Classical and Quantum Statistical Mechanics of Permutation Representations

Paul Seagraves

Fabien Paillusson

This chapter follows on from the previous chapter in quantum statistical mechanics but specialising on systems with identical particles. Using Gibbs prescritpion on generic states from Chapter 3, the occupation number representation is introduced. Constraints imposed on statistics by irreducible representations of the permutation group are discussed. These group-theoretic considerations are used to justify the use of Gentile’s parastatistics. Fermions and bosons are introduced as special cases of Gentile’s statistics, corresponding to the trivial representation of the permutation group for bosons, and the sign representation of the permutation group for fermions. Basic applications to fermions and bosons is given, including the Fermi–Dirac and Bose–Einstein statistics. A detailed expose of why photons are said to have zero chemical potential is also proposed.

Phil Attard

The seventh chapter is concerned with the formulation of quantum statistical mechanics from first principles. It is shown that interactions with the reservoir or environment cause the subsystem wave function to collapse into a mixture of pure entropy states, which obey the axioms of classical probability theory. A dual representation in position and momentum states allows a permutation loop expansion for particle symmetrisation, and a second expansion accounting for the non-commutativity of the position and momentum operators. The leading term gives classical statistical mechanics, and higher order terms gives successive quantum corrections that become increasingly important for low temperatures, high densities, and small masses. From the permutation loop expansion, explicit results for the quantum ideal gas and the quantum harmonic oscillator are derived. The role of entropy in the classical manifestation of the universe given the underlying quantum mechanical wave equation is discussed.

Massimo V. Fischetti, William G. Vandenberghe

Simon W. Saunders

This chapter analyzes permutation symmetry, focusing on the proper understanding of pchapter indistinguishability in classical statistical mechanics and in quantum theory. It shows that it is possible to treat the statistical mechanical statistics for classical pchapters as invariant under permutation symmetry and argues that while the concept of indistinguishable, permutation invariant, classical pchapters is coherent, it is in contradiction with many claims found in the literature. The chapter also contends that the concept of permutation symmetry should be considered in the same level as other symmetries and invariances of physical theories.

D. I. Blokhint︠s︡ev