Equilibrium thermodynamics motion

It should be realized that unlike the study of equilibrium thermodynamics for which a model is often mapped onto Ising system, elementary mechanism of atomic motion plays a deterministic role in the kinetic study. In an actual alloy system, diffusion of an atomic species is mainly driven by vacancy mechanism. The incorporation of the vacancy mechanism into PPM formalism, however, is not readily achieved, since the abundant freedom of microscopic path of atomic movement demands intractable number of variational parameters. The present study is, therefore, limited to a simple spin kinetics, known as Glauber dynamics [14] for which flipping events at fixed lattice points drive the phase transition. Hence, the present study for a spin system is regarded as a precursor to an alloy kinetics. The limitation of the model is critically examined and pointed out in the subsequent sections. 

We can specially show that the main principles of nonequilibrium thermodynamics (the Onsager relations, the Prigogine theorem, symmetry principle) and other theories of motion (for example, theory of dynamic systems, synergetics, thermodynamic analysis of chemical kinetics) are observed in the MEIS-based equilibrium modeling. In order to do that, we will derive these statements from the principles of equilibrium thermodynamics. 

The next sphere of competition between equilibrium and nonequilibrium thermodynamics is the analysis of irreversible trajectories. A popular opinion about the possibility for the equilibrium thermodynamics only to determine admissible directions of motion for nonequilibrium processes was already mentioned in Introduction. However, the more... 

As may be seen from arguments like those that led from Eqs. (A. 15) to (A. 16), Onsager s eqs. (A.38) and (A.40) properly reduce to equilibrium thermodynamics. Thus, Eqs. (A.38) and (A.40) predict that after the final equilibrium state Tp is reached, all motions of the unconstrained A s cease, and that these parameters remain forever fixed at their final state values A. ... 

In summary, Onsager did succeed in finding a nonequilibrium extension of equilibrium thermodynamics. However, to resolve the dual problem of formulating a thermodynamic equation of motion and of choosing the thermodynamic forces, he was obliged to make limiting slow variable assumptions. Thus his central Eq. (A.53) models actual macroscopic parameters motions in a highly simplified way, namely, as coupled diffusive motions in the equilibrium potential oc S Tp A) of Eq. (A.36). 

Relations between the theories of states and trajectories and capabilities of equilibrium thermodynamic analysis to study reversible and irreversible kinetics can be more fully revealed by considering another type of models of extreme intermediate states, namely MEIS of hydraulic circuits (Gorban et al., 2001, 2006 Kaganovich et al., 1997, 2007, 2010). Convenience and clearness of using these models to describe the considered problems are determined by the fact that they are intended to study an essentially irreversible process, i.e. motion of a viscous fluid. Besides, they can be treated as models of the mechanism of fluid transportation from the specified source nodes of a hydraulic system to the specified consumption nodes. The major variable of the hydraulic circuit theory (Khasilev, 1957,1964 Merenkov and Khasilev, 1985), i.e. continuous medium flow, has an obvious kinetic sense. 

Rapid progress in computer technology that contributes to enhancement of competitiveness for simple algorithms that require multiple increase in computations and at the same time immeasurably decrease the labor input in creation of software and preparation of initial information gives a weighty support in the required proofs. The algorithms constructed on the basis of simple and universal prerequisites of equilibrium thermodynamics are obviously simple. They are used for stepwise description of trajectories that does not require derivation of "underivable" equations. The preparation process of initial experimental and theoretical data needed for their application becomes sharply easier. In this case the characteristics of rest states, whose sequences are applied to describe steps in decision making, are measured and calculated rather easier than characteristics of motion. 

While integrating the classical (Newtonian) equations of motion provides information regarding the constant-energy surface, one may wish to explore the equilibrium thermodynamic properties of a system. If a microscopic dynamic variable A takes on values A t ) along the trajectory at the time step t , then the time average... 

On the other hand, we may look at thermal feeling of our planetary ecosystems, which, in general, is influenced by positioning in interstellar space [9,191]. No planet is a closed system neither is it in contact with two or more thermal baths. Each planet is in contact with a hot radiator (sun 5800 K) and cold radiation sinks (outer space 2.75 K). Each planet therefore realizes a kind of a specific cosmological engine. Being a sphere is crucial because of its rotation and revolution modes with an inclined axis, which are responsible for the richness and complexity in the behavior induced by the solar influx. The atmospheric fluid motion adds the spice of chaos so that the thermodynamic events, which take place within a planet, are sustained by non-equilibrium flows, which must obey the fundamental laws of non-equilibrium thermodynamics. 

Consider an equilibrium thermodynamic ensemble, say a set of atomic systems characterized by the macroscopic variables T (temperature), 2 (volume), andiV (number of particles). Each system in this ensemble contains N atoms whose positions and momenta are assigned according to the distribution function (5.2) subjected to the volume restriction. At some given time each system in this ensemble is in a particular microscopic state that corresponds to a point (r, p ) in phase space. As the system evolves in time such a point moves according to the Newton equations of motion, defining what we call a phase space trajectory (see Section 1.2.2). The ensemble corresponds to a set of such trajectories, defined by their starting point and by the Newton equations. Due to the uniqueness of solutions of the Newton s equations, these trajectories do not intersect with themselves or with each other. 

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