Disordered systems, models and replica method. Markov chains and Monte Carlo method. Brownian motion, Langevin and Fokker-Planck eqs. Fluctuation dissipation theorem and linear response theory. Nonequilibrium phase transitions and applications. Scale invariance in growth processes. Edwards-Wilkinson and Kardar-Parisi-Zhang equations. Pattern formation in nonequilibrium systems. Multiple scale analysis.
More informations, with hyperlinks to online material, are available at the page:
http://www.fi.isc.cnr.it/users/paolo.politi/didattica_2009-2010.html
[Barabasi-Stanley] A.-L. Barabasi and H.E. Stanley, Fractal concepts in surface growth, Cambridge University Press
[Castellano] C. Castellano, S. Fortunato, V. Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81, 591 (2009).
[Chandler] David Chandler, Introduction to modern statistical mechanics (Oxford University Press, 1987).
[Dotsenko] Viktor Dotsenko, Introduction to the Replica Theory of Disordered Statistical Systems (Cambridge University Press, 2001). Grosse parti del libro sono riportate in un suo articolo di rivista: V.S. Dotsenko, Physics of the spin-glass state, Physics Uspekhi 36 (6), 455 (1993).
[Fischer&Hertz] K. H. Fischer & J. A. Hertz, Spin Glasses (Cambridge University Press, 1993).
[Gardiner] C. W. Gardiner, Handbook of Stochastic Methods, 3rd edition (Springer, 2003).
[Hoyle] Rebecca Hoyle, Pattern Formation - An introduction to methods, Cambridge University Press.
[Krug] J. Krug, Origins of scale invariance in growth processes, Advances in physics, vol. 46, pag. 139-282 (1997)
[Lavenda] Bernard H. Lavenda, Il moto browniano da Einstein a oggi (Le Scienze, febbraio 1983).
[LiviMB] Roberto Livi, Note sul moto browniano.
[LiviNE] Roberto Livi, Note su fenomeni stazionari di non equilibrio.
[Manneville] Paul Manneville, Instabilities, chaos and turbulence, Imperial College Press.
[Peliti] Luca Peliti, Appunti di meccanica statistica (Boringhieri, 2003).
[Pimpinelli-Villain] Alberto Pimpinelli and Jacques Villain, Physics of crystal growth, Cambridge University Press.
[Sonoda] H. Sonoda, Derivation of the Nyquist relation using a random electric field (sul suo sito web).
[YoungMC] A. Peter Young, Monte Carlo Simulations in Statistical Physics (sul suo sito web).
[YoungRS] A. Peter Young, Phase Transitions in Random Systems (sul suo sito web).
Learning Objectives
Knowledge acquired:
Objective of the course is to widen the knowledge towards more advanced topics, as the statistical mechanics of disordered or out of equilibrium systems. There is also the objective to acquire some theoretical bases for numerical simulations.
Competence acquired:
Ability to apply the acquired knowledge to models not explicitly treated in the course. Some examples: mean field analysis for the “bridge” model; linear stability analysis; multiscale analysis for any anharmonic oscillator.
Skills acquired (at the end of the course):
Ability to deal with the bibliography and the starting calculations for the development of a Thesis with important aspects in Statistical Mechanics.
Prerequisites
Courses required:
Courses recommended: Meccanica Statistica I
Teaching Methods
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...):
150
Hours reserved to private study and other individual formative activities: 96
Contact hours for: Lectures (hours): 54
Further information
Office hours:
Contacts:
Paolo Politi
Istituto dei Sistemi Complessi, CNR
Edificio B, stanza B-128, first floor
Via Madonna del Piano, 10
50019 Sesto Fiorentino
Telefono: 055 522 6686
Paolo.Politi@isc.cnr.it
Office hours:
Every day, on phone appointment or on e-mail appointment.
Type of Assessment
Oral examination.
Course program
Disordered systems
Extensivity, additivity, self-averaging. Frustration and disorder. Quenching and annealing. Non frustrated disordered ferromagnet: site dilution. Harris criterion. Random Field Ising Model and Imry and Ma criterion. Introduction to spin glasses. Susceptibility and Edwards-Anderson order parameter. Self-averaging for frustrated systems.
Ergodicity breaking. State superposition and function P(q). Hierarchical structure of spin glass states. Replica trick and replica symmetry breaking. Sherrington and Kirkpatrick model. p-spin model and Random Energy Model. Random Energy Model: thermodynamical treatment. Random Energy Model and replica method.
Bibliografia:
[Peliti] section 10.3 (10.2: outline on percolation)
[Dotsenko] chapters 1,2 and 4
[YoungRS]
[Fischer&Hertz] chapter 2 e and section 3.6
Online resources:
Percolazione
Molecular Dynamics and Monte Carlo method
Molecolar dynamics. Outline on the theory of stochastic processes. Markov chains and Master equation. Detailed balance, updating algorithms and Young proof of convergence. General proof of convergence. Statistical errors and thermodynamical limit
Bibliografy:
[Peliti] chapter 8 and appendix D
[YoungMC]
Online resources:
Molecular dynamics with con Lennard-Jones potential
Monte Carlo method: Ising 2D
Brownian motion, Langevin and Fokker-Planck equations
Introduction to brownian motion.Entropy mixing and osmotic pressure. Brownian motion: Einstein treatment. n(z) profile in presence of gravity. Brownian motion: Langevin treatment. Green-Kubo relation. Overdumped brownian motion: random walk. Fokker-Planck equation: two derivations.Examples of the FP eq.: diffusion, diffusion with force. FP eq. for many stochastic variables. FP eq. for the not overdumped brownian motion.
Bibliografy:
[Gardiner] chapter 3
[Huang] section 2.4 and 2.5
[Lavenda]
[LiviMB]
[Peliti] chapter 9
Online resources:
Brownian motion, one and two
Fluctuation-dissipation theorem and linear response theorem
Onsager regression hypothesis. Fluctuation-dissipation theorem. Brownian motion and electric noise (Nyquist relation). Definition and properties of the response function. Time-independent perturbation and susceptibility vs fluctuations relation.
Bibliografy:
[Chandler] chapter 8
[Peliti] chapter 9
[Sonoda]
Non equilibrium phase transitions
Out of equilibrium systems, master equation and detailed balance. Ising model and lattice gas. External field and breaking of detailed balance. TASEP model: defintion. TASEP with input/ouptut of particles. Mean field solution. BRIDGE model and sponstaneous symmetry breaking.
Bibliografy:
[LiviNE]
Growth processes, Edwards-Wilkinson (EW) e Kardar-Parisi-Zhang (KPZ) equations
Introduction to interface dynamics. Different sources of noise. Interface in the Ising model with external field. Rughness exponents. Scale transformation and critical dimensions. Random deposition model. General continuum equations and power counting. Derivation of EW e KPZ equations. EW: power counting, exact results and FP equation. KPZ: power counting, tilt transformation and Fokker-Planck equation. Deterministic KPZ. Comparison between discrete and continuum growth models.
Bibliografy:
[Barabasi-Stanley]
[Pimpinelli-Villain]
[Krug]
Instability and pattern formation: linear and nonlinear analysis
Examples of instabilities and pattern formin systems. Linear stability analysis for the Rayleigh-Bénard cell. Introduction to multiple scales method for the anharmonic oscillator. Application to the Swift-Hohenberg equation. Outline of primary and secondary instabilities.
Bibliografy:
[Hoyle]
[Manneville]
Statistical mechanics and social dynamics
Historical outline and general considerations. Opinion dynamics and Random Field Ising Model. Voter model.
Bibliografia:
[Castellano]