Nonequilibrium Method

Free energy differences can also be computed from nonequilibrium simulations switching between two Hamiltonians, using measurements of the work Wo i performed on the system during the switching process [41-46]. The Jarzynski relation [41] 

Jarzynski relation can be seen as a generalization of EXP, and Crooks relation can be used to derive the BAR equation for nonequilibrium work [28,42]. A multi-state 

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]. 

To compute free energy differences between states with little overlap, it will usually be more efficient to compute the free energy along a pathway of intermediate states. Free energy calculations can be made significantly more efficient by optimizing the choice of intermediate states for increased phase space overlap [36, 59-61], 

The simplest path to construct is linear in the two end point Hamiltonians  

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Due to the small emission and absorption cross sections of Er +, a high Er density is needed to reach reasonable values of optical gain. Typically Er densities are between 0.1% and 1.0% (10 -10- Er/cm- ). These values are far beyond the equilibrium solubility limits of Er in silicon. Therefore, nonequilibrium methods have to be used, such as ion implantation. Er implantation in crystalline silicon leads to amorphization, and additional annealing (600°C) is required to... 

Despite its apparent simplicity, this calculation is relatively difficult to perform. Gomez etal. [18] demonstrated that ABF performs much better than both slow-growth and fast-growth implementations of the nonequilibrium method of Jarzynski and Crooks (see the next chapter). 

Nonequilibrium Methods for Equilibrium Free Energy Calculations... 

In this chapter, we will show how nonequilibrium methods can be used to calculate equilibrium free energies. This may appear contradictory at first glance. However, as was shown by Jarzynski [1, 2], nonequilibrium perturbations can be used to obtain equilibrium free energies in a formally exact way. Moreover, Jarzynski s identity also provides the basis for a quantitative analysis of experiments involving the mechanical manipulation of single molecules using, e.g., force microscopes or laser tweezers [3-6]. 

Computational efficiency remains a central question. For a given amount of computer time, how good are nonequilibrium estimates of free energy differences compared to estimates from equilibrium methods Overall, evidence is mounting that nonequilibrium methods are less efficient than equilibrium methods [13, 20, 38], However, new approaches have been suggested that use long time steps [25]. For relatively slow transformations, it has been shown [20] that for a given amount of computer simulation time, one obtains more-accurate results for few slow transformations than for many fast transformations. At the other extreme, i.e., in the limit of... 

The methods that do not contain steps to ensure the establishment of equilibrium can be considered nonequilibrium methods. In the past few years, several methods commonly used for solubility measurements in the early discovery setting have been reported (Curatolo, 1996 Lipinski et al., 1997 Pan et al., 2001). These methods typically begin with dimethylsulfoxide (DMSO) solutions or with amorphous material. Turbidity and ultraviolet detection are commonly used because they easily can be designed into high-throughput instrumentation. 

Compared to nonequilibrium methods, equilibrium methods tend to be simpler, less expensive, more selective, therefore require less cleanup, require determination of preequilibrium/equilibrium status, are time, temperature, and matrix dependent, and require internal standards for calibration... 

Christiansen et al. (54) applied the Naphtali-Sandholm method to natural gas mixtures. They replaced the equilibrium relationships and component vapor rates with the bubble-point equation and total liquid rate to get practically half the number of functions and variables [to iV(C + 2)]. By exclusively using the Soave-Redlich-Kwong equation of state, they were able to use analytical derivatives of revalues and enthalpies with respect to composition and temperature. To improve stability in the calculation, they limited the changes in the independent variables between trials to where each change did not exceed a preset maximum. There is a Naphtali-Sandholm method in the FraChem program of OLI Systems, Florham Park, New Jersey CHEMCAD of Coade Inc, of Houston, Texas PRO/II of Simulation Sciences of Fullerton, California and Distil-R of TECS Software, Houston, Texas. Variations of the Naphtali-Sandholm method are used in other methods such as the homotopy methods (Sec. 4,2.12) and the nonequilibrium methods (Sec. 4.2.13). 

Nonequilibrium methods attempt to get around the difficulty of predicting efficiencies by doing away with the equilibrium-stage concept. Instead, they apply a transport phenomena approach for predicting mass transfer rates. The mass transfer rates are calculated continuously along the column length and not in discrete equilibrium stages. This process is similar to the transfer unit concept (Sec. 10.3.1). 

For development of nonequilibrium methods to continue, the calculations for mass transfer coefficients and interfacial areas required by these models will have to be added to physical property packages. Krishnamurthy and Taylor (89) present methods and recommendations for calculating the mass and energy transfer coefficients and rates. Help may be available from published manuals or supplier literature. 

Nonequilibrium methods (Sec. 4,2.13) tend to be global Newton methods extended to solve mass-transfer-inhibited systems. Nonequilibrium methods are not yet completely extended to more common systems, but these methods should see the greatest amount of development in distillation modeling. 

E j Net energy gain by a phase in a nonequilibrium method, stage,/,... 

Net energy or mass transferred between bulk phases in a nonequilibrium method, Sec. 4.2,13,... 

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