Stochastic processes in natural sciences 2024/2025

Maciej Lisicki & Miłosz Panfil

Trajectories of particles in the Vicsek model. Simulations by R.J. Budzyński

Mid-term (late November)

Exam (February)

Possible topics:

Pawula’s theorem: explained and illustrated with examples
Quantum non-equilibrium dynamics: master equation and Fermi’s golden rule
Lotka-Volterra: from mean field solution of the master equation to the equations for the averages, numerical solutions, etc.
Kinetic description of coagulation in suspensions – Smoluchowski’s theory, hierarchy of kinetic equations and dynamics
Fokker-Planck description of collective motion of active matter – kinetic equations, stability
First passage times – definition and methods of computation of FPTs for Brownian systems
Dynamic Light Scattering (DLS) in colloidal suspensions – stochastic description of scattered light intensity and relationship with diffusive properties
The Black-Scholes equation and the role of stochastic processes in economic models
Turing’s model of pattern formation – principles and resulting patterns
Holtsmark's distribution and the statistics of the gravitational field arising from a random distribution of stars <\ul>

Class materials: (updated systematically)
Lecture notes

Further reading:
- Stochastic Problems in Physics and Astronomy, S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). [random walk in 1D]
- Ito versus Stratonovich, N.G. van Kampen