Many processes in physics appear to behave randomly. The occurence of randomness is intrinsically linked to thermal or quantum fluctuations. For instance, a colloid in a liquid undergoes a continuous random motion known as Brownian motion which is the simplest form of a continuous time stochastic process. We will see how stochastic processes can be used to model a huge variety of processes from physics, chemistry and biology (and even economics where they are used to study stock marketmovements). The probability distributions of many stochastic processes obey the Fokker-Planck equation. This equation can be used to find the steady state distribution or other quantities such as survivalprobabilities. Discrete systems, for instance Ising spins or particles on a lattice, also have dynamics which can be described by Markov chain. Physically the systems evolution in the future only depends on itscurrentstate and not all of its past history. The idea of a Markov chain is vital for numerical simulation of discrete interacting systems, where we cannot compute the thermodynamic properties analytically, and is employed in Monte Carlo simulations.

 

The idea of this course is to give a general introduction to stochastic processes which will be useful in a wide variety of scientific areas for both pure and applied research.

1 ) Stochastic calculus and Langevin equations

1.1 Discrete time continuous space stochastic processes

1.2 The Ito Stochastic Calculus

1.3 Examples of Stochastic Differential Equations - underdamped Brownian motion and taking the over damped limit

1.4 The Generator and the Forward Fokker-Planck Equation

1.5 Links with physical descriptions of diffusion, Fick’s law.

1.6. First passage times.

1.7 Transport properties of a colloid in spatially varying potential.

1.8. Reduction of underdamped equations to over damped equation - the method of projection operators.

1.9 Stochastic processes in Fourier space - correlation functions.

1.10 Partially damped simple harmonic oscillator in the Langevin treatment, fluctuation dissipation theorem and KramersKronigTheorem.

 

2) Markov chains

2.1 Basic definitions and applications

2.2 Master equations for Markov chains

2.3 Detailed balance and the principle of Monte Carlo simulations for equilibrium statistical physics systems, sampling questions for Monte Carlo simulations

2.4 Glauber solution for the dynamics of 1D-Ising Model

2.5 Correlation and response functions for Markov chains - generalised proof of fluctuation dissipation theorem -applications, for example conductivity of metals