Multiscale Theory

TL;DR

This paper develops an abstract framework connecting various mesoscopic dynamical theories through a generalized time evolution equation based on the Boltzmann kinetic equation, highlighting links and open problems.

Contribution

It introduces a novel abstract formulation of the Boltzmann kinetic equation connecting different mesoscopic theories and discusses related open problems.

Findings

Unified framework for mesoscopic theories

Representation of kinetic equations as abstract evolution equations

Identification of open research questions

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.

Equations

$$\dot{x}=L{E}_x(x)+{\Xi }_{x^{\ast }}(x,X(x^{\ast }))$$

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

$$\Xi (x,0)=0\forall x$$

$$\Xi reachesitsminimumatX=0\forall x$$

$$\Xi (x,X)isaconvexfunctionofXinaneighborhoodofX=0\forall x$$

$$E=E(x);N=N(x);S=S(x)$$

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

$$x^{\ast }=S_x$$

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

$$\dot{N}=0;\dot{E}=0;\dot{S}=<x^{\ast },{\Xi }_{x^{\ast }}>{|}_{x^{\ast }=S_x}\ge 0$$

$${\Phi }_x(x,{\mathbf{E}}^{\ast },{\mathbf{N}}^{\ast })=0$$

$${\mathbf{S}}^{\ast }({\mathbf{E}}^{\ast },{\mathbf{N}}^{\ast })=\Phi (\hat{x}({\mathbf{E}}^{\ast },{\mathbf{N}}^{\ast }),{\mathbf{E}}^{\ast },{\mathbf{N}}^{\ast })$$

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

$$\dot{X}=-\mathbb{G}{\Psi }_X(X,J)$$

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

$${\Psi }_X(X,J)=0$$

Full text

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

[TABLE]

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

[TABLE]

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:

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

and with the thermodynamic force

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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 $X$

[TABLE]

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

[TABLE]

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

[TABLE]

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.