  The widespread popularity of replica exchange and expanded ensemble
algorithms for simulating complex molecular systems in chemistry and biophysics
has generated much interest in enhancing phase space mixing of these protocols,
thus improving their efficiency. Here, we demonstrate how both of these classes
of algorithms can be considered a form of Gibbs sampling within a Markov chain
Monte Carlo (MCMC) framework. While the update of the conformational degrees of
freedom by Metropolis Monte Carlo or molecular dynamics unavoidably generates
correlated samples, we show how judicious updating of the thermodynamic state
indices---corresponding to thermodynamic parameters such as temperature or
alchemical coupling variables---associated with these configurations can
substantially increase mixing while still sampling from the desired
distributions. We show how state update methods in common use lead to
suboptimal mixing, and present some simple, inexpensive alternatives that can
increase mixing of the overall Markov chain, reducing simulation times
necessary to obtain estimates of the desired precision. These improved schemes
are demonstrated for several common applications, including an alchemical
expanded ensemble simulation, parallel tempering, and multidimensional replica
exchange umbrella sampling.
