Mesoscopic equilibrium thermodynamics

Question Where does the fundamental thermodynamic relation s = s(y) come from Given a physical system, what is the fundamental thermodynamic relation representing it in equilibrium thermodynamics  

We shall give two answers in equilibrium theories and another answer in nonequilibrium theories (discussed in Sections 3 and 4). 

The second answer is found by taking a more microscopic (i.e., more detailed) view, called a mesoscopic view, that we shall now present. 

Let the state variables (1) be replaced by a state variable xeM corresponding to a more detailed (more microscopic) view than the one taken in classical equilibrium thermodynamics. Several examples of x are discussed below in the examples accompanying this section. Following closely Section 2.1, we shall now formulate equilibrium thermodynamics that we shall call a mesoscopic equilibrium thermodynamics. 

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In this section we limit ourselves to equilibrium. The time evolution that is absent in this section will be taken into consideration in the next two sections. We begin the equilibrium analysis with classical equilibrium thermodynamics of a one-component system. The classical Gibbs formulation is then put into the setting of contact geometry. In Section 2.2 we extend the set of state variables used in the classical theory and introduce a mesoscopic equilibrium thermodynamics. 

The above realization of the abstract mesoscopic equilibrium thermodynamics is called a Canonical-Ensemble Statistical Mechanics. We shall now briefly present also another realization, called a Microcanonical-Ensemble Statistical Mechanics since it offers a useful physical interpretation of entropy. 

Since around the mid-1990s, there has been a proliferation of hydrate time-dependent studies. These include macroscopic, mesoscopic, and molecular-level measurements and modeling efforts. A proliferation of kinetic measurements marks the maturing of hydrates as a field of research. Typically, research efforts begin with the consideration of time-independent thermodynamic equilibrium properties due to relative ease of measurement. As an area matures and phase equilibrium thermodynamics becomes better defined, research generally turns to time-dependent measurements such as kinetics and transport properties. 

The review is organized as follows In Section 2 we present the multiscale equilibrium thermodynamics in the setting of contact geometry. The time evolution (multiscale nonequilibrium thermodynamics) representing approach of a mesoscopic level LmeSoi to the level of equilibrium thermodynamics Leth is discussed in Section 3. A generalization in which the level Leth is replaced by another mesoscopic level LmesoZ is considered in Section 4. The notion of multiscale thermodynamics of systems arises in the analysis of this type of time evolution. 

Summing up, we have arrived at the mesoscopic time evolution Equation (55) by extending the geometrical structure of equilibrium thermodynamics to the time evolution. A few observations are now in order ... 

Experimental observations of the time evolution of externally unforced macroscopic systems on the level meSo l show that the level eth of classical equilibrium thermodynamics is not the only level offering a simplified description of appropriately prepared macroscopic systems. For example, if Cmeso is the level of kinetic theory (Sections 2.2.1, starting point. In order to see the approach 2.2.2, and 3.1.3) then, besides the level, also the level of fluid mechanics (we shall denote it here Ath) emerges in experimental observations as a possible simplified description of the experimentally observed time evolution. The preparation process is the same as the preparation process for Ath (i.e., the system is left sufficiently long time isolated) except that we do not have to wait till the approach to equilibrium is completed. If the level of fluid mechanics indeed emerges as a possible reduced description, we have then the following four types of the time evolution leading from a mesoscopic to a more macroscopic level of description (i) Mslow/ (ii) Aneso 2 -> Ath, (ui) Aneso l -> Aneso 2, and (iv) Aneso i —> Aneso 2 —> Ath- The first two are the same as (111). We now turn our attention to the third one, that is,... 

For a chemical reaction, for example, non-equilibrium thermodynamics formulates a linear relationship between the reaction rate and the affinity, which constitute only the first term in the development of the law of mass action. To obtain the full law, one has to take into account not only the initial and final states of the kinetics but all intermediate configurations, i.e. one has to introduce a mesoscopic degree of freedom accounting for the different molecular configurations. When this is done, in the framework of mesoscopic non-equilibrium thermodynamics, one arrives at the law of mass action governing the kinetics for arbitrary values of the thermodynamic... 

The Mesoscopic Non-Equilibrium Thermodynamics Approach to Polymer Crystallization... 

Nucleation and growth are the most important processes in determining the final morphology of the crystallized polymer and consequently the properties of the material. Both processes can be quite conveniently described within the framework of mesoscopic non-equilibrium thermodynamics. 

Mesoscopic non-equilibrium thermodynamics provides a description of activated processes. In the case considered here, when crystallization proceeds by the formation of spherical clusters, the process can be characterized by a coordinate y, which may represent for instance the number of monomers in a cluster, its radius or even a global-order parameter indicating the degree of crystallinity. Polymer crystallization can be viewed as a diffusion process through the free energy barrier that separates the melted phase from the crystalline phase. From mesoscopic non-equilibrium thermodynamics we can analyze the kinetic of the process. Before proceeding to discuss this point, we will illustrate how the theory applies to the study of general activated processes. 

The formalism introduced in the previous subsections is able to incorporate the effect of these influences in the crystallization kinetics, thus providing a more realistic modeling of the process, which is mandatoiy for practical and industrial purposes. Due to the strong foundations of our mesoscopic formalism in the roots of standard non-equilibrium thermodynamics, it is easy to incorporate the influence of other transport processes (like heat conduction or diffusion) into the description of crystallization. In addition, our framework naturally accounts for the couplings between all these different influences. 

We have shown that mesoscopic non-equilibrium thermodynamics satisfactorily describes the dynamics of activated processes in general and that of polymer crystallization in particular. Identification of the different mesoscopic configurations of the system, when it irreversibly proceeds from the initial to the final phases, through a set of internal coordinates, and application of the scheme of non-equilibrium thermodynamics enable us to derive the non-linear kinetic laws governing the behavior of the system. 

This description has to be compared with that proposed by non-equilibrium thermodynamics in terms of only two states, corresponding to the melted and crystallized phases in the example we are discussing, from which only one may account for the linear domain, when the chemical potentials at the wells are not very different. This feature imposes serious limitations in the application of NET to activation processes since that condition is rarely encountered in experimental situations and has therefore restricted its use to only transport processes. The mesoscopic version of non-equilibrium thermodynamics, on the contrary, circumvents the difficulty offering a promising general scenario useful in the characterization of the wide class of activated processes, which appear frequently in systems outside equilibrium of different nature. 

Bedeaux, D., Mazur, P. (2001). Mesoscopic non-equilibrium thermodynamics for quantum systems. Physica A. 298 81-100. 

This new theory of the non-equilibrium thermodynamics of multiphase polymer systems offers a better explanation of the conductivity breakthrough in polymer blends than the percolation theory, and the mesoscopic metal concept explains conductivity on the molecular level better than the exciton model based on semiconductors. It can also be used to explain other complex phenomena, such as the improvement in the impact strength of polymers due to dispersion of rubber particles, the increase in the viscosity of filled systems, or the formation of gels in colloids or microemulsions. It is thus possible to draw valuable conclusions and make forecasts for the industrial application of such systems. 

The concepts of meso-thermodynamics can be extended to some non-equilibrium phenomena. In particular, like the thermodynamic properties, transport coefficients, such as the diffusion coefficient, become spatially dependent at meso-scales. Moreover, away from equilibrium, generic long-range correlations emerge even in simple molecular fluids, making the famous concept of local equilibrium, at least, questionable. In this section we focus only on one application of mesoscopic nonequilibrium thermodynamics in fluids fluid phase separation. 

Discussing the problem of the adequate use of the Second Law for biological systems, McClare indicated that from the point of conventional equilibrium thermodynamics, entropy is a macroscopic function of the system s state. Entropy change determines the direction of spontaneous irreversible processes in the whole system. At the intermediate level of structural organization, described within the approach of statistical mechanics, entropy is a mesoscopic value which is determined by the probability partition function. On the other hand, because of the reversibility of physical processes at the microscopic level, entropy cannot be a microscopic value. This statement had been clearly argued as early as 1912 by Paul and Tatyana Ehrenfest [8]. Following this line of argument, McClare deduced that entropy cannot be a characteristic function of molecules at the microscopic level. 

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