By C. Dellago, P.G. Bolhuis, P.L. Geissler (auth.), Mauro Ferrario Professor, Giovanni Ciccotti Professor, Kurt Binder Professor (eds.)
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Extra resources for Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology Volume 1
Dellago et al. is sampled. Since this procedure generates a random walk that visits conﬁgurations according to their statistical weight, it is also called importance sampling. Monte Carlo procedures can be applied very generally to sample probability distributions . In particular, Monte Carlo techniques can also be used to sample ensembles of pathways. In this case a random walk is carried out in the space of trajectories instead of conﬁguration space. The basic step of this procedure consists of generating a new path, x(n) (T ), from an old one, x(o) (T ).
Other examples of deterministic dynamics include the extended Lagrangian dynamics of Car and Parrinello  and various dynamics with thermostats such as the Nose-Hoover thermostat  or the Gaussian isokinetic thermostat . Also for these types of dynamics the above deﬁnition of the transition path ensemble applies. Brownian Dynamics As an example for a stochastic process consider a system evolving according to the Langevin equation in the high friction limit where inertial eﬀects can be neglected and momenta are not required for the description of the system: mγ r˙ = − ∂V (r) +F .
The partition function in the numerator is the one for the ensemble of pathways that start in A and end in B. In a sense, this ratio of partition functions measures the volume in trajectory space occupied by reactive trajectories relative to that of the trajectories starting in A but without restriction on the ending point. This perspective suggests to determine the correlation function C(t) via calculation of ∆F (t) = FAB (t) − FA , the diﬀerence of the free energies related to the partition functions in the numerator and denominator.