Course program
1) Lagrangian and Hamiltonian mechanics
N particles systems; generalized coordinates and conjugate momenta; phase space: micro-state and macro-state of a system; Lagrangian function and the Lagrangian equations of motion. The Hamiltonian function and the Hamiltonian equations of motion.
2) Introduction to statistical mechanics
Statistical ensamble; the Gibbs hypothesis; distribution function; the measure of a physical quantity; the Liouville theorem and its consequences.
Microcanonical ensamble: distribution and partition functions; entropy; therma, mechanical and chemical equilibrium; statistical macroscopic quantities: temperature, pressure, chemical potential. Intensive and extensive observables; the link with classical physics. Canonical ensamble: distribution and partition functions; energy probability density and density of states; the partition function of an ideal gas; the Helmoltz free energy; Maxwell and Maxwell-Boltzmann distributions; The equipartition theorem; Grand-Canonical Ensamble: Distribution and partition functions; the ideal gas in the granmd-canonical ensamble; quantum partition functions; the Bose-Einstein and the Fermi-Dirac distributions.
3) Monte-Carlo and statistical mechanics
Monte-Carlo integration; pseudo-random numbers. Random numbers generators, casuality tests; The Lehmer generator; importance sampling: inversion and rejection; Markov chains; the Metropolis-Monte-Carlo algorithm; technicalities: interaction potentials; periodic boundary conditions; truncation and shift etc.; orientational moves for polyatomic molecules; Metropolis Monte-Carlo in the micro-canonical and grand canonical ensambles.
4) Classical Molecular Dynamics
Equilibration and initialization; integrators: Verlet, Velocity Verlet, Gear Predictor –Corrector etc; the Verlet list; Lyapunov instability. Observables. Temperature, pressure and the Virial Theorem; energy conservation; structural observables: the Radial Distribution Function; Diffusion: the Green-Kubo formula; Molecular Dynamics in the canonical ensamble: the fluctuation of temperature; Thermostats: velocity rescaling; Andersen; the Nosè-Hoover thermostat. Barostats.
5) Potenziali classici
Fitting procedures; two bodies potentials; many-bodies potentials;Potentials for metals, semiconductors and soft-matter.
6) Introduction to quantum theory of many-body systems
The Schrödinger equations of many-bodies systems. frozen-core and Born-Oppenheimer approximations; the single electron approximation; the variational Rayleigh-Ritz principle; The Hartree equations; the Pauli exclusion principle; the Slater determinant and the Hartree-Fock equations; tight binding molecular dynamics; the tight binding method and the Bloch theorem; Linear Combination of Atomic Orbitals; parametrization and transferability; the Hamiltonian matrix and its diagonalization; the Hellmann-Feynman and the forces. The density functional theory. The Hohenberg-Kohn theorems and the Kohn-Sham equations;
Prerequisites
As a prerequisite it is mandatory that the students have attended the first module the course and have strong knowledge of physics and mathematics
Books
Class notes and textbook from the teacher.
a deeper understanding from:
1) first and second chapters: C. Kittel “Elementary Statistical Physics” John Wiley & Sons
2) third and fourth chapters Frenkel-Smit “Understanding Molecular Simulation”; Academic Press; Allen; Tildsley “Computer Simulation of Liquids”; Ed. Oxford Science;
3) fifth chapter: F: Ercolessi “A molecular dynamics primer”
WWW: http://www.fisica.uniud.it/~ercolessi/
4) sixth chapter: M.P. Marder “ Condensed Matter Physics” (Cap. 6; 7; 8; 9) John Wiley & Sons. R. M. Martin “Electronic Structure: Basic Theory and Practical Methods”, Cambridge University Press.
Teaching mode
The Course is divided into a series of lectures. The usage of slides is not considered because all the arguments are entirely developed in front of the students using the blackboard or some auxiliary electronic devices.
Frequency
Beside not being formally mandatory, attending the class is strongly recommended due to the intrinsic difficulty of the themes treated.
Exam mode
The examination consists in an oral interview and shall ensure that the following objectives are met:the students must demonstrate to have acquired the basic knowledge of the atomistic simulation theory and techniques.
The evaluation is based on:
verifying of the acquired knowledge (60%) ("Knowledge and understanding" and "Applying knowledge and understanding")
verifying of language property and clarity of presentation (20%) ("Communication skills")
verifying of the ability to apply the acquired knowledge in the field of nanotechnology (20%) ("Learning skills")
Bibliography
1) Per il primo e il secondo capitolo: C. Kittel “Elementary Statistical Physics” John Wiley & Sons
2) per il terzo e il quarto capitolo Frenkel-Smit “Understanding Molecular Simulation”; Academic Press; Allen; Tildsley “Computer Simulation of Liquids”; Ed. Oxford Science;
3) per il quinto capitolo: F: Ercolessi “A molecular dynamics primer”
WWW: http://www.fisica.uniud.it/~ercolessi/
4) per il sesto e il settimo capitolo: M.P. Marder “ Condensed Matter Physics” (Cap. 6; 7; 8; 9) John Wiley & Sons. R. M. Martin “Electronic Structure: Basic Theory and Practical Methods”, Cambridge University Press.
Lesson mode
The Course is divided into a series of lectures. The usage of slides is not considered because all the arguments are entirely developed in front of the students using the blackboard or some auxiliary electronic devices.
