Statistical mechanics
Statistical mechanics, quantitative study of systems consisting of a large number of interacting elements, such as the atoms or molecules of a solid, liquid, or gas, or the individual quanta of light making up electromagnetic radiation. Although the nature of each individual element of a system and the interactions between any pair of elements may both be well understood, the large number of elements and possible interactions can present an almost overwhelming challenge to the investigator who seeks to understand the behavior of the system. Statistical mechanics provides a mathematical framework upon which such an understanding may be built. Since many systems in nature contain large number of elements, the applicability of statistical mechanics is broad. In contrast to thermodynamics, which approaches such systems from a macroscopic, or large-scale, point of view, statistical mechanics usually approaches systems from a microscopic, or atomic-scale, point of view.
The foundation of statistical physics was laid towards the end of the nineteenth century by James Clerk Maxwell, Ludwig Boltzmann, Josiah Willard Gibbs and largely completed by Albert Einstein in 1905. Maxwell's kinetic theory of gases can be said to represent the starting point. Boltzmann made the argument more general and introduced the concept of ensembles. Instead of considering a single system he considered a large number of equivalent systems which had been prepared in the same way. He obtained probabilities for the possible states by calculating the relative frequency that a given state would occur in the ensemble. Gibbs followed up by establishing the equivalence of statistical physics and thermodynamics. He did this by stressing an analogy with classical mechanics, which was the best understood branch of theoretical physics at the time. Finally Einstein rounded out the picture by his theory of fluctuations, diffusion and Brownian motion.
These developments happened before the advent of quantum mechanics. Einsteins theory of the photoelectric effect only appeared in 1905, and a comprehensive theory of quantum mechanics only became available two decades later. However, statistical physics becomes simpler if one can appeal to some quantum concepts. In classical mechanics we describe a microscopic system by specifying the coordinates and momenta of the particles. The allowed value of these form a continuum. The procedure of counting requires, however discrete states. The modern way of getting around this difficulty is to consider classical mechanics as a limiting case of quantum mechanics, and we will take this approach, rather than following the historical route.
The language of statistical mechanics has changed enormously since the work of Boltzmann late in the 19th century. Old books and papers are hard to read as both the terminology and the notation are foreign. Anyone interested in the development of classical mechanics needs to know that it grew primarily out of the kinetic theory of gases and was developed into statistical mechanics first by Ludwig Boltzmann and then by J. Willard Gibbs. A review of Boltzmann's work was written by Paul and Tatiana Ehrenfest and was published in the German Encyclopedia of Mathematical Sciences in 1912. This is not an easy book for novices for the reasons given above, but it repays close study. It also contains a bibliography of the literature up to 1912 that is no longer well known. The English translation is titled "The Conceptual Foundations of the Statistical Approach in Mechanics" was published by Dover in various editions.