Courses

Rigorous Results in Statistical Mechanics II: Nonequilibrium

642:564

Rigorous Results in Statistical Mechanics II: Nonequilibrium
Joel Lebowitz
Subtitle: Emergent Phenomena
Text: None

Prerequisites: talk with me

Description: Statistical Mechanics: Exact Results

Emergent Phenomena in Multicomponent Systems
Statistical mechanics successfully explains how properties of macroscopic systems, such as a glass of water, originate in the cooperative behavior of atoms and molecules, the microscopic constituents of all matter. Some of the observed phenomena are paradigms of emergent behavior,
having no direct counterpart in the properties or dynamics of individual atoms.

Particularly fascinating and important examples of such emergent phenomena are phase transitions. These correspond to abrupt changes in the behavior of a macroscopic system as some parameter is changed across some specified value. A familiar example is the melting of ice or the boiling of water at a precise value of the temperature, depending on the pressure. These would (or should) be astonishing if they were not so familiar.

Fortunately, many of these striking features of macroscopic systems can be obtained from simplified microscopic models. These can be treated in a mathematically rigorous way.  Surprisingly these models are also applicable to systems where the microscopic entities are not atoms or molecules but viruses, fish, or Wall Street traders.

The course will develop the concepts and necessary mathematical tools for describing cooperative phenomena in systems consisting of many components.

For requirements please contact me: This email address is being protected from spambots. You need JavaScript enabled to view it.: Room 612, Hill Center

Rigorous Results in Statistical Mechanics I: Equilibrium

Course ID/Name 

16:642:563

Description:

The course will cover traditional areas of statistical mechanics with a mathematical flavor. It will describe exact results where available and heuristic physical arguments where applicable. A rough outline is given below:

Course outline:

  1. Overview: microscopic vs. macroscopic descriptions; microscopic dynamics and thermodynamics.
  2. Energy surface; microcanonical ensemble; ideal gases; Boltzmann’s entropy, typicality.
  3. Alternate equilibrium ensembles; canonical, grand-canonical, pressure, etc. Partition functions and thermodynamics.
  4. Thermodynamic limit; existence; equivalence of ensembles; Gibbs measures.
  5. Cooperative phenomena: phase diagrams and phase transitions; probabilities, correlations and partition functions. Law of large numbers, fluctuations, large deviations.
  6. Ising model, exact solutions. Griffith’s, FKG and other inequalities; Peierle’s argument; Lee-Yang theorems.
  7. High temperature; low temperature expansions; Pirogov-Sinai theory.
  8. Fugacity and density expansions.
  9. Mean field theory and long range potentials.
  10. Approximate theories: integral equations, Percus-Yevick, hypernetted chain. Debye-Hückel theory.
  11. Critical phenomena: universality, renormalization group.
  12. Percolation and stochastic Loewner evolution.

If you have any questions about the course please email me: This email address is being protected from spambots. You need JavaScript enabled to view it.. We can then set up a time to meet.