Course Title: Statistical Mechanics From Microscopic Dynamics to Macroscopic Behavior

Course Number: 16:642:653 Statistical Mechanics I

Instructor: Professor Joel L. Lebowitz

Course Description:

A). Statistical mechanics tries to explain the behavior of macroscopic systems, such as the boiling and freezing of liquids, in terms of the dynamics of their microscopic constituents: atoms and molecules. Crucial to the explanation is the very, very large number of these quasi-independent microscopic entities, typically of order 1020 or more, making up any macroscopic object. This brings in the "statistics" and permits the autonomous description of the behavior of such systems via thermodynamics and transport equations. These utilize the emergent macroscopic concepts of temperature, pressure, etc.

The subject was developed initially within the framework of classical mechanics, where the dynamics are given by Hamiltonian equations of motion. The transition from Newtonian dynamics to quantum mechanics has, surprisingly, left the basic framework of the subject intact.

There have also been many extensions of statistical mechanical ideas to cooperative phenomena in systems composed of many complex interacting individual entities. These entities can be viruses, plants, humans, stars or anything in between. Striking features of such collective phenomena are phase transitions. These correspond to discontinuous changes in the properties or behavior of such a system as some parameter is changed similar in some way to the melting of ice or the boiling of water. Such transitions correspond to epidemics and revolutions, etc. The mathematical equations which best describe the time evolution of such interacting systems may be deterministic, stochastic, or a combination of the two.

B). Possible topics to be covered:
1.Historical background: thermodynamics, entropy, kinetic theory.
2.Time evolution: resolution of apparent conflict between microscopic reversibility and macroscopic irreversibility, "typical" behavior and the approach to equilibrium.
3.Equilibrium formalism: micro, macro and grand canonical ensembles, ideal gases, lattice models, fugacity and virial expansion, harmonic crystals.
4.Emergent phenomena: Phase transitions, van der Waals equation of state, mean field theories.
5.Exact results; Ising model, Peirles' argument, Onsager solution, inequalities, Potts model, percolation.
6.Quantum systems: Equilibrium formalism, ideal gases, photons, phonons, Bose-Einstein condensation, liquid helium.
7.Transport: Brownian motion, diffusion equation, Navier-stokes equations, Boltzmann equations.
8. Stochastic evolution of lattice systems: Glauber and Kawasaki dynamics, Cellular automata.
9. Non-equilibrium stationary states in open and driven systems.
10. Applications of statistical mechanical ideas to cooperative phenomena in ecological, biological and social systems: contact processes, voter models, traffic models, etc.

Requirements:

The course will be informal and interactive. If interested, please contact me.

Established class schedule:

Monday: 3:20pm - 4:40pm, Hill 423
Thursday: 5:00pm - 6:20pm, Hill 423