# Statistical and Thermal Physics 2E: Statistical Mechanics I

## Chapter 4: Methodology

We combine ideas from thermodynamics and probability theory to introduce *statistical mechanics*, which is the study of how microscopic behavior leads to macroscopic behavior in systems of many constituents. There only are a few models, such as the ideal gas whose macroscopic averages can be calculated analytically starting from the microscopic model.

In this chapter we use the computer to illustrate the counting of microstates and the behavior of some many particle systems.

## Chapter 5: Magnetic Systems

We apply the tools of statistical mechanics to magnetic systems. The most important and simplest model containing interactions, is the Ising model defined by the energy of a microstate $$E = -J \sum_{\lt ij \gt} s_i s_j - H \sum_i s_i,$$ where $J$ is the energy of interaction and is positive for a ferromagnetic interaction, $s_i = \pm 1$ is the spin of a spin 1/2 particle such as an electron, $\lt ij \gt$ denotes nearest-neighbor pairs of spins located on a lattice, and $H$ is the applied external magnetic field.

In addition to studying the Ising model and its variations, we consider various techniques such as transfer matrices, mean-field theories, and computer simulations, which have become essential for our understanding of magnetic systems.

## Chapter 6: Many-Particle Systems

We apply the techniques of statistical mechanics to many particle systems. Our focus will be on systems where we can treat the particles as noninteracting. The first half of the chapter discusses classical and semiclassical systems, and the second half discusses systems where quantum mechanics is essential.

The key results of classical systems are the Maxwell velocity and speed distributions and the equipartition theorem, which are valid for gases, liquids, and solids.

Quantum systems that are discussed include blackbody radiation (considered as a collection of photons), the Debye model of solids (considering quantized sound waves or normal modes as phonons), Fermi systems such as electrons in metals, neutron stars, and white dwarfs, and Bose gases which can condense into a single ground state at low enough temperatures. To study these systems we use the grand canonical ensemble to treat the indistinguishability of the particles, and introduce the concepts of single particle density of states and the mean occupation of each single particle state (distribution function) at a temperature $T$ and chemical potential $\mu$.