Lectures on Statistical Mechanics

The following sections are included:

• Dedication
• Preface
• Contents
• Glossary

Statistical Mechanics is concerned with physical systems that contain so many degrees of freedom that a detailed description is both impossible and useless. As Loschmidt established, there are about 10²⁰ molecules in a few cubic centimeters (or milliliters) of air at the usual room conditions, and each molecule needs six coordinates to specify its position and its orientation in space, plus as many velocities. Not only is it hopeless to attempt a description that takes full account of all these ~ 10²¹ coordinates, it is also pointless because we would not have any use of such a lot of information if it were made available…

After this review of some aspects of thermodynamics, we now turn to statistical mechanics proper. It deals with the molecular, microscopic foundation of thermodynamics, with the goal of understanding the rather simple properties of large systems — simple in the sense that we can characterize them with a small number of parameter values — as a consequence of the physical laws obeyed by the very many (~ 10²⁰ for a millimole) atomic particles that compose the system…

The following sections are included:

• Getting started
• Occupation numbers; bosons; fermions
• Average occupation numbers
  • Bosons
  • Fermions
  • Dilute-gas limit
• Photon gas
• Phonon gas
• Bose–Einstein condensation
• Fermion gas
• Electron gas in a metal
• Pressure and energy density
• The Third Law

The following sections are included:

• Thomas–Fermi model
• One-dimensional Ising model
• Renormalization group method (Ising chain)
• Renormalization group method (Ising square lattice)
• Order-disorder phase transition
• Onsager’s solution
• Three-dimensional Ising model. Spontaneous magnetization
• Mean-field approximation

The following sections are included:

• Large and small contributions
• Gibbs–Bogolyubov inequality
  • Example: Ising model (mean-field approximation)
  • Quantum aspects
• Peierls inequality
• Minimum principle for the free energy
• Ising chain with next-next-neighbors interaction
• Ising model and two-state quantum system

The following sections are included:

• Interacting atoms
• Clusters
• Cluster expansion; virial expansion
• Finite volume, finite number of particles
• Second virial coefficient: van der Waals gas

We turn our attention to stochastic processes, for which Brownian motion is a prime example. The original observation was about the erratic motion of pollen immersed in water, as viewed under a microscope, a never-ending random walk of the pollen grains resulting — as we now know — from the collision of water molecules with the much larger grains. These collisions transfer momentum between projectile (water) and target (grain) and so lead to sudden changes of the velocity vector and thus to the erratic motion of the pollen grain.

The following sections are included:

• Exercises for Chapters 1–7
• Hints

The following section is included:

• Index