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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.
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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(t0)=q0 1. Brackets denote average over noise 2. Vector q is argument of P. No relation to function q(t)
Formally integrate Langevin equation • qi 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<q2(t)>/dt well defined 2<qdq/dt> 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(t0)=q0, 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, xI=x(t); Stratonovich chooses xS=x(t+ε/2) • Resolve by working with the discrete Langevin system
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 Equation 2 • Stationary solution: