Aim of the course: 
The course represents the natural continuation of the courses on Elementary 
Statistical Mechanics taught in BSc programs, that usually discuss the general 
principles on which Statistical Mechanics is based and the most elementary 
applications (typically ideal-gas systems), as these are the only ones that 
can be studied analytically. Analytic studies of more complex systems are 
not feasible, hence one must resort to numerical methods. 
This the subject of the course: to present the conceptual foundations of the 
different methods that are presently used in the study of statistical systems. 
The emphasis is on the methods rather than on the systems. Hence the course is 
suitable for people interested in simulations of lattice QCD, spin systems, 
spin glasses and molecular systems.


Program:

(a) STATISTICAL MECHANICS AND THERMODYNAMICS  (8 hours)

Review of thermodynamics. First and second law of thermodynamics. 
Thermodynamic potentials. 
Statistical mechanics and classical mechanics. Paradoxes: mechanical
reversibility and thermodynamic irreversibility, Poincare' recurrence time.
Probabilistic interpretation of statistical mechanics and 
definition of the concept of ensemble. Definitions of the different 
ensembles and connection with thermodynamics. Examples: the ideal gas and 
the one-dimensional spin chain. 
Correlation functions, structure factor and connection with scattering 
experiments on liquids, magnets, etc.
Molecular systems: Ideal and excess properties. Configurational integral and
reduced probabilities. The radial distribution function. Henderson theorem.

Bibliography: Any book on statistical mechanics. 
Huang, Statistical Mechanics
Tuckerman, Statistical Mechanics
are appropriate choices. For a discussion of the connections between 
statistical and classical mechanics see also Falcioni, Vulpiani,
Meccanica Statistica (in Italian).

(b) MONTE CARLO METHODS  (10 hours)
Monte Carlo method in statistical mechanics. Random sequences and Markov 
chains. Asymptotic behavior of stationary Markov chains. 
Microscopic reversibility, Metropolis-Hastings and heat-bath algorithms. 
Statistical errors, jackknife method, Monte Carlo autocorrelation times. 
Simulations of disorder systems. 
Examples: Lennard-Jones gas simulations in the different ensembles, 
Ising model and spin-glass simulations.

Bibliography: I will follow my notes "Introduction on Monte Carlo methods",
Scuola Nazionale di Fisica Teorica, Parma, 1991 and the material on
"Foundations of Monte Carlo algorithms", Summer School, Regensburg, 2012.
[copies will be provided].
Examples will be taken from
Understanding Molecular Simulation, D.  Frenkel & B. Smit, Academic Press (AP) 
and from 
A Guide to Monte Carlo Simulations in Statistical Physics, Landau and Binder,
Cambridge Univ. Press

(c) BIASED METHODS  (6 hours)

Biased sampling and reweighting methods. Umbrella sampling and 
simulated tempering. Piccioni theorem. Parallel tempering.

Bibliography: A. Pelissetto and F. Ricci-Tersenghi,
Large Deviations in Monte Carlo Methods,
in Large Deviations in Physics: The Legacy of the Law of Large
Numbers, Lecture Notes in Physics (2014) [copies will be provided].
Examples will also be taken from
Understanding Molecular Simulation, D.  Frenkel & B. Smit, Academic Press (AP) 
and from 
A Guide to Monte Carlo Simulations in Statistical Physics, Landau and Binder,
Cambridge Univ. Press

(d) SOME TRICKS OF THE TRADE: (6 hours)

Volume effects, boundary conditions, Percus-Lebowitz general theorems.
Dealing with long-range forces: the Ewald method for Coulombic systems.

Some implementation details: Verlet lists and cells for molecular simulations; 
bitmaps and hash tables for lattice systems.

(e) MOLECULAR DYNAMICS  (12 hours)

Review of classical mechanics: Hamilton equations and symplectic nature 
of the evolution.  Discretization of the evolution equation. 
Force calculation. Integration of the evolution equations. 
Calculation of thermodynamic quantities.
Molecular dynamics in the canonical and isobaric ensemble: thermostats and 
barostats. 
Classical mechanics for systems with holonomic constraints. 
Molecular dynamics in the presence of constraints

Bibliography:
Tuckerman, Statistical Mechanics, Oxford University Press (OUP); 
Understanding Molecular Simulation, D.  Frenkel & B. Smit, Academic Press (AP) 

(f) ADVANCED TOPICS (6 hours) 

Time correlation functions at equilibrium and their properties. 
Onsager-Kubo relations and linear response theory. 
Einstein relationship and Ohm law.
Brownian motion, Langevin and Fokker-Planck equations.

Bibliography:
Statistical Mechanics, M. E.  Tuckerman, Oxford University Press (OUP)