It is time now to review the progress we have made so far. Our starting point was the fundamental atomic nature of matter. We also assumed that interactions between individual atoms and molecules are governed by the laws of mechanics, either classical or quantum depending on the particular circumstances. In the first chapter we developed a simple qualitative picture of the way molecules interact in a complex system. This qualitative picture allowed us to describe classical thermodynamics. In particular we were able to introduce the key concepts of equilibrium, temperature, entropy, and we were able to point out that complex systems in equilibrium can be well described with only a very small number of state variables. Given that matter is made of very large numbers of independent atoms or molecules this is an extraordinary result.
In the second chapter we began the process of formalizing the qualitative link from mechanics to thermodynamics. The formal development starts of course with mechanics. Mechanics on its own, however, is not enough, as it does not contain the concept of thermal equilibrium. The solution we presented was to define thermal equilibrium probabilistically, and the theory which results is statistical mechanics. This solution requires a fundamentally new idea which is not present in mechanics. Once this idea is accepted, the further development of the subject is straightforward if perhaps technically challenging.
Statistical mechanics is a very successful physical theory. In this book, we have applied it to a variety of systems including non-interacting and interacting gases, paramagnetic and spin systems, quantum systems with both Bose and Fermi statistics, astrophysics, helium superfluids and solids.