  A variety of enhanced statistical and numerical methods are now routinely
used to extract comprehensible and relevant thermodynamic information from the
vast amount of complex, high-dimensional data obtained from intensive molecular
simulations. The characterization of kinetic properties, such as diffusion
coefficients, of molecular systems with significantly high energy barriers, on
the other hand, has received less attention. Among others, Markov state models,
in which the long-time statistical dynamics of a system is approximated by a
Markov chain on a discrete partition of configuration space, have seen
widespread use in recent years, with the aim of tackling these fundamental
issues. Here we propose a general, automatic method to assess multidimensional
position-dependent diffusion coefficients within the framework of Markovian
stochastic processes and Kramers-Moyal expansion. We apply the formalism to
one- and two-dimensional analytic potentials and data from explicit solvent
molecular dynamics simulations, including the water-mediated conformations of
alanine dipeptide. Importantly, the developed algortihm presents significant
improvement compared to standard methods when the transport of solute across
three-dimensional heterogeneous porous media is studied, for example, the
prediction of membrane permeation of drug molecules.
