# Random physics

Alberto Verga, research notebook

$$\newcommand{\I}{\mathrm{i}} \newcommand{\E}{\mathrm{e}} \newcommand{\D}{\mathop{}\!\mathrm{d}} \newcommand{\Dd}{\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}#1}}$$

Work in progress

# Lectures on Statistical Physics

## Bibliography

• Landau, D. and Lifshitz, E., Statistical Physics (Pergamon Oxford, 1980). The best classical text on statistical physics.
• Kardar, M., Statistical Physics, vol. 1 Particles, vol. 2 Fields (Cambridge, 2007). Clear presentation, interesting applications.
• Sethna, J. P., Statistical Mechanics, Entropy, Order Parameter and Complexity (Oxford, 2006). Very intersting and up to date text with a wealth of applications; intermediate level.
• Schwabl, F., Statistical Mechanics (Springer, 2006). Excellent presentation of the basis of statistical physics at a conceptual level. This book, primary addressed to undergraduate students, deal with more advanced subjects through well chosen applications. Highly recommended.

## Syllabus (2018-2019)

This course addresses to students who followed a first statistical physics course at the level of the classical book by Kittel, “Thermal physics” (1980).

In addition to pleanry lectures, the course proposes complementary exercises (marked EX), and a series of “applications”, sensed give you a grasp of the diversity of domains for which statistical mechanics methods are useful (even outside the traditional physical field), including problems whose solution needs numerical methods.

A list of homeworks is given in “Problems, exercises and applications”. Selected solution and complements can be found in

### Principles

• Statistical ensemble: from the microscopic states to thermodynamics
• Density matrix and the microcanonical distribution of an isolated system
• Gibbs ensembles
• Gibbs distribution (classical and quantum); the partition function and the free energy
• Thermodynamic quantities

### Noninteracting systems

• Energy equipartition
• Ideal gas
• Rotation and vibration of molecules
• Bose distribution, photons and phonons, Debye, Bose condensation
• Fermi distribution, degenerated electron gas
• Pauli paramagnetism and mean field ferromagnetism
• Landau diamagnetism

### Interactions

• Ising model, low and high energy expansions in 2D
• Cumulant expansion and virial coefficients
• Van der Waals equation and liquid-gas transition
• Bose and Fermi liquids

### Phase transitions and fluctuations

• Linear response and correlations
• Phenomenology and scaling laws
• Order parameter and Landau free energy
• Symmetry breaking

### Applications

• From the binomial distribution to the large number and central limit theorems
• Chaos, ergodicity and mixing
• Mixing entropy and the Gibbs paradox
• Maxwell demon and information theory
• Two level systems
• Quantum oscillator
• Monte Carlo for the ising model
• 2x2 random matrices, Wiener surmise
• Lenard-Jones potential and molecular dynamics
• White dwarf stars
• Black hole entropy
• Yang-Lee theory of phase transitions
• Quantum spin in a transverse field