Multiscale Mesoscopic Entropy of Driven Macroscopic Systems

Existence or absence of a sufficiently general principle guiding behavior of driven macroscopical systems has yet to be proven. Arguments for the existence have been introduced for instance in [1,2,3,4,5,6,7,8], arguments for the absence in [9]. All suggested general principles that we know of are formulated in the setting of classical equilibrium thermodynamics and equilibrium statistical mechanics with the classical concept of equilibrium or local equilibrium entropy. Such setting limits interactions between the macroscopic systems and their surroundings and excludes investigations of driven systems requiring nano-scale microscopic physics. The arguments against a potential-based general principle have arisen in [9] in a detailed analysis of driven systems exhibiting emergence of highly organized patterns.

In both pro- and con-oriented investigations the following question arises: What if the analysis were carried on a (mesoscopic) level of description involving more details, or alternatively, what if the analysis were carried out simultaneously on several different levels of description? The mesoscopic level involving more details could be a level that includes extra state variables addressing microscopic details playing an important role for instance in biology or nanotechnology (such state variables are absent in [1,2,3,4,5,6,7,8]) or it also could be a level involving state variables with which details of mesoscopic structures emerging for instance in turbulent flows can be described (such state variables are absent in the formulations discussed in [9]). Since all the general principles suggested so far involve a potential that is closely related to entropy or its production, the more narrowly focused but closely related is then the question of what is the entropy of driven systems on a general mesoscopic (i.e., beyond equilibrium and local equilibrium) levels of description. This is exactly the question that we investigate in this paper.

The equilibrium entropies (i.e., those arising in the classical or local equilibrium thermodynamics or those on mesoscopic levels of description that are reduced from the Gibbs entropy arising on the microscopic level of equilibrium statistical mechanics) have a limited legitimacy outside equilibrium. Multiscale mesoscopic dynamics (MM dynamics) developed in [10,11] provides a setting for a new approach to defining nonequilibrium entropy (we shall call it mM entropy) on mesoscopic levels of description. In this paper we apply it to externally driven macroscopic systems. In Section 2 we recall MM dynamics and the mM entropy arising in it. In Section 3 we show that in the setting of equilibrium or local equilibrium thermodynamics the mM entropy coincides with the equilibrium or the local equilibrium entropy. Finally, in Section 4, we provide a specific example of the mM entropy.

We begin by recalling that the physics taking place in macroscopic systems can be expressed on different levels of description. The levels have emerged in the effort to organize and understand experience collected in certain types (different for different levels) of experimental observations. The levels are initially independent and autonomous. For example water can be modeled on the completely microscopic level on which it is seen as being composed of atoms (we shall denote this levels by the symbol L_MICRO), or on more macroscopic levels as e.g. the level of kinetic theory (denoted L_KT), the level of classical fluid mechanics (denoted L_leq), and the level of classical thermodynamics (denoted L_eq).

We shall now regard macroscopic systems simultaneously on two levels. We assume that both levels have been found to provide a setting for describing the experimentally observed behavior. Let L_meso be the more macroscopic (i.e., involving the least amount of details) and L_MESO the more microscopic level. The time evolution on L_MESO is seen to proceed in two stages. The first one brings the detailed description on the level L_MESO to a less detailed description on the level L_meso. This stage is then followed by the second stage in which the time evolution continues on the level L_meso. We shall refer to the first stage in the time evolution taking place on L_MESO as the fast time evolution and to the second as the slow time evolution since the time interval needed to approach the level L_meso is small relative to the time intervals needed to observe changes due to the time evolution on the level L_meso. The existence of the split of the time evolution taking place on L_MESO into fast and slow is a consequence of the experimentally observed fact that our externally driven macroscopic system under consideration is found to be well described on both levels L_meso and L_MESO.

The pairs of levels (L_meso, L_MESO) can be for example (L_eq, L_MICRO), (L_leq, L_KT). We shall discuss several examples below in Section 3 and Section 4.

Now, we continue with our general observations about macroscopic time evolution. Having recognized the existence of two types of the time evolution (namely the fast and the slow), we can now say that our interest in the behavior of driven systems is in fact an interest in the slow time evolution. We shall argue however that the fast time evolution is still very pertinent since it provides potentials (namely the energy and the entropy) needed in the anticipated optimization strategy of responses of macroscopic systems to driving forces. We thus turn our attention first to the fast time evolution.

By analyzing many examples of relations inside pairs (L_meso, L_MESO), a common structure of the fast time evolution has emerged (see [10] and references cited therein). The elements of the structure that are pertinent to our interests in this paper are the following:

(i)

The fast time evolution bringing L_MESO into L_meso is generated by two potentials, one is E^(MESO) and the other S^{(MESO→meso)}. The former is physically interpreted as the energy on the level L_MESO. The latter does not decrease during the fast time evolution and is physically interpreted as the entropy on the level L_MESO. As the time goes to infinity, S^{(MESO→meso)} reaches its maximum that we denote S^{(meso←MESO)}. This is a new potential (a new entropy) on the level L_meso.

(ii)

In the enlarge space that involves the state variables, their conjugates, and the generating potential (combining L_MESO E^(MESO) and S^{(MESO→meso)}), the fast time evolution appears to be a sequence (parametrized by time) of Legendre transformations.

(iii)

The fast time evolution is distinguished among all time evolutions during which the level L_MESO is being reduced to a less microscopic (i.e., involving less details) level L_meso by the following two properties: (a) S^{(MESO→meso)} does not decrease; and (b) the total entropy S created in the passage from L_MESO to L_meso (i.e., S = ʃdtd(S^{(MESO→meso)}/dt) reaches its extremum. If we realize that the state at which the system is found to be well described on the level L_meso is in fact a stationary state of the fast time evolution we can also formulate this result (for its proof, see [10]) as follows. Any open system develops toward a stationary state following a path on which the entropy generation reaches its extremum. The entropy is the potential generating the fast time evolution, the stationary state is a state that remains unchanged in time only with respect to the fast time evolution but, in general, changes in time (in a slower pace) in the slow time evolution.

Before proceeding with our investigation, we put the results that we have just recalled into the context of results of a similar nature that have been obtained previously. The fact that entropy depends on the level of description is well known (see for example [12,13,14,15]). In our setting, the entropy does not depend only on the level L_MESO on which the time evolution is originally formulated but also on the more macroscopic level L_meso to which L_MESO reduces during the time evolution. Every pair of well established levels (i.e., levels well tested with experimental observations) is assocoated with one entropy S^{(MESO→meso)} on level L_MESO and one entropy S^{(meso←MESO)} on level L_meso. In most previous investigations the level L_meso was taken to be usually the level of classical equilibrium thermodynamics. But even in the investigations in which L_meso is a general mesoscopic level (for example those reported in [14,15]) the two entropies S^{(MESO→meso)} and S^{(meso←MESO)} have not been identified. We briefly explain why. The analysis of the time evolution on the level L_MESO is in [14,15] inspired by the Chapman-Enskog analysis of solutions to the Boltzmann equations and proceeds as follows. First, an entropy on the level L_MESO is suggested (it is for example the Boltzmann H-function in the context of the Boltzmann equation) and by its maximization subject to constraints corresponding to the level L_meso a manifold of states representing in L_MESO the state space used in L_meso is identified (in the context of the Chapman-Enskog method for solving the Boltzmann equation it is the manifold formed by local Maxwellian distribution functions). The time evolution taking place in L_MESO that is restricted to this manifold represents an initial approximation of the reduced time evolution taking place on the level L_meso. In order to improve this initial approximation the manifold is deformed (in the context of the Chapman-Enskog method for solving the Boltzmann equation such deformation is achieved by making the so called first Chapman-Enskog approximation). Many ingenious methods for making this type of deformations in a rather general setting are suggested in [14,15]. Having the deformed manifold, we are making a next step. We identify a deformation of the initial entropy (i.e., deformation of the H-function in the case of the Boltzmann equation) that corresponds to the deformed manifold in the sense that the deformed manifold is the manifold obtained by its maximization (in the same way as the initial manifold is obtained by maximiaing the initial entropy). Such deformed entropy is then the entropy S^{(MESO→meso)}. Beside this new feature of our analysis a whole new physics and mathematics enters our investigation through the contact-structure geometry formulation.

Now we return to the problem that is in the center of our interest in this paper and formulate a conjecture about responses of macroscopic systems to external forces.

We now illustrate the general considerations presented above on the example in which L_meso is taken to be the level of the classical equilibrium (denoted L_eq). First, we recall that on this level the macroscopic systems are at thermodynamic equilibrium. This means that their states are characterized by two real numbers (e.g., (e,v), where e is the specific internal energy and v specific volume—for the sake of simplicity we restrict ourselves to one component systems) and that they are surrounded by thermodynamic walls. A state of thermodynamic equilibrium is a state reached in a preparation process in which the macroscopic system under consideration is left sufficiently long time surrounded by unchanging thermodynamic walls. The thermodynamic walls are defined as modes of interaction with surrounding that make possible to measure the conjugate state variables. We recall that the conjugate of e is the temperature T and the conjugate of v is the pressure P. The measurement process is based on the existence of a potential s = s(e,v), called an entropy, that is required to reach maximum if the systems involved in the measurement interact through the thermodynamic walls. The interaction through e-passing thermodynamic wall (also called heat-passing) leads to equilibration of temperatures on both sides of the wall and the v-passing thermodynamic wall (a mechanical piston) leads to equilibration of pressures on both sides of the wall. The entropy s = s(e,v) is the potential S^(eq←MESO) inherited from an analysis of the fast time evolution that is, in this example, the time evolution taking place in the process needed to prepare the macroscopic systems to thermodynamic equilibrium states. The level L_MESO can be any level involving internal time evolution.

It can be for instance the completely microscopic level L_MICRO (i.e., the level on which the macroscopic systems are seen as composed of atoms). In this case the setting for the analysis of L_MICRO → L_eq becomes the setting of Gibbs’ equilibrium statistical mechanics. We recall that in Gibbs’ analysis the time evolution on the level L_MICRO is drastically reduced, with the help of the ergodicity-type hypothesis, to only global conservation laws of classical mechanics.

Another example of the level L_MESO that involves more details than the level L_eq is the level of local equilibrium L_leq. It differs from L_eq by allowing the thermodynamic equilibrium states to be different from one place to another (i.e., to be local). The state variables are the same as on L_eq except that they are local (i.e., instead of real number they are fields) and a field having the physical interpretation of the flow velocity is adopted as a new state variable. The entropy is assumed to be the same as on the level L_eq except that it is local.

The time evolution on the level L_eq, i.e., the slow time evolution, is no time evolution except the time evolution reflecting imposed time changes in thermodynamic walls. The slow time evolution on the level L_leq is the time evolution studied in the classical fluid mechanics.

The classical example of a driven system on the level L_eq is Carnot’s engine. Its input is the heat flow induced by imposed temperature gradient and its output is a mechanical work. In this example the physical interpretation of the entropy as a measure of organization inside the macroscopic systems is revealed. In order to transform the macroscopically disorganized internal energy into macroscopically organized mechanical energy one needs to begin with an organization of the internal energy. This is done in Carnot’s engine by sustaining heat flux. But in order to sustain the heat flux, it is obviously impossible to transform all the disorganized internal energy into the organized mechanical energy. A part of the energy flux is used in the organization process (see more in [8])

Importance of the entropy (or the entropy generation) in many other examples of macroscopic systems on the levels L_eq and L_leq that are open to the environment through thermodynamic walls has been revealed in [1,2,3,4,5,6,7,8]. The investigations and applications reported in these papers (see also references cited therein) provide support for the existence of a universal principle guiding the behavior of driven macroscopic systems and for the conjecture formulated in Section 2.

Many practical applications require however formulations on levels L_meso that are more detailed (more microscopic) than the levels L_eq and L_leq. We illustrate this need on the example of Rayleigh-Bénard system (a horizontal layer of a fluid subjected to the gravitation force and temperature gradient). This macroscopic system is open to its surrounding through thermodynamic walls (as the systems discussed in the previous section are) and in addition it is subjected to a force field (the gravitational force field). Experimental observations show the following behavior. If the imposed temperature gradient is small, the fluid, that is initially at rest, remains at rest and only heat flows through the layer. This behavior is found to be well described by the classical fluid mechanics (i.e., on the level L_leq). We now gradually increase the temperature gradient ∆T. When it reaches a certain value ((∆T)_crit, called a critical value), the fluid inside the layer begins to flow in macroscopically organized manner. Emergence of this new macroscopic pattern is seen in experimental observations and it is also seen in the mathematical formulation as a bifurcation in solutions to the governing equations of fluid mechanics. From the physical point of view, the emergence of an internal macroscopic motion inside the fluid (that becomes the turbulent motion for large temperature gradients) transforms the originally simple fluid into a complex fluid with an internal structure. The level L_leq of classical fluid mechanics that is well suited for describing the behavior of simple fluids (i.e., the behavior seen if the imposed temperature gradient is smaller than (∆T)_crit) is not anymore sufficient to describe the time evolution of such complex fluid. This also means that the classical local entropy, playing the important role described in Section 1 in many applications discussed in [1,2,3,4,5,6,7,8], is not playing such role in complex fluids. We however anticipate that the potential, S^(cf←MESO) where L_cf is the level suitable for describing the complex fluid under investigation and L_MESO is a more microscopic level of description, will assume the role that the local entropy plays for simple fluids. We hope to identified an appropriate level L_MESO on which we formulate dynamics of the Rayleigh-Bénard system and extract from it the fast time evolution leading to S^(cf←MESO) in a future paper.

Any past or future formulation of thermodynamics of open systems has to address the concept of entropy outside its natural realm of equilibrium thermodynamics and equilibrium statistical mechanics. While in many applications of considerable practical interest (see e.g., [1,2,3,4,5,6,7,8]) the equilibrium entropy (including the local equilibrium entropy and the entropies arising in equilibrium statistical mechanics) is found to be sufficient, important applications in nano-technology and bio-technology have to face this problem. In this paper, following [10,11], we suggest and briefly explore a systematic path leading from the classical equilibrium entropy to entropies of driven systems on mesoscopic levels of description. Starting point is a dynamical viewpoint of the classical equilibrium entropy. We see the origin of the entropy in the process required to prepare systems to states in which classical equilibrium thermodynamics is applicable. The time evolution taking place in the preparation process is generated by a potential that then, if evaluated at the states reached in the preparation process, becomes the entropy. This dynamical point of view of entropy is then shown to extend naturally to different levels of description and to mesoscopic descriptions of driven macroscopic systems. Entropy arises in the comparison of two well established (i.e., well tested with experimental observations) levels of description one of which involves more details than the other. There is therefore as many entropies as there is pairs of such levels. Beside bringing a clear definition of the general concept of entropy, this viewpoint also puts into focus the importance of multiscale mesoscopic investigations of complex systems encountered in nano- and bio-technologies. The general discussion is illustrated in this paper on the example of heat conduction.