Gibbs-Boltzmann equilibrium

The diffusive relaxation of the probability density p(R, t) of an ensemble of systems towards the Gibbs-Boltzmann equilibrium distribution is described by the Smoluchowski equation... 

A recent work has demonstrated that the formulation of reaction-diffusion problems in systems that display slow diffusion within a continuous-time random walk model with a broad waiting time pdf of the form (6) leads to a fractional reaction-diffusion equation that includes a source or sink term in the same additive way as in the Brownian limit [63], With the fractional formulation for single-species slow reaction-diffusion obtained by the authors still being linear, no pattern formation due to Turing instabilities can arise. This is due to the fact that fractional systems of the type (15) are close to Gibbs-Boltzmann thermodynamic equilibrium as shown in the next section. 

In the 19th century the variational principles of mechanics that allow one to determine the extreme equilibrium (passing through the continuous sequence of equilibrium states) trajectories, as was noted in the introduction, were extended to the description of nonconservative systems (Polak, 1960), i.e., the systems in which irreversibility of the processes occurs. However, the analysis of interrelations between the notions of "equilibrium" and "reversibility," "equilibrium processes" and "reversible processes" started only during the period when the classical equilibrium thermodynamics was created by Clausius, Helmholtz, Maxwell, Boltzmann, and Gibbs. Boltzmann (1878) and Gibbs (1876, 1878, 1902) started to use the terms of equilibria to describe the processes that satisfy the entropy increase principle and follow the "time arrow."... 

Most investigations on nonequilibrium systems were initially carried out in the NESS. It is widely believed that NESSs are among the best candidate nonequilibrium systems to possibly extend the Boltzmann-Gibbs ensemble theory beyond equilibrium [50, 51]. 

In Eq. (1.36), Nj is the equilibrium number of point defects, N is the total number of atomic sites per volume or mole, Ej is the activation energy for formation of the defect, is Boltzmann s constant (1.38 x 10 J/atom K), and T is absolute temperature. Equation (1.36) is an Arrhenius-type expression of which we will see a great deal in subsequent chapters. Many of these Arrhenius expressions can be derived from the Gibbs free energy, AG. 

Here the pre-exponential factor At is the product of a temperature-dependent constant (ksT/h) = 2 X 10 °Ts where ke and h are the Boltzmann and Planck constants, and a solvent-specific coefficient that relates to both the solvent viscosity and to its orientational relaxation rate. This coefficient may be near unity for very mobile solvent molecules but may be considerably less than unity for viscous or orientationally hindered highly stractured solvents. The exponential factor involves the activation Gibbs energy that describes the height of the barrier to the formation of the activated complex from the reactants. It also describes temperature and pressure dependencies of the reaction rate. It is assumed that the activated complex is in equilibrium with the reactants, but that its change to form the products is rapid and independent of its environment in the solution (de Sainte Claire et al., 1997). 

In the following, we first describe (Section 13.3.1) a statistical mechanical formulation of Mayer and co-workers that anticipated certain features of thermodynamic geometry. We then outline (Section 13.3.2) the standard quantum statistical thermodynamic treatment of chemical equilibrium in the Gibbs canonical ensemble in order to trace the statistical origins of metric geometry in Boltzmann s probabilistic assumptions. In the concluding two sections, we illustrate how modem ab initio molecular calculations can be enlisted to predict thermodynamic properties of chemical reaction (Sections 13.3.3) and cluster equilibrium mixtures (Section 13.3.4), thereby relating chemical and phase thermodynamics to a modem ab initio electronic stmcture picture of molecular and supramolecular interactions. 

The articles of L. S. Ornstein (1908-1909) are, first, very valuable contributions to the clarification of the foundations of Gibbs s treatment. In addition, they show (1) that operations with canonical ensembles occasionally furnish a more convenient computational scheme for the treatment of complicated problems of equilibrium (e.g., the equilibrium in capillary transition layers) than does Boltzmann s procedure (2) the reason why the two methods give the same result in a large group of equilibrium problems. 

The equilibrium constants are related to the thermodynamic Gibb s free energy in the regular manner ksTlnKn = AG ° = Af/ - TASj, where kg is Boltzmann s constant and T is the temperature, and ideal mixing (no volume change) is assumed. The chemical equilibrium constants can also be written in terms of the activities of the different species, defined in terms of the chemical potential, of specie k, as a/ expi/hJkgT). Thus, at equilibrium... 

The general mathematical formulation of the equilibrium statistical mechanics based on the generalized statistical entropy for the first and second thermodynamic potentials was given. The Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles were investigated as an example. It was shown that the statistical mechanics based on the Tsallis statistical entropy satisfies the requirements of equilibrium thermodynamics in the thermodynamic limit if the entropic index z=l/(q-l) is an extensive variable of state of the system. 

In modem physics, there exist alternative theories for the equilibrium statistical mechanics [1, 2] based on the generalized statistical entropy [3-12]. They are compatible with the second part of the second law of thermodynamics, i.e., the maximum entropy principle [13-14], which leads to uncertainty in the definition of the statistical entropy and consequently the equilibrium probability density functions. This means that the equilibrium statistical mechanics is in a crisis. Thus, the requirements of the equilibrium thermodynamics shall have an exclusive role in selection of the right theory for the equilibrium statistical mechanics. The main difficulty in foundation of the statistical mechanics based on the generalized statistical entropy, i.e., the deformed Boltzmann-Gibbs entropy, is the problem of its connection with the equilibrium thermodynamics. The proof of the zero law of thermodynamics and the principle of additivity... 

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

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