Nonequilibrium Work Free Energy Methods

This equality holds even if the transitions from one state to the other (also referred to as switching) take a finite time. The NEW method for calculating the free energy using Jarzynski s equality (6.10) has rapidly gained popularity it can be applied to a broad range of processes, including both computer simulations and experimental studies. 

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These bounds are the nonequilibrium equivalents of the Gibbs-Bogoliubov bounds discussed in Chap. 2. Having the free energy now bounded from above and below already demonstrates the power of using both forward and backward transformations. Moreover, as was shown by Crooks [18, 19], the distribution of work values from forward and backward paths satisfies a relation that is central to histogram methods in free energy calculations... 

In this chapter, we will examine in depth the characteristic errors of two free energy techniques and present improved methods based on a better understanding of their behavior. The two techniques examined are free energy perturbation (FEP) [2] and nonequilibrium work (NEW) based on Jarzynski s equality [3-6]. These techniques are discussed in Chaps. 2 and 5. The FEP method is one of the most popular approaches for computing free energy differences in molecular simulation see, e.g., [1, 7-10]. The recently developed NEW method, which is closely related to FEP, is gaining popularity in both simulation [11-18] and experimental applications [19-21],... 

As discussed in detail in Chap. 5, free energy differences can be calculated from the statistics of the work carried out during nonequilibrium transformations. The basis for this method, also known as fast switching , is an identity derived in 1997 by... 

A number of sfraightforward applications of fhe Jarzynski relafion (offen termed "fast growth") soon followed [44,48-50]. If was quickly realized, however, that the poorly sampled tails of the work distribution in Eq. (7) contribute substantially to the free energy, posing difficulfies in convergence similar to those that occur with EXP [45,48,51]. Indeed, convergence was found fo require a number of simulations that increased exponentially in the typical dissipative work of the reverse process [45]. In practice, comparisons of fasf growfh mefhods and more standard equilibrium simulations seem to show that application of nonequilibrium measurements were often less efficient than equilibrium methods [39,52]. 

A second class of models directly relates flow to blend structure without the assumption of an ellipsoidal droplet shape. This description was initiated by Doi and Ohta for an equiviscous blend with equal compositions of both components [34], Coupling this method with a constraint of constant volume of the inclusions, leads again to equations for microstructural dynamics in blends with a droplet-matrix morphology [35], An alternative way to develop these microstructural theories is the use of nonequilibrium thermodynamics. This way, Grmela et al. showed that the phenomenological Maffettone-Minale model can be retrieved for a specific choice of the free energy [36], An in-depth review of the different available models for droplet dynamics can be found in the work of Minale [20]. 

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