Boltzmann-weighted trajectory

The Boltzmann weight appears implicitly in the way the states are chosen. The form of the above equation is like a time average as calculated in MD. The MC method involves designing a stochastic algorithm for stepping from one state of the system to the next, generating a trajectory. This will take the form of aMarkov chain, specified by transition probabilities which are independent of the prior history of the system. 

This identity [3, 15] between a weighted average of nonequilibrium trajectories (r.h.s.) and the equilibrium Boltzmann distribution (l.h.s.) is implicit in the work of Jarzynski [2], and is given explicitly by Crooks [16]. The average (... is over an ensemble of trajectories starting from the equilibrium distribution at / 0 and... 

In some cases, one may not be able to collect enough trajectories for a histogram analysis. Instead, one can use moments [15]. In particular, the derivative of the free energy is approximately given by the average force (weighted by the Boltzmann factor of the accumulated work)... 

The scaling factor A, is chosen to satisfy Eq. (23). The number of time slices is N, and Q is the number of degrees of freedom in a single time slice. The code in MOIL is producing complete trajectories in any of the steps following the dynamics of Eqs. (22) and (23). The trajectories so obtained have the Boltzmann-like weight for a trajectory, expected from formula (21). 

The equation above comes with normalized initial distribution p x,0) = po(x). In equilibrium, the diffusion coefficient D and the mobility p are related by the Einstein equation D = Tp, where T is the temperature of the surroundings with Boltzmann s constant set to unity to make entropy dimensionless. The dynamics is described by assigning a weight to each path or trajectory ... 

The activation free energy AA can be used to compute the TST approximation of the rate constant = Ce, where C is the preexponential factor. Because not every trajectory that reaches the transition state ends up as products, the actual rate is reduced by a factor k (the transmission coefficient) as described earlier. The transmission coefficient can be calculated using the reactive flux correlation function method. " " " Starting from an equilibrated ensemble of the solute molecules constrained to the transition state ( = 0), random velocities in the direction of the reaction coordinate are assigned from a flux-weighted Maxwell-Boltzmann distribution, and the constraint is released. The value of the reaction coordinate is followed dynamically until the solvent-induced recrossings of the transition state cease (in less than 0.1 ps). The normalized flux correlation function can be calculated using " ... 

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