  Deterministic dynamics is an essential part of many MCMC algorithms, e.g.
Hybrid Monte Carlo or samplers utilizing normalizing flows. This paper presents
a general construction of deterministic measure-preserving dynamics using
autonomous ODEs and tools from differential geometry. We show how Hybrid Monte
Carlo and other deterministic samplers follow as special cases of our theory.
We then demonstrate the utility of our approach by constructing a continuous
non-sequential version of Gibbs sampling in terms of an ODE flow and extending
it to discrete state spaces. We find that our deterministic samplers are more
sample efficient than stochastic counterparts, even if the latter generate
independent samples.
