  We study analytically the computational cost of the Generalised Hybrid Monte
Carlo (GHMC) algorithm for free field theory. We calculate the Metropolis
acceptance probability for leapfrog and higher-order discretisations of the
Molecular Dynamics (MD) equations of motion. We show how to calculate
autocorrelation functions of arbitrary polynomial operators, and use these to
optimise the GHMC momentum mixing angle, the trajectory length, and the
integration stepsize for the special cases of linear and quadratic operators.
We show that long trajectories are optimal for GHMC, and that standard HMC is
more efficient than algorithms based on Second Order Langevin Monte Carlo
(L2MC), sometimes known as Kramers Equation. We show that contrary to naive
expectations HMC and L2MC have the same volume dependence, but their dynamical
critical exponents are z = 1 and z = 3/2 respectively.
