Learning outcomes of the course unit
To provide the students with the basic theoretical concepts in statistical mechanics and with the methods to calculate the thermodynamical properties of macroscopic systems at equilibrium from the statistical distribution of microscopic variables in phase space
Course contents summary
The statistical description of a macroscopic system. Systems with many degrees of freedom and classical mechanics. Phase space and microscopic Hamiltonian dynamics. Average values without dynamics: statistical ensembles and probability measures. Liouville theorem. The problems of the microscopic approach. Temporal averages and the ergodic hypothesis. Recurrence times and macroscopic variables. How and if equilibrium is reached.
Microcanonical, canonical and gran canonical ensembles
Brief review of thermodynamics: extensive and intensive variables, thermodynamic potentials, Legendre transformations, response functions. The statistical ensembles in the thermodynamic limit and how one gets thermodynamics back: the partition functions and the thermodynamic observables. Number of states and entropy. Fluctuations and response functions.
Entropy and information theory: Shannon’s entropy and the probability distributions of the three ensembles.
Dynamics vs statistical mechanics in computer simulations: molecular dynamics and Metropolis Monte Carlo. Detailed balance.
The partition functions in the statistical ensembles and counting the number of states: independent systems and occupation numbers, integrals and discrete sums. Examples: the classical perfect gas and the Gibbs paradox. The crystal of classical harmonic oscillators. The Maxwell distribution. Magnetic gases and spin models on lattices. The Ising model. Problems and paradoxes in classical statistical mechanics: equipartion and specific heats. Quantum statistical mechanics.
Recent topics in statistical mechanics
Phase transitions and universality. Counting methods for states: applications to combinatorial problems, cost functions, optimization problems and algorithmic complexity. Entropy of sequences of characters.
L.Peliti, Appunti di Meccanica Statistica, Bollati Boringhieri (2003)
L. Landau, Lifsitz, Fisica Statistica, Editori Riuniti, (1963)
K. Huang, Statistical Mechanics, Wiley & Sons (1963)
D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press (1987)