Course topics and techniques in a hierarchical structure

Topics

0. Thermodynamics potentials (Thermo I material)
1.1 Extremum principles
1.2 Response functions and thermodynamic stability
1.3 Relations between the potentials / Legendre transforms

1. Phase-transitions
1.1 Phenomenology, classification and terminology
1.1.1 What is phase and a phase-transition?
1.1.2 Types of phase-transitions (*):
1.1.2.1 Quantum phase transitions
1.1.2.2 Thermal phase transitions
1.1.2.3 Topological phase transitions
1.1.2.4 Dynamics phase transitions
1.1.3 Importance of the thermodynamic limit and interactions
1.1.4 Ehrenfest and modern classifications of thermal phase transitions
1.1.5 Order-to-disorder transitions
1.1.5.1 Order parameter, ordered and disordered phases
1.1.5.2 Long-range order
1.1.5.3 Spontaneous symmetry breaking
1.1.6 First-order transitions
1.1.6.1 Isotherms
1.1.6.2 Coexistence region
1.1.6.3 Coexistense curves and Clausius-Clayperon equation
1.1.6.4 S-T curve
1.1.6.5 Latent heat and its relation to intermolecular interactions
1.1.6.6 Helmholtz and Gibbs free energies
1.2 Mean field theory of phase-transitions
1.2.1 Introduction to the many-body problem
1.2.2 van der Waals mean-field theory of the liquid-gas transition
1.2.2.1 Helmholtz free energy and the equation of state
1.2.2.2 Spindal and binodal lines
1.2.2.3 Critical point and equation of corresponding states
1.2.2.3 Maxwell construction
1.2.2.4 Surface effects: superheating and supercooling
1.2.3 Mean field theory of magnetic phase transition
1.2.3.1 Origin of magnetism (*)
1.2.3.2 Weiss mean-field treatment
1.2.3.3 Models of magnetic materials: Heisenberg / Ising
1.2.3.4 Symmetries and spontaneous symmetry breaking
1.2.3.5 Gibbs Free energy and the equation of state
1.2.3.6 Analogy between the liquid-gas and magnetic transitions 
1.3 Critical phenomena
1.3.1 Critical exponents and universality
1.3.2 Spatial correlations
1.3.2.1 Order parameter and long-range order
1.3.2.3 Lower critical dimension: absence of long-range order in lower dimensions
1.3.2.3 Correlation length and its critical exponent
1.3.3 Failures and successes of the mean-field approximation, the upper critical dimension
1.3.4 Scale invariance, and the origin of universality
1.3.5 Irrelevance of quantum effects (*)

2. Nonequilibrium statistical mechanics
2.1 Kinetic theory and transport
2.1.1 Origin of pressure
2.1.2 Maxwell-Boltzmann distribution
2.1.3 Mean-free path and its distribution
2.1.4 Relaxation time and its distribution
2.1.5 Naive derivation of transport coefficients
2.1.6 The role of conservation laws
2.2 Stochastic methods
2.2.1 Einstein approach
         2.2.1.1 Markov chains / random walk
         2.2.1.2 Master equation
         2.2.1.3 Fokker-Planck (*) and diffusion equations
         2.2.1.4 Stationary states and the principle of detailed balance
2.2.2 Langevin approach
         2.2.2.1 Standard and differential Langevin equations
         2.2.2.2 Langevin-Einstein equivalence and stationary state
         2.2.2.3 Brownian motion / Orenstein-Uhlenbeck process
         2.2.2.4 Fluctuation-dissipation relations (Einstein relation)
2.3 Thermalization
2.3.1 Lioville's theorem
2.3.2 Ergodicity and mixing
2.3.3 BGKKY heirarchy (*)
2.3.4 The Boltzmann equation
2.3.4.1 Relaxation time approximation and transport coefficients
2.3.4.2 Stationary, Maxwell-Boltzmann distribution
2.3.4.3 H-theorem and Loschmidt and Zemerlo objections
2.3.4.4 Collisional invariants (*)
2.3.4.5 Connection to hydrodynamics: Euler and Navier-Stokes equations (*)
2.3.5 Quantum thermalization (*)

(*) Sections which are marked by a star, are advanced material which is not part of the formal course syllabus

Techniques

0. Thermo I
    1. Non-interacting partition functions
    2. Calculations and changes of thermodynamics potentials
    3. Equations of state
    4. Reconstruction of thermodynamic potentials from equations of state
    5. Legendre transformations
    6. Variational principle in statistical mechanics
1. Phase transitions
    1. Calculations in phase coexistence (using Clausius-Clapeyron)
    2. Exact solution in 1D: Transfer matrix
    3. Calculations using the vdW equation
    4. Obtainment of the free energy by variation (Alben model)
    5. Graphical solution of a self-consistent equation
    6. Calculation of critical temperature and critical exponents
    7. Behavior of thermodynamic quantities near criticality (spinodal & binodal curves, order parameter, etc.)
    8. Applying mean-field theory (for continuous as well as discrete models)
    9. Obtaining and using scaling relations between critical exponents
2. Nonequilibrium statistical mechanics
    1. Usage of the Maxwell-Boltzmann distribution
    2. "Naïve" calculation of linear transport coefficients
    3. Calculation of moments for different random walk models
    4. Solving the diffusion equation with various initial and boundary conditions
    5. Markov chains: analysis and modelling using a Markov chain
    6. Markov chains: recognizing the existence a stationary state and finding it
    7. Master equation: modelling using a master equation
    8. Master equation: calculating the average of a one-step process
    9. Master equation: exact solution of system of linear ODEs
    10. Detailed balance (check whether satisfied and\or use)
    11. Langevin approach: model a given system and obtain different moments (mean, variance, correlations), relate to physical
    12. Calculation of linear transport coefficients using the relaxation time approximation of the Boltzmann equation

Last modified: Wednesday, 24 June 2020, 5:17 PM