main

In this section, we provide a brief overview of HMC as well as the specific rendition, split HMC [92]. Given "position" variables x and "momentum" variables v, we define the Hamiltonian for a dynamical system as H(x, v) which can usually be written as U(x)+K(v), where U(x) is the potential energy and K(v) is the kinetic energy. We then simulate the Hamiltonian, which is given by the partial differential equations: ẋ = @H @H, v = @v @x. Of course, this must be done in discrete time for most Hamiltonians that are not perfectly integrable. One notable exception is when x is Gaussian, in which case the dynamical system corresponds to the evolution of a simple harmonic oscillator (i.e. a spring-mass system).