Roles of energy and entropy in multiscale dynamics and thermodynamics

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


Introduction
Our point of departure is a pair of autonomous levels of description of macroscopic systems.Both have arisen from certain type of experimental observations.The types of observations are different different levels.One level, that we call starting level, is based on more detailed observations than the second level called target level.We emphasize that both levels are autonomous in the sense that both levels separately provide a good description of the experimentally observed behavior without a need of other levels.In particular, we investigate the levels of particle mechanics, kinetic theory and fluid mechanics as starting levels.All these three levels are paired with the level of the classical equilibrium thermodynamics as the target level.In Section 4 we also investigate pairs of levels in which both the starting and the target levels are mesoscopic levels involving time evolution.
Since we assume that both the starting and the target levels are autonomous and well established (i.e.predictions of the theory agree with results of observations), one observation (among all observations made on the starting level) has to be an observation of the approach to the target level.In other words, the autonomous existence of both levels guarantees the possibility to prepare macroscopic systems for the target level and the visibility of the preparation process in the time evolution observed on the starting level.We emphasize that in passing from the starting level to a target level that involves less details we loose details but gain emerging overall features.
We use the following terminology.Solutions of the governing equations on the starting level are trajectories, their collection is called a phase portrait.We distinguish three types of energy.Sources of the external energy are outside the macroscopic system under investigation.The potential energy in gravitational field is an example of the external energy.The inner energy is the energy that can be expressed in terms of the state variables used on the chosen level of description.The total energy in the Gibbs theory or the kinetic energy of a fluid in fluid mechanics are examples of the inner energy.The former is expressed in terms of the n-particle distribution function that serves as the state variable in the Gibbs theory and the latter in terms of the velocity and mass fields that serve as state variables in fluid mechanics.The internal energy is the energy that cannot be expressed in terms of the state variables that are used on the chosen level of description.Microscopic details that are not seen on the chosen level of description are needed to express it.
The objective of this paper is to investigate the roles that the energy and that entropy play in the passage from the starting level to the target level.
Energy provides the force generating the phase portrait.

Entropy plays two roles: (Ent I) it expresses the internal energy, (Ent II) it makes patterns, emerging in the phase portrait during the time evolution, manifestly visible by sweeping away unimportant details. The patterns represent the target levels inside the starting level.
We explore this viewpoint of energy and entropy in the setting of GENERIC time evolution (a combination of Hamiltonian and gradient dynamics) describing the passage from a starting level to a target level.The "universal competition" between energy and entropy is also discussed, but with a different perspective, in [1].
Before starting our discussion we briefly recall history of GENERIC dynamics.The first step in its formulations, made by Vladimir Arnold [2], was casting the Euler fluid mechanics into the form of noncanonical Hamiltonian dynamics.Vladimir Arnold has also realized connection with older results obtained in investigations of Lie groups.Such connection then allowed to formulate nondissipative parts of kinetic [3] and other mesoscopic time evolution equations as noncanonical Hamilton's equations.In proceedings of the conference devoted to this subject (organized by Jerrold Marsden in the summer of 1983 in Boulder, Colorado) the formulation of the complete Boltzmann kinetic equation (that includes the dissipative collision term) as a combination of Hamiltonian and generalized gradient dynamics has appeared [4].Many other mesoscopic time evolution equations have been put into such form in [5], [6], [7], [8], [9], [10].In [11], [12] the combination of Hamiltonian and generalized gradient dynamics has been called GENERIC (an acronym for General Equation for Non Equilibrium Reversible Irreversible Coupling).The natural geometrical setting for the Hamilton-gradient dynamics is contact geometry [13], [14].

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

Static theory (MaxEnt)
The state variable chosen on the starting level is denoted by the symbol x, the energy is E(x), the entropy S(x), and the number of moles N (x).In the Gibbs theory x is the n-particle distribution function (n ∼ 10 23 ), in the Boltzmann theory, discussed in Section 3.1, the state variable x is the one particle distribution function, in fluid mechanics theories discussed in Section 3.3 the state variable x is a collection of hydrodynamic fields, and in the classical equilibrium thermodynamics the state variable x = (E, N, V ), where E is the energy, N number of moles, and V is the volume.
The input into the passage starting level −→ level of equilibrium thermodynamics is the fundamental thermodynamic relation on the starting level consisting of three real valued functions S = S(x); y = y(x) where y = E N are state variables on the target level that is in this section always the level of the classical equilibrium thermodynamics, E is the equilibrium thermodynamic energy and N is the equilibrium thermodynamic number of moles.By y(x) = E(x) N (x) we denote the energy and the number of moles given in the fundamental thermodynamic relation ( 1) on the starting level.Both functions S(x) and y(x) are assumed to be sufficiently regular and the entropy S(x) is moreover assumed to be concave.
With (1) we construct thermodynamic potential The covector y * = (E * , N * ) denotes the Lagrange multipliers.We require In the standard equilibrium thermodynamic notation E * = 1 T and N * = − µ T , where T is the absolute equilibrium thermodynamic temperature in energy units and µ is the equilibrium thermodynamic chemical potential.The requirement (3) is thus the requirement that the absolute temperature T is positive.By <, > we denote pairing in the equilibrium thermodynamic state space; < y * , y >= E * E + N * N .
With the thermodynamic potential Φ(x, y * ) we are now in position to make the passage staring level → target level.The passage is made by is denoted x eq (y * ) and called equilibrium states.We use a shorthand notation Φ x = ∂Φ ∂x .If x is an element of an infinite dimensional space (e.g. a distribution function) then the derivative is an appropriate functional derivative.
The Legendre transformation S * (y * ) (that belongs to the target level) of S(x) (that belongs to the starting level) is given by The function S * (y * ) is a Legendre transformation of S(y) that is the entropy on the target level implied by the entropy S(x) on the starting level.Explicitly, where Φ * (y * , y) = −S * (y * )+ < y * , y > and y * ex (y) is a solution of Φ * y * (y * , y) = 0.So far we have not addressed yet the region Ω ⊂ R 3 in which the macroscopic system under investigation is confined.We do it now but only on the target level.We characterize Ω only by its volume V .We take it into account by extending S(y) introduced in (7) into S = S(y, V ).We require that both S and y are extensive state variables in the sense that S(λE, λN, λV ) = λS, where λ ∈ R. From the Euler relation we have S =< ∂S ∂y , y > + ∂S ∂V V .From (7) we have ∂S ∂y = y * .The Euler relation takes the form S =< y * , y > +V * V , where V * = ∂S ∂V .In the standard notation of the classical thermodynamics V * = P T , where P is the equilibrium pressure.Consequently, Summing up, we have passed from the starting level with state variables x and the fundamental thermodynamic relation (1) to the target level with state variables y.The input is the fundamental thermodynamic relation (1), the output is the equilibrium state f eq (y * ) and the fundamental thermodynamic relation S = S(y); y = y The equilibrium state f eq (y * ) places the level of equilibrium thermodynamics inside the starting level and S(y) is the fundamental thermodynamic relation on the level of equilibrium thermodynamics that is inherited from the fundamental thermodynamics relation on the starting level.We note that the passage starting level → target level made by MaxEnt ( 4) is in fact a reducing Legendre transformation.
The conjugate variables y * introduced in the thermodynamic potential (2) belong to the target level (that is in this section the level of equilibrium thermodynamics).One of them, namely e * = 1 T , relates energy to entropy and is therefore of particular interest in this paper.We shall discuss it in more detail in Section 2.3 below.

Dynamic theory
The postulated maximization of the entropy (4) is replaced in the dynamic formulation by the time evolution equation where is called, in accordance with the standard terminology (see ( [1]), an available free energy.The maximization of the entropy S(x) subjected to constraints E(x), N (x) is made by following the time evolution governed by (10).The asymptotic solution t → ∞ of ( 10) is the equilibrium state x eq (y * ) that is also a solution to (5).
We now explain the meaning of the symbols introduced in the GENERIC equation (10) and show that its solutions indeed approach x eq (y * ).The first term on the right hand side of (10) is the Hamiltonian vector field.It is the covector E x transformed into vector by the Poisson bivector L(x).This bivector is defined by the bracket {A, B} =< A x , LB x > that is required to be a Poisson bracket satisfying the following properties: antisymmetry {A, B} = −{B, A} Jacobi identity {A, {B, C}} + {B, {C, A}} + {C, {A, B}} = 0 (12) A(x) and B(x) are real valued and sufficiently regular function of x.From the physical point of view, the Poisson bracket expresses the kinematics of the state variable x.
Next, we turn to the second term on the right hand side of (10).The symbol Ξ stands for a real valued function of (x, x * ), called a dissipation potential, satisfying the following properties We note that for small x * (i.e. in a small neighborhood of equilibrium states) all dissipation potentials are quadratic functions of x * .
With the requirements ( 12), ( 13), (15) Eq.( 10) implies This inequality makes the thermodynamic potential (2) the Lyapunov function (provided ( 3) is taken into account and provided not only −S(x) but also Φ(x) is a convex function) indicating the approach to x eq (y * ).A rigorous proof of the approach requires an additional analysis.We shall comment about it at the end of Section 3.1.
If we equip (10) with an extra structure LS x = 0; LN x = 0 Ξ depends on x * only through the dependence on the thermodynamic f orce X = Kx * where K is a linear operator satisf ying KE x = 0; KN x = 0 (15) then (10) turns into the familiar GENERIC equation ( see e.g.[18]) where Ξ = 1 e * Ξ. Equation ( 16) with the structure (15 then implies is a stronger property than (14).The result (17) implies ( 14) but gives a more information about solutions to (16).

Roles of Energy and Entropy
The roles that the energy and the entropy play on various starting levels depend on the levels.They will be discussed below.Here we recall the roles that they play on the target level that is in this section the level of the classical equilibrium thermodynamics.On this level the entropy plays the role of one of the state variables.There is no inner energy, the only energy is the internal energy.Moreover, the internal energy is in one-to-one relation with the entropy.The entropy thus plays only the role (Ent I ).
There is no time evolution on the level of equilibrium thermodynamics.The entropy therefore does not play the role (Ent II).However, equilibrium thermodynamics considers processes in which states change.The changes are due to interactions through walls surrounding the macroscopic systems.The walls can pass or prevent passing the internal energy, can expand or shrink the system, and can enlarge or diminish the number of moles.An investigation of the time evolution involved in such changes would require to step outside the level of equilibrium thermodynamics on a level involving more details.When remaining inside the equilibrium level, the processes are considered as sequences of equilibrium states.Their final outcome is determined by MaxEnt.In this sense the entropy plays on the level of equilibrium thermodynamics also the role (Ent II).The unimportant details that are swept away in this pattern recognition process are details of initial arrangements of the subsystems.
Since the relation between internal energy and entropy is invertible their roles as state variables can be exchanged.The fundamental thermodynamic relation E = E(S, N, V ) can be replaced by In addition, the set of state variables (S, N, V ) can be enlarged by adopting variables characterizing overall states of macroscopic systems.For example it can be the overall velocity or coordinates characterizing placement of the macroscopic system in an imposed force field (e.g.gravitational field).In such case the total energy is a sum of the inner energy (the energy that can be expressed in terms of the newly adopted state variables, for example the overall kinetic and/or potential energy) and the remaining internal energy expressed in terms of (S, N, V ).
Because our objective in this paper is to investigate relations between energy and entropy, a particularly important concept is the temperature.On the level of equilibrium thermodynamics the temperature is defined by 1 T = S E (see (7)).MaxEnt principle implies that two systems, that are connected by a wall that freely passes the internal energy E and both are surrounded by a wall that prevents such passing approach an equilibrium state at which both systems, have the same temperature.This is the way the temperature on the level of equilib-rium thermodynamics is measured.One of the two systems is a thermometer for which the fundamental thermodynamic relation is known and thus its temperature can be read in other quantities (e.g.volume).A general temperature can be defined [19] as a measure of internal energy, its measurements involve a process of equilibration.Different meanings that can be given to "internal energy" and "equilibration" lead to different meanings of the temperature.

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

Boltzmann's kinetic theory
Boltzmann's investigation [15] of dynamics of ideal gases was the first step towards understanding the approach of macroscopic systems to equilibrium.The insight allowing to recognize the equilibrium pattern in the phase portrait is the realization that collisions of gas particles are the source of unimportant details that have to be swept away in order that the equilibrium pattern is revealed.We present below the Boltzmann theory as a particular realization of the multiscale formulation of starting level → level of equilibrium thermodynamics.
Before placing ourselves on the level of kinetic theory, we recall that on the level of equilibrium thermodynamics the individual nature of an ideal gas is expressed in the equilibrium-thermodynamics fundamental thermodynamic relation where k B is the Boltzmann constant and R the universal gas constant.On the level of equilibrium thermodynamics, this relation is obtained from experimental observations of the behavior of ideal gases.Our objective is to get (19) from investigating the passage level of kinetic theory→level of equilibrium thermodynamics.Our objective is to introduce a particular realization of (10) which implies (19).
We begin with the static theory.The state variable on the level of kinetic theory is the one particle distribution function We put the mass of one particle equal to one, r is the position coordinate and v momentum of one particle.The ideal gas under investigation is assumed to be confined in the region Ω ⊂ R 3 with periodic boundary conditions (i.e.all integrals over the boundary that arise in by parts integrations equal zero).
The fundamental thermodynamic relation of an ideal gas on the level of kinetic theory is The energy E(f ) is the kinetic energy.The physical interpretation of f (r, v) as a distribution function leads directly to the number of moles N (f ), and the entropy S(f ) is postulated.
The MaxEnt principle ( 4) leads (see ( 5)) to the equilibrium state and the ideal gas fundamental thermodynamic relation (19) on the level of equilibrium thermodynamics.
Next, we turn to the dynamic theory.The postulated MaxEnt principle (4) is replaced by the Boltzmann kinetic equation that is a particular realization of ( 16) with the Poisson bracket (where A(f ), B(f ) are real valued sufficiently regular functions of f (r, v)) and the dissipation potential where In addition, W is symmetric with respect to the interchange of v → v 1 and v ′ → v ′ 1 and with respect to the interchange of (v, v 1 ) and (v ′ , v ′ 1 ).The thermodynamic force X is given by The Poisson bracket (24) expresses kinematics of the one particle distribution function f (r, v) (see e.g.[18]).Regarding the dissipation potential (25), a direct verification proves that it satisfies the required properties (13).The degeneracy requirements (15) can also be proved by a direct verification.All functions C(f ) = dr dvζ(f (r, v)) where ζ : R → R satisfy {A, C} = 0 ∀A.Such functions are called Casimirs of the bracket {A, B}.Consequently, the Poisson bivector L defined by ( 24) satisfies thus the degeneracy requirement (15).The required degeneracy of the dissipation potential (25) can also be directly verified.
With the above specification of the building blocks of (10), the thermodynamic potential (2) plays the role of the Lyapunov function for the time evolution governed by Eq.( 23).Maximization of the entropy subjected to constraints of energy and number of moles is thus made by following the time evolution governed by ( 23) to its conclusion.
There are still missing pieces in a rigorous proof of the approach of solutions of (23) to the equilibrium state (22).In particular it is the existence of solutions of the Boltzmann equation and an additional analysis (in addition to identifying the Lyapunov function) needed to prove the approach to the equilibrium state.The former, provided in [20], demonstrates agreement with experimental observations.The time evolution of ideal gases is seen in experimental observations to exist.Solutions to the Boltzmann kinetic equation describing the time evolution of ideal gases in their mathematical representation is proven in [20] to exist.Results proven in [21], [22], [23] are even more physically significant.Solutions to the Boltzmann equation with the Hamiltonian term missing approach the local equilibrium (that is the equilibrium state (22) in which n and T are unspecified functions of the position coordinate r).It is the coupling with the Hamiltonian term that brings solutions to the total equilibrium (22).The Hamiltonian term by itself does not produce any dissipation.The enhancement of dissipation, that is due to the coupling of gradient dynamics with Hamiltonian dynamics, is called Grad-Villani-Desvillettes enhancement of dissipation.Very likely this is the principal mechanism making the transformation of time reversible and nondissipative Hamiltonian time evolution of ∼ 10 23 particles to time irreversible and dissipative GENERIC time evolution.A very small instability on the microscopic level may be enhanced by the Grad-Villani-Desvillettes enhancement to macroscopic dissipation bringing macroscopic systems to equilibrium states.So far only the damping that brings the local equilibrium to the global equilibrium in the Boltzmann dynamics [21], [22] and the Landau damping occurring in the Vlasov dynamics [23], [24] have been proven rigorously.An argument supporting the general importance of the Grad-Villani-Desvillettes enhancement of dissipation is presented in Section 3.3.2.

Roles of Energy and Entropy
Ideal gas particles do not interact among themselves.Their kinetic energy, that is expressed in terms of f (r, v) and is thus an inner energy, is the only energy.The entropy plays only the role (Ent II).In the process of its maximization (or equivalently in the process of following solutions of the Boltzmann kinetic equation to t → ∞) unimportant details in the phase portrait are swept away and the equilibrium pattern in the phase portrait emerges.The entropy is not postulated in the Boltzmann theory.It arises from investigating solutions of the Boltzmann kinetic equation.The individual nature of ideal gases is expressed only in the energy (that is the inner energy).The entropy is universal.We also note that on the starting level there is no direct relation between the energy and the entropy.Both are functions of f (r, v) but f (r, v) cannot be eliminated between them.On the other hand, on the target level the energy and the entropy are directly related, there is a one-to-one relation between them.
There is no temperature on the level of kinetic theory because there is no internal energy.Only after the inner energy has been transformed by MaxEnt (or equivalently by following the time evolution generated by the Boltzmann equation), to the internal energy on the level of equilibrium thermodynamics, the temperature can be defined on the submanifold composed of the equilibrium states f eq (y) 3  2 T = ( dr dvf ) −1 dr dv v 2 2 .We can also introduce local temperature that arises on the manifold composed of the local equilibrium states. .Before leaving the Boltzmann theory, we note that the abstract multiscale thermodynamics in Section 2 is in fact an extraction of the mathematical structure that is present in the Boltzmann equation.By transforming the original Boltzmann theory into the form presented in Section 2 we have in fact extended its applicability to general macroscopic systems.

Gibbs equilibrium statistical mechanics
. Gibbs' investigation [16] is limited to static situations but is applicable to all macroscopic systems.The state variable is chosen to be n-particle distribution function f (z), where n ∼ 10 23 is a fixed number of microscopic particles composing macroscopic systems, where z i = (r i , v i ), is the position coordinate and the momentum if i − th particle, i = 1, 2, ..., n, n ∼ 10 23 .The Gibbs equilibrium statistical mechanics is traditionally introduced (see e.g.[25]) in two steps.The equilibrium distribution function f eq (z) in the first step, the entropy in the second step.Gibbs assumes that f eq (z) depends only on constants of motion (i.e. on the energy and the number of moles).Other details of their trajectories do not enter f eq (z) due to the ergodic hypothesis according to which the particle trajectories are uniformly spread.The exponential dependence of f eq (z) on the energy and the number of moles then follows from noting that the energy and the number of moles of two independent subsystems is a sum of their energies and the numbers of moles while f (z) is a multiplication of their two distribution functions.The Gibbs entropy appears by requiring that f eq (z) arises in MaxEnt.
Below, we present the Gibbs theory as a particular realization of the static multiscale formulation of level of particle mechanics−→ level of equilibrium thermodynamics in Section 2.1.We shall also supplement the static theory with a corresponding to it dynamic theory that is a particular realization of the multiscale dynamic theory in Section 2.2.
The state variable on the starting level of the Gibbs theory is the n-particle distribution function The fundamental thermodynamic relation is where h(z) is the particle Hamiltonian and k B is the Boltzmann constant.With these specifications the MaxEnt principle leads to the equilibrium state and the fundamental thermodynamic relation.
Now we turn to the dynamic theory.The Gibbs theory addresses the time evolution only in the energy conservation and in the ergodic hypothesis.Inspired by Boltzmann, we suggest a particular realization of (10) replacing the ergodic hypothesis by the time evolution.
The time evolution of f (z) (a lift of the Hamiltonian and reversible time evolution of z to the time evolution of real valued functions of z [18]) is governed by the Liouville equation We note that this equation is a Hamilton equation ∂f (z) ∂t = L(f )E(f ) with the energy E(f ) given in (28) and the Poisson bivector L(z) given by the Poisson bracket [18] {A, ∂A f ∂v iα (32) This means that the Liouville equation is a particular realization of ( 10) with the second term on its right hand side missing.We also note that the lift from the time evolution of z to the time evolution of f (z) is a linearization.Typically very nonlinear equations governing the time evolution of z become linear Liouville equations (31) governing the time evolution of f (z).
Can we modify the Liouville equation (31) in such a way that: (i) it becomes a particular realization of (10), and (ii) the time evolution that it generates maximizes the Gibbs entropy subjected to constraints of the energy and the number of moles (specified in (28)).Our goal is to make the emergence of the equilibrium pattern in the Liouville phase portrait (i.e.collection of solutions to (31)) manifestly visible in the phase of appropriately modified Liouville equation.The Gibbs equilibrium phase portrait will appear as an attractive fixed point in its phase portrait.
The modification of Eq.( 31) that we search should ideally result from a thorough analysis if its solutions, in particular then from an analysis of overall features of its solutions.We do not follow this path.Instead, we suggest a formal modification of (31) by adding to its right hand side an appropriate (inspired by the Boltzmann collision term) particular realization of the second term on the right hand side of (16).When the macroscopic system is an ideal gas then, following Boltzmann, a separate treatment (separate from the free flow of particles) of binary collisions leads to such modification.For general macroscopic systems we replace binary collisions with transformations z ↔ z ′ that preserve (i.e.h(z) = h(z ′ )).We introduce thermodynamic force where f * (z) = S f (z) (f ) with S(f) given in (28).With the dissipation potential the modified Liouville equation becomes The function W in ( 34) is required to satisfy the following three properties: 35) (already suggested in [26]) is indeed a particular realization of (16).Also the degeneracy requirement ( 15) is clearly satisfied.Its asymptotic solution is the equilibrium distribution function ( 29) of the Gibbs theory.We note that the linearity of the Liouville equation has not been erased in its modification.The modified Liouville equation ( 35) remains a linear equation.
The dynamic formulation (35) of the static Gibbs theory remains formal.Both the physical basis of the transformation z → z ′ and a detailed analysis of solutions to (35) needed to complete the proof of the approach of its solutions to f eq (z) remain open.This type of investigation reaches beyond the scope of this paper.

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

Euler-Navier-Stokes-Fourier fluid dynamics
Starting levels in the two previous realizations of the mesoscopic thermodynamics had no internal energy.Only after the passage to the target level their initial starting-level inner energy turned to the target-level internal energy on the level of equilibrium thermodynamics.In this section we choose the level of fluid mechanics as the starting level.This level has both the inner energy and the internal energy.The level of fluid mechanics is an extension of the level of equilibrium thermodynamics to spatially inhomogeneous fluids.Fluids are composed of fluid particles.A fluid particle of unit volume at r ∈ R 3 has the momentum u(r), an internal structure characterized by the internal energy ǫ(r), and the mass ρ(r).The energy E that serves as the state variable on the level of equilibrium thermodynamics turns in the extension to the local internal energy ǫ(r), the number of moles N to 1 M (mol) ρ(r), where M (mol) is the molar mass.The volume V on the level of equilibrium thermodynamics enters the extension in assigning unit volume to fluid particles and by requiring that the energy, the mass and the entropy are extensive variables.The most important new feature in the extension is the motion of the fluid particles characterized by the new state variable u(r).The motion of the fluid particles then brings new contribution u 2 (r) ρ(r) to the energy (the kinetic energy) that is the inner energy since it is expressed in terms of the state variables.The total energy is E = E (in) + E (int) , where E (in) = dr u 2 2ρ is the total kinetic energy and E (int) = drǫ(r) the total internal energy.
Global conservation dA dt = 0 of a quantity A becomes in the extension to the local field a(r) (that is related to A by A = dra(r)) the local conservation ∂r , where J (a) (a(r)) is a flux of the field a(r).The local conservation of A requires thus to specify the flux J (a) and boundary conditions.As to whether the local conservation implies the global one is decided by the boundary conditions.In this paper we always assume periodic boundary conditions that make integrals over boundary equal zero.The local conservation thus always implies in this paper the global conservation.
Fluids that we investigate in this section on the level of fluid mechanics are characterized on the level of equilibrium thermodynamics by fundamental thermodynamic relations (18).Our objective is to make the passage level of fluid mechanics→level of equilibrium thermodynamics which ends up with a placement of the equilibrium thermodynamic state space inside the fluid mechanics state space (i.e. with the fluid-mechanics equilibrium state) and the equilibrium fundamental thermodynamic relation (18) with which the extension started.In other words, we begin with fluids that are characterized on the level of equilibrium thermodynamics by (18), extend their investigation to fluid mechanics and then by following the time evolution end up with the fundamental thermodynamic relation (18) with which the extension started.Our focus is put again on the roles that energy and entropy play in such passage.

The variables
x = (s(r), ρ(r), u(r)) playing the role of state variables on the level of fluid mechanics are physically interpreted as local entropy per unit volume, local mass per unit volume, and local momentum respectively.
The fundamental thermodynamic relation is The energy E (in) (ρ, u) is the kinetic energy that is in the classical hydrodynamics the inner energy, i.e. the part of the energy that can be expressed in terms of state variables excluding the entropy.Moreover, s ǫ (r) = 1 τ (r) is physically interpreted as a local temperature and is assumed to be positive (an extension of (3) to local fields).If the function ǫ(s, ρ; r) is assumed to be the same as on the level of equilibrium thermodynamics then such assumption is called an assumption of local equilibrium.Identification of τ (r) with the local temperature is a weaker form of the local equilibrium assumption.The positivity of the local temperature implies that there is a one-to-one relation between the energy field e(r) and the entropy field s(r).For later use we recall the relations With the fundamental thermodynamic relation (37) and the local equilibrium assumption, the MaxEnt principle leads to the equilibrium state u eq (r) = 0; (s e (r)) eq = 1 and the fundamental thermodynamic relation (9).With this result we end the static theory.Now we turn to the dynamical theory.We look for a particular realization of ( 16) that replaces the postulated maximization in the MaxEnt principle..We know already the potentials (37) entering ( 16), we only need to identify the Poisson bracket expressing kinematics of (36) and a dissipation potential.
Motion of continuum are transformations R 3 → R 3 .These transformation form a Lie group.The momentum field u is an element of the dual of the algebra corresponding to the group.The structure of the group manifests itself in the dual of the Lie algebra that corresponds to the group as the Poisson bracket {A, B} = dru i ∂Au i ∂rj B uj − ∂Bu i ∂rj A uj [28], [18] (the convention of summation over repeated indices is used).The remaining two fields ρ(r) and s(r) are let to be passively advected by u(r).The Poisson bracket that expresses kinematics of the state variables ( 36) is [18] {A, B} = dr u i ∂A ui ∂r j B uj − ∂B ui ∂r j +ρ ∂A ρ ∂r i B ui − ∂B ρ ∂r i A ui +s ∂A s ∂r i B ui − ∂B s ∂r i A ui The Hamilton time evolution equation ∂x ∂t = LE x with x given in (36) and L in (40) are the Euler equations where We note in particular that the degeneracy requirements (15) hold and that the entropy contributes, through its presence in the local pressure p(r), to drive the time reversible Hamiltonian time evolution.The gradient of the local pressure is partially an entropic force.Entropy takes this new role because of the presence of the internal energy.In the absence of such energy, as it is in the case of the Boltzmann and the Gibbs theories, the entropy does not participate in the Hamiltonian part of the time evolution.
The forces that drive fluids to thermodynamic equilibrium are the Fourier force X F and the Navier Stokes forces X N S and X N Svol X N S ij (r) = ∂e ui ∂r j + ∂e uj ∂r i X Dissipation potential (44) generates the Navier Stokes Fourier gradient terms supplementing the Euler equations ( 41) The coefficients λ > 0, η > 0, η (vol) > 0 introduced in the dissipation potential (44) are functions of (36).The calculations that are needed to turn (45) into the familiar Navier Stokes Fourier equations use the relations (38) (see [18] for details of the calculations) .

Extended fluid mechanics
The level of kinetic theory playing the role of the starting level in Section 3.1 involves more details than the level of fluid mechanics that plays the role of the starting level in this section.Indeed, the physical interpretation of the one particle distribution function f (r, v) and of the hydrodynamic fields (u(r), ρ(r)) implies that the hydrodynamic fields can be expressed in terms of f (r, v) as its moments ρ(r) = dvf (r, v) and u(r) = dv vf (r, v).Are there autonomous levels that lie between the level of kinetic theory and fluid mechanics?Such levels, if they exist, are then expected to provide a theoretical framework for complex fluids (as for example viscoelastic polymeric fluids or suspensions) that are found to be outside the domain of applicability of the classical fluid mechanics.
Intermediate levels can be constructed in two ways: bottom up and top down.The former is an extension of the level of fluid mechanics by adopting extra fields (e.g. higher order moments of f (r, v) or fields characterizing an internal structure) as extra state variables.The latter as a reduction from the level of the kinetic theory or other theories involving more details than fluid mechanics.We first discuss the former and then the latter constructions.
There are two kinds of the bottom up extensions.The state variables in the first are x = (u(r), ρ(r), a(r)) = (u(r), ρ(r), a 1 (r), ..., a k (r)) and in the second x = (u(r), ρ(r), s(r), a(r)).In the first extension we proceed as in the Boltzmann and the Gibbs theories but with the fields (u(r), ρ(r), a(r)) replacing the distribution function f (r, v) (or f (z) in the Gibbs theory).We need to specify the fundamental thermodynamic relation (i.e.we need to specify S = drs(u(r), ρ(r), a(r)), E = dre(u(r), ρ(r), a(r)), N = drn(u(r), ρ(r), a(r)), Poisson bracket expressing kinematics of the fields (u(r), ρ(r), a(r)), thermodynamic forces, and dissipation potential.As in the Boltzmann and the Gibbs theory, there is no internal energy.In the second extension we proceed as in fluid mechanics but with the kinetic energy replaced by an enlarged inner energy involving the extra fields a(r).The total energy involves an internal energy that is expressed in terms of the entropy field that serves as one of the state variables.
The most familiar example of the top down extension is the reformulation of the Boltzmann kinetic equation ( 23) into Grad's hierarchy [29].The one particle distribution function f (r, v) is replaced by infinite number of fields a(r) (∞) (r) = (a 1 (r), ...) that are moments of f (r, v) in the momentum v.There are two ways to rewrite ( 23) into an equation governing the time evolution of a (∞) (r).First, it is a direct method [29] consisting of multiplying ( 23) by v i and integrating over v, then by v i v j and integrating over v, and repeating this process for all moments.In the second method we regard (23) as a particular realization of the GENERIC equation (10) and rewrite into moments separately all its building blocks (for example the Poisson bracket ( 24) is rewritten into Grad's moments in [30]).The two methods lead to two different infinite hierarchies [30].
If our objective is to obtain an extended fluid mechanics that involves a finite number of fields a(r), that addresses physics of complex fluids, and that is compatible with equilibrium thermodynamics in the sense of Section 2 then, in both bottom up and top down approaches, we need to identify the finite number of fields a(r) = (a 1 (r), ..., a k (r)) joining the hydrodynamic fields (ρ(r), u(r)) and corresponding to them building blocks of GENERIC.An inspiration can come from an insight into the physics of the internal structure (e.g.various models of macromolecules composing polymeric fluids [31], [32], [33]), from an insight into the geometrical structure of continuum dynamics ( [34], [35], and from hierarchy reformulations of kinetic equations [29], [36], [37], [38].Poisson brackets expressing kinematics of some of extended sets of hydrodynamic fields (u(r), ρ(r), a(r)) are identified in [13], [39].
In the top down approach to extensions the passage from an infinite number of fields a (∞) (r) to a finite number a(r) = (a 1 (r), ..., a k (r)) is called a closure of the infinite hierarchy.Difficulties encountered in finding appropriate closures are well illustrated in the problem of finding Poisson brackets expressing kinematics of a finite number of moments a(r) = (a 1 (r), ..., a k (r)) in the Grad hierarchy.Two arguments developed in [40] suggest that there are only three autonomous fluid mechanics theories that can be based on the Grad hierarchy: FM ∞ that is fluid mechanics with all infinite number of moments playing the role of state variables (such fluid mechanics is equivalent to kinetic theory), FM 5 that is the classical fluid mechanics with the classical hydrodynamic fields playing the role of state variables, and FM ∞−5 that is fluid mechanics in which all Grad moments except the hydrodynamic fields play the role of state variables.The first argument is based on the analysis of the Lie algebra of moments, the second argument is physical.We recall the latter argument.After the onset of turbulence the originally simple fluid whose behavior is well described by the classical hydrodynamic fields becomes a complex fluid with an internal structure that needs higher Grad moments to describe its behavior.From the Kolmogorov cascade we know that the complexity of the flow passes gradually to smaller and smaller scales until the molecular scale is reached.At the molecular scale the inner energy (i.e. the energy expressed in terms of a(r)) of the complex turbulent flow turns into an internal energy.If there were an autonomous fluid mechanics with a finite number k of Grad moments then the Kolmogorov cascade would have a plateau.The turbulent decay would stop on the k-scale (i.e. the scale associated with k-th Grad moments), the flow would become k-laminar (i.e.laminar in the setting of the extended fluid mechanics in which k Grad moments play the role of the state variables) and then a k-onset of turbulence would be needed to continue the turbulent decay to the molecular scale.No such plateau in the Kolmogorov cascade is, to the best of our knowledge, observed.

Roles of Energy and Entropy
Fluids are composed of fluid particles that have an internal structure and associated with it an internal energy.Its presence distinguishes the level of fluid mechanics from two levels discussed in the two previous illustrations of mesoscopic thermodynamics.The total energy e(r) of the fluid particle at the position r is a sum of the inner energy (that is the kinetic energy u 2 (r) 2ρ(r) ) and the internal energy ǫ(s, ρ; r).The total energy E = dre(r) of the fluid is then the sum of the energy of all fluid particles.The energy that generates the Hamiltonian part of the time evolution involves entropy and thus the entropy participates in the Hamiltonian time evolution.Specifically, the gradient of the local pressure p(r) is the entropic force.The entropy generates in addition also the dissipative forces driving the fluids to total equilibrium at which the fluid particles do not move and their internal structures are identical.The entropy plays thus on the starting level both roles (Ent I) and (Ent II).A weak local equilibrium assumption is carrying the one-to-one relation between internal energy and entropy from the level of equilibrium thermodynamics to the level of fluid mechanics, a complete local equilibrium assumption is carrying the complete equilibrium thermodynamic relation to individual fluid particles.
An interesting insight into the dependence of the passages starting level → level of equilibrium thermodynamics on the starting level can be gained by comparing the passages with kinetic theory and fluid mechanics playing the role of starting levels.For all starting levels the time evolution making the passages are governed by (10).The differences are in the strength of the dissipation in the second term on the right hand side of (10).In the context of fluid mechanics the passage from local equilibrium to the total thermodynamics equilibrium is made by the Navier-Stokes-Fourier dissipative forces (43) and the dissipation potential (44).In the context of the Boltzmann kinetic theory the passage from local Maxwellian distribution functions (that we can see as states corresponding to local equilibrium states in fluid mechanics) to the total Maxwellian distribution functions (22) does not require an extra explicit dissipation.It is made simply by coupling the dissipation driving to the local Maxwellian distribution functions with the Hamiltonian completely nondissipative part of the time evolution (by the Grad-Villani-Desvillettes enhancement of dissipation [22]).More precisely, solutions of the Boltzmann equation get in the course of the time evolution only close to the local equilibrium and reach it only in the total equilibrium (22).This comparison supports the conjecture that a very weak dissipation (instability) on the microscopic level is sufficient to grow by the Grad-Villani-Desvillettes enhancement of dissipation to macroscopic dissipation towards the level equilibrium thermodynamics.
Temperature on the level of fluid mechanics has the same meaning as the temperature on the level of the equilibrium thermodynamics.The situation is different if the fluid mechanics is extended by adopting fields a(r) as extra state variables.If the state variables of the extended fluid mechanics do not include the entropy field s(r) then such setting is essentially the same as the setting of the Boltzmann and the Gibbs theory.There is no internal energy and no temperature on the starting level.If the entropy field s(r) is included in the set of state variables then the energy that cannot be expressed in terms of (u(r), ρ(r), s(r), a(r)) is the internal energy.But such internal energy is different from the internal energy in fluid mechanics in which the extra fields a(r) are missing in the set of state variables.The part of the energy ǫ(a(r)) that is expressed in terms of a(r) is excluded from the internal energy in the extended fluid mechanics.The meaning of temperature depends on what kind of walls we use in its measurement and on what kind of control we have over the fields a(r).The walls can either pass or prevent passing the energy ǫ(r) − ǫ(a(r)) (W alls (extf m) ) or they are the same as the ones used in the classical fluid mechanics in which the fields a(r) are not included in the set of state variables (W alls (f m) ).With W alls (f m) and no control over a(r) the measured temperature is the same as in the classical fluid mechanics.With W alls (extf m) and with control over a(r) the measured temperature is a different temperature.

van der Waals gas
In this section we extend the kinetic theory discussed in Section 3.1.The extension is not made by adopting extra distribution functions as state variables (the one particle distribution function f (r, v) remains the only state variable) but by introducing an internal energy.From the microscopic point of view, the van der Waals (vdW) gas particles become interacting particles.Two types of interactions are considered: a long range attraction and a short range hard-core repulsion.The former is considered as an inner energy and the latter as an internal energy.The inner energy is the mean-field type Vlasov energy, the internal energy is expressed in a modification of the Boltzmann entropy.The hard core repulsion is replaced by an excluded-volume type constraint.From the point of view of hydrodynamic and equilibrium-thermodynamic type observations the vdW gas differs from the ideal gas by experiencing transition from gas to liquid.The Vlasov gas shows the approach to spatially homogeneous distribution (Landau damping).How do the roles of the energy and the entropy change when ideal gas becomes vdW or Vlasov gas?.
Before discussing the vdW gas on the level of kinetic theory, we recall the vdW fundamental thermodynamic relation on the level of equilibrium thermodynamics We note that ( 46) is a two parameter (a, b) deformation of the ideal gas fundamental thermodynamic relation (19) on the level of equilibrium thermodynamics.If a = 0 and b = 0 then (46) reduces to (19).On the level of equilibrium thermodynamics, the relation ( 46) is based on heuristic arguments and experimental observations.
As in the previous sections we begin with the static theory.The state variable is the same one particle distribution function (20) as in the Boltzmann theory.The fundamental thermodynamic relation extending ( 21) is [42] where n(r) = dvf (r, v).MaxEnt principle implies the equilibrium state where n eq (r) is a solution to and the fundamental thermodynamic relation (46) on the level of equilibrium thermodynamics.
Gas-liquid phase transition manifests itself in (48) (as the familiar P-V-T curves -see e.g [41]) and also in the manifold of equilibrium states (48) (as a bifurcation in solutions to (49) -see [42], [43], [18]).Now we turn to the dynamic theory.Our objective again is to replace the maximization of the entropy postulated in the static theory with the time evolution describing the experimentally observed approach to equilibrium.How does the kinetic equation ( 23) change when the ideal gas change into the vdW gas?There are at least two answers to this question.First it is a particular realization of (10) with the potentials (47) and the second is the Boltzmann kinetic equation with two changes: the Vlasov term expressing the influence of the attractive force is added to its right hand side and the Boltzmann collisions of point particles are replaced by Enskog collisions of particles having a finite size.The modified kinetic equation (called Enskog Vlasov equation [44]) seems to be very natural from the physical point of view but its compatibility with the static theory presented above as well as with the van der Waals fundamental thermodynamic relation (46) requires additional assumptions that do not appear to be natural from the physical point of view [45], [43].Below, we follow the path (already suggested in [46]) on which the kinetic equation emerges as a particular realization of (10) with potentials (47).
The state variable is the same as in the Boltzmann kinetic theory, its physical interpretation is the same, and thus also its kinematics expressed in the Poisson bracket (24) is the same.We leave also the same thermodynamic forces (26) as well as the thermodynamic potential (25).The binary collisions in which the entropy plays the role ((ENT II) are thus left in the vdW gas the same as in the ideal gas.The finite size of the particles is taken into account in the Hamiltonian part of the time evolution by changing the available free energy E(f ) (47).With the building blocks specified above, Eq.( 10) becomes Using the thermodynamic potential (47), the kinetic equation ( 50) gets the form Solutions to this kinetic equation describe approach to the equilibrium vdW theory.Equation ( 51) is a particular realization of the GENERIC equation (10).
It shares the GENERIC structure with all well established equations describing approach to the level of equilibrium thermodynamics.The particularly interesting and new feature of ( 51) is that vdW equilibrium theory addresses gas-liquid phase transitions and consequently (51) addresses its dynamical aspects.In order to see them we need detailed solution to (51).We hope to present them in a future paper.

Roles of Energy and Entropy
When comparing the static kinetic theory of ideal gases (Section 3.1) with the static kinetic theory of the vdW gas, the most important new feature in the vdW theory is the presence of internal energy that is expressed as a modification (47) of the Bolktzmann entropy (21).From the physical point of view, the hard-core repulsive potential cannot be expressed in terms of only one particle distribution function (i.e. it is not an inner energy).At least two particle distribution function is needed to express it.In the setting of the kinetic theory in which only one particle distribution function serves as the state variable the hard core repulsive potential has to be considered as an internal energy.Its expression in terms of entropy then follows from replacing the hard-core potential with the excluded-volume type constraint [42].We note that if we do not modify the entropy in (47) (i.e. if we keep only the Boltzmann entropy ( 21)) then there is no modification of the inner energy E(f ) in ( 47) that would imply the vdW fundamental thermodynamic relation (46) on the level of equilibrium thermodynamics.This result is just another manifestation of a well known result from equilibrium statistical mechanics, namely that the Gibbs equilibrium theory does not provide a setting for phase transitions unless it is somehow extended (e.g. by carrying it to thermodynamic limit N → ∞; V → ∞; N V = const.)[47].An internal energy that modifies the Gibbs entropy (in the setting of the Gibbs theory) or the Boltzmann entropy (in the setting of the Boltzmann theory) is needed to make phase transitions visible in geometrical features of the manifold of equilibrium states and geometrical features of the fundamental equilibrium thermodynamics relation, both obtained by MaxEnt.
In the dynamical view of the kinetic theory of the vdW gas we keep the setting of the static theory.As in fluid mechanics (Section 3.3), the internal energy (expressed in terms of the entropy in kinetic theory in (47)) generates one of the forces that drive the reversible Hamiltonian mechanics.Unlike fluid mechanics, the total entropy is not conserved.Only the available free energy (11) is.We note that if the extra term f (r, v) ln(1−bn(r))) in the entropy in (47) were replaced by f (r, v) ln(1−bf (r, v))) then the modified entropy would remain a Casimir of the Poisson bracket (24) and the total entropy would be conserved in the Hamiltonian part of the time evolution.With such modification the equilibrium states f eq (r, v) as well as the implied fundamental thermodynamic relation on the level equilibrium thermodynamics would be however modified.In particular, the dependence of f eq (r, v) on v would be a modified Maxwell distribution.From the physical point of view (i.e. in order to guarantee the experimentally observed approach to equilibrium) the conservation of E(f ) and not the conservation of S(f ) is essential in the Hamiltonian part of the time evolution.

starting level −→ target level with dynamics
In this section we change the target level.The level of equilibrium thermodynamics is replaced by a level involving less details than the starting level but still involving the time evolution.Not only the roles of the energy and the entropy change but also the energy and the entropy themselves are replaced by different potentials.The well known and thoroughly investigated passage from kinetic theory to fluid mechanics will serve us as a guide.There are essentially two points of view of the passage a mesoscopic level involving time evolution−→ another mesoscopic level that involves less details but still involves time evolution.In the first we regard it as an intermediate step in the passage a mesoscopic level involving time evolution−→ level of equilibrium thermodynamics.In the second we regard it in the space of vector fields rather than in the state space.
The first viewpoint is, roughly speaking, the viewpoint of Chapman and Enskog [48], [49] in their investigation of level of kinetic theory−→ level of fluid mechanics.The idea is to look in the state space for an invariant (or approximately invariant) attractive manifold.The time evolution describing approach to the invariant manifold is then the time evolution describing the passage a mesoscopic level involving time evolution−→ another mesoscopic level that involves less details but still involves time evolution.This viewpoint has two disadvantages.The first is that passages to mesoscopic levels with time evolution exist also when the level of equilibrium thermodynamics is not well established (due to the presence of external forces) and consequently the passage a mesoscopic level involving time evolution−→ level of equilibrium thermodynamics does not exist.The second disadvantage is that the thermodynamics based on the approach to fixed points (Section 2) does not directly extend to thermodynamics based on the approach to invariant (or approximately invariant) submanifolds.
On the other hand, when we lift the time evolution in the state space to the time evolution in the tangent (or alternative cotangent) space, the approach involved in the passage a mesoscopic level involving time evolution−→ another mesoscopic level that involves less details but still involves time evolution becomes the approach to fixed point (approach to vector fields governing the time evolution on the target level).The viewpoint of thermodynamics presented in Section 2 becomes directly applicable.Moreover, this second viewpoint does not require that the approach to equilibrium states exists and thus is directly applicable to externally forced systems.In the context of the extensive literature devoted to investigations of the passage level of kinetic theory−→ level of fluid mechanics the second viewpoint of the approach to mesoscopic levels with the time evolution is essentially the one used in the Grad type investigations [29].
We follow below the second viewpoint.Our main objective is to show that the passage a mesoscopic level involving time evolution→ another mesoscopic level that involves less details but still involves time evolution leads to thermodynamics but with entropy and energy replaced by, roughly speaking, their rates.We shall restrict our investigation in this section to working out one particular example taken from chemical kinetics.
(n p ) eq    is governed by a particular realization of ( 10) The potential Φ(n) is a thermodynamic potential and Ξ a dissipation potential.
We make a few observations about Eq.( 54).
(i) If the dissipation potential Ξ depends on n * only through its dependence on the thermodynamic force X = γ i n * i (called in chemical kinetics chemical affinity) then (54) takes the form where and (ii) Particular forms of the potentials Φ and Ξ for which (54) becomes the familiar Guldberg-Waage mass action law can be found in [50].In our analysis we can keep Φ and Ξ unspecified.
(iii) Since ( 54) is a particular realization of (10) its solutions approach (as t → ∞) equilibrium n eq that is a solution to Φ n = 0.

Starting level: Level of extended chemical kinetics (ExtChemKin)
Chemical reactions, seen on the scale of molecules, are complex processes described by quantum mechanics.Is there an intermediate level between the level on which the mass action law is formulated and the level on which the chemical reactions are seen on the scale of molecules?We follow here [50] where the extension of ( 54) is made by including chemical inertia.The chemical flux J introduced in ( 57) is adopted as an extra state variable.It has been argued in [51] that this extension indeed carries the mass action law (54) towards the microscopic level.We assume that the extended mass action law described below represents a well established level.
With the state variables the particular realization of (10) governing the time evolution takes the form where are the stoichiometric matrices, Θ is a dissipation potentiasl (its relation to the dissipation potential Ξ appering in ( 54) is given later in (65)), and , where Φ (ext) (n, J) is a thermodynamic potential extending the thermodynamic potential Φ(n) introduced in (54).
Regarding solutions to (59), we note that this equation is a particular realization of ( 10) and thus the thermodynamic potential Φ (ext) (n, J) plays the role of the Lyapunov function indicating approach to equilibrium state n J eq that is a solution to Φ = 0. Static view of (59) is the MaxEnt passage to the equilibrium state n J eq .Other properties of solutions to Eq.( 59) that address its relation to (54) are discussed in the next section.

(ExtChemKin)−→ (ChemKin)
Now we are in position to investigate in the setting of chemical kinetics described above the passage a mesoscopic level involving time evolution−→ another mesoscopic level that involves less details but still involves time evolution.The target level is the level represented by (59) and the starting level is represented by (54).
If the term Θ J * is dominant in the second equation (59) then J evolves faster that n.As a first approximation, we can solve (59) by following first the time evolution governed by the second equation in (59) to its conclusion and then follow the time evolution governed by the first equation in (59) in which J * is replaced by J * stat (n * ) that is a solution to Solution of (61) is where Θ † ((J * ) † ) is the Legendre transformation of Θ(J * ).
After J reached the state at which J * = J stat (n * ) the time evolution continues as the time evolution of n governed by the first equation in (59) in which J * is replaced by J * stat (n * ) given in (63).The time evolution of n is thus governed by By comparing this equation with Eq.( 54) we see that the two dissipation potentials Ξ and Θ are related by Summing up, we have reformulated (59) into two equations: Eq.( 54) and In (66) we have used J * = GJ, where G = Φ JJ , and (65) in (54).The three equations ( 59), ( 54) and (66) describe three reductions.We emphasize again that the existence of all three reductions is guaranteed by our assumption that all three levels (i.e. the level of equilibrium thermodynamics, the level of the Guldberg-Waage chemical kinetics, and the level of the extended Guldberg-Waage chemical kinetics) are autonomous well established levels.All tree equations (59), (54), and (66) are particular realization of (10).The first two describe approach to the level of equilibrium thermodynamics and the third approach to the level of the Guldberg-Waage chemical kinetics.The first two equations are thus two additional examples of the time evolution equations investigated in Section 2, the third is new.We shall now discuss it in a more detail.
First, we establish terminology.In order to make a clear distinction between passages starting level →level of equilibrium thermodynamics and starting level →target level with target levels that are different from the level of equilibrium thermodynamics we use the adjective "rate" in investigations of the latter.We call the thermodynamic potential Ψ in (62) a rate thermodynamic potential, Θ in (66) a rate entropy, the time evolution governed by (66) a rate time evolution.The adjective "rate" points to the fact that the space in which the time evolution takes place is the tangent (or alternatively cotangent) space.
Both the entropy S (ext) (n.J) and the rate entropy Θ(n, J * ) appear in both equations (59) and (66).Their roles are however very different.In Eq.(59) (governing the reduction to the level of equilibrium thermodynamics) the entropy S (ext) (n.J) drives the reduction and the rate entropy Θ(n, J * ) plays the role of the geometrical structure transforming gradient of the entropy into a vector.On the contrary, in Eq.(66) (governing the reduction to the level of the Guldberg-Waage chemical kinetics) the rate entropy Θ(n, J * ) drives the reduction and the entropy S (ext) (n.J) (in the form of the Hessian G = Φ (ext) JJ ) is the geometrical structure in the vector field.
In a small vicinity of thermodynamic equilibrium states the rate entropy Θ can be seen, roughly speaking, as entropy production in the Guldberg-Waage time evolution.This follows from: (i) the relation (65) between the dissipation potential Ξ and the rate entropy, (ii) from dΦ dt = − < n * , Ξ n * > | n * =Φ n implied by (54), from the fact that close to equilibrium all dissipation potentials are quadratic (see the text following (13)), (iii) and from the fact that Legendre transformations of quadratic potentials as well as the scalar product of the variable with gradient of the potential remain quadratic potentials.. Skeptical views of the relevance of the entropy production in mesoscopic dynamics, occasionally appearing in the literature, originate most likely from unsuccessful attempts to see the entropy production as Lyapunov-like function in the time evolution equations describing approach to thermodynamic equilibrium states.The entropy production (more precisely the rate entropy) does play such role but in different time evolution equations.Equations describing the approach to equilibrium stats are replaced by equations describing the approach to lower level (i.e.involving less details than the dynamics on the starting level) mesoscopic dynamics.The rate entropy has moreover a more general applicability than the entropy since the approach to a lower level dynamics exists also when the approach to thermodynamic equilibrium states does not exist (e.g.due to presence of external forces).
Finally, we make an observation about the static version of (66).We recall that the static version of (54) is the MaxEnt principle.The passage from the starting level to the level of equilibrium thermodynamics, made by following the time evolution governed by (54), is made in the static version of (66) simply by maximizing the entropy subjected to constraints (Φ n = 0).Analogically, the static version of (66) is minimization of the rate entropy subjected to constraints (i.e. by solving Ψ J * = 0).This view of (66) is in fact the Onsager variational principle [52], [53], [54], [55] applied to chemical kinetics.The time evolution equation (66) can be seen thus as an extension of the Onsager variational principle to chemical kinetics similarly as ( 54) is an extension of MaxEnt to dynamics.When seeing (66) in the context of the Onsager variational principle, the rate thermodynamic potential Ψ is called Rayleighian.We note that the rate entropy Θ decreases to its minimum in the time evolution governed by (66) .We can therefore call the minimization of the rate entropy MinRent similarly as we call the maximization of the entropy MaxEnt.

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