MODELS OF THEORETICAL PHYSICS

Introduction; "The Unreasonable Effectiveness of Mathematics in the Natural Sciences
(Wigner 1959)"; Gaussian integrals Wick theorem
Perturbation theory connected contributions Steepest descent
Legendre transformation Characteristic/Generating functions of general probability
distributions/measures
The Wiener integral geometric characteristics of Brownian paths and Hausdorff/fractal
dimension
Brownian paths and polymer physics biopolymer elasticity. The random walk
generating function, the Gaussian field theory and coupled quantum harmonic
oscillators
Levy walks violation of universality
Field theories as models of interacting systems
O(n) symmetric Phi^4– theory. The large n limit: Spherical (Berlin-Kac) model and 1/n
expansion.

Perturbative expansion. Introduction to renormalization group techniques and universality.

Generalized diffusion and stochastic differential equations. The Feynman-Kac formula: diffusion with sinks and sources
Feynman path integrals and the quantum version of the Feynman-Kac formula.
Quantum mechanics (solvable model: free particle, harmonic oscillator)
Quantum vs stochastic phenomena: quantum tunneling and stochastic tunneling
Stochastic amplification and stochastic resonance
Non-perturbative methods: instantons
Diffusion in random media and anomalous diffusion
Quantum Mechanics in a random potential localization and random matrices
Statistical physics of random spin systems and the machine-learning problem
Random energy model, replica trick
Cavity method, Random Field Ising Model