Equilibrium statistical mechanics thermodynamics

Introduction.—Statistical physics deals with the relation between the macroscopic laws that describe the internal state of a system and the dynamics of the interactions of its microscopic constituents. The derivation of the nonequilibrium macroscopic laws, such as those of hydrodynamics, from the microscopic laws has not been developed as generally as in the equilibrium case (the derivation of thermodynamic relations by equilibrium statistical mechanics). The microscopic analysis of nonequilibrium phenomena, however, has achieved a considerable degree of success for the particular case of dilute gases. In this case, the kinetic theory, or transport theory, allows one to relate the transport of matter or of energy, for example (as in diffusion, or heat flow, respectively), to the mechanics of the molecules that make up the system. 

The aim of this section is to give the steady-state probability distribution in phase space. This then provides a basis for nonequilibrium statistical mechanics, just as the Boltzmann distribution is the basis for equilibrium statistical mechanics. The connection with the preceding theory for nonequilibrium thermodynamics will also be given. 

An analogy may be drawn between the phase behavior of weakly attractive monodisperse dispersions and that of conventional molecular systems provided coalescence and Ostwald ripening do not occur. The similarity arises from the common form of the pair potential, whose dominant feature in both cases is the presence of a shallow minimum. The equilibrium statistical mechanics of such systems have been extensively explored. As previously explained, the primary difficulty in predicting equilibrium phase behavior lies in the many-body interactions intrinsic to any condensed phase. Fortunately, the synthesis of several methods (integral equation approaches, perturbation theories, virial expansions, and computer simulations) now provides accurate predictions of thermodynamic properties and phase behavior of dense molecular fluids or colloidal fluids [1]. 

Experiments indicate that the smooth variations of thermodynamic properties (e.g., V, Ky, and the specific heat at constant pressure Cp) with temperature are intermpted by the kinetic process of glass formation, leading to cooling rate dependent kinks in these properties as a function of temperature. In our view, these kinks cannot be described by an equilibrium statistical mechanical theory, but rather are a challenge for a nonequilibrium theory of glass formation. Nonetheless, some insight into the origin of these kinks and the qualitative... 

Refs. [i] Prigogine I-Autobiography http //nobelprize.org/chemistry/ laureates/1977/prigogine [ii] Prigogine I (1967) Introduction to the thermodynamics of irreversible processes. Wiley Interscience, New York [iii] Prigogine I (1962) Non-equilibrium statistical mechanics. Wiley Interscience, New York [iv] Glansdorff P, Prigogine I (1971) Thermodynamic theory of structure stability and fluctuations. Wiley, London... 

Both diffusion and conduction are nonequilibrium (irreversible) processes and are therefore not amenable to the methods of equilibrium thermodynamics or equilibrium statistical mechanics. In these latter disciplines, the concepts of time and change are absent. It is possible, however, to imagine a situation where the two processes oppose and balance each other and a pseudoequilibrium obtains. This is done as follows (Fig. 4.62). 

We have also pointed out here the formal connection between our formalism and the existing numerical algorithms in special cases (CP-algorithm and time-dependent optimized potential) as well as avenues to go beyond these to include non-adiabatic processes. It should be stressed that unlike the existing theories, our framework is based on a stationary action principle, which facilitates incorporation of the initial constraint of thermodynamic equilibrium. This development is made feasible by working in a superspace formalism. This work thus provides a practical theoretical framework for studying the non-equilibrium statistical mechanics of systems initially in thermodynamic equilibrium. 

In making models, we must respect the principles of equilibrium statistical mechanics, but cannot wholly rely on them, since we have every reason to believe that in polymer crystallization we have only a frustrated approach towards thermodynamic equilibrium. Most of the models are in some way or another founded on their authors conceptions of the nature of the process of high-polymer crystallization from the melt. That is a process on which direct detailed information is hard to get, and some imaginative extrapolation from what one knows about related problems is almost unavoidable. 

The Blume-Emery-Griffiths (BEG) model is one of the well-known spin lattice models in equilibrium statistical mechanics. It was originally introduced with the aim to account for phase separation in helium mixtures [30]. Besides various thermodynamic properties, the model has been extended to study the structural phase transitions in many bulk systems. By... 

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 structure of the chapter is as follows. In Section 2, we review the basic postulates of the equilibrium thermodynamics. The equilibrium statistical mechanics based on generalized entropy is formulated in a general form in Section 3. In Section 4, we describe the Tsallis statistics and analyze its possible connection with the equilibrium thermodynamics. The main conclusions are summarized in the final section. 

In comparison with the equilibrium thermodynamics, the system in the equilibrium statistical mechanics is described by two additional elements the microstates of the system and the... 

In the equilibrium statistical mechanics, the unknown probabilities of microstates p, are found from the second part of the second law of thermodynamics, i.e., from the constrained extremum of the thermodynamic potential (Eq. (29)) as a function of the variables (pv pw) under the condition that the variables (pv. .., pw) satisfy Eq. (27). Moreover, it is supposed that the value of the entropy in the i th microstate of the system is a function of the probability pt of this microstate, i.e., =Sf=Sf(pf). Then to determine the unknown probabilities [pt] at... 

Finally, it should be mentioned that the equilibrium statistical mechanics is thermodynamically self-consistent if the statistical variables (x1,. .., xn), the potentials (/, g,. ..), and the variables (u1,. ..,un) are homogeneous variables of the first- or zero-order satisfying Eqs. (21)-(25). 

In conclusion, let us summarize the main principles of the equilibrium statistical mechanics based on the generalized statistical entropy. The basic idea is that in the thermodynamic equilibrium, there exists a universal function called thermodynamic potential that completely describes the properties and states of the thermodynamic system. The fundamental thermodynamic potential, its arguments (variables of state), and its first partial derivatives with respect to the variables of state determine the complete set of physical quantities characterizing the properties of the thermodynamic system. The physical system can be prepared in many ways given by the different sets of the variables of state and their appropriate thermodynamic potentials. The first thermodynamic potential is obtained from the fundamental thermodynamic potential by the Legendre transform. The second thermodynamic potential is obtained by the substitution of one variable of state with the fundamental thermodynamic potential. Then the complete set of physical quantities and the appropriate thermodynamic potential determine the physical properties of the given system and their dependences. In the equilibrium thermodynamics, the thermodynamic potential of the physical system is given a priori, and it is a multivariate function of several variables of state. However, in the equilibrium... 

Statistical mechanics is normally further divided into two branches, one dealing with equilibrium systems, the other with non-equilibrium systems. Equilibrium statistical mechanics is sometimes called statistical thermodynamics [70]. Kinetic theory of gases is a particular field of non-equilibrium statistical mechanics that focuses on dilute gases which are only slightly removed from equilibrium [28]. 

The fundamental problem in classical equilibrium statistical mechanics is to evaluate the partition function. Once this is done, we can calculate all the thermodynamic quantities, as these are typically first and second partial derivatives of the partition function. Except for very simple model systems, this is an unsolved problem. In the theory of gases and liquids, the partition function is rarely mentioned. The reason for this is that the evaluation of the partition function can be replaced by the evaluation of the grand canonical correlation functions. Using this approach, and the assumption that the potential energy of the system can be written as a sum of pair potentials, the evaluation of the partition function is equivalent to the calculation of... 

In describing thermodynamic and equilibrium statistical-mechanical behaviors of a classical fluid, we often make use of a radial distribution function g r). The latter for a fluid of N particles in volume V expresses a local number density of particles situated at distance r from a fixed particle divided by an average number density p = NjV), when the order of IjN is negligible in comparison with 1. Various thermodynamic quantities are related to g(r). For a single-component monatomic system of particles interacting with a pairwise additive potential 0(r), the relationship connecting the pressure P to g(r) is the virial theorem, ... 

In equilibrium statistical mechanics, one is concerned with the thermodynamic and other macroscopic properties of matter. The aim is to derive these properties from the laws of molecular dynamics and thus create a link between microscopic molecular motion and thermodynamic behaviour. A typical macroscopic system is composed of a large number N of molecules occupying a volume V which is large compared to that occupied by a molecule ... 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The connection between statistical mechanics and thermodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a harmonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

This section summarizes the classical, equilibrium, statistical mechanics of many-particle systems, where the particles are described by their positions, q, and momenta, p. The section begins with a review of the definition of entropy and a derivation of the Boltzmann distribution and discusses the effects of fluctuations about the most probable state of a system. Some worked examples are presented to illustrate the thermodynamics of the nearly ideal gas and the Gaussian probability distribution for fluctuations. 

The equilibrium state in thermodynamics is characterized by detailed balance. When there is detailed balance in a system, the self-organization far from equilibrium associated with non-equilibrium statistical mechanics cannot occur. 

Kinetic theory, non-equilibrium statistical mechanics and non-equilibrium molecular dynamics (NEMD) have proved to be useful in estimating both straight and cross-coefficients such as thermal conductivity, viscosity and electrical conductivity. In a typical case, cross-coefficient in case of electro-osmosis has also been estimated by NEMD. Experimental data on thermo-electric power has been analysed in terms of free electron gas theory and non-equilibrium thermodynamic theory [9]. It is found that phenomenological coefficients are temperature dependent. Free electron gas theory has been used for estimating the coefficients in homogeneous conductors and thermo-couples. 

The presence of / > in Eq. (3) gives rise to oscillatory, nondissipative dynamics for A(/), while F gives rise to decay and dissipation. Since iClyX and ia> are composed entirely of equal-time correlation functions, these quantities are, in principle, available from equilibrium statistical mechanics. On the other hand, irreversible thermodynamics gives no expression for L, aside from certain symmetry requirements, the Onsager relations. 

In the classical free statistics, the number of the functional groups on the surface of a tree-like cluster is of the same order of that of the groups inside the cluster, so that a simple thermodynamic limit without surface term is impossible to take. The equilibrium statistical mechanics for the polycondensation was refined by Yan [14] to treat surface correction in such finite systems. He found the same result as Ziff and Stell. Thus the treatment of the postgel regime is not unique. The rigorous treatment of the problem requires at least one additional parameter defining relative probability of occurrence of infra- and intermolecular reactions in the gel. 

In equilibrium statistical mechanics, and more particularly in the theory of phase transitions, the thermodynamic limit is considered to eliminate size effects (such as the rounding-off of isotherms) which tend to blur the essential features we are trying to understand. The same situation occurs in nonequilibrium statistical mechanics here we mainly want to get rid of recurrences, which are not only physically undesirable but also present hopelessly entangled situations (e.g., the mean recurrence time of the dog-flea model becomes a meaningless concept in the limit of large systems). 

The Langmuir isotherm can also be derived by other methods including statistical mechanics, thermodynamics, and chemical reaction equilibrium. The last approach is especially straightforward and useful, and it is developed as follows. For nondissociative chemisorption, the adsorption step is represented as a reaction, i.e., for an adsorbing gas-phase molecule. A, which adsorbs on a site, ... 

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