Skewing Momenta Distributions to Enhance Free Energy Calculations from Trajectory Space Methods

8 Skewing Momenta Distributions to Enhance Free Energy Calculations from Trajectory Space Methods 

The identification of the slow manifold introduced in the previous section for the MEHMC method turns out to be effective not only for enhanced thermodynamic 

For the purposes of the present treatment, we wish to rewrite this trajectory average as an average over the initial, equilibrium distribution. If the system evolves according to deterministic (e.g., Hamiltonian) dynamics, each trajectory is uniquely determined by its initial point, and (8.46) can be written without modification as an average over the canonical phase space distribution. 

If the system evolves according to some stochastic scheme, each initial point can lead to a multitude of trajectories. We note, however, that as long as each trajectory is initiated from an equilibrium distribution, (8.46) can still be rewritten as an average over the initial distribution  

The quantity Wt is the irreversible work done along a particular trajectory in time t, defined by Wt = f ) drrfyf ((r(r), t)/ )t. Because the system s Hamiltonian evolves at a finite rate, the dynamical flow of the system no longer preserves an equilibrium distribution. Moreover, Wt is a functional of the trajectory, rather than a function of the end points. 

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