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According to classical mechanics, equations of motion supplemented by initial conditions uniquely determine the subsequent evolution of a given system. For typical systems of our interest, however, the number of mechanical degrees of freedom is of the order of 1024. One cannot possibly write down 1024 equations of motion, much less solve them. It is also impossible to specify the initial conditions for such a system with a required accuracy. Moreover, even if we could somehow accomplish all of this, it would be entirely impossible to comprehend the resulting list of coordinates and momenta at any instant. Despite a hopeless scenario this observation might suggest, behavior of a macroscopic system is surprisingly regular as we have seen in thermodynamics. It is as if laws governing behavior of a macroscopic system are quite different from those governing its behavior at a microscopic level. In this chapter, we examine the connection between these two distinct ways of looking at a macroscopic system.
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Gibbs J W (1981) Elementary principles in statistical mechanics. Ox Bow Press, Connecticut Our Sect. 3.9 followed Chap. 4 of the book, where the canonical ensemble was introduced. The same chapter demonstrates the relationship between statistical mechanics and thermodynamics and provides further motivation for the canonical ensemble.
Lanczos C (1986) The variational principles of mechanics. Dover, New York The phrase “Euler force” is introduced in p. 103.
Landau L D, Lifshitz E M (1980) Statistical physics: Part 1, 3rd edn. Pergamon Press, New York In developing the basic principles of statistical mechanics, we loosely followed Chap. 1 of the book. Our treatment of a rotating body is based on their treatment of the subject, in particular, Sects. 26 and 34.
Schrödinger E (1989) Statistical thermodynamics. Dover, New York Chapter 3 of the book provides a detailed discussion on the meaning of setting the constant term in (3.67) to zero. He uses Boltzmann’s entropy formula rather than that of Gibbs. But, the former can be derived from the latter as we shall see later.
Tolman R C (1979) The principles of statistical mechanics. Dover, New York For an extended discussion on the statistical approach and its validity, see Chap. 3. A brief summary is in Sect. 25.
Whitaker S (1992) Introduction to fluid mechanics. Krieger Publishing Company, Florida It is far more natural and physically compelling to write down laws of physics, such as the conservation of mass, Newton’s equation of motion, and the first law of thermodynamics, for a (possibly moving and deforming) body than to do so for a control volume fixed in space. A derivation of the equation of continuity from this more satisfactory stand point is found in Chap. 3.
- Classical Statistical Mechanics