Statistical mechanical foundations

MSN.53.1. Prigogine and F. Henin, Kinetic Equation, Quasiparticles and Entropy, in Proceedings, lUPAP Meeting on Statistical Mechanics, Foundations and Applications, Copenhagen 1966, Benjamin, New York, 1967. 

In order to introduce basic equations and quantities, a preliminary survey is made in Section II of the statistical mechanics foundations of the structural theories of fluids. In particular, the definitions of the structural functions and their relationships with thermodynamic quantities, as the internal energy, the pressure, and the isothermal compressibility, are briefly recalled together with the exact equations that relate them to the interparticular potential. We take advantage of the survey of these quantities to introduce what is a natural constraint, namely, the thermodynamic consistency. 

Bennema, P. On the Crystallographic and Statistical Mechanical Foundation of the Hartman Perdok Theory. In Crystal Growth of Organic Materials. Conference Series, USA. 1996 15-21. 

A. D. Sokal, Monte Carlo Methods in Statistical Mechanics Foundations and New Algorithms, Cours de Troisieme Cycle de la Physique en Suisse Romande (Lausanne, June 1989). 

At its foundation level, statistical mechanics mvolves some profound and difficult questions which are not fiilly understood, even for systems in equilibrium. At the level of its applications, however, the rules of calculation tliat have been developed over more than a century have been very successfLil. 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics 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 hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

Gibbs, J.W. (1902) Elementary Principles in Statistical Mechanics, Developed with Special Reference to the Rational Foundations of Thermodynamics (Yale University Press, New Haven). 

The work on gas theory had many extensions. In 1865 Johann Josef Loschmidt used estimates of the mean free path to make the first generally accepted estimate of atomic diameters. In later papers Maxwell, Ludwig Boltzmann, and Josiah Willard Gibbs extended the rrratherrratics beyorrd gas theory to a new gerreralized science of statistical mechanics. Whenjoined to quantum mechanics, this became the foundation of much of modern theoretical con-derrsed matter physics. 

This volume also contains four appendices. The appendices give the mathematical foundation for the thermodynamic derivations (Appendix 1), describe the ITS-90 temperature scale (Appendix 2), describe equations of state for gases (Appendix 3), and summarize the relationships and data needed for calculating thermodynamic properties from statistical mechanics (Appendix 4). We believe that they will prove useful to students and practicing scientists alike. 

There are two other methods in which computers can be used to give information about defects in solids, often setting out from atomistic simulations or quantum mechanical foundations. Statistical methods, which can be applied to the generation of random walks, of relevance to diffusion of defects in solids or over surfaces, are well suited to a small computer. Similarly, the generation of patterns, such as the aggregation of atoms by diffusion, or superlattice arrays of defects, or defects formed by radiation damage, can be depicted visually, which leads to a better understanding of atomic processes. 

J. Koi3fta, Principles of Electrochemistry, Wiley, New York, 1987 J. Goodisman, Electrochemistry Theoretical Foundations, Quantum and Statistical Mechanics, Thermodynamics, the Solid State, Wiley, New York, 1987 G. Battistuzzi, M. Bellei, and M. Sola, J. Biol. Inorganic Chem. 11, 586-592 (2006) R. Heyrovska, Electroanalysis, 18, 351-361 (2006) G. Battistuzzi, M. Borsari, G. W. Ranters, E. de Waal, A. Leonard , and M. Sola, Biochemistry 41, 14293-14298 (2002). 

MSN.62. I. Prigogine, Dynamic foundations of thermodynamics and statistical mechanics, in A Critical Review of Thermodynamics, E. B. Stuart, B. Gal-Or, and A. Brainard, eds., Mono Book Corp., Baltimore, 1970, pp. 1-18. 

MSN.93. I. Prigogine and A. P. Grecos, Topics in nonequilibrium statistical mechanics, in Problems in the Foundations of Physics, LXXII Corso, Soc. Ital. Fisica. 

Hunter, R. J., Foundations of Colloid Science, Vol. 2, Clarendon Press, Oxford, England, 1989. (Undergraduate and graduate levels. Along with Volume 1, these two volumes cover almost all the topics covered in the present chapter at a more advanced level. Volume 1 discusses DLVO theory and thermodynamic approaches to polymer-induced stability or instability and is at the undergraduate level. Volume 2 presents advanced topics (e.g., statistical mechanics of concentrated dispersions, rheology of dispersions, etc.).)... 

It is noteworthy that Gibbs himself was acutely aware of the qualitative failures of 19th-century molecular theory (as revealed, for example, by erroneous classical predictions of heat capacities Sidebar 3.8). In the preface to his Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics (published in the last year of his life), Gibbs wrote ... 

The importance of this fact for statistical mechanics was stressed by A.J. Khinchin, Mathematical Foundations of Statistical Mechanics (G. Gamow, transl., Dover Publications, New York 1949) p. 63. But he called A a sum function only if n = 1. 

This is a simple analogy of Birkhoff s ergodic theorem for dynamical systems, see A.I. Khinchin, Mathematical Foundation of Statistical Mechanics (Dover, New York 1949) L.E. Reichl, A Modern Course in Statistical Physics (University of Texas Press, Austin, TX 1980) ch. 8. 

The theoretical foundation for reaction dynamics is quantum mechanics and statistical mechanics. In addition, in the description of nuclear motion, concepts from classical mechanics play an important role. A few results of molecular quantum mechanics and statistical mechanics are summarized in the next two sections. In the second part of the book, we will return to concepts and results of particular relevance to condensed-phase dynamics. 

After Flory 46 and Stockmayer1471 published books describing the statistical mechanics of chain molecules and molecular fluids, respectively, which further laid the foundation of modern polymer analyses, theoretical examination of branched macromolecules continued to advance. 

Gibbs, J.W. Elementary principles in statistical mechanics, developed with essential reference to the rational foundation of thermodynamics. N.Y. (1902). 

Jaynes, E. T., Foundations of probability theory and statistical mechanics, in "Delaware Seminar in the Foundation of Physics" (M. Bunge, ed.). Springer, New York (1967). 

At least in this respect, therefore, a further development of the foundations of statistical mechanics has unquestionably become necessary. 

In this concise classic, Paul Ehrenfest, one of the 20th century s greatest physicists, reformulated the foundations of the statistical approach in mechanics. Originally published in 1912 as an article for the German Encyclopedia of Mathematical Sciences , it has lost little of its scientific and didactic value, and no serious student of statistical mechanics can afford to remain ignorant of this great work. 

---