Molecular dynamics models resolve material properties in terms of pairwise interactions between N particles. These can be characterized via a partial differential equation for the N-particle probability distribution function. The BBGKY hierarchy represents the coupled interaction of the partially integrated marginals of the probability distribution. Under ideal circumstances, as N grows the effective interactions can be modeled by coupling a single particle to a mean-field generated by the other particles – in this situation the joint probability distributions reduce to tensor products of a single particle distribution. This represents a tremendous reduction in complexity. This mean-field limit is not universal, and it generally breaks down when small groups of particles have successive interactions with each other such as between the water molecules in a hydration shell of an ion. These repeated interactions generate correlations described by 2 and 3 particle joint distributions. We will review the BBGKY framework, and the impact of correlation on the hydrodynamic limits that provide a further reduction to continuum systems. Our focus will be on models of interactions of particles with dipole moments, such as water.
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