Statistical physics

Abstract

Thermodynamics is a phenomenological theory for interacting many-body systems, whose empirical laws are taken from experiments. The aim of statistical physics is to derive these phenomenological laws starting from a true microscopic description of the many-body system. It is then needed to perform an appropriate average over the many microscopic degrees of freedom. The fundamental assumption of statistical physics is that every microscopic state that is accessible to the system is equally probable. Therefore, the problem of finding the correct probability distribution over which we have to average reduces to the problem of finding the total number of states of the system.

In this chapter, we discuss the basic concepts and techniques that are used in statistical physics to describe many-body systems. In particular, we briefly consider the three most frequently used statistical ensembles, namely the micro-canonical, the canonical and the grand-canonical ensemble. In the thermodynamic limit these ensembles become essentially equivalent, such that we can choose the ensemble that is most convenient to work with. In our case, this is nearly always the grand-canonical ensemble, which we then apply to the study of the ideal gases. We treat the classical gas that obeys Maxwell-Boltzmann statistics, the Bose gas that obeys Bose-Einstein statistics and the Fermi gas that obeys Fermi-Dirac statistics. At low temperatures, the ideal Bose gas undergoes a phase transition better known as Bose-Einstein condensation. A thorough knowledge of the ideal gases is important for understanding the interacting quantum gases, which is the topic of the second part of this book.