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Geometry of mesoscopic dynamics and thermodynamics
J. Non-Newtonian Fluid Mech. 120 (2004) 137–147
Geometry of mesoscopic dynamics and thermodynamics Miroslav Grmela∗ Ecole Polytechnique de Montreal, C.P. 6079 suc. Centre-Ville, Montreal, Que. H3C3A7, Canada Received 16 September 2003; received in revised form 19 November 2003; accepted 19 November 2003 This article is part of a Special Volume containing papers from the 3rd International Workshop on Nonequilibrium Thermodynamics and Complex Fluids
Abstract Contact geometry provides a natural setting for classical thermo-dynamics. In this paper we use it to derive the structure of mesoscopic dynamics (GENERIC) expressing its compatibility with thermodynamics. In the second part we derive kinematics (Poisson brackets) of a large family of mesoscopic state variables. © 2004 Elsevier B.V. All rights reserved. Keywords: Thermodynamics; Dynamics; Complex fluids; Contact geometry
1. Introduction Geometry has proven to be very useful in microscopic physics and relativity theory. We may recall for example the role that the geometrical formulation of classical mechanics played in the development of its extensions to quantum mechanics and to the gauge field formulations. In this paper we turn our attention to the role that geometry plays (or could play) in mesoscopic dynamics and thermodynamics. Before embarking on this subject we shall clarify the concept of a mesoscopic level of description. We begin by listing some examples of well established levels: quantum chromodynamics, quantum mechanics, classical mechanics, classical hydrodynamics, classical thermodynamics. We note that all these levels have arisen as a consequence of an experience collected by observing (with a certain class of observational tools specific to each level) a class of systems (again specific to each level) and an attempt to organize and explain results of the observations. Consequently, the levels are self-contained and independent. Neither of them needs the others to be clearly formulated and used. For example, in order to apply hydrodynamics we do not need to involve classical or quantum mechanics. On the other hand, we expect that the levels are also related. To establish the relation, we begin by comparing the quantities used to describe states (termed state variables). We say that a level (Level 1) is more microscopic than another ∗ Corresponding author. Tel.: +1-514-340-4711x4627; fax: +1-514-340-4159. E-mail address: [email protected] (M. Grmela).
0377-0257/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2003.11.009
level (Level 2) if the state variables used on Level 1 contain more details than those used on Level 2. In other words, the state variables used on Level 2 can be obtained as certain projections of the state variables used on Level 1. We expect that Level 2 can be derived (reduced) from Level 1. The reduction process can be intuitively seen as follows. First, we solve the time evolution equations on both Level 1 and Level 2 for all initial conditions and for a family of the parameters used to express the individual nature of the system under consideration on Level 1 and Level 2 respectively. As a result, we obtain on Level 1 as well as on Level 2 a set of trajectories. We shall term them a phase portrait on Level 1 and a phase portrait on Level 2. An identification of the phase portrait on Level 2 as a pattern inside the phase portrait on Level 1 constitutes a reduction of Level 1 to Level 2. Let us consider three levels: Level 1, Level 2, and Level 3. Let Level 2 be more macroscopic than Level 1 and Level 3 more macroscopic than both Level 1 and Level 2. Let, moreover, the behavior of the systems under consideration be found to be well described on all three levels. This then means, in particular, that the phase portrait on Level 3 has to be recognizable in both phase portraits obtained on Level 1 and Level 2. Consequently, solutions of the governing equations on Level 1 and Level 2 have to share some common features. This fact should be then a consequence of a common structure of the governing equations on Level 1 and Level 2. The above argument thus indicates that the governing equations on different levels share a common structure as a consequence of the existence (experimentally established) of relations among them. Our first illustration of the role of geometry in mesoscopic physics is an identification of the
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common structure in the case when Level 3 is the level of classical thermodynamics.
2. Mesoscopic dynamics The systems that are externally unforced and free from internal constraints (see Section 2.4) are seen to reach states, called thermodynamic equilibrium states, at which their behavior is found to be well described by equilibrium thermodynamics. This universal experimental observation is made on all mesoscopic levels that are more microscopic than the level of classical thermodynamics. Consequently, solutions to the governing equations on mesoscopic levels have a common property, which then means that the governing equations have to share a common structure. Identification of such structure has always been the main goal of non-equilibrium thermodynamics. One element of the structure is known to be the Onsager–Casimir symmetry of the time evolution equations on mesoscopic levels of description [1,2]. In the setting for the reduction process sketched in the introduction, Onsager derived this symmetry for Level 1 being the level of classical mechanics, Level 2 being the mesoscopic level, and Level 3 being the level of classical thermodynamics. The symmetry of the time evolution equations on Level 2 appears as a result of the time reversibility on Level 1 and the assumption that the time evolution of fluctuations about thermodynamic equilibrium states, in a small neighborhood of the equilibrium states, follows the mesoscopic time evolution. Below, we shall present a structure that extends the Onsager–Casimir symmetry. First, we shall introduce it by using geometrical considerations. Later, we shall also recall briefly other arguments leading to it. 2.1. Classical thermodynamics Since it is classical thermodynamics that we want to recover on mesoscopic levels involving time evolution, we begin with the geometry of classical thermodynamics. Its importance has already been recognized by Gibbs [3]. The titles of the first two papers (published in 1873) in which Gibbs completed the formulation of classical thermodynamics are: “Graphical methods in the thermodynamics of fluids” and “A method of geometrical representation of the thermodynamic properties of substances by means of surfaces”. We shall now present Gibbs’ geometrical formulation of thermodynamics. Let E and N denote the energy and the number of moles per unit volume. We choose them to be the universal state variables of classical thermodynamics of one component systems. According to Gibbs, thermodynamics is introduced as a function S = S(E, N), called a fundamental thermodynamic relation. By the symbol S we denote the entropy per unit volume. The fundamental thermodynamic relation can also be seen (again following Gibbs) as a two dimensional surface imbedded in a three dimensional space with coor-
dinates (E, N, S), i.e. as the image of the mapping (E, N) H → (E, N, S(E, N)). This two dimensional surface will be termed a Gibbs manifold. The Gibbs manifold does not however represent a complete formulation of thermodynamics. What is missing is the statement that the equilibrium states are those that maximize the entropy subjected to constraints expressing the conditions in which the systems under consideration are held. In other words, the fundamental role that the Legendre transformations play in thermodynamics, a very important part of the formulation of thermodynamics, remains still outside of the geometrical viewpoint. Hermann [4] was the first to realize that the importance of Legendre transformations can be incorporated into the geometrical formulation by making the one-jet extension of the Gibbs manifold, i.e., by introducing a manifold, called the Gibbs–Legendre manifold, Mth as the image of the mapping (E, N) → (E, N, SE (E, N), SN (E, N), S(E, N)), where SE (E, N): = ∂S/∂E evaluated at (E, N), similarly SN (E,N): = ∂S/∂N evaluated at (E, N). The Gibbs–Legendre manifold is again a two dimensional manifold but now imbedded in a five dimensional space with coordinates (E, N, E*, N*, S). The Legendre transformation are brought into the geometrical formulation by providing the space with coordinates (E, N, E*, N*, S) with the contact structure, i.e. with the one-form dS – E*dE – N*dN. The Gibbs–Legendre manifold is then distinguished by the property that it is a manifold on which the contact one-form vanishes. Manifolds having such property are called in geometry Legendre submanifolds. The Legendre transformations are the transformation leaving the contact structure (i.e. the contact one-form) invariant. With the contact structure, the formulation of thermodynamics is starting to look like the modern formulations of theories in microscopic physics. The first question in such formulations is always: what are the transformations of fundamental importance? In the context of thermodynamics, we answer that they are the Legendre transformations. The next question is then: what is the space in which the transformations act naturally? We answer that it is a space equipped with the contact structure since the contact structure remains invariant under the Legendre transformations. A specific physical system is represented in thermodynamics as a specific Legendre submanifold. At this point we introduce an additional notation. The space whose elements are (E, N) will be denoted by the symbol Mth (i.e. (E, N) ∈ Mth ), the space whose elements are ˆ th (i.e. (E, N, E*, N*, S) (E, N, E*, N*, S) by the symbol M ˆ ∈ Mth ), the one-form dS – E*dE – N*dN is denoted ϑth , and the Gibbs–Legendre manifolds by the symbol Mth . The subscript “th” stands for “thermodynamic”. 2.2. Mesoscopic thermodynamics Now we leave classical thermodynamics and turn our attention to mesoscopic levels. The state variable will be denoted by the symbol x, the state space by M (i.e. x ∈ M).
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The choice of x is not universal, it depends on the level as well as on the systems under investigation and our objectives. As an example, we can think of x as representing the one particle distribution function. Our goal now is to introduce into M a time evolution that will agree with the experimentally observed approach to the level of equilibrium thermodynamics. We shall proceed as follows: first (in this section), we equip M with a structure that naturally extends to mesoscopic levels the structure of equilibrium thermodynamics presented in Section 2.1, second (in Section 2.3), we introduce into M the time evolution that is compatible with the structure. The two dimensional space Mth is equipped with two distinct coordinates: E and N. In the mesoscopic state space, denoted by the symbol M, thisstructure takesthe form of the bundle structure M = (M, , Mth ) where : M → Mth is the bundle projection: E = E(x), N = N(x). Mth is the base space of the bundle. The fundamental thermodynamic relation (called now a mesoscopic fundamental thermodynamic relation) is a function H: M → R, H = H(x). To avoid a possible misunderstanding, we call H(x) an eta-function, not an entropy. H(x) becomes the entropy S appearing in equilibrium thermodynamics only if evaluated at the equilibrium state xth that is the state approached as the time t → ∞ (see Eqs. (1)-(6) and Section 2.3). To give an example of the eta-function, we take x to be one particle distribution function and H(x) the Boltzmann H-function. The eta-function is assumed to be sufficiently regular (so that we can differentiate it as many times as it is necessary) and convex (so that the Legendre transformations, see Eqs. (1)-(6) below, are well defined). Following the formulation of classical equilibrium thermodynamics where we passed from the two dimensional space Mth with coordinates (E, N) to the five dimensional ˆ th with coordinates (E, N, E*, N*, S), we now pass space M ˆ = T ∗ M × R. By T*M we denote the cotanfrom M to M ˆ are (x, x*, z), x ∈ M, gent bundle of M. The elements of M x∗ ∈ Tx∗ M, z ∈ R. By Tx∗ M we denote the cotangent space ˆ is equipped with a canonical contact over x. The space M structure, namely with the one-form ϑ = dz − x∗ dx. The mesoscopic Gibbs–Legendre manifold, denoted by the symbol M, is denned as the image of the mapping x → (x, Hx (x), ˆ th = T ∗ Mth × R. H(x)), where Hx : = ∂H/∂x. Note that M The passage M → Mth , i.e. the passage from mesoscopic to classical thermodynamics, is achieved by a Legendre transformation Lth ◦ L, where L and Lth are two Legendre transformations defined as follows. The Legendre transformation L is generated by the thermodynamic potential ∗
∗
∗
∗
Φ(x, E , N ) = −H(x) + E E(x) + N N(x).
(1)
We pass from Φ(x, E*, N*) to another potential that we denote S* (E*, N*) by solving first Φx = 0,
(2)
and then inserting the solution to Eq. (2), denoted xeq (E*, N*), into Φ, i.e.
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S ∗ (E∗ , N ∗ ) = Φ(xeq (E∗ , N ∗ ), E∗ ,N ∗ ).
(3)
The second Legendre transformation Lth is generated by the thermodynamic potential Φ∗ (E∗ , N ∗ , E, N) = −S ∗ (E∗ , N ∗ ) + EE∗ + NN∗ .
(4)
We pass from S* (E*, N*) to S (E, N) by solving first Φ∗E
= 0;
Φ∗N ∗ = 0,
(5)
and then inserting the solution, denoted (E*, N*)eq (E, N), into Φ* (E*, N*), i.e. S(E, N) = Φ∗ ((E∗ , N ∗ )eq (E, N), E, N).
(6)
2.3. Mesoscopic time evolution Contrary to the equilibrium thermodynamic state variable (E, N), the mesoscopic state variable x evolves in time. The structure that we have introduced into the mesoscopic state space in the previous section would be of no use if it would not be preserved in the time evolution. We thus require that the time evolution preserves the contact structure ϑ, takes place on the mesoscopic Gibbs–Legendre manifold M, and brings M into Mth (i.e. M → Mth as t → ∞). We thus require that the Legendre transformation Lth ◦ L (see Eqs. (1)-(6)) carrying M into Mth arises as a result of the mesoscopic time evolution. ˆ It is well known (see e.g. [5]) that the time evolution in M preserving the contact one-form ϑ = dz−x∗dx is generated by the vector field: dx = Ψx∗ dt dx∗ = −Ψx dt dz = −Ψ + x∗ , Ψx∗ , dt
(7)
where Ψ (x, x*) is the so called contact Hamiltonian. (Note that we assume that Ψ is independent of z). By Ψ x we denote, following the notation used throughout this paper, the derivative of Ψ with respect to x. Similarly, Ψx∗ is a derivative of Ψ with respect x*. By , we denote the pairing with the conjugate variables. The problem of determining the mesoscopic time evolution is now reduced to determining the contact Hamiltonian Ψ . Physical considerations (see [6,7]) lead to Ψ(x, x∗ ) = −(Ξ(x∗ ) − Ξ(Φx )) +
1 ∗ x , L(x)Φx . E∗
(8)
where Φ is the thermodynamic potential (1); , called a dissipation potential, is a real valued function satisfying the following four properties: (i) Ξ(0) = 0, (ii) reaches its minimum at 0, (iii) is convex in a neighborhood of 0, (iv) is degenerate in the sense that Ex , Ξx∗ = Nx , Ξx∗ = 0; L is a Poisson bivector expressing mathematically the kinematics of the mesoscopic state variable x. We say that L is a
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bivector if {A, B}:= Ax , L(x) Bx is a Poisson bracket (i.e. {A, B} = −{B, A} and {A, B} satisfies the Jacobi identity). A, B are sufficiently regular real-valued functions of x. We require moreover that L is degenerate in the sense that {A, H} = {A, N} = 0 for all sufficiently regular functions A. It is now easy to verify that the time evolution generated by the vector field (7) with the contact Hamiltonian (8) leaves the mesoscopic Gibbs–Legendre manifold M invariant and M → Mth as t → ∞. The time evolution Eqs. (7) and (8) evaluated on the mesoscopic Gibbs–Legendre manifold M takes the form dx = L(x)Ex + ΞHx . dt
(9)
We note that the contact Hamiltonian Ψ vanishes on M. This is indeed one of the general properties of contact Hamiltonians (see e.g. [5]): they vanish on Legendre submanifolds (i.e the submanifolds on which the contact one-form ϑ vanishes). As we see from the third equation in Eq. (7), the second term in the contact Hamiltonian, i.e. the term involving the Poisson bivector, does not contribute to rate of the eta-function. The first term on the right hand-side of (8), if inserted into x∗ , Ψx∗ , implies that dz/dt ≥ 0. We can regard the contact Hamiltonian (8) as a “rate” thermodynamic potential. If we compare (8) with the thermodynamic potential (1), we see that the eta-function H is replaced by (Ξ(x∗ ) − Ξ(Φx )) (a quantity closely related, as we have noted above, to the rate of the eta-function), the energy E by x*, L(x)x (a quantity closely related to the rate of energy), and E* by its inverse. In order to bring the previous considerations into the relation with the Onsager–Casimir symmetry, we have to introduce into the mesoscopic state space M an additional structure, namely the parity operator J: M → M satisfying J ◦ J = identity. For example, in the case x = one particle distribution function f(X, π), where X is the position coordinate and π the momentum, J f (X, π) = f(X, −π). The potentials and are required to be invariant with respect to J and L (Jx) = −L(x). The Eq. (9) linearized about an equilibrium state xeq (see Eq. (2) for its definition) becomes the linear time evolution equation investigated by Onsager and Casimir. The symmetric linear operator appearing in the Onsager–Casimir theory arises in the nonlinear time evolution Eq. (9) as a linear part of the gradient of the convex dissipation potential . The antisymmetric linear operator appearing the Onsager–Casimir theory arises in Eq. (9) as the linear part of the Poisson bivector L (x) (see more in [6]). 2.4. Externally driven and internally constrained systems There are physical systems that do not approach equilibrium states and thus the level of equilibrium thermodynamics is not admissible to them. Since the considerations that we have followed until this point were based on the existence of the approach to equilibrium, they cannot be directly applied to such systems. A small modification described be-
low will render them applicable however. The reason why some physical systems do not approach equilibrium states can be an applied external force (e.g. a horizontal layer of a fluid heated from below, i.e. the Rayleigh–Benard system, is a well studied example of driven systems) or internal constraints created during the approach to equilibrium (this is for example the situation arising during glass formation). To begin the adaptation, we replace the experimental observation of the approach to the equilibrium thermodynamic description, which served us as the basis in the previous sections, by a more general experimental observation of the approach of a level of description to a more macroscopic level of description. For example, the Rayleigh–Bénard problem is found to be well described on the level of hydrodynamics. This means that solutions to the governing equations describing the Rayleigh–Bénard problem on the levels that are more microscopic than the level of hydrodynamics (for example the level of kinetic theory) have to show approach to the hydrodynamic description. We now adapt Sections 2.2 and 2.3 to this change. Let M0 be the mesoscopic state space on which the behavior of the externally driven or internally constrained system under investigation is found to be well described (in the example of the Rayleigh–Bénard problem it is the state space of hydrodynamics). We now equip the state space M, that is moremicroscopic than M0 , with the bundle structure M = (M, , M0 ). If we compare this bundle with the bundle appearing in Section 2.2, we see that we have replaced Mth with the space M . By we denote the bundle projection 0 : M → M0 . The rest of the geometrical structure of M consists of (as in Sections 2.2 and 2.3) the eta-function H (x), the dissipation potential (x*), and the Poisson bivector L(x). All these three quantities are now different however from those used in the case when Mth serves as the base space. Only their general properties, with the exception of the degeneracy requirements on and L, remain unchanged. The degeneracy required in Section 2.3 guarantees that: (i) the energy is conserved, (ii) the eta-function does not decrease in time, and (iii) no time evolution takes place in Mth . Now, we require the degeneracy that guarantees (i), (ii), and (iii) but with M0 replacing Mth . The manifold constructed in the same way as Mth in Sections 2.2 and 2.3, but now with the bundle structure (M, , M0 ) replacing (M, , Mth ), will be denoted by the symbol M0 . The time evolution (7) (or Eq. (9)) describes the approach to the level of description that uses M0 as its state space (we recall that, for example, in the case of the Rayleigh–Bénard problem it is the state space of classical hydrodynamics). It carries M to M0 . Summing up, the time evolution in M is split into a “fast” time evolution generated by Eq. (7) and taking place on M (it carries M to M0 ) and the “slow” time evolution that follows it and that takes place on M0 . Neither the total time evolution in M nor the slow time evolution in M0 describe an approach to a reduced description and cannot be thus cast into the form of Eqs. (7) or (9). It is only the generator of the fast time evolution in M, describing the approach to the
M. Grmela / J. Non-Newtonian Fluid Mech. 120 (2004) 137–147
slow time evolution in M0 , that has always the form Eq. (7). The setting presented in Sections 2.2 and 2.3 can be seen now as a special case of the setting described in this section. What corresponds in Section 2.3 to the “slow” time evolution is no time evolution (recall that no time evolution takes place in Mth or Mth ). The only time evolutixon in Section 2.3 is the “fast” time evolution. Eqs. (1)-(6) in which Mth is replaced by M0 represent a passage from the eta-function H (x) to the entropy S (x0 ), x0 ∈ M0 . This is the way that thermodynamics (the fundamental thermodynamic relation) is introduced into the state space M0 on which the “slow” time evolution describing externally driven or internally constrained systems takes place. Some illustrations have been worked out in Refs. [7–9]. Finally, we recall that the decomposition of the time evolution into “slow” and “fast” has been the main reduction technique since Chapman and Enskog successfully reduced the Boltzmann kinetic equation to the hydrodynamic equations [10]. The essence of the analysis that follows Chapman and Enskog (see [11] for very clear presentation and new developments) is a search for invariant (or more frequently “approximately” invariant) manifolds Minv : M0 → Minv ⊂ M. The slow time evolution in M0 is then an appropriate projection of the time evolution on Minv . The main new feature of the approach described in this section is that we do not search for Minv but for the approximately invariant Gibbs–Legendre manifold Minv :M0 → Minv ⊂ M. By knowing Minv we know Minv but, in addition, we also know the fundamental thermodynamic relation on M0 implied by the fast time evolution. We shall illustrate this point in a simple example in which M0 ≡ Mth (i.e. the slow time evolution is no time evolution), elements of M are one particle distribution functions, and the time evolution in M is generated by the Boltzmann equation. In this case Minv is the manifold composed of Maxwellian distribution functions. The knowledge of Minv leaves us however still without the knowledge of the fundamental thermodynamic relation in Mth (in this case it is the fundamental thermodynamic relation of an ideal gas). It is Minv that captures both Minv and the fundamental thermodynamic relation in Mth implied by the Boltzmann H-function. 2.5. Historical remark The investigation of the Hamiltonian nature of nondissipative time evolution equations arising in mesoscopic physics has been started by Clebsch [12]. He asked the following question: The Euler hydrodynamic equation is a version of Newton’s law, so what is its Hamiltonian formulation? A modern mathematical analysis of this question has been provided by Arnold [13]. His analysis stimulated an interest in this subject among both mathematicians and physicists (for example, Dzyaloshinskii and Volovick [14] applied techniques developed in quantum mechanics, Morrison and Greene [15] focused on the governing equations arising in plasma physics). In the summer of 1983 Jerrold Marsden
141
organized a conference on the subject [16] attended by both mathematicians and physicists. The usefulness of Hamiltonian formulations in the study of dynamics and thermodynamics of complex fluids has been first noted in [17]. The progress in applications to complex fluids has been summarized in [18–23]. Dissipation has been incorporated into the Hamiltonian formulation in [14,24,25]. Geometrical interpretation of the mesoscopic dynamics has been developed in [6,26]. Eq. (9) has been gradually introduced in [17–25] as a common structure of well established mesoscopic dynamical theories. It has been called GENERIC in [19,20].
3. Mesoscopic state variables and their kinematics Let the problem put in front of us be the following. We have collected information, both experimental and theoretical, about a specific complex fluid (we can think of, say, suspension of solid particles in a fluid). We want now to organize the experience and the insight that we have acquired. We thus look for a mathematical model. Eqs. (7) or (9) can serve as a very useful pivotal point about which construction of the governing equations of the model can be organized. What we are looking for is a specific realization of Eqs. (7) or (9). We say that we have found a realization of Eqs. (7) or (9) if we have specified the state variables x, found their kinematics (expressed in the Poisson bracket {,}), specified the fibration (i.e. the functions E(x) and N(x)), specified the eta-function H(x), and the dissipation potential (x*). We shall now make a few comments about the realization of Eqs. (7) or (9). First, we note that instead of looking for one realization we can look for several realizations on different levels of description. This is in particular pertinent if the experimental observations at our disposal have been made on several levels (e.g. the classical rheological measurements and microscopic measurements as such made in slow-neutron-scattering experiments). In the case of suspensions, we can try to express our insight and experimental evidence on the level on which correlations in both velocities and positions are seen and also on the level on which only correlations in the position coordinate serves to characterize the microstructure. The “multilevel modelling” allows us to express our “multilevel insights” and “multilevel experimental evidence” that we often acquire. Moreover, we gain an additional understanding of the physics involved by investigating mathematically the relations among the models in the family. In the rest of this paper we shall comment only about the state variables x and about their kinematics. As we have already noted in Section 2.2, the choice of state variables depends on the specificity of the system under investigation and also on our objectives. Here we shall try to identify a rather large pool of candidates for the role of microstructural state variables.
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We begin with the microscopic state variables characterizing states of n structureless classical particles:
3.1. Liouville description
xparticles = (X1 , . . . , Xn , π1 , . . . , πn ),
Standard geometrical considerations lead us naturally to the Liouville reformulation of particle dynamics [28]. Let Mparticles denote the state space of classical mechanics (i.e. xparticles ∈ Mparticles ). We begin by constructing a bundle with Mparticles as its base space and fibers consisting of 6n-forms (i.e. volume elements). Sections of such a bundle, fn (xparticles )dxparticles , are the new state variables. We shall denote it by the symbol xLnp . The upper index stands for “Liouville description of n particles”. The kinematics of xLnp is induced, in a standard way (see [29]), from the kinematics (12) in the base space: {A, B}Lpn = dxparticles fn {Afn , Bf }particles , (14)
(10)
where i = 1, 2, . . . , n; Xi ∈ Q ⊂ R3 are the position coordinates, and πi ∈ Tx∗i ; i = 1, 2, . . . , n are the momenta. By T*Q we denote the cotangent bundle of Q; Tx∗i Q is the cotangent space over Xi . The 6n dimensional space whose elements are (X1 , . . . , Xn , π1 , . . . , πn ) will be denoted by the symbol M(particles) . We shall also consider particles with an internal structure. In particular, we shall consider fibers. Their states are characterized by: xfibers = (X1 , . . . , Xn , π1 , . . . , πn , p1 , . . . , pn , m1 , . . . , mn ),
(11)
where Xi and πi , i = 1, 2, . . . , n have the same meaning as in Eq. (10), p1 , . . . , pn are the unit vectors along the fibers, and m1 , . . . , mn are the angular momenta of the fibers in the coordinate system rotated with the fibers. The quantities p1 , . . . , pn and m1 , . . . , mn describe the internal structure of the fibers. Kinematics of xpartlcles is expressed in the canonical Poisson bracket {A, B}particles =
n
(AXiα Bπiα − BXiα Aπiα ),
(12) A large family of very useful mesoscopic state variables can be now constructed by applying various projections on xLnp or xLnf . Let gk = dxχk (x)fn (x); k = 1, 2, . . . , m, (16)
i=1
and of xfibers in the Poisson bracket [27] {A, B}fibers = {A, B}particles + +
n i=1
n
where A,B are sufficiently regular real-valued functions (functionals) of xLnp , Afn (xLnp ) := δA/δfn (xLnp ) is the Volterra functional derivative. We can interpret Eq. (14) as an average of Eq. (12) similarly as, say, energy ELpn can be interpreted as an average of the particle energy Lpn Lpn Eparticles = Efn : ELpn = dxparticles fn Efn . In exactly the same way we can introduce the field state variables xLnf := fn (xfibers )dxfibers and their kinematics {A, B}Lpf = dxfibers fn {Afn , Bfn }fibers . (15)
miα (Ami × Bmi )α
i=1
piα [(Ami × Bpi ) − (Bmi × Api )]α . (13)
A and B are sufficiently regular real valued functions of xparticles in Eq. (12) and of xfibers in (13). The state variables (10), (11) are not themselves the mesoscopic state variables. By passing to field descriptions, they can however lead us to a large family of state variables that are suitable for mesoscopic levels. This can be done by following three routes: Liouville route, Langevin route, and the route of continuum mechanics. The state variables and their kinematics that arise on the Liouville route are recalled in Section 3.1. The Langevin route consists of: (i) upgrading the status of the state variables (10),(11) (xparticles and xfibers become random variables), (ii) adding to the right hand-side of the time evolution equations a noise, and (iii) reformulating the dynamics in terms of probability distribution functions. We shall not explore this route in this paper. The route of continuum mechanics will be followed in Section 3.2. We shall see that this route offers a very interesting new class of mesoscopic state variables.
be an example of such a projection. By x in Eq. (16) we denote either xparticles or xfibers ; χ1 (x), . . . , χm (x) given functions of x. In order to obtain the Poisson bracket expressing kinematics of the mesoscopic state variables g1 , . . . , gm from the Poisson bracket (14) (or (15)), we proceed as follows: First, we restrict the functions A and B in Eq. (14) (or in (15)) to those that depend on fn only through their dependence on χ1 (x), . . . , χm (x). As a consequence of such a restriction, Afn (x) =
m
χj (x)Aχj .
(17)
j=1
Next, we insert Eq. (17) into (15) (or (16)) and obtain in this way a bracket involving the state variables g1 , . . . , gm . It will be a Poisson bracket if we succeed to express everything in the bracket in terms of g1 , . . . , gm . This is clearly not always possible. There are nevertheless many particular cases in which it is possible. In fact, these particular cases provide the most frequently used mesoscopic state variables. Among them we mention: (1) Reduced distribution functions (for example a one particle distribution function f1 (X, π) = dxparticles
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fn (xparticles )χ(X, π, xparticles ), where χ(X, π ,xparticles ) = (δ(X−X1 )δ(π−π1 ) . . . δ(X−Xn )δ(π−πn ))), de-notes the Dirac delta-function. (2) Classical hydrodynamic fields ρ(X) = dπf1 (X, π), u(X) = dππf1 (X, π), where ρ is the mass density, u is the momentum field, s is the entropy field, kB is theBoltzmann constant). (3) Conformation tensors describing states of macromolecules, particles in suspensions, interfaces in immiscible blends and other complex fluids (conformation tensors are moments of two or higher order distribution functions).
Eq. (9) since we are limiting ourselves in this section only to the nondissipative part of the time evolution) becomes
3.2. Continuum description
where
∂p = 0, ∂t ∂s = 0, ∂t ∂Xα = Eπα , ∂t (p−cont)
∂ταβ ∂πα =− ∂t ∂aβ (p−cont)
In this section we pass from Eq. (10) or (11) to fields by replacing the (discrete) particle label i = 1, 2, . . . n by an element of a continuum that we shall call a material space, Mmatertal . The particle coordinates thus become fields (i.e. functions of the elements of the material space). Below, we shall discuss three cases of the material space: (i) Mmaterial = R3 ; i.e. the classical continuum (Section 3.2.1), (ii) Mmaterial = R3 × R3 ; we call it a dumbbell-continuum (Section 3.2.2), and (iii) Mmaterial = R3 × S2 ; we call it a fiber-continuum (Section 3.2.3), S2 denotes the two-dimensional sphere of radius one. The first attempt to introduce generalized continua has been made by Cosserat brothers in [30]. We suggest therefore to call the continua discussed in Sections 3.2.2 and 3.2.3, Cosserat-like continua. Some of the new mesoscopic state variables that arise in this section can be transformed (see Section 3.2.1) into the state variables introduced in the context of the Liouville approach in Section 3.1. The novel Cosserat-like continuum theories are expected to be particularly useful in discussing extensions of classical elasticity to complex materials exhibiting elasto-plastic behavior. 3.2.1. Particle continuum The material space is chosen to be Mmaterial = R3 ; its elements are denoted by the symbol a, i.e. a ∈ R3 . The state variables that we shall use in this section are the particle coordinates X(a), the momenta π (a) and two scalar fields ρ (a), and s (a). The field ρ has the physical meaning of the mass density and s of the total entropy density: xp−cont = (X(a), π(a), ρ(a), s(a)).
(18)
The upper index “p-cont” stands for the particle continuum. The kinematics of Eq. (18) is expressed by the continuum version of Eq. (12): {A, B}p−cont = da(AXα (a) Bπα (a) − BXα (a) Aπα (a) ). (19) The time evolution Eq. (9) corresponding to the Poisson bracket (19) (without the last term on the right-hand side of
ταβ
,
= −E ∂Xα . ∂aβ
(20)
(21)
E (xp−cont ) denotes, as always, the energy. We have assumed that the energy E depends only on the gradient of X(a) and not on X(a) itself (i.e. EX␣ =−∂/∂a E∂X␣ /∂a ).  Eqs. (20) and (21) are the well known governing equations of the classical elasticity theory. We have shown that they arise in a straightforward manner from the Hamiltonian continuum kinematics. 3.3. Eshelbian reformulation We now interpret a ∈ Mmaterial as an initial condition for X. We regard thus the vector valued field X(a) as a one-to-one map a → X.
(22)
We shall use this map to project the time evolution (20), (21) of the state variables (18) to the material space Mmaterial . In order to do this, we note first that the momentum πi of particles (see (10)) is an element of the cotangent space attached to the configuration space whose coordinates are Xi . This remains true also in the continuum theory where the particle label i = 1,2, . . . , n is replaced by the vector a ∈ R3 . The time evolution (20), (21) is not therefore constructed solely on the material space Mmaterial . In order to make it as such, we have to transform the field (a) from the cotangent space of the position coordinate space to the cotangent space of the material space. Since it is the transformation (22) that relates the material and the coordinate space, the transformation (X(a), π(a), ρ(a), s(a)) → (X(a), P(a), ρ(a), s(a)), (23) where Pγ = −
∂Xα πα ∂aγ
(24)
achieves this objective. The state space whose elements are (X(a), P(a), ρ(a), s(a)) is called the Eshelby space. We shall denote it by the symbol M(Eshelby) .
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Our next task is to derive the time evolution in M(Eshelby) from the time evolution (20), (21). First we turn our attention to kinematics, i.e. to the Poisson bracket. By applying the transformation (23), (24) to Eq. (19) we obtain ∂Xα Eshelby {A, B} = da − (AXα BPγ ∂aγ ∂Pβ −BXα APγ ) − (APγ BPβ − BPγ APβ ) . ∂aγ (25) The time evolution Eq. (9) (with the second term on its right hand-side absent) corresponding to the Poisson bracket (25) becomes ∂ρ = 0, ∂t ∂s = 0, ∂t ∂Xα ∂Xα =− EPγ , ∂t ∂aγ ∂Xγ ∂ ∂Pα =− (E∂Xγ /∂a∈ ) + ∂t ∂aα ∂a∈ If we now use
Ecin =
da
∂Pα ∂P∈ − ∂a∈ ∂aα
EP∈ . (26)
E = Ecin + Epot =
da ecin +
(27)
∂Xγ ∂ − (E∂Xγ /∂aε ) ∂aα ∂aε ∂Xγ ∂e ∂ −eδαε + E∂Xγ /∂aε − , =− ∂aε ∂aα inhom ∂aα ∂Pα ∂Pε ∂ ∂ecin EPε = − − (ecin ) + , (28) ∂aε ∂aα ∂aα ∂aα inhom Eq. (26) takes the form ∂ρ =0 ∂t ∂s =0 ∂t ∂Xα 1 = πα ∂t ρ
Eshelby
−
∂(epot − ecin ) ∂aγ
,
(29)
∂Xε E∂Xε /∂aγ δaα
(30)
inhom
where Eshelby
ταγ
= (−epot + ecin )δαγ +
Can the mesoscopic state variables arising in continuum descriptions be related to the mesoscopic state variables that we have seen in Section 3.1 to arise in the Liouville description? Below, we shall show one example of such relation. We shall relate Eq. (18) to the classical hydrodynamic state variables (see the last paragraph of Section 3.1 for their relation to the particle distribution function). Following [34], the hydrodynamic fields are related to the state variables (18) by: ρ(x) = dr δ(r − X−1 (x))ρ(X(r)), s(x) =
and the identities
∂ταγ ∂Pα = ∂t ∂aγ
3.4. Eulerian reformulation
da epot ,
1 π γ πγ , 2ρ
is the Eshelby extra stress tensor [31]. A density field, for example e(a), depends on a through its dependence on the fields (X (a), P(a), ρ(a), s(a)) included in the set of state variables and also through an explicit dependence (due to the inhomogeneity). By (∂e/∂a)inhom we denote the change of e due to the inhomogeneity. The classical continuum time evolution (20), (21) has been projected onto the material space in [32]. The fact that the projected time evolution coincides with the Eshelbian time evolution developed by Eshelby [31] on the basis of other considerations (discussion of the time evolution of material discontinuities) has been realized by Maugin [33].
u(x) =
dr δ(r − X−1 (x))s(X(r)), dr δ(r − X−1 (x))π(X(r)),
(31)
where (ρ, s, u) are the classical hydrodynamic fields of the mass density, the total entropy, and the momentum respectively. The mapping (31) is a projection of the same type as the projection (16). We now apply it to the Poisson bracket (19). Straightforward calculations show that all terms in the projected bracket can be eventually expressed only with the hydrodynamic fields. We thus arrive in this way at a Poisson bracket expressing kinematics of the set of classical hydrodynamic fields. The Poisson bracket obtained in this way turns out to be exactly the same as the one that arise by applying the projection (16) (with the functions χk specified in the point (2) in text following Eq. (17)) on the Poisson bracket (14). The passage from X(a), π (a) to the Euler hydrodynamic fields and the bracket can also be seen as a symmetry reduction (the Marsden–Weinstein reduction [29]). The symmetry involved is the particle-relabelling symmetry of the Euler hydrodynamic fields. By applying the symmetry-reduction technique one obtains again the same Poisson bracket expressing kinematics of the Euler hydrodynamic fields. Summing up, we see that we can arrive at the classical hydrodynamic state variables and at their kinematics by two routes: (i) By following the Liouville description where
M. Grmela / J. Non-Newtonian Fluid Mech. 120 (2004) 137–147
we pass from the n-particle distribution function to the one particle distribution function and subsequently to its five moments defining the hydrodynamic fields in terms of the particle distribution functions, (ii) By following the route of continuum mechanics where we pass from the state variables (18) to the hydrodynamic fields by the transformation (31). There are, of course, other hydrodynamics-type state variables that can be extracted from the continuum state variables (18), e.g. it is the coordinate in the material space seen as a hydrodynamic field: a(x) = dr δ(r − X−1 (x))r (32) The extended hydrodynamics in which the fields (ρ(x), s(x), u(x), a(x)) serve as state variables has been introduced in [35]. Its formulation as a particular realization of (9) has been worked out in [36,37]. The extended hydrodynamics has been called Lin hydrodynamics in [36] since the nondissipative part of the equation governing the time evolution of the field a(x) is the Lin constraint introduced in the context of the variational formulations of hydrodynamics in [38], see also [39]. Gradients of the field a(x) have the physical interpretation of the conformation tensors arising in the context of the Liouville approach to state variables (see the point (3) in the last paragraph of Section 3.1). Finally, we note that the same type of considerations that led us in this section from Eq. (20) to the Euler hydrodynamics can also be used to reformulate the Eshelbian Eq. (29) into the Eulerian form. This reformulation and its usefulness in applications will be explored in a subsequent paper. 3.4.1. Dumbbell continuum We turn now our attention to complex materials. Instead of structureless particles composing the material (the case considered above), we consider now pairs of particles (dumbbells) as the principal constituents of the material. We shall choose thus the material space to be: Mmaterial = R3 × R3 . Its elements are denoted by the symbol (a1 , a2 ); i.e. (a1 , a2 ) ∈ R3 × R3 . The state variables that we shall use in this section are the particle coordinates (X1 (a1 , a2 ), X2 (a1 , a2 )), the momenta (π1 (a1 , a2 ), π2 (a1 , a2 )), and two scalar fields ρ (a1 , a2 ) and s(a1 , a2 ). The fields ρ and s have the same physical meaning as in Section 3.2.1. The set of state variables is thus x
d−cont
= (X1 (a1 , a2 ), X2 (a1 , a2 ), π1 (a1 , a2 ), π2 (a1 , a2 ), ρ(a1 , a2 ), s(a1 , a2 )).
(33)
The upper index “d − cont” stands for the dumbbell continuum. The kinematics of Eq. (33) is the continuous version of (12): d−cont
{A, B}
=
da1
da2
2
(AXiα Bπiα − BXiα Aπiα ).
i=1
(34)
145
The time evolution (9) corresponding to the Poisson bracket (34) (the last term on the right hand-side of Eq. (9) is omitted due to our limitation to the nondissipative time evolution) becomes ∂ρ = 0, ∂t ∂s = 0, ∂t ∂X1α = Eπ1α , ∂t ∂X2α = Eπ2α , ∂t (d−cont)
∂τ11αβ ∂π1α =− ∂t ∂a1β
(d−cont)
−
(d−cont)
∂τ21αβ ∂π2α =− ∂t ∂a1β
∂τ12αβ
,
∂a2β (d−cont)
−
∂τ22αβ
∂a2β
,
(35)
where (d−cont)
= −E∂X1α /∂a1β ;
τ12αβ
(d−cont)
= −E∂X2α /∂a1β ;
τ22αβ
τ11αβ τ21αβ
(d−cont)
= −E∂X1α /∂a2β ,
(d−cont)
= −E∂X2α /∂a2β , (36)
E(xd −cont ) denotes the energy. We have assumed that the energy E depends only on the gradients of X1 (a1 , a2 ) and X2 (a1 , a2 ) and not on X1 and X2 themselves. The Eshelbian and Eulerian reformulations of the dumbbell-continuum dynamics (35) will be presented in a separate publication. 3.4.2. Fiber continuum The particles composing the material are now seen as fibers. The material space is thus chosen to be Mmaterial = R3 × S 2 , its elements are denoted by the symbol (a, b), a ∈ R3 , b ∈ S2 . By S2 we denote the two dimensional sphere of radius one. The variables describing states of the fiber continuum are xf −cont = (X(a, b), p(a, b), π(a, b), m(a, b), ρ(a, b), s(a, b)).
(37)
The upper index “f − cont” stands for the fiber continuum. The kinematics of Eq. (37) is expressed in the continuous version of the Poisson bracket (13): f −cont p−cont {A, B} = {A, B} + da dbmα (Am × Bm )α + da dbpα [(Am × Bp ) − Bm × Ap )]α (38)
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The time evolution Eq. (9) corresponding to the Poisson bracket (38) (the last term on the right hand-side of Eq. (9) is omitted due to our limitation to the nondissipative time evolution) becomes ∂ρ = 0, ∂t ∂s = 0, ∂t ∂Xα = Eπα , ∂t (fa−cont)
∂ταβ ∂πα =− ∂t ∂aβ
(i) Search for approximately invariant mesoscopic Gibbs–Legendre manifolds (rather than for just approximately invariant mesoscopic manifolds) in the analysis of reductions to more macroscopic levels of description (see Section 2.4). The extra information involved in the Gibbs–Legendre manifolds has the physical meaning of the mesoscopic fundamental thermodynamic relation on mesoscopic levels. This is the way to introduce clearly thermodynamics of driven systems. (ii) Novel continuum-like theories suitable for complex materials exhibiting elasto-plastic behavior (see Section 3.2).
(fb−cont)
−
∂ταβ
,
∂bβ
Acknowledgements
∂pα = −(p × Em )α , ∂t
This work has been partially supported by the Natural Sciences and Engineering Research Council of Canada. (a)
(b)
∂τγk ∂τγk ∂mα = −(m × Em )α − εαβγ pβ − εαβγ pβ , ∂t ∂ak ∂bk
References (39)
where (fa−cont)
ταβ
(a)
(fb−cont)
= −E∂Xα /∂aβ ;
ταβ = −E∂pα /∂aβ ;
ταβ (b)
= −E∂Xα /∂bβ ,
ταβ = −E∂pα /∂bβ .
(40)
ε is the alternating tensor ((m × Em )α = εαβγ mβ Emγ ), and E(xf −cont ) denotes the energy. We have assumed that the energy E depends only on the gradient of X(a, b) and p(a, b) and not on X and p themselves. The Eshelbian and Eulerian reformulations of the fiber-continuum dynamics (39) will be presented in a separate publication.
4. Concluding remarks By following the Gibbs geometrical viewpoint of thermodynamics to mesoscopic levels of description, we have arrived at a general framework for the mesoscopic time evolution that is compatible with thermodynamics. If the mesoscopic state space is equipped with the structure of thermodynamics then there is essentially only one time evolution that fits into it. The framework identified with the help of geometry appears to be the same as the one that has been distilled previously as a common structure of well-established mesoscopic dynamical theories (as for example the Boltzmann kinetic equation and the classical hydrodynamic equations). Methods of geometry led us also (Section 3) to a large class of well known and new mesoscopic state variables and to their kinematics (expressed in Poisson brackets). Among the new avenues that open with geometry and provide an interesting new perspective we mention two:
[1] L. Onsager, Reciprocal relations in irreversible processes, Phys. Rev. 37 (1931) 405–426; 38 (1931) 2265–2279 [2] H.B.G. Casimir, On Onsager’s principle of microscopic reversibility, Rev. Mod. Phys. 17 (1945) 343–350. [3] J.W. Gibbs, Collected Works, Longmans, Green and Comp., New York, 1984. [4] R. Hermann, Geometry, Physics and Systems, Marcel Dekker, New York, 1984. [5] V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1989. [6] M. Grmela, Reciprocity relations in thermodynamics, Phys. A 309 (2002) 304–328. [7] M. Grmela, Reductions in dynamical theories, J. Chem. Phys. 85 (1986) 5689–5697. [8] M. Grmela, Thermodynamics of driven systems, Phys. Rev. 48 (1993) 919–930. [9] M. Grmela, A. Cournoyer, On thermodynamics of glasses, Open Syst. Inf. Dyn. 5 (1988) 67–88. [10] S. Chapman, T.S. Cowling, Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1958. [11] A.N. Gorban, I.V. Karlin, Thermodynamic parametrization, Phys. A 190 (1992) 393–404. [12] A. Clebsch, J. Reine Angew. Math. (Crelle) 56 (1895) 1. [13] V.I. Arnold, Sur la gomtrie dinrentielle des groupes de Lie de dimension infini et ses applications l’hydrodynamique des fluides parfaits, Ann. Ins. Fourier 16 (1966) 319–361. [14] I.E. Dzyaloshinski, G.E. Volovick, Poisson bracket in condensed matter physics, Ann. Phys. (NY) 125 (1980) 67–97. [15] P.J. Morrison, J.M. Greene, Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics, Phys. Rev. Lett. 45 (1980) 790–794. [16] Proceedings of the Workshop on Hamiltonian systems, Contemporary Mathematics, vol. 28, 1984. [17] M. Grmela, Hamiltonian dynamics of incompressible elastic fluids, Phys. Lett. A 130 (1988) 81–86. [18] A.N. Beris, B.J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford University Press, Oxford, 1994. [19] M. Grmela, H.C. Oettinger, Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Phys. Rev. E 56 (1997) 6620–6632.
M. Grmela / J. Non-Newtonian Fluid Mech. 120 (2004) 137–147 [20] H.C. Oettinger, M. Grmela, Dynamics and thermodynamics of complex fluids. Part II. Illustrations of a general formalism, Phys. Rev. E 56 (1997) 6632–6655. [21] R.J.J. Jongschaap, H.C. Oettinger, Proceedings of the Second International Workshop on Nonequilibrium Thermodynamics and Complex Fluids, J. Non-Newtonian Fluid Mech. 96, 2001. [22] B.J. Edwards, A.N. Beris, Proceedings on the Third International Workshop on Nonequilibrium Thermodynamics and Complex Fluids, J. Non-Newtonian Fluid Mech. (2004) (in press). [23] M. Grmela, Complex fluids subjected to external influences, J. NonNewtonian Fluid. Mech. 96 (2001) 221–254. [24] M. Grmela, Particle and bracket formulations of kinetic equations, Contemp. Math. 28 (1984) 125. [25] A.N. Kaufman, Dissipative Hamiltonian systems, Phys. Lett. A 100 (1984) 419–422. [26] M. Grmela, in: J. Harnad (Eds.), Proceedings of the Workshop on Hamiltonian Systems Transfprmation Groups and Spectral Transform Methods, J.E. Marsden Publishers, CRM Universite de Montreal, 1990. [27] E.C.G. Sudarshan, N. Mukunda, Classical Mechanics-Modern Perspectives, Wiley, New York, 1974. [28] J. Liouville, J. Math. Pur. et Appl. 3 (1838) 342. [29] J.E. Marsden, Lectures on Mechanics, Cambridge University Press, Cambridge, 1992.
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[30] E. Cosserat, F. Cosserat, Théorie des corps deformables, Hermann, Paris, 1909. [31] J.D. Eshelby, Force on an elastic singularity, Phil. Trans. Soc. Lond. A 244 (1951) 87–112. [32] J.C. Simo, J.E. Marsden, P.S. Kirshnaprasad, Hamiltonian structure of nonlinear elasticity: The material and and convective representations of solids, rods and plates, Arch. Rat. Mech. Anal. 104 (1988) 125– 183. [33] G.A. Maugin, Material forces: concepts and applications, Appl. Mech. Rev. 48 (1995) 213–245. [34] H.D.I. Abarbanel, R. Brown, Y.M. Yang, Hamiltonian formulation of inviscid flows with free boundaries, Phys. Fluid 31 (1988) 2802– 2809. [35] H. Temmen, H. Pleiner, M. Liu, H.R. Brand, Phys. Rev. Lett. 84 (2000) 3228. [36] M. Grmela, Lagrange hydrodynamics as extended Euler hydrodynamics: Hamiltonian and GENERIC structures, Phys. Lett. A 296 (2002) 97–104. [37] M. Grmela, A framework for elasto-plastic hydrodynamics, Phys. Lett. A 312 (2003) 136–146. [38] C.C. Lin, Hydrodynamics of helium II, Proc. Int. Sch. Phys. New York Academic Press XXI (1963) 93–146. [39] R. Salmon, Hamiltonian fluid mechanics, Ann. Rev. Fluid Mech. 20 (1988) 225–256.
Geometry of mesoscopic dynamics and thermodynamics Miroslav Grmela∗ Ecole Polytechnique de Montreal, C.P. 6079 suc. Centre-Ville, Montreal, Que. H3C3A7, Canada Received 16 September 2003; received in revised form 19 November 2003; accepted 19 November 2003 This article is part of a Special Volume containing papers from the 3rd International Workshop on Nonequilibrium Thermodynamics and Complex Fluids
Abstract Contact geometry provides a natural setting for classical thermo-dynamics. In this paper we use it to derive the structure of mesoscopic dynamics (GENERIC) expressing its compatibility with thermodynamics. In the second part we derive kinematics (Poisson brackets) of a large family of mesoscopic state variables. © 2004 Elsevier B.V. All rights reserved. Keywords: Thermodynamics; Dynamics; Complex fluids; Contact geometry
1. Introduction Geometry has proven to be very useful in microscopic physics and relativity theory. We may recall for example the role that the geometrical formulation of classical mechanics played in the development of its extensions to quantum mechanics and to the gauge field formulations. In this paper we turn our attention to the role that geometry plays (or could play) in mesoscopic dynamics and thermodynamics. Before embarking on this subject we shall clarify the concept of a mesoscopic level of description. We begin by listing some examples of well established levels: quantum chromodynamics, quantum mechanics, classical mechanics, classical hydrodynamics, classical thermodynamics. We note that all these levels have arisen as a consequence of an experience collected by observing (with a certain class of observational tools specific to each level) a class of systems (again specific to each level) and an attempt to organize and explain results of the observations. Consequently, the levels are self-contained and independent. Neither of them needs the others to be clearly formulated and used. For example, in order to apply hydrodynamics we do not need to involve classical or quantum mechanics. On the other hand, we expect that the levels are also related. To establish the relation, we begin by comparing the quantities used to describe states (termed state variables). We say that a level (Level 1) is more microscopic than another ∗ Corresponding author. Tel.: +1-514-340-4711x4627; fax: +1-514-340-4159. E-mail address: [email protected] (M. Grmela).
0377-0257/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2003.11.009
level (Level 2) if the state variables used on Level 1 contain more details than those used on Level 2. In other words, the state variables used on Level 2 can be obtained as certain projections of the state variables used on Level 1. We expect that Level 2 can be derived (reduced) from Level 1. The reduction process can be intuitively seen as follows. First, we solve the time evolution equations on both Level 1 and Level 2 for all initial conditions and for a family of the parameters used to express the individual nature of the system under consideration on Level 1 and Level 2 respectively. As a result, we obtain on Level 1 as well as on Level 2 a set of trajectories. We shall term them a phase portrait on Level 1 and a phase portrait on Level 2. An identification of the phase portrait on Level 2 as a pattern inside the phase portrait on Level 1 constitutes a reduction of Level 1 to Level 2. Let us consider three levels: Level 1, Level 2, and Level 3. Let Level 2 be more macroscopic than Level 1 and Level 3 more macroscopic than both Level 1 and Level 2. Let, moreover, the behavior of the systems under consideration be found to be well described on all three levels. This then means, in particular, that the phase portrait on Level 3 has to be recognizable in both phase portraits obtained on Level 1 and Level 2. Consequently, solutions of the governing equations on Level 1 and Level 2 have to share some common features. This fact should be then a consequence of a common structure of the governing equations on Level 1 and Level 2. The above argument thus indicates that the governing equations on different levels share a common structure as a consequence of the existence (experimentally established) of relations among them. Our first illustration of the role of geometry in mesoscopic physics is an identification of the
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common structure in the case when Level 3 is the level of classical thermodynamics.
2. Mesoscopic dynamics The systems that are externally unforced and free from internal constraints (see Section 2.4) are seen to reach states, called thermodynamic equilibrium states, at which their behavior is found to be well described by equilibrium thermodynamics. This universal experimental observation is made on all mesoscopic levels that are more microscopic than the level of classical thermodynamics. Consequently, solutions to the governing equations on mesoscopic levels have a common property, which then means that the governing equations have to share a common structure. Identification of such structure has always been the main goal of non-equilibrium thermodynamics. One element of the structure is known to be the Onsager–Casimir symmetry of the time evolution equations on mesoscopic levels of description [1,2]. In the setting for the reduction process sketched in the introduction, Onsager derived this symmetry for Level 1 being the level of classical mechanics, Level 2 being the mesoscopic level, and Level 3 being the level of classical thermodynamics. The symmetry of the time evolution equations on Level 2 appears as a result of the time reversibility on Level 1 and the assumption that the time evolution of fluctuations about thermodynamic equilibrium states, in a small neighborhood of the equilibrium states, follows the mesoscopic time evolution. Below, we shall present a structure that extends the Onsager–Casimir symmetry. First, we shall introduce it by using geometrical considerations. Later, we shall also recall briefly other arguments leading to it. 2.1. Classical thermodynamics Since it is classical thermodynamics that we want to recover on mesoscopic levels involving time evolution, we begin with the geometry of classical thermodynamics. Its importance has already been recognized by Gibbs [3]. The titles of the first two papers (published in 1873) in which Gibbs completed the formulation of classical thermodynamics are: “Graphical methods in the thermodynamics of fluids” and “A method of geometrical representation of the thermodynamic properties of substances by means of surfaces”. We shall now present Gibbs’ geometrical formulation of thermodynamics. Let E and N denote the energy and the number of moles per unit volume. We choose them to be the universal state variables of classical thermodynamics of one component systems. According to Gibbs, thermodynamics is introduced as a function S = S(E, N), called a fundamental thermodynamic relation. By the symbol S we denote the entropy per unit volume. The fundamental thermodynamic relation can also be seen (again following Gibbs) as a two dimensional surface imbedded in a three dimensional space with coor-
dinates (E, N, S), i.e. as the image of the mapping (E, N) H → (E, N, S(E, N)). This two dimensional surface will be termed a Gibbs manifold. The Gibbs manifold does not however represent a complete formulation of thermodynamics. What is missing is the statement that the equilibrium states are those that maximize the entropy subjected to constraints expressing the conditions in which the systems under consideration are held. In other words, the fundamental role that the Legendre transformations play in thermodynamics, a very important part of the formulation of thermodynamics, remains still outside of the geometrical viewpoint. Hermann [4] was the first to realize that the importance of Legendre transformations can be incorporated into the geometrical formulation by making the one-jet extension of the Gibbs manifold, i.e., by introducing a manifold, called the Gibbs–Legendre manifold, Mth as the image of the mapping (E, N) → (E, N, SE (E, N), SN (E, N), S(E, N)), where SE (E, N): = ∂S/∂E evaluated at (E, N), similarly SN (E,N): = ∂S/∂N evaluated at (E, N). The Gibbs–Legendre manifold is again a two dimensional manifold but now imbedded in a five dimensional space with coordinates (E, N, E*, N*, S). The Legendre transformation are brought into the geometrical formulation by providing the space with coordinates (E, N, E*, N*, S) with the contact structure, i.e. with the one-form dS – E*dE – N*dN. The Gibbs–Legendre manifold is then distinguished by the property that it is a manifold on which the contact one-form vanishes. Manifolds having such property are called in geometry Legendre submanifolds. The Legendre transformations are the transformation leaving the contact structure (i.e. the contact one-form) invariant. With the contact structure, the formulation of thermodynamics is starting to look like the modern formulations of theories in microscopic physics. The first question in such formulations is always: what are the transformations of fundamental importance? In the context of thermodynamics, we answer that they are the Legendre transformations. The next question is then: what is the space in which the transformations act naturally? We answer that it is a space equipped with the contact structure since the contact structure remains invariant under the Legendre transformations. A specific physical system is represented in thermodynamics as a specific Legendre submanifold. At this point we introduce an additional notation. The space whose elements are (E, N) will be denoted by the symbol Mth (i.e. (E, N) ∈ Mth ), the space whose elements are ˆ th (i.e. (E, N, E*, N*, S) (E, N, E*, N*, S) by the symbol M ˆ ∈ Mth ), the one-form dS – E*dE – N*dN is denoted ϑth , and the Gibbs–Legendre manifolds by the symbol Mth . The subscript “th” stands for “thermodynamic”. 2.2. Mesoscopic thermodynamics Now we leave classical thermodynamics and turn our attention to mesoscopic levels. The state variable will be denoted by the symbol x, the state space by M (i.e. x ∈ M).
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The choice of x is not universal, it depends on the level as well as on the systems under investigation and our objectives. As an example, we can think of x as representing the one particle distribution function. Our goal now is to introduce into M a time evolution that will agree with the experimentally observed approach to the level of equilibrium thermodynamics. We shall proceed as follows: first (in this section), we equip M with a structure that naturally extends to mesoscopic levels the structure of equilibrium thermodynamics presented in Section 2.1, second (in Section 2.3), we introduce into M the time evolution that is compatible with the structure. The two dimensional space Mth is equipped with two distinct coordinates: E and N. In the mesoscopic state space, denoted by the symbol M, thisstructure takesthe form of the bundle structure M = (M, , Mth ) where : M → Mth is the bundle projection: E = E(x), N = N(x). Mth is the base space of the bundle. The fundamental thermodynamic relation (called now a mesoscopic fundamental thermodynamic relation) is a function H: M → R, H = H(x). To avoid a possible misunderstanding, we call H(x) an eta-function, not an entropy. H(x) becomes the entropy S appearing in equilibrium thermodynamics only if evaluated at the equilibrium state xth that is the state approached as the time t → ∞ (see Eqs. (1)-(6) and Section 2.3). To give an example of the eta-function, we take x to be one particle distribution function and H(x) the Boltzmann H-function. The eta-function is assumed to be sufficiently regular (so that we can differentiate it as many times as it is necessary) and convex (so that the Legendre transformations, see Eqs. (1)-(6) below, are well defined). Following the formulation of classical equilibrium thermodynamics where we passed from the two dimensional space Mth with coordinates (E, N) to the five dimensional ˆ th with coordinates (E, N, E*, N*, S), we now pass space M ˆ = T ∗ M × R. By T*M we denote the cotanfrom M to M ˆ are (x, x*, z), x ∈ M, gent bundle of M. The elements of M x∗ ∈ Tx∗ M, z ∈ R. By Tx∗ M we denote the cotangent space ˆ is equipped with a canonical contact over x. The space M structure, namely with the one-form ϑ = dz − x∗ dx. The mesoscopic Gibbs–Legendre manifold, denoted by the symbol M, is denned as the image of the mapping x → (x, Hx (x), ˆ th = T ∗ Mth × R. H(x)), where Hx : = ∂H/∂x. Note that M The passage M → Mth , i.e. the passage from mesoscopic to classical thermodynamics, is achieved by a Legendre transformation Lth ◦ L, where L and Lth are two Legendre transformations defined as follows. The Legendre transformation L is generated by the thermodynamic potential ∗
∗
∗
∗
Φ(x, E , N ) = −H(x) + E E(x) + N N(x).
(1)
We pass from Φ(x, E*, N*) to another potential that we denote S* (E*, N*) by solving first Φx = 0,
(2)
and then inserting the solution to Eq. (2), denoted xeq (E*, N*), into Φ, i.e.
139
S ∗ (E∗ , N ∗ ) = Φ(xeq (E∗ , N ∗ ), E∗ ,N ∗ ).
(3)
The second Legendre transformation Lth is generated by the thermodynamic potential Φ∗ (E∗ , N ∗ , E, N) = −S ∗ (E∗ , N ∗ ) + EE∗ + NN∗ .
(4)
We pass from S* (E*, N*) to S (E, N) by solving first Φ∗E
= 0;
Φ∗N ∗ = 0,
(5)
and then inserting the solution, denoted (E*, N*)eq (E, N), into Φ* (E*, N*), i.e. S(E, N) = Φ∗ ((E∗ , N ∗ )eq (E, N), E, N).
(6)
2.3. Mesoscopic time evolution Contrary to the equilibrium thermodynamic state variable (E, N), the mesoscopic state variable x evolves in time. The structure that we have introduced into the mesoscopic state space in the previous section would be of no use if it would not be preserved in the time evolution. We thus require that the time evolution preserves the contact structure ϑ, takes place on the mesoscopic Gibbs–Legendre manifold M, and brings M into Mth (i.e. M → Mth as t → ∞). We thus require that the Legendre transformation Lth ◦ L (see Eqs. (1)-(6)) carrying M into Mth arises as a result of the mesoscopic time evolution. ˆ It is well known (see e.g. [5]) that the time evolution in M preserving the contact one-form ϑ = dz−x∗dx is generated by the vector field: dx = Ψx∗ dt dx∗ = −Ψx dt dz = −Ψ + x∗ , Ψx∗ , dt
(7)
where Ψ (x, x*) is the so called contact Hamiltonian. (Note that we assume that Ψ is independent of z). By Ψ x we denote, following the notation used throughout this paper, the derivative of Ψ with respect to x. Similarly, Ψx∗ is a derivative of Ψ with respect x*. By , we denote the pairing with the conjugate variables. The problem of determining the mesoscopic time evolution is now reduced to determining the contact Hamiltonian Ψ . Physical considerations (see [6,7]) lead to Ψ(x, x∗ ) = −(Ξ(x∗ ) − Ξ(Φx )) +
1 ∗ x , L(x)Φx . E∗
(8)
where Φ is the thermodynamic potential (1); , called a dissipation potential, is a real valued function satisfying the following four properties: (i) Ξ(0) = 0, (ii) reaches its minimum at 0, (iii) is convex in a neighborhood of 0, (iv) is degenerate in the sense that Ex , Ξx∗ = Nx , Ξx∗ = 0; L is a Poisson bivector expressing mathematically the kinematics of the mesoscopic state variable x. We say that L is a
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bivector if {A, B}:= Ax , L(x) Bx is a Poisson bracket (i.e. {A, B} = −{B, A} and {A, B} satisfies the Jacobi identity). A, B are sufficiently regular real-valued functions of x. We require moreover that L is degenerate in the sense that {A, H} = {A, N} = 0 for all sufficiently regular functions A. It is now easy to verify that the time evolution generated by the vector field (7) with the contact Hamiltonian (8) leaves the mesoscopic Gibbs–Legendre manifold M invariant and M → Mth as t → ∞. The time evolution Eqs. (7) and (8) evaluated on the mesoscopic Gibbs–Legendre manifold M takes the form dx = L(x)Ex + ΞHx . dt
(9)
We note that the contact Hamiltonian Ψ vanishes on M. This is indeed one of the general properties of contact Hamiltonians (see e.g. [5]): they vanish on Legendre submanifolds (i.e the submanifolds on which the contact one-form ϑ vanishes). As we see from the third equation in Eq. (7), the second term in the contact Hamiltonian, i.e. the term involving the Poisson bivector, does not contribute to rate of the eta-function. The first term on the right hand-side of (8), if inserted into x∗ , Ψx∗ , implies that dz/dt ≥ 0. We can regard the contact Hamiltonian (8) as a “rate” thermodynamic potential. If we compare (8) with the thermodynamic potential (1), we see that the eta-function H is replaced by (Ξ(x∗ ) − Ξ(Φx )) (a quantity closely related, as we have noted above, to the rate of the eta-function), the energy E by x*, L(x)x (a quantity closely related to the rate of energy), and E* by its inverse. In order to bring the previous considerations into the relation with the Onsager–Casimir symmetry, we have to introduce into the mesoscopic state space M an additional structure, namely the parity operator J: M → M satisfying J ◦ J = identity. For example, in the case x = one particle distribution function f(X, π), where X is the position coordinate and π the momentum, J f (X, π) = f(X, −π). The potentials and are required to be invariant with respect to J and L (Jx) = −L(x). The Eq. (9) linearized about an equilibrium state xeq (see Eq. (2) for its definition) becomes the linear time evolution equation investigated by Onsager and Casimir. The symmetric linear operator appearing in the Onsager–Casimir theory arises in the nonlinear time evolution Eq. (9) as a linear part of the gradient of the convex dissipation potential . The antisymmetric linear operator appearing the Onsager–Casimir theory arises in Eq. (9) as the linear part of the Poisson bivector L (x) (see more in [6]). 2.4. Externally driven and internally constrained systems There are physical systems that do not approach equilibrium states and thus the level of equilibrium thermodynamics is not admissible to them. Since the considerations that we have followed until this point were based on the existence of the approach to equilibrium, they cannot be directly applied to such systems. A small modification described be-
low will render them applicable however. The reason why some physical systems do not approach equilibrium states can be an applied external force (e.g. a horizontal layer of a fluid heated from below, i.e. the Rayleigh–Benard system, is a well studied example of driven systems) or internal constraints created during the approach to equilibrium (this is for example the situation arising during glass formation). To begin the adaptation, we replace the experimental observation of the approach to the equilibrium thermodynamic description, which served us as the basis in the previous sections, by a more general experimental observation of the approach of a level of description to a more macroscopic level of description. For example, the Rayleigh–Bénard problem is found to be well described on the level of hydrodynamics. This means that solutions to the governing equations describing the Rayleigh–Bénard problem on the levels that are more microscopic than the level of hydrodynamics (for example the level of kinetic theory) have to show approach to the hydrodynamic description. We now adapt Sections 2.2 and 2.3 to this change. Let M0 be the mesoscopic state space on which the behavior of the externally driven or internally constrained system under investigation is found to be well described (in the example of the Rayleigh–Bénard problem it is the state space of hydrodynamics). We now equip the state space M, that is moremicroscopic than M0 , with the bundle structure M = (M, , M0 ). If we compare this bundle with the bundle appearing in Section 2.2, we see that we have replaced Mth with the space M . By we denote the bundle projection 0 : M → M0 . The rest of the geometrical structure of M consists of (as in Sections 2.2 and 2.3) the eta-function H (x), the dissipation potential (x*), and the Poisson bivector L(x). All these three quantities are now different however from those used in the case when Mth serves as the base space. Only their general properties, with the exception of the degeneracy requirements on and L, remain unchanged. The degeneracy required in Section 2.3 guarantees that: (i) the energy is conserved, (ii) the eta-function does not decrease in time, and (iii) no time evolution takes place in Mth . Now, we require the degeneracy that guarantees (i), (ii), and (iii) but with M0 replacing Mth . The manifold constructed in the same way as Mth in Sections 2.2 and 2.3, but now with the bundle structure (M, , M0 ) replacing (M, , Mth ), will be denoted by the symbol M0 . The time evolution (7) (or Eq. (9)) describes the approach to the level of description that uses M0 as its state space (we recall that, for example, in the case of the Rayleigh–Bénard problem it is the state space of classical hydrodynamics). It carries M to M0 . Summing up, the time evolution in M is split into a “fast” time evolution generated by Eq. (7) and taking place on M (it carries M to M0 ) and the “slow” time evolution that follows it and that takes place on M0 . Neither the total time evolution in M nor the slow time evolution in M0 describe an approach to a reduced description and cannot be thus cast into the form of Eqs. (7) or (9). It is only the generator of the fast time evolution in M, describing the approach to the
M. Grmela / J. Non-Newtonian Fluid Mech. 120 (2004) 137–147
slow time evolution in M0 , that has always the form Eq. (7). The setting presented in Sections 2.2 and 2.3 can be seen now as a special case of the setting described in this section. What corresponds in Section 2.3 to the “slow” time evolution is no time evolution (recall that no time evolution takes place in Mth or Mth ). The only time evolutixon in Section 2.3 is the “fast” time evolution. Eqs. (1)-(6) in which Mth is replaced by M0 represent a passage from the eta-function H (x) to the entropy S (x0 ), x0 ∈ M0 . This is the way that thermodynamics (the fundamental thermodynamic relation) is introduced into the state space M0 on which the “slow” time evolution describing externally driven or internally constrained systems takes place. Some illustrations have been worked out in Refs. [7–9]. Finally, we recall that the decomposition of the time evolution into “slow” and “fast” has been the main reduction technique since Chapman and Enskog successfully reduced the Boltzmann kinetic equation to the hydrodynamic equations [10]. The essence of the analysis that follows Chapman and Enskog (see [11] for very clear presentation and new developments) is a search for invariant (or more frequently “approximately” invariant) manifolds Minv : M0 → Minv ⊂ M. The slow time evolution in M0 is then an appropriate projection of the time evolution on Minv . The main new feature of the approach described in this section is that we do not search for Minv but for the approximately invariant Gibbs–Legendre manifold Minv :M0 → Minv ⊂ M. By knowing Minv we know Minv but, in addition, we also know the fundamental thermodynamic relation on M0 implied by the fast time evolution. We shall illustrate this point in a simple example in which M0 ≡ Mth (i.e. the slow time evolution is no time evolution), elements of M are one particle distribution functions, and the time evolution in M is generated by the Boltzmann equation. In this case Minv is the manifold composed of Maxwellian distribution functions. The knowledge of Minv leaves us however still without the knowledge of the fundamental thermodynamic relation in Mth (in this case it is the fundamental thermodynamic relation of an ideal gas). It is Minv that captures both Minv and the fundamental thermodynamic relation in Mth implied by the Boltzmann H-function. 2.5. Historical remark The investigation of the Hamiltonian nature of nondissipative time evolution equations arising in mesoscopic physics has been started by Clebsch [12]. He asked the following question: The Euler hydrodynamic equation is a version of Newton’s law, so what is its Hamiltonian formulation? A modern mathematical analysis of this question has been provided by Arnold [13]. His analysis stimulated an interest in this subject among both mathematicians and physicists (for example, Dzyaloshinskii and Volovick [14] applied techniques developed in quantum mechanics, Morrison and Greene [15] focused on the governing equations arising in plasma physics). In the summer of 1983 Jerrold Marsden
141
organized a conference on the subject [16] attended by both mathematicians and physicists. The usefulness of Hamiltonian formulations in the study of dynamics and thermodynamics of complex fluids has been first noted in [17]. The progress in applications to complex fluids has been summarized in [18–23]. Dissipation has been incorporated into the Hamiltonian formulation in [14,24,25]. Geometrical interpretation of the mesoscopic dynamics has been developed in [6,26]. Eq. (9) has been gradually introduced in [17–25] as a common structure of well established mesoscopic dynamical theories. It has been called GENERIC in [19,20].
3. Mesoscopic state variables and their kinematics Let the problem put in front of us be the following. We have collected information, both experimental and theoretical, about a specific complex fluid (we can think of, say, suspension of solid particles in a fluid). We want now to organize the experience and the insight that we have acquired. We thus look for a mathematical model. Eqs. (7) or (9) can serve as a very useful pivotal point about which construction of the governing equations of the model can be organized. What we are looking for is a specific realization of Eqs. (7) or (9). We say that we have found a realization of Eqs. (7) or (9) if we have specified the state variables x, found their kinematics (expressed in the Poisson bracket {,}), specified the fibration (i.e. the functions E(x) and N(x)), specified the eta-function H(x), and the dissipation potential (x*). We shall now make a few comments about the realization of Eqs. (7) or (9). First, we note that instead of looking for one realization we can look for several realizations on different levels of description. This is in particular pertinent if the experimental observations at our disposal have been made on several levels (e.g. the classical rheological measurements and microscopic measurements as such made in slow-neutron-scattering experiments). In the case of suspensions, we can try to express our insight and experimental evidence on the level on which correlations in both velocities and positions are seen and also on the level on which only correlations in the position coordinate serves to characterize the microstructure. The “multilevel modelling” allows us to express our “multilevel insights” and “multilevel experimental evidence” that we often acquire. Moreover, we gain an additional understanding of the physics involved by investigating mathematically the relations among the models in the family. In the rest of this paper we shall comment only about the state variables x and about their kinematics. As we have already noted in Section 2.2, the choice of state variables depends on the specificity of the system under investigation and also on our objectives. Here we shall try to identify a rather large pool of candidates for the role of microstructural state variables.
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We begin with the microscopic state variables characterizing states of n structureless classical particles:
3.1. Liouville description
xparticles = (X1 , . . . , Xn , π1 , . . . , πn ),
Standard geometrical considerations lead us naturally to the Liouville reformulation of particle dynamics [28]. Let Mparticles denote the state space of classical mechanics (i.e. xparticles ∈ Mparticles ). We begin by constructing a bundle with Mparticles as its base space and fibers consisting of 6n-forms (i.e. volume elements). Sections of such a bundle, fn (xparticles )dxparticles , are the new state variables. We shall denote it by the symbol xLnp . The upper index stands for “Liouville description of n particles”. The kinematics of xLnp is induced, in a standard way (see [29]), from the kinematics (12) in the base space: {A, B}Lpn = dxparticles fn {Afn , Bf }particles , (14)
(10)
where i = 1, 2, . . . , n; Xi ∈ Q ⊂ R3 are the position coordinates, and πi ∈ Tx∗i ; i = 1, 2, . . . , n are the momenta. By T*Q we denote the cotangent bundle of Q; Tx∗i Q is the cotangent space over Xi . The 6n dimensional space whose elements are (X1 , . . . , Xn , π1 , . . . , πn ) will be denoted by the symbol M(particles) . We shall also consider particles with an internal structure. In particular, we shall consider fibers. Their states are characterized by: xfibers = (X1 , . . . , Xn , π1 , . . . , πn , p1 , . . . , pn , m1 , . . . , mn ),
(11)
where Xi and πi , i = 1, 2, . . . , n have the same meaning as in Eq. (10), p1 , . . . , pn are the unit vectors along the fibers, and m1 , . . . , mn are the angular momenta of the fibers in the coordinate system rotated with the fibers. The quantities p1 , . . . , pn and m1 , . . . , mn describe the internal structure of the fibers. Kinematics of xpartlcles is expressed in the canonical Poisson bracket {A, B}particles =
n
(AXiα Bπiα − BXiα Aπiα ),
(12) A large family of very useful mesoscopic state variables can be now constructed by applying various projections on xLnp or xLnf . Let gk = dxχk (x)fn (x); k = 1, 2, . . . , m, (16)
i=1
and of xfibers in the Poisson bracket [27] {A, B}fibers = {A, B}particles + +
n i=1
n
where A,B are sufficiently regular real-valued functions (functionals) of xLnp , Afn (xLnp ) := δA/δfn (xLnp ) is the Volterra functional derivative. We can interpret Eq. (14) as an average of Eq. (12) similarly as, say, energy ELpn can be interpreted as an average of the particle energy Lpn Lpn Eparticles = Efn : ELpn = dxparticles fn Efn . In exactly the same way we can introduce the field state variables xLnf := fn (xfibers )dxfibers and their kinematics {A, B}Lpf = dxfibers fn {Afn , Bfn }fibers . (15)
miα (Ami × Bmi )α
i=1
piα [(Ami × Bpi ) − (Bmi × Api )]α . (13)
A and B are sufficiently regular real valued functions of xparticles in Eq. (12) and of xfibers in (13). The state variables (10), (11) are not themselves the mesoscopic state variables. By passing to field descriptions, they can however lead us to a large family of state variables that are suitable for mesoscopic levels. This can be done by following three routes: Liouville route, Langevin route, and the route of continuum mechanics. The state variables and their kinematics that arise on the Liouville route are recalled in Section 3.1. The Langevin route consists of: (i) upgrading the status of the state variables (10),(11) (xparticles and xfibers become random variables), (ii) adding to the right hand-side of the time evolution equations a noise, and (iii) reformulating the dynamics in terms of probability distribution functions. We shall not explore this route in this paper. The route of continuum mechanics will be followed in Section 3.2. We shall see that this route offers a very interesting new class of mesoscopic state variables.
be an example of such a projection. By x in Eq. (16) we denote either xparticles or xfibers ; χ1 (x), . . . , χm (x) given functions of x. In order to obtain the Poisson bracket expressing kinematics of the mesoscopic state variables g1 , . . . , gm from the Poisson bracket (14) (or (15)), we proceed as follows: First, we restrict the functions A and B in Eq. (14) (or in (15)) to those that depend on fn only through their dependence on χ1 (x), . . . , χm (x). As a consequence of such a restriction, Afn (x) =
m
χj (x)Aχj .
(17)
j=1
Next, we insert Eq. (17) into (15) (or (16)) and obtain in this way a bracket involving the state variables g1 , . . . , gm . It will be a Poisson bracket if we succeed to express everything in the bracket in terms of g1 , . . . , gm . This is clearly not always possible. There are nevertheless many particular cases in which it is possible. In fact, these particular cases provide the most frequently used mesoscopic state variables. Among them we mention: (1) Reduced distribution functions (for example a one particle distribution function f1 (X, π) = dxparticles
M. Grmela / J. Non-Newtonian Fluid Mech. 120 (2004) 137–147
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fn (xparticles )χ(X, π, xparticles ), where χ(X, π ,xparticles ) = (δ(X−X1 )δ(π−π1 ) . . . δ(X−Xn )δ(π−πn ))), de-notes the Dirac delta-function. (2) Classical hydrodynamic fields ρ(X) = dπf1 (X, π), u(X) = dππf1 (X, π), where ρ is the mass density, u is the momentum field, s is the entropy field, kB is theBoltzmann constant). (3) Conformation tensors describing states of macromolecules, particles in suspensions, interfaces in immiscible blends and other complex fluids (conformation tensors are moments of two or higher order distribution functions).
Eq. (9) since we are limiting ourselves in this section only to the nondissipative part of the time evolution) becomes
3.2. Continuum description
where
∂p = 0, ∂t ∂s = 0, ∂t ∂Xα = Eπα , ∂t (p−cont)
∂ταβ ∂πα =− ∂t ∂aβ (p−cont)
In this section we pass from Eq. (10) or (11) to fields by replacing the (discrete) particle label i = 1, 2, . . . n by an element of a continuum that we shall call a material space, Mmatertal . The particle coordinates thus become fields (i.e. functions of the elements of the material space). Below, we shall discuss three cases of the material space: (i) Mmaterial = R3 ; i.e. the classical continuum (Section 3.2.1), (ii) Mmaterial = R3 × R3 ; we call it a dumbbell-continuum (Section 3.2.2), and (iii) Mmaterial = R3 × S2 ; we call it a fiber-continuum (Section 3.2.3), S2 denotes the two-dimensional sphere of radius one. The first attempt to introduce generalized continua has been made by Cosserat brothers in [30]. We suggest therefore to call the continua discussed in Sections 3.2.2 and 3.2.3, Cosserat-like continua. Some of the new mesoscopic state variables that arise in this section can be transformed (see Section 3.2.1) into the state variables introduced in the context of the Liouville approach in Section 3.1. The novel Cosserat-like continuum theories are expected to be particularly useful in discussing extensions of classical elasticity to complex materials exhibiting elasto-plastic behavior. 3.2.1. Particle continuum The material space is chosen to be Mmaterial = R3 ; its elements are denoted by the symbol a, i.e. a ∈ R3 . The state variables that we shall use in this section are the particle coordinates X(a), the momenta π (a) and two scalar fields ρ (a), and s (a). The field ρ has the physical meaning of the mass density and s of the total entropy density: xp−cont = (X(a), π(a), ρ(a), s(a)).
(18)
The upper index “p-cont” stands for the particle continuum. The kinematics of Eq. (18) is expressed by the continuum version of Eq. (12): {A, B}p−cont = da(AXα (a) Bπα (a) − BXα (a) Aπα (a) ). (19) The time evolution Eq. (9) corresponding to the Poisson bracket (19) (without the last term on the right-hand side of
ταβ
,
= −E ∂Xα . ∂aβ
(20)
(21)
E (xp−cont ) denotes, as always, the energy. We have assumed that the energy E depends only on the gradient of X(a) and not on X(a) itself (i.e. EX␣ =−∂/∂a E∂X␣ /∂a ).  Eqs. (20) and (21) are the well known governing equations of the classical elasticity theory. We have shown that they arise in a straightforward manner from the Hamiltonian continuum kinematics. 3.3. Eshelbian reformulation We now interpret a ∈ Mmaterial as an initial condition for X. We regard thus the vector valued field X(a) as a one-to-one map a → X.
(22)
We shall use this map to project the time evolution (20), (21) of the state variables (18) to the material space Mmaterial . In order to do this, we note first that the momentum πi of particles (see (10)) is an element of the cotangent space attached to the configuration space whose coordinates are Xi . This remains true also in the continuum theory where the particle label i = 1,2, . . . , n is replaced by the vector a ∈ R3 . The time evolution (20), (21) is not therefore constructed solely on the material space Mmaterial . In order to make it as such, we have to transform the field (a) from the cotangent space of the position coordinate space to the cotangent space of the material space. Since it is the transformation (22) that relates the material and the coordinate space, the transformation (X(a), π(a), ρ(a), s(a)) → (X(a), P(a), ρ(a), s(a)), (23) where Pγ = −
∂Xα πα ∂aγ
(24)
achieves this objective. The state space whose elements are (X(a), P(a), ρ(a), s(a)) is called the Eshelby space. We shall denote it by the symbol M(Eshelby) .
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Our next task is to derive the time evolution in M(Eshelby) from the time evolution (20), (21). First we turn our attention to kinematics, i.e. to the Poisson bracket. By applying the transformation (23), (24) to Eq. (19) we obtain ∂Xα Eshelby {A, B} = da − (AXα BPγ ∂aγ ∂Pβ −BXα APγ ) − (APγ BPβ − BPγ APβ ) . ∂aγ (25) The time evolution Eq. (9) (with the second term on its right hand-side absent) corresponding to the Poisson bracket (25) becomes ∂ρ = 0, ∂t ∂s = 0, ∂t ∂Xα ∂Xα =− EPγ , ∂t ∂aγ ∂Xγ ∂ ∂Pα =− (E∂Xγ /∂a∈ ) + ∂t ∂aα ∂a∈ If we now use
Ecin =
da
∂Pα ∂P∈ − ∂a∈ ∂aα
EP∈ . (26)
E = Ecin + Epot =
da ecin +
(27)
∂Xγ ∂ − (E∂Xγ /∂aε ) ∂aα ∂aε ∂Xγ ∂e ∂ −eδαε + E∂Xγ /∂aε − , =− ∂aε ∂aα inhom ∂aα ∂Pα ∂Pε ∂ ∂ecin EPε = − − (ecin ) + , (28) ∂aε ∂aα ∂aα ∂aα inhom Eq. (26) takes the form ∂ρ =0 ∂t ∂s =0 ∂t ∂Xα 1 = πα ∂t ρ
Eshelby
−
∂(epot − ecin ) ∂aγ
,
(29)
∂Xε E∂Xε /∂aγ δaα
(30)
inhom
where Eshelby
ταγ
= (−epot + ecin )δαγ +
Can the mesoscopic state variables arising in continuum descriptions be related to the mesoscopic state variables that we have seen in Section 3.1 to arise in the Liouville description? Below, we shall show one example of such relation. We shall relate Eq. (18) to the classical hydrodynamic state variables (see the last paragraph of Section 3.1 for their relation to the particle distribution function). Following [34], the hydrodynamic fields are related to the state variables (18) by: ρ(x) = dr δ(r − X−1 (x))ρ(X(r)), s(x) =
and the identities
∂ταγ ∂Pα = ∂t ∂aγ
3.4. Eulerian reformulation
da epot ,
1 π γ πγ , 2ρ
is the Eshelby extra stress tensor [31]. A density field, for example e(a), depends on a through its dependence on the fields (X (a), P(a), ρ(a), s(a)) included in the set of state variables and also through an explicit dependence (due to the inhomogeneity). By (∂e/∂a)inhom we denote the change of e due to the inhomogeneity. The classical continuum time evolution (20), (21) has been projected onto the material space in [32]. The fact that the projected time evolution coincides with the Eshelbian time evolution developed by Eshelby [31] on the basis of other considerations (discussion of the time evolution of material discontinuities) has been realized by Maugin [33].
u(x) =
dr δ(r − X−1 (x))s(X(r)), dr δ(r − X−1 (x))π(X(r)),
(31)
where (ρ, s, u) are the classical hydrodynamic fields of the mass density, the total entropy, and the momentum respectively. The mapping (31) is a projection of the same type as the projection (16). We now apply it to the Poisson bracket (19). Straightforward calculations show that all terms in the projected bracket can be eventually expressed only with the hydrodynamic fields. We thus arrive in this way at a Poisson bracket expressing kinematics of the set of classical hydrodynamic fields. The Poisson bracket obtained in this way turns out to be exactly the same as the one that arise by applying the projection (16) (with the functions χk specified in the point (2) in text following Eq. (17)) on the Poisson bracket (14). The passage from X(a), π (a) to the Euler hydrodynamic fields and the bracket can also be seen as a symmetry reduction (the Marsden–Weinstein reduction [29]). The symmetry involved is the particle-relabelling symmetry of the Euler hydrodynamic fields. By applying the symmetry-reduction technique one obtains again the same Poisson bracket expressing kinematics of the Euler hydrodynamic fields. Summing up, we see that we can arrive at the classical hydrodynamic state variables and at their kinematics by two routes: (i) By following the Liouville description where
M. Grmela / J. Non-Newtonian Fluid Mech. 120 (2004) 137–147
we pass from the n-particle distribution function to the one particle distribution function and subsequently to its five moments defining the hydrodynamic fields in terms of the particle distribution functions, (ii) By following the route of continuum mechanics where we pass from the state variables (18) to the hydrodynamic fields by the transformation (31). There are, of course, other hydrodynamics-type state variables that can be extracted from the continuum state variables (18), e.g. it is the coordinate in the material space seen as a hydrodynamic field: a(x) = dr δ(r − X−1 (x))r (32) The extended hydrodynamics in which the fields (ρ(x), s(x), u(x), a(x)) serve as state variables has been introduced in [35]. Its formulation as a particular realization of (9) has been worked out in [36,37]. The extended hydrodynamics has been called Lin hydrodynamics in [36] since the nondissipative part of the equation governing the time evolution of the field a(x) is the Lin constraint introduced in the context of the variational formulations of hydrodynamics in [38], see also [39]. Gradients of the field a(x) have the physical interpretation of the conformation tensors arising in the context of the Liouville approach to state variables (see the point (3) in the last paragraph of Section 3.1). Finally, we note that the same type of considerations that led us in this section from Eq. (20) to the Euler hydrodynamics can also be used to reformulate the Eshelbian Eq. (29) into the Eulerian form. This reformulation and its usefulness in applications will be explored in a subsequent paper. 3.4.1. Dumbbell continuum We turn now our attention to complex materials. Instead of structureless particles composing the material (the case considered above), we consider now pairs of particles (dumbbells) as the principal constituents of the material. We shall choose thus the material space to be: Mmaterial = R3 × R3 . Its elements are denoted by the symbol (a1 , a2 ); i.e. (a1 , a2 ) ∈ R3 × R3 . The state variables that we shall use in this section are the particle coordinates (X1 (a1 , a2 ), X2 (a1 , a2 )), the momenta (π1 (a1 , a2 ), π2 (a1 , a2 )), and two scalar fields ρ (a1 , a2 ) and s(a1 , a2 ). The fields ρ and s have the same physical meaning as in Section 3.2.1. The set of state variables is thus x
d−cont
= (X1 (a1 , a2 ), X2 (a1 , a2 ), π1 (a1 , a2 ), π2 (a1 , a2 ), ρ(a1 , a2 ), s(a1 , a2 )).
(33)
The upper index “d − cont” stands for the dumbbell continuum. The kinematics of Eq. (33) is the continuous version of (12): d−cont
{A, B}
=
da1
da2
2
(AXiα Bπiα − BXiα Aπiα ).
i=1
(34)
145
The time evolution (9) corresponding to the Poisson bracket (34) (the last term on the right hand-side of Eq. (9) is omitted due to our limitation to the nondissipative time evolution) becomes ∂ρ = 0, ∂t ∂s = 0, ∂t ∂X1α = Eπ1α , ∂t ∂X2α = Eπ2α , ∂t (d−cont)
∂τ11αβ ∂π1α =− ∂t ∂a1β
(d−cont)
−
(d−cont)
∂τ21αβ ∂π2α =− ∂t ∂a1β
∂τ12αβ
,
∂a2β (d−cont)
−
∂τ22αβ
∂a2β
,
(35)
where (d−cont)
= −E∂X1α /∂a1β ;
τ12αβ
(d−cont)
= −E∂X2α /∂a1β ;
τ22αβ
τ11αβ τ21αβ
(d−cont)
= −E∂X1α /∂a2β ,
(d−cont)
= −E∂X2α /∂a2β , (36)
E(xd −cont ) denotes the energy. We have assumed that the energy E depends only on the gradients of X1 (a1 , a2 ) and X2 (a1 , a2 ) and not on X1 and X2 themselves. The Eshelbian and Eulerian reformulations of the dumbbell-continuum dynamics (35) will be presented in a separate publication. 3.4.2. Fiber continuum The particles composing the material are now seen as fibers. The material space is thus chosen to be Mmaterial = R3 × S 2 , its elements are denoted by the symbol (a, b), a ∈ R3 , b ∈ S2 . By S2 we denote the two dimensional sphere of radius one. The variables describing states of the fiber continuum are xf −cont = (X(a, b), p(a, b), π(a, b), m(a, b), ρ(a, b), s(a, b)).
(37)
The upper index “f − cont” stands for the fiber continuum. The kinematics of Eq. (37) is expressed in the continuous version of the Poisson bracket (13): f −cont p−cont {A, B} = {A, B} + da dbmα (Am × Bm )α + da dbpα [(Am × Bp ) − Bm × Ap )]α (38)
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M. Grmela / J. Non-Newtonian Fluid Mech. 120 (2004) 137–147
The time evolution Eq. (9) corresponding to the Poisson bracket (38) (the last term on the right hand-side of Eq. (9) is omitted due to our limitation to the nondissipative time evolution) becomes ∂ρ = 0, ∂t ∂s = 0, ∂t ∂Xα = Eπα , ∂t (fa−cont)
∂ταβ ∂πα =− ∂t ∂aβ
(i) Search for approximately invariant mesoscopic Gibbs–Legendre manifolds (rather than for just approximately invariant mesoscopic manifolds) in the analysis of reductions to more macroscopic levels of description (see Section 2.4). The extra information involved in the Gibbs–Legendre manifolds has the physical meaning of the mesoscopic fundamental thermodynamic relation on mesoscopic levels. This is the way to introduce clearly thermodynamics of driven systems. (ii) Novel continuum-like theories suitable for complex materials exhibiting elasto-plastic behavior (see Section 3.2).
(fb−cont)
−
∂ταβ
,
∂bβ
Acknowledgements
∂pα = −(p × Em )α , ∂t
This work has been partially supported by the Natural Sciences and Engineering Research Council of Canada. (a)
(b)
∂τγk ∂τγk ∂mα = −(m × Em )α − εαβγ pβ − εαβγ pβ , ∂t ∂ak ∂bk
References (39)
where (fa−cont)
ταβ
(a)
(fb−cont)
= −E∂Xα /∂aβ ;
ταβ = −E∂pα /∂aβ ;
ταβ (b)
= −E∂Xα /∂bβ ,
ταβ = −E∂pα /∂bβ .
(40)
ε is the alternating tensor ((m × Em )α = εαβγ mβ Emγ ), and E(xf −cont ) denotes the energy. We have assumed that the energy E depends only on the gradient of X(a, b) and p(a, b) and not on X and p themselves. The Eshelbian and Eulerian reformulations of the fiber-continuum dynamics (39) will be presented in a separate publication.
4. Concluding remarks By following the Gibbs geometrical viewpoint of thermodynamics to mesoscopic levels of description, we have arrived at a general framework for the mesoscopic time evolution that is compatible with thermodynamics. If the mesoscopic state space is equipped with the structure of thermodynamics then there is essentially only one time evolution that fits into it. The framework identified with the help of geometry appears to be the same as the one that has been distilled previously as a common structure of well-established mesoscopic dynamical theories (as for example the Boltzmann kinetic equation and the classical hydrodynamic equations). Methods of geometry led us also (Section 3) to a large class of well known and new mesoscopic state variables and to their kinematics (expressed in Poisson brackets). Among the new avenues that open with geometry and provide an interesting new perspective we mention two:
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