Free Energies from Transition Path Sampling Simulations

4 Free Energies from Transition Path Sampling Simulations 

As discussed in previous chapters, one is often interested in calculating the free energy as a function of a given reaction coordinate . Such a free energy profile A( ) is defined as 

5 In most applications of transition path sampling it is sufficient to define the stable regions si and 88 in terms of configurational coordinates without reference to the momenta. The transition path sampling formalism, however, can be also applied to situations in which si and 88 also depend on the atomic momenta. 

In this case the shooting and shifting procedure may be viewed as a particular move in a Monte Carlo simulation similar to hybrid Monte Carlo [30]. 

For Newtonian dynamics and a canonical distributions of initial conditions one can reject or accept the new path before even generating the trajectory. This can be done because Newtonian dynamics conserves the energy and the canonical phase-space distribution is a function of the energy only. Therefore, the ratio plz ]/p z at time 0 is equal to the ratio p[.tj,n ]/p z ° at the shooting time and the new trajectory needs to be calculated only if accepted. For a microcanonical distribution of initial conditions all phase-space points on the energy shell have the same weight and therefore all new pathways are accepted. The same is true for Langevin dynamics with a canonical distribution of initial conditions. 

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The transition path sampling techniques developed for the study of rare events can also be used to improve the calculation of free energies from non-equilibrium transformations on the basis of Jarzynski s theorem. In this approach equilibrium free energies are linked to the statistics of work performed during irreversible transformations. When applying these ideas in computer simulations, one has to deal with rare events (of a different kind, however) and transition path sampling can help to solve this problem. Due to space limitations we cannot comprehensively discuss all aspects of transition path sampling in this chapter. For further information the reader is referred to several recent review articles [11-15]. Of these articles, [12] is the most detailed and comprehensive. 

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