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Contents
Contents
Notes on Statistical and Computational Physics
Dario Alfè
Date:
2021/2022
Contents
Introduction
The first law
The second law
A toy model
Physical systems
Equilibrium in isolated systems
The entropy
Temperature
Pressure and chemical potential
Fundamental thermodynamic relation
Heat capacity at low temperature
Equilibrium, heat bath
The Boltzmann distribution
Energy fluctuations
The Gibbs distribution
Particle number fluctuations
Isobaric-isothermal ensemble
General form for the entropy
Third law of thermodynamics
The Helmholtz and the Gibbs free energies
Boltzmann, alternative derivation
Free expansion of perfect gas
Equivalent formulations of the second law
Maxwell relations
The Perfect Classical gas
The partition function
The Gibbs paradox
The classical limit
Maxwell velocity distribution
Regime of validity classical limit
The equation of state
The heat capacity
Interacting systems
General form of the partition function for a classical system
The equipartition of the energy
Classical and quantum distributions
The Maxwell-Boltzman distribution
Quantum distributions
The Fermi-Dirac distribution
The Bose-Einstein distribution
Bose-Einstein condensation
Equilibrium between phases
The Clausius-Clapeyron relation
Chemical equilibrium in a multispecie system
The vapour pressure
Vapour pressure of H
in equilibrium with MgH
Applications
Harmonic crystalline solids
Linear chain
Alternative derivation of the dispersion relation
Three dimensional crystals
The small displacement method
Anharmonicity
Perturbative expansion
Molecular Dynamics
Ensemble averages
Reblocking
Thermostats
Mean square displacement
Density Functional theory
Hohenberg-Kohn theorems
The Kohn-Sham method
The local density approximation
Limitations of the LDA
Crystals
Plane waves basis functions
Tutorial thermal expansion
Perfect crystal
Vibrating crystal
Harmonic approximation
Primitive cell
Super-cell generation
Improving the calculations
Convergence with respect to
k
-point sampling
Convergence with respect to the size of the super-cell
Convergence with respect to the size of the displacement
Thermal expansion of Al
Anharmonicity
Tutorial: Transition pressure in MgO
The density of states
Bloch theorem
Preliminary remarks and notation
Commuting operators
Translation operators
Bloch theorem
Time reversal
Constrained extremes
Functionals
Stirling approximation
Kinetic integrals
Physical constants
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