Splitting Methods for Langevin Dynamics

We will now consider a family of iterations for molecular systems which are constructed from simple building blocks based on the splitting of an SDE, specifically Langevin dynamics  

Just as in the deterministic setting (for Newton s equations of motion) we construct such splitting methods by way of an additive decomposition of the vector field, where the differential equations corresponding to any individual piece can be solved exactly. Methods are built from a sequence of updates corresponding to an exact solve of each piece. By exact solve we here mean the construction of a method of exactly sampling the distribution generated by the corresponding component. The 

Other than the Ornstein-Uhlenbeck process, the remaining piece of the splitting corresponds to Newtonian constant-energy (microcanonical) Hamiltonian dynamics. Since Hamiltonian dynamics leaves invariant any function of the energy, its corresponding Fokker-Planck operator (in this case the Liouvillian, = — //) will preserve distributions that are functions of the Hamiltonian H. This implies in particular that it preserves pp, which is proportional to exp(— 6//). Thus the forward propagator associated to the Hamiltonian system automatically preserves the Gibbs distribution, and we have 

Formally, we could combine the flow map of Hamiltonian dynamics with an exact solve of the OU process to create a sampling iteration which preserves pp, as we also have 

To make this practical in the general case, where the Hamiltonian system is not integrable (and we cannot explicitly compute its flow map), we need to be able to compute (or approximate) the maps involved. 

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