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 is 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. Although we will mostly work with simple monatomic gases, the course is also suitable for people interested in simulations of lattice QCD, spin systems, spin glasses and molecular systems. During the course students will be required to perform some numerical work at home and present it in a short paper (examples are given in the e-learning site of the course): (a) error analysis and determination of autocorrelations times; (b) MC simulation of a monoatomic gas; (c) MD simulation of a monoatomic gas. Program: (a) STATISTICAL MECHANICS AND THERMODYNAMICS 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 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 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: 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 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)

Tuckerman, Statistical Mechanics, Oxford University Press (OUP); Understanding Molecular Simulation, D. Frenkel & B. Smit, Academic Press (AP) Notes provided on the e-learning site of the course https://elearning.uniroma1.it/course/view.php?id=2878