In case anybody is interested in the (very entertaining) lecture `Advanced Statistical Mechanics' by Jens Eisert, here is a short summary of the content. In fact, this is part of my exam preparation ;) I omitted most of the introduction on basic quantum mechanics, as this knowledge is required for the course anyway.

# Introduction

- Classification:
*isolated thermodynamic system*: no exchange of energy or matter.*closed system*: no exchange of matter.*open system*: the rest. - extensive quantities scale with the system size, intensive quantities do not
- Law of Boyle-Mariotte:
- Maxwell's demon

# Thermodynamics

- differential form and exact differential
- syntactic sugar: (explicit version of partial derivative)
- First law: (or , , depending on control parameter)
- adiabatic process:
- with heat flow:
- ideal gas:
- thermal equilibrium as definition of the temperature
- control parameter spaces are added for joint thermodynamic systems
- definition heat bath
- definition perpetuum mobile (first kind: efficiency > 1, second kind: work from inner energy)
- empirical entropy relevant for reversibility of processes: means reversible
- Carnot process: isothermal and adiabatic steps starting with low entropy and low temperature
- Carnot engine either heat engine or heat pump
- Carnot efficiency:
- Carnot process as a basis for absolute temperature scale
- entropy only grows with
- Gibbs' fundamental equation:
- side note: Lagrange parameters
- free energy
- chemical potential
- Gibbs-Duhem equation:
- free enthalpy:
- inner energy of joint systems is sub-additive, entropy is super-additive
- definition convex functions:
- f is concave iff -f is convex. Iff f is both concave and convex, it is an affine function
- definition convex hull
- Legendre transform
- van der Waals gas:
- Guggenheim-Quadrat: "Suff hilft Fhysikern pei großen Taten."

# Classical statistics

- phase space
- Liouville measure:
- definition energy surface: or
- expectation value:
- variance:
- definition statistically independent: (only for non-interacting Hamiltonians)
- micro canonical, canonical and grand-canonical ensemble
- ensemble necessary in order to find equilibrium states due to "Umkehreinwand" and "Wiederkehreinwand"
- definition ergodic: systems without other constants of motion without
- expectation value micro canonical ensemble:
- expectation value canonical ensemble:
- canonical ensemble inverse temperature :
- ensembles different, but mathematically equal for
- the canonical ensemble minimizes the free energy
- equipartition theorem:
- equipartition theorem gives fkT/2 for kinetic energy

# Quantum statistical physics

- bounded operator:
- Pauli matrices
- trace is cyclic
- expectation value:
- density operator for pure states: ,
- density operator for mixed states: ,
- von Neumann equation:
- ,
- product states:
- microcanonical ensemble:
- canonical ensemble: ,
- free energy:
- von Neumann entropy:
- for pure states, the entropy vanishes, entropy monotonous function of the temperature
- for , the entropy converges to 0
- definition symplectic transformations:
- Hamiltonian of the harmonic chain:
- continuum limit:
- canonical partition function for both Bose and Fermi gas
- grand canonical ensemble: ,
- (minus sign for bosons)
- Fermi pressure due to quantum effects
- BEC

# Lattice models and phase transitions

- local Hamiltonians
- low temperatures: lattice tends to be ordered
- definition contour, length
- 1D Ising model: no phase transition due to vanishing order paramter
- mean field approximation: good results for high dimensions, allows for phase transitions