  Monte Carlo sampling of the canonical distribution presents a formidable
challenge when the potential energy landscape is characterized by a large
number of local minima separated by high barriers. The principal observation of
this work is that the multiple local minima and energy barriers in a landscape
can often occur as a result of discrete symmetries in the potential energy
function. A new Monte Carlo method is proposed, group action Markov chain Monte
Carlo (GA-MCMC), which augments more conventional trial moves (e.g. random
jumps, hybrid Monte Carlo, etc.) with the application of a group action from a
well-chosen generating set of the discrete symmetry group; the result is a
framework for symmetry-adapted MCMC. It is shown that conventional trial moves
are generally optimal for "local mixing" rates, i.e. sampling a single energy
well; whereas the group action portion of the GA-MCMC trial move allows the
Markov chain to propagate between energy wells and can vastly improve the rate
of "global mixing". The proposed method is compared with standard jumps and
umbrella sampling (a popular alternative for energy landscapes with barriers)
for potential energies with translational, reflection, and rotational
symmetries. GA-MCMC is shown to consistently outperform the considered
alternatives, even when the symmetry of the potential energy function is
broken. The work culminates by extending GA-MCMC to a clustering-type algorithm
for interacting dielectric polymer chains. Not only does GA-MCMC again
outperform the considered alternatives, but it is the only method which
consistently converges for all of the cases considered. Some new and
interesting phenomena regarding the electro-elasticity of dielectric polymer
chains, unveiled via GA-MCMC, is briefly discussed.
