Stochastic processes in natural sciences 2024/2025

Maciej Lisicki & Miłosz Panfil

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

Lectures & classes:
Wednesdays 11:15 - 13:00
Thursdays 14:15 – 16:00

---

Mid-term: 11 Nov

Exam: 27/28 Jan

---

---

Group projects

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]
  • Ruchy Browna, B. Cichocki, Delta 4/1983 (in Polish)
  • Ruchy Browna (II), B. Cichocki, Delta 5/1983 (in Polish)
  • Albert Einstein – praca o ruchach Browna z 1905 roku, B. Cichocki, Delta 6/2005 (in Polish)
  • Ito versus Stratonovich, N.G. van Kampen