Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

Appendix AStochastic calculus in outlineThis appendix states some results needed to deal with random terms in differential equations.Proofs of these statements are in [61], with further explanation in [22].A.1 The Fokker-Planck equationConsider a system of stochastic variables x, distributed according to a probability distributionP (x,t). In many cases, the time evolution of P (x,t) is governed by the Fokker-Planck(FP) equation:∂∂t P = − ∑ i∂A i (x,t)P + 1 ∑∂x i 2ij∂∂x i∂∂x j[B(x,t)B T (x,t) ] ij P(A.1)where A is vector and B is a matrix. The first-order derivative terms in Eq. (A.1), knownas drift terms, govern the deterministic evolution of x, while the second-order derivativeterms govern the stochastic or diffusive behaviour of x.An explicit expression for the evolution of individual trajectories of x(t) is possiblewith the Wiener increment dW :dx i (t) =A i (x,t)dt + ∑ jB ij (x,t)dW j (t)(A.2)which should be interpreted according to the Ito calculus.A.2 The Wiener processThe well known Wiener process W (t) is governed by a FP equation in which the driftterm is zero and the diffusion coefficient is equal to one.Its random trajectories are169