Stochastic Differential Equations Langevin equations Fokker Planck equations Equilibrium distributions correlation functions Purely dissipative Langevin equation Simple example Generalised stochastic Markov processes Discretized Langevin equations

Simplest form of Langevin equation Consider only Gaussian white noise Related to Markov processes Ω characterises width of noise Alternative definition via 1 and 2 point correlation functions (Wick’s theorem):

Probability distribution Given value of q at time t=0, q(t 0 )=q 0 1. Brackets denote average over noise 2. Vector q is argument of P. No relation to function q(t)

Fokker Planck equation

Identity

Application

Formally integrate Langevin equation q i only depends on times t>t’. Causality dictates we integrate only over range t’ – t. But we require only limit t=t’ which is ill- defined  Problem of Langevin equation  d /dt well defined 2 is not  Circumvent by discretizing time or ‘smearing out delta function in the noise function  Use symmetry of η(t) with respect to time to obtain θ(0)  1/2

Fokker Planck Equation Taking account of initial condition q(t 0 )=q 0, this is identical to Schrödinger equation for matrix elements of, H (generally non-Hermitean) Hamiltonian Formal relation between stochastic differential equations and Euclidean quantum mechanics Averaging over noise yields same results as QM using FP Hamiltonian

Dissipative Langevin equation Introduce Transform F-P equation into:

Hamiltonian Evolution operator in imaginary time

Generalised Markov Process NB: Ambiguity in choosing the particular time for x in the second term on the RHS. Ito chooses t at the beginning of the time step, x I =x(t); Stratonovich chooses x S =x(t+ε/2) Resolve by working with the discrete Langevin system

Ito & Stratonovich

Discrete Langevin Equation Average over noise v is at time t; average over anterior times is performed by integration over q(t). Noise, v(t) and q(t) are uncorrelated, as consequence of Langevin equ. The q(t) distribution is P(q,t) and noise distribution is:

Fokker Planck 1

Fokker Planck Equation 2 Stationary solution:

Interpretation