(e^{X}+e^{-X}-2\right)$
(11)

where $W\geq 0,W\neq 0$ 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$}}^{\prime}_{1})^{2}$.
Moreover, $W$ 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
fundamental thermodynamic relation (10) implied by the above constitutive
relation is indeed the fundamental thermodynamic relation representing
$\mathbb{V}^{(bottom)}$ an ideal gas. Details of calculation can be found in
[17].

Rate Generic equation describing
$\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$

The vector field generating the time evolution (1) involves thermodynamic
forces $X$. In order that Eq.(1) be an autonomous time evolution equation the
forces $X$ have to be expressed in terms of $x$. In other words, Eq.(1) has to
be closed. We look for an equation that governs the time evolution of $X$. Its
solution will be the closure. From the physical point of view, making the lift
means turning to the stage of the time evolution that precedes the time
evolution of $x$ governed by the closed Eq.(1).

The equation governing the time evolution of $X$

$\dot{X}=-\mathbb{G}\Psi_{X}(X,J)$ (12)

arises in lifts of the pre-Generic equation (1) to higher dimensional spaces
in which $x^{*}$ and $X$ are considered to be independent state variables.
Geometrical as well as the physical aspects of the lifting process are
discussed in [32], [33]. Here we only briefly indicate the passage from (1) to
(12). For the sake of simplicity, we take $X=x^{*}$. Equation (1) can be cast
into the form $\dot{x}=\left(<x^{*},LE_{x}>+\Xi(x,x^{*})\right)_{x^{*}}$. By
replacing $\dot{x}$ on the left hand side of this equation with
$\mathbb{G}^{-1}\dot{x}^{*}$, we arrive at (12) with $J=LE_{x}$. The resulting
equation (12) is then considered to be the equation governing the time
evolution of $x^{*}$. The quantity $x$ in it is considered to be a parameter.
We note that if $x^{*}=S_{x}$, i.e. if $x^{*}$ is the conjugate state
variable, then $\mathbb{G}=-(S_{xx})^{-1}$, where $S_{xx}$ is the Hessian of
the entropy $S(x)$, and the rate Generic equation is an equivalent
reformulation of (1).

Since $X$ is a force entering the vector field generating the time evolution
of $x$, we call (12) a rate Generic equation. The rate thermodynamic potential
$\Psi(X,J)$ appearing in (12) is given by

$\Psi(X,J)=-\mathfrak{S}(X)+<X,J>$ (13)

where the rate entropy $\mathfrak{S}(x,X)=\Xi(x,X)$. The fluxes $J$ play the
role of Lagrange multipliers (the same role as $E^{*}$ and $N^{*}$ play in the
thermodynamic potential (5)). The quantities used as state variables in
$\mathbb{V}^{(down)}$ are entering in $J$. The operator $\mathbb{G}$ is a
positive definite operator.

In the particular case of the Boltzmann equation,
$X:\mathbb{R}^{12}\rightarrow\mathbb{R};({\boldmath\mbox{$v$}},{\boldmath\mbox{$v$}}_{1},{\boldmath\mbox{$v$}}^{\prime},{\boldmath\mbox{$v$}}^{\prime}_{1})\mapsto
X$, the dissipation potential $\Xi$ is given in (11), and
$\mathbb{G}==\mathbb{K}G\mathbb{K}^{T}$, where $-G$ is the inverse of the
Hessian of the Boltzmann entropy. The term $<X,J>$ is the power associated
with the collisions.

The rate thermodynamic potential $\Psi$ plays in the rate Generic equation
(12) the same role as the thermodynamic potential $\Phi$ plays in the Generic
equation (1). First, $\Psi$ plays the role of the Lyapunov function for the
approach of $X$ to the rate equilibrium states $\widehat{X}(J)$ that are
solutions to

$\Psi_{X}(X,J)=0$ (14)

This means that if our interest is limited to the vertex $\mathbb{V}^{(down)}$
obtained in the passage $\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$
then it is enough to know the rate thermodynamic potential $\Psi$ and to solve
(14). This static version of the link
$\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$, known as Onsager’s
variational principle, has arisen previously [28], [29], [31] in attempts to
express the observation that macroscopic systems subjected to external forces
reach states at which effect of the forces is minimal. In our analysis this
principle arises as a static version of the 0-th law of multiscale dynamics
which states that both $\mathbb{V}^{(up)}$ and $\mathbb{V}^{(doen)}$ exist as
autonomous theories. If the the macroscopic system is not prevented by
external forces or internal constraints to reach the vertex
$\mathbb{V}^{(bottom)}$ then the thermodynamic potential $\Xi$ is closely
related to the entropy production $<x^{*},\Xi_{x^{*}}>$ (see (8)) and
consequently, the extremization of the rate thermodynamic potential $\Psi$ is
a minimization of the entropy production subjected to constraints. Note that
the switch from maximization to minimization when we switch from entropy to
rate entropy (production of entropy) is a consequence of the negative
definiteness of the Hessian of the entropy (concavity of the entropy).

The second role that the thermodynamic potential plays is to provide the
vertex $\mathbb{V}^{(down)}$ with the fundamental thermodynamic relation
$\mathfrak{S}^{*}(J)=\Psi(\widehat{X}(J),J)$. Its possible applications and
its relation to the fundamental thermodynamic relation in
$\mathbb{V}^{(bottom)}$ remain to be investigated.

##  3 Open Problems

Keeping with the tradition of IWNET meetings, we turn at the end to the future
and suggest open problems. Almost all the problems listed below have already
been mentioned in the cited references but their all need further
clarifications and illustrations.

Boundary conditions

Macroscopic physical systems that we have considered so far in this Note were
either infinite or surrounded by boundaries with periodic boundary conditions.
If the boundaries are present then it is necessary to take into account the
physics taking place on them. The boundaries and their interaction with the
bulk play the role of external forces acting on the bulk. The physics taking
place on the boundaries is expressed in an upper vertex
$\mathbb{V}^{(boundary)}$, the physics taking place in the bulk in a lower
vertex $\mathbb{V}^{(bulk)}$. The boundary conditions addressing the boundary-
bulk coupling are expressed in the rate link
$\mathbb{V}^{(boundary)}\longrightarrow\mathbb{V}^{(bulk)}$. The link is a
rate link since boundaries act on the time evolution in the bulk as forces.
They are entering the bulk in vector fields. The time evolution on the
boundary will be thus governed by (12) and the Lagrange coefficients in the
rate thermodynamic potential entering it will be the boundary conditions
(fluxes on the boundaries). The static version of this viewpoint of boundary
conditions is the Onsager variational principle adapted to boundary
conditions. In other words, we suggest to regard the time evolution equation
in $\mathbb{V}^{(bulk)}$ equipped with boundary conditions as a closed part of
a hierarchy addressing both the bulk and the boundary physics. The discarded
part of the hierarchy (addressing the boundary physics and its coupling with
the bulk) has the form of (12) and its solution is the closure (the boundary
conditions).

We emphasize that if the existence of boundary conditions is well established
(we can call it 0-th law of boundary thermodynamics) then the above view of
boundary conditions is as well founded as, for instance, the classical
equilibrium thermodynamics (that is based on the 0-th law of equilibrium
thermodynamics). The 0-th law of boundary thermodynamics in the mesoscopic
theory $\mathbb{V}^{(bulk)}$ is the experimentally observed existence of
boundary conditions. The boundary conditions exist if the boundary physics and
its coupling to the bulk can be expressed by supplying the governing equations
in $\mathbb{V}^{(bulk)}$ with boundary conditions and if their solutions are
found to agree with results of experimental observations.

The static version of this formulation of boundary conditions has already been
introduced and illustrated on specific examples in [34], [35].

Temperature

The importance of the role that the temperature plays in our everyday life
makes the word ”temperature” appear also, for instance, in sentences like ”the
meeting of two leaders lowered the temperature”. Can we define the temperature
in a way that embraces all such meanings? We suggest the following definition.
Temperature is 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.

For the equilibrium (or local equilibrium) internal energy, the equilibration
made by maximizing the equilibrium (or local equilibrium) entropy, and the
thermodynamic walls that freely pass or prevent passing the internal energy,
the above definition becomes the standard definition of the temperature. The
macroscopic system whose standard temperature is measured is put into the
contact with another macroscopic system called a thermometer. The two systems
are separated by the thermodynamic wall that freely passes the internal energy
and both are separated from the exterior by the thermodynamic wall that
prevents its passing. The thermometer is a system with a known fundamental
thermodynamic relation. This means that the thermometer is able to translate
the temperature into another quantity that can be directly seen or felt. For
example if the thermometer is a human body then the temperature is read in the
intensity of the nervous reactions. The process of equilibration is an
essential part of the notion of the temperature. It brings into the definition
the entropy and distinguishes the internal energy from the total energy. The
internal energy is the part of the total energy that is out of our direct
control (except by the thermodynamic walls that can pass it or prevent its
passing). In the standard meaning of the temperature, the overall volume, the
number of moles, and the overall macroscopic velocity are considered to be
directly controlled from exterior. The macroscopic work and the overall
macroscopic kinetic energy are thus not included in the internal energy. In
the system that is surrounded by a wall that prevents passing the internal
energy, its evolution is determined only by the equilibration (by the
evolution of the entropy).

The meaning of the temperature can be changed by changing the internal energy
and the equilibration. The internal energy can be changed for instance by
getting a part of the internal structure (e.g. the conformation of
macromolecular chains) under our direct control. The new internal energy is
then the original internal energy minus the energy that is expressed in terms
of the internal structure that we can control. Measurements of the new
temperature would still however need walls that pass or prevent passing only
the new internal energy.

Other interesting changes in the meaning of the temperature could be those
caused by changing the equilibration. The equilibration and the entropy
associated with it that we considered above was the equilibration involved in
the link $\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(bottom)}$. We have seen
however that the link $\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$ ,
where both vertices represent well established autonomous theories, involves
also an equilibration. For example, let $\mathbb{V}^{(up)}$ be kinetic theory
and $\mathbb{V}^{(down)}$ hydrodynamics and
$\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$ be the rate link. The
internal energy in this case is a rate energy and the entropy the rate
entropy. Since the equilibration involved in
$\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$ occurs also in open,
externally driven, and far from equilibrium systems, this temperature is well
suited for such systems. Finding the walls that pass and or prevent passing
the rate energies remains still however a challenge.

An extensive review of temperatures in which the equilibration is generated by
standard entropies can be found in [36]. A possibility to introduce
temperatures measuring time variations of the internal energy and the
equilibrations that occur also in externally driven systems and are driven by
rate entropies has been suggested previously in [37], and in [38].

Finally, we return to the beginning of this comment and suggest a possible
”social temperature”. The internal energy is a collection of conspiracy
theories, the thermometer is a group selected for polling, the thermodynamic
walls are those that freely pass or prevent passing the gossip (electronic
messages), and the fundamental thermodynamic relation that makes the
temperature visible are answers to polling questions (questions about
happiness, political leaning and similar questions about the state of mind
influenced by the conspiracy theories). The equilibration is made by spreading
the gossip about the conspiracy theories. The entropy in this illustration is
a measure of information about the conspiracy theories.

Criticality

Let the macroscopic system under investigation be fixed. For example let it be
one liter of water. Some mesoscopic theories are applicable (they faithfully
describe the observed behavior of the water) and some are not applicable.
Among the former are the theories represented by the vertices
$\mathbb{V}^{top}$, $\mathbb{V}^{bottom}$, and hydrodynamics. Among the latter
is the Boltzmann kinetic theory and solid mechanics. In order to make the
distinction visible in the graph, we imagine the vertices representing the
theories that are applicable as illuminated and the remaining vertices as
dark. Let now the system under investigation be subjected to external
influences. For example, we begin to heat the water. Eventually, the water
changes into gas. When observing the phase change in the phase portraits of
the illuminated vertices, we see in them very large changes. In particular, we
note two phenomena. First, the entropy generating the phase portrait inside
the vertex hydrodynamics becomes to loose its concavity. Second, we see a
significant increase of fluctuations indicating shortening of the links. We
recall that the length of the links is a measure of the separability of the
mesoscopic theories. In addition, we begin to see also changes in the
lightening of the vertices. The vertex representing hydrodynamics becomes dark
and the Boltzmann kinetic theory becomes lit up. In the critical region where
the liquid and gas become essentially indistinguishable a large number of
vertices in the graph become slightly illuminated. In general, by approaching
the critical point we expect to see the graph shrinking and uniformly
illuminated.

We now explore consequences of these observations. First, we turn to the loss
of concavity of the entropy seen in the mathematical description of the
approach to the critical region. With many difficulties that such loss brings
there is one emerging advantage. In the critical region the entropy is
universal. Outside the critical region both the entropies and the rate
entropies are in mesoscopic theories different for different systems. They are
in fact the quantities, among others like for instance energies, in which the
individual nature of the systems is expressed. But in the vicinity of critical
points all entropies or rate entropies are essentially the same. This
advantage has been foreseen by Landau [39] and was used to formulate his
theory of critical phenomena. The universality of the functions near critical
points has been rigorously proven by [40].

The second observation made in critical regions is the loss of autonomy of
vertices in the graph. The loss of autonomy is manifested by a significant
increase of fluctuations. This observation offers a new definition of critical
points. Critical points are defined as points at which no pattern in phase
portraits and rate phase portraits can be recognized . This truly multiscale
definition of critical points is the definition that emerged in the
renormalization group theory of critical points [41]. The motivation for the
renormalization group theory was the finding that the implications of the
Landau theory mentioned in the previous paragraph do not exactly agree with
experimental observations of the critical behavior.

The renormalization group theory of critical phenomena and associated with it
the lack-of-pattern definition of critical points has been originally
formulated [41] inside the setting of the Gibbs equilibrium statistical
mechanics. The formulation inside the setting of the Landau theory has been
introduced in [42], [43]. We briefly describe the essential points. First, the
static MaxEnt link $\mathbb{V}^{(up)}\rightarrow\mathbb{V}^{bottom}$ is
formulated in a neighborhood of the critical point where the
$\mathbb{V}^{(up)}$-entropy looses its convexity; $\mathbb{V}^{(up)}$ is a
vertex representing a theory involving more details than the vertex
$\mathbb{V}^{(bottom)}$. Let $P^{(up)}$ be the set of $\mathbb{V}^{(up)}$
state variables as well as the parameters entering the
$\mathbb{V}^{(up)}$-entropy. In the second step we do the same thing but with
the vertex $\mathbb{V}^{(upp)}$ replacing $\mathbb{V}^{(up)}$. The vertex
$\mathbb{V}^{(upp)}$ represents a theory that is an upper theory vis-à-vis the
theory represented by the vertex $\mathbb{V}^{(up)}$. It is a refinement of
the $\mathbb{V}^{((up))}$-theory. For example in the illustration worked out
in [42], [43] a one component system is made into two component system by
simply putting another colour on some of the particles. The colour does not
change in any way their properties. What changes is only our perception of the
system. We are taking a sharper view but the system remains exactly the same.
The energy remains unchanged, only the entropy changes. In the third step we
make the static MaxEnt link $\mathbb{V}^{(upp)}\rightarrow\mathbb{V}^{(up)}$
(again only in a neighborhood of the critical point where
$\mathbb{V}^{(upp)}$-entropy looses its convexity). As a result, we arrive
back at $\mathbb{V}^{(up)}$-theory but with $\overline{P^{(up)}}$ replacing
$P^{(up)}$. If $\overline{P^{(up)}}$ is the same as $P^{(up)}$ then this means
that there is no pattern and thus that the point is critical in the sense of
the lack-of-pattern definition of critical points. Details and a specific
illustration in the context of the van der Waals theory can be found in [42],
[43].

The observation of large fluctuations and thus the lack of autonomy of
vertices in the critical region makes also possible to address the relation
between the loss of concavity of the entropy and the loss of convexity of the
rate entropy in the critical region. As explained in Section 2, the rate time
evolution driven by the rate entropy precedes the time evolution driven by the
entropy. We conjecture that the critical behavior will be seen in both the
time evolution and the rate time evolution, in both the entropy and the rate
entropy. In externally driven systems that are prevented from reaching
equilibrium states and thus the vertex $\mathbb{V}^{(bottom)}$ for such
systems remains dark. The critical behavior manifests itself in externally
driven systems as an occurrence of a bifurcation in the vertex $\mathbb{V}$.
We conjecture that the bifurcation seen in $\mathbb{V}$ will also be seen in
the loss of convexity in the rate entropy.

Hierarchies

We have already noted in Section 2 that casting the upper vector fields into
hierarchies can be interpreted as the first step in formulating rate links
directed from upper theories to lower theories. The part of the hierarchy that
is cut off plays the role of the rate Generic equation that represents the
dynamical version of the Onsager variational principle. Its solution is the
closure of the part of the hierarchy that is retained. We shall make here
additional comments.

Structure preserving hierarchies

If the upper vector field $(vf)^{up}$ has a structure involving several
elements then we can put into the hierarchy form only some of the elements and
keep the remaining intact. The resulting hierarchy then clearly keeps the
structure of $(vf)^{up}$ and may also be more suitable for the pattern
recognition in its phase portrait. For example, let $(vf)^{up}$ be the
Hamiltonian vector field (i.e. $(vf)^{up}=LE_{x}$; $L$ is the upper Poisson
bivector, $E(x)$ is the upper energy, $x$ is the upper state variable). We can
put into the hierarchy form only the Poisson bivector $L(x)$ and leave the
energy $E(x)$ untouched. Such structure-hierarchy form of the BBGKY hierarchy
is worked out in [44] and of the Grad hierarchy in [45].

The process of casting $L$ into the hierarchy is as easy (if not easier) as
casting the whole vector field $LE_{x}$ into the hierarchy. We illustrate it
on the example of the Grad hierarchy. The projection $M^{up}\rightarrow
M^{down}$ is in this example:
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})\mapsto{\boldmath\mbox{$c$}}({\boldmath\mbox{$r$}}))=(c^{(0)}({\boldmath\mbox{$r$}}),c^{(1)}_{i}({\boldmath\mbox{$r$}}),c^{(2)}_{ij}({\boldmath\mbox{$r$}}),c^{(3)}_{ijk}({\boldmath\mbox{$r$}}),...)$
= $(\int d{\boldmath\mbox{$v$}}f,\int d{\boldmath\mbox{$v$}}v_{i}f,\int
d{\boldmath\mbox{$v$}}v_{i}f,\int d{\boldmath\mbox{$v$}}v_{i}v_{j}f,\int
d{\boldmath\mbox{$v$}}v_{i}v_{j}v_{k}f,....);i,j,k=1,2,3$ . The standard Grad
hierarchy is constructed by multiplying the Hamiltonian part
$-\frac{\partial(v_{l}f)}{\partial r_{l}}$ of the Boltzmann vector field by
$v_{i},v_{i}v_{j},...$ and integrating the products over $v$. The structure-
Grad hierarchy is obtained by substituting $\frac{\delta}{\delta
f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})}$ in the Poisson bracket
$\\{A,B\\}=\int d{\boldmath\mbox{$r$}}\int
d{\boldmath\mbox{$v$}}f\left(\frac{\partial A_{f}}{\partial
r_{l}}\frac{\partial B_{f}}{\partial v_{l}}-\frac{\partial B_{f}}{\partial
r_{l}}\frac{\partial A_{f}}{\partial v_{l}}\right)$ with
$\left(\frac{\delta}{\delta c^{(0)}}+v_{i}\frac{\delta}{\delta
c^{(1)}_{i}}+v_{i}v_{j}\frac{\delta}{\delta c^{(2)}_{ij}}+...\right)$. As a
result we arrive at the bracket $\\{A,B\\}$ in which $A$ and $B$ are real
valued functions of the Grad moments. With such bracket we then write
$LE_{{\boldmath\mbox{$c$}}}$, where $E({\boldmath\mbox{$c$}})$ is an energy (a
function of $c$) that we are free to choose to express the individual nature
of the system under investigation. Details of calculations can be found in
[45].

The structure-Grad hierarchy has at least two advantages. First, the energy in
the Boltzmann equation as well as in its standard-Grad hierarchy reformulation
is only the kinetic energy. From the physical point of view, the investigation
is thus limited to ideal gases. On the other hand, the choice of the energy
$E({\boldmath\mbox{$c$}})$ in the structure-Grad hierarchy is unconstrained.
Any energy expressed in terms of the 1-particle distribution function
$f({\boldmath\mbox{$r$}},{\boldmath\mbox{$v$}})$ or in terms of the low state
variable ${\boldmath\mbox{$c$}}({\boldmath\mbox{$r$}})$ in its reformulation
into the structure-Grad hierarchy can be chosen.

The second advantage is that both the free flow term in the Boltzmann equation
and its structure-hierarchy reformulations possess manifestly the Hamiltonian
structure.

Domains of applicability

The domains of applicability of the unclosed and the closed hierarchies can be
different. For example the standard infinite Grad hierarchy is an equivalent
reformulation of the Boltzmann kinetic equations and thus its domain of
applicability is an ideal gas. The domain of applicability of the closed
5-moment hierarchy is much larger (it includes simple fluids). Such change of
domains of applicability can happen because the patterns recognized in the
upper phase portrait are often determined by only some features of the upper
vector fields. The same patterns may arise in many very different upper phase
portraits. For example, in the case of the Grad hierarchy, the 5-moment
equations are essentially the same in the standard Grad hierarchy and in the
structure-Grad hierarchy. This means that the determining feature of the
closed equation is the kinematics (the Poisson vector $L$) and not the
particular choice of the energy.

Existence of closures

Another interesting question about the closures is the question about their
existence. Do patterns in the upper phase portrait exist? In the Grad
hierarchy the closure to the first five moments exists. Its existence is
supported by mathematical results about the Lie group expressing kinematics
[46] and also by the existence of hydrodynamics as an autonomous dynamical
theory that has arisen on the basis of its own experimental observations. On
the other hand, the closure into a larger than five moments does not have to
exist. Mathematical results about the kinematics [46] in fact indicate it.
Also, there is no well established autonomous hydrodynamics involving higher
order Grad fields. There are, of course, well established generalized
hydrodynamic theories of complex fluids that involve extra fields and
distribution functions. But the complexity in such fluids is due to the
presence of an internal structure. The kinetic-theory origin of such theories
is the Kirkwood kinetic theory [47], [48] (see also [49]) that unfolds from
the BBGKY hierarchy and not from the Boltzmann kinetic theory. The complexity
expressed in the higher moments of the Grad moments
${\boldmath\mbox{$c$}}({\boldmath\mbox{$r$}})$ are complexities of velocity
correlations. They arise and become visible in complex flows, in particular
then in turbulent flows. There is no autonomous hydrodynamics-like theory of
complex flows. Observations of turbulent flows do not indicate existence of
any autonomous extended hydrodynamics. For example the Kolmogorov cascade of
energies of turbulent flows does not show anything like it.

Two classes of Grad’s moments

The fields serving as state variables in the classical hydrodynamics divide
naturally into two classes. In the first class are fields of the mass density
$\rho({\boldmath\mbox{$r$}})$ and the momentum
${\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}})$ (entering the macroscopic
mechanics) and in the second class is the field of the internal energy
$\epsilon({\boldmath\mbox{$r$}})$ (entering the microscopic internal
structure). It seems natural to keep this division also in extensions of the
classical hydrodynamics. For example, in the Cattaneo extension [50] the state
variables split into
$(\rho({\boldmath\mbox{$r$}}),{\boldmath\mbox{$u$}}({\boldmath\mbox{$r$}}))$
and
$(\epsilon({\boldmath\mbox{$r$}}),{\boldmath\mbox{$q$}}({\boldmath\mbox{$r$}}))$,
where ${\boldmath\mbox{$q$}}({\boldmath\mbox{$r$}})$ is the heat flux. With
such split it is then possible to formulate a Lagrangian version of the
Cattaneo hydrodynamics as motion of two fluid particles. One is mechanical and
the other caloric (see details in [51]). Grad’s hierarchy with the extra
structure provided by splitting Grad’s moments into two classes is
systematically investigates in [52].

##  4 Concluding Remarks

The founding stone of the multiscale theory is the 0-th law of multiscale
dynamics. The 0-th law of equilibrium thermodynamics guarantees the existence
of equilibrium states and equilibrium thermodynamics. The 0-th law of
multiscale dynamics extends it to the existence of autonomous mesoscopic
dynamical theories and multiscale dynamics. The multiscale theory organizes
the autonomous dynamical theories into an oriented graph. The vertices
$\mathbb{V}$ in the graph are the autonomous dynamical theories. Every two
vertices are connected by a link. Every link
$\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$ has an orientation, it
is directed from the vertex $\mathbb{V}^{(up)}$ representing a theory
involving more details to the vertex $\mathbb{V}^{(down)}$ representing a
theory involving less details. Every link has also a length that is a measure
of the separability of the two vertices that it connects.

From the physical point of view, the link
$\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$ represents the process
(the time evolution taking place in the state space of $\mathbb{V}^{(up)}$) in
which the macroscopic systems under investigation are prepared for being
investigated in the theory represented by the vertex $\mathbb{V}^{(down)}$. If
the time evolution is lifted from the state space to the time evolution in the
vector fields on the state space then the link is called a rate link. While
the vertices are all different one from the other, the links share a universal
structure. The time evolution that they represent is in all links generated by
an entropy (a rate entropy in rate links). The concept of the entropy has thus
its origin in the 0-th law of multiscale dynamics. The entropy enters the time
evolution represented by the links
$\mathbb{V}^{(up)}\longrightarrow\mathbb{V}^{(down)}$ as an ”ambassador” of
the pattern representing the vertex $\mathbb{V}^{(down)}$ in the
$\mathbb{V}^{(up)}$ phase portrait. In particular, if $\mathbb{V}^{(down)}$ is
$\mathbb{V}^{(bottom)}$, the entropy generating the preparation process, if
evaluated at the pattern reached in the preparation process, becomes the
equilibrium entropy introduced originally in the equilibrium thermodynamics in
its 2-nd law.

Acknowledgement

I would like to thank Oǧul Esen, Václav Klika, Hans Christian Öttinger, Michal
Pavelka, and Henning Struchtrup for stimulating discussions.

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BRX TH-????

CALT-TH 2022-???

The Anthropic (and Mis-) Principle revisited, Steven Weinberg in memoriam

S. Deser

Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, CA 91125;
Physics Department, Brandeis University, Waltham, MA 02454
<EMAIL_ADDRESS>
###### Abstract

I review the role and meaning of the Anthropic Principle, particularly in its
relevance to particle physics as that subject evolves. This is neither a
technical paper nor a resource letter, but rather an easy introduction.

##  1 Introduction

Recent tributes to Steven Weinberg, our greatest living — now alas gone —
theoretical physicist, have emphasized his use of this idea, among so many
others. He made it a mainstream physics notion, worthy of discussing in a
wider context, if only because of (despite?) its somewhat pompous title.
Indeed, one might flippantly dismiss it on the grounds that, like the Holy
Roman Empire’s name, it is neither Anthropic nor a Principle, but rather a
tautology. But despite this perhaps misleading fact, it is shorthand for a
particular web of observational data, neither a panacea nor an act of
desperation. To summarize, the Principle states the seemingly obvious
requirement that the laws of Nature must permit the existence of some sort of
sentient life, therefore our chemistry and biology: Like any experimental
datum, this one puts stringent limits on our choice of (effective low energy,
if not final) laws and their various constants such as masses, charges,
Planck’s and Newton’s, as necessary conditions for organic chemistry, say, to
be possible. Thus the word Anthropic is too narrow —natural selection might
well have evolved other forms of intelligence, as indeed it did until
stumbling on us, and we might well not last long, either—but here we are and
so must be accounted for! Also it provides a fascinating window on how our
requirements change with time in the Wilsonian progression to higher energies.
By the 1890s, organic chemistry was a flourishing autonomous discipline, but
of course did not explain, but rather restated the Principle. Then the Bohr
atom of 1915 gave it a microscopic basis; by 1926, Dirac could claim all of
chemistry was subsumed by QM. The physics of, say, 1932, when the photon,
electron and its neutrino, proton, neutron were in hand, as well as the
nuclear models of the era could be said to provide a full basis. Indeed, Occam
would be happy to stop here! But then came the modern standard model, which if
anything, overshot the 1932 requirements, with its new particles, symmetries
and confinement. Yet the principle is too tight in that it requires all the
constants, such as masses, charges, etc., to be bounded, whereas we have no
idea at all as to why they are what they are! The above microscopic
constraints are only some necessary conditions. There are also cosmological
ones [1]; there too, Weinberg has contributed significantly [2], especially to
tight limits on the value of the Cosmological Constant. The existence of sun-
like stars with satellites of appropriate orbits is too far outside our scope
to be useful, though stellar formation and stability are nontrivially
relevant, while planets and their orbits are too “accidental” to be of use.
Indeed, new light has just now been shed [3] on our sun’s origins and
stability, a step forward. As for the word Principle, it is a bit too august —
all physics is based on Principle in the sense of agreement with the observed
world. Descartes’s famous cry was already proto-Anthropic, though alas only
sufficient, not necessary — as we know too well! In the late Fifties, Fred
Hoyle and Willie Fowler explored the necessity of carbon energy levels (and
iron’s for the formation of the heavy elements). Note that I do NOT imply
anything as silly as a pre-established harmony between our brains and Nature’s
laws.

##  2 Details

That was the short version of a complicated, and now enormously studied,
subject, one worthy of wider intellectual appreciation. It is also at the
heart of how science progresses. We look for data experimental or
observational — on which to base our models, as we modestly call our theories.
This is not a tendentious process: it simply puts limits on their range. The
Principle then is a name for one set of such data, namely and somewhat
loosely, those that are required in order that our universe allows for our
existence, which means more concretely two different sets: The first requires
that the laws of gravity and matter make it possible for chemistry, in
particular organic — involving the element carbon — to exist and evolve into
biology via ever- more complicated molecular structures. The latter moves us
into the second, very different arm, namely accidental, cosmological and
galactic effects, such as the cosmological constant’s value, formation and
stability of stellar objects like our sun, ones that have its radiative
properties and can live in that state for a long enough time before dying out
or being involved in some galactic catastrophe, and even more fortuitously, to
eject planets some of whose orbits permit the thermodynamic properties that
make life possible, if not necessary. This latter is a very tall order indeed,
as the vain search for exoplanets with these properties testifies. Not all
necessary conditions can be fruitfully exploited! The main result here was to
put fairly limits on the value of the cosmological constant.

Fortunately, to borrow a sentence from the law, de minimis non curat physics:
We are neither interested in, nor capable of calculating, such contingent
events, as long as they are not manifestly forbidden by our models. So, being
opportunistic is what the scientific method reduces to — in this case finding
any necessary conditions to explain what we observe, or discover upon
experimentation guided by previous insights… and luck. The Principle’s special
place is that it seemingly introduces a human element into the game. Yes, but
— really — no more than limitations from entirely inhuman data! As a final
thought, however, the above apparently trivial solar butterfly effect is why
we Anthropos are so rare, if not unique, in our universe. I call it the Mis-
Anthropic Principle: optimal planets and their orbits are forbiddingly
unlikely to occur! It no way invalidates the Principle’s usefulness in
bounding acceptable physical models; both are necessary.

##  3 Summary

Science proceeds using any tools it can; the Anthropic Principle is an example
summarizing a web of observed regularities, both cosmological and microscopic,
that any physical models must have room for, but in a way that evolves with
theoretical progress, as we have illustrated. Undoubtedly more such changes
will appear in due time, perhaps an endless process!

## Acknowledgements

This work was supported by the U.S. Department of Energy, Office of Science,
Office of High Energy Physics, Award No. de-sc0011632.

## References

  * [1] See, for example, Ellis George F. R., arXiv:1105.2462.
  * [2] Weinberg S., Phys. Rev. Lett., 59 (1987) 2607.
  * [3] New York Times Science, January 20, 2022.