Numerical stochastic integration for quasi-symplectic flows

Algorithms for the numerical integration of Langevin equations obeying detailed balance are introduced. The algorithms are derived fulfilling the requirements that they should become symplectic in the deterministic frictionless limit, and should reproduce the equilibrium distributions to some higher order in the integration time step when the equilibrium distribution exists. Extensions to the case when the system volume or pressure are kept constant are discussed. Comparisons with other integration schemes are carried out both for static and dynamical quantities.