Syllabus
Phenomenology of critical phenomena, order parameter, critical temperature, singular behaviour near critical temperature, critical exponents, universality. Scaling hypothesis and scaling relations, universality classes.
Ising model and equivalent models, Bragg-Williams mean field approximations, mean field critical exponents, exact solution in 1D. High temperature expansions, analysis of series.
Markov process, stochastic differential equations, Fokker-Planck equation, Langevin equation, kinetic Ising model, phase ordering, Glauber and Kawasaki dynamics.
Dynamic scaling: examples of time evolution of interfaces in experiments and discrete models, roughness, growth and dynamical exponents. Dynamical universality classes in growth: random deposition, Edwards-Wilkinson equation, Kardar-Parisi-Zhang equation.
Cellular automata and self-organized criticality, game of life, sand piles, BTW model, asymmetric exclusion model and other traffic problems.
Network theory: Erdös-Rényiho model, small worlds, scale-free networks, robustness of networks, examples: internet, social networks, power grids, multi-agent systems.
Combinatorial Optimization: P-NP-NP complete problems, simulated annealing. Applications: spin glasses, traveling salesman problem, K-SAT.
Annotation
We introduce new trends in applications of equilibrium and nonequilibrium statistical physics, which applies also in many non-traditional areas which are usually called "complexity science". First we explain the critical behavior in the equilibrium case, including the methods of calculations for model systems. After explaining the fundamentals of stochastic processes, we will deal with selected problems of nonequilibrium statistical physics and complex systems: dynamical scaling, cellular automata, random networks, optimization problems.
For the 4th and 5th year of the TF study.