Nonequilibrium approaches

The next three chapters deal with the most widely used classes of methods free energy perturbation (FEP) [3], methods based on probability distributions and histograms, and thermodynamic integration (TI) [1, 2], These chapters represent a mix of traditional material that has already been well covered, as well as the description of new techniques that have been developed only recendy. The common thread followed here is that different methods share the same underlying principles. Chapter 5 is dedicated to a relatively new class of methods, based on calculating free energies from nonequilibrium dynamics. In Chap. 6, we discuss an important topic that has not received, so far, sufficient attention - the analysis of errors in free energy calculations, especially those based on perturbative and nonequilibrium approaches. 

Using a nonequilibrium approach, strong binding can be studied (ligand-receptor complex) (43). However, of particular interest in ACE and MACE is the characterization of weak interactions, since the rate of complex formation and the exchange of solute between aqueous and micellar phase could be too fast to be studied with conventional structure determination methods (MS, NMR). The alternative to those methods, namely, to measure in an equilibrium state, makes MACE highly attractive. Thus, weak bond strengths (acid-base and complex/partition equilibria) are measurable. 

This partition and the subsequent nonequilibrium approach were originally formulated and commonly applied to electronic processes (for example solute electronic transitions) as well as to the evaluation of solute response to external oscillating fields [41], Such phenomena are discussed elsewhere in this book suffice it to say that in these cases the fast term is connected to the polarization of the electron clouds and the slow contribution accounts for all the nuclear degrees of freedom of the solvent molecules. 

Comparison of computational efficiency of equilibrium and nonequilibrium approaches... 

An important feature of protein crystal growth experiments is the need to carry out crystallization trials with very small quantities of scarce and expensive materials. When experiments are carried out in such small volumes (typically, 5—100 ju.1), it becomes difficult to define and control solution properties. The situation becomes particularly complicated when vapor diffusion or other nonequilibrium approaches to crystal growth are used, as these produce different and changing conditions throughout the small volumes involved. 

J. P. Ryckaert, A. Bellemans, G. Ciccotti, and G. V. Paolini, Phys. Rev. A, 39, 259 (1989). Evaluation of Transport Coefficients of Simple Fluids by Molecular Dynamics Comparison of Green-Kubo and Nonequilibrium Approaches for Shear Viscosity. 

As mentioned above, the poor stability of nitrate compounds leads to the use of nonequilibrium approaches for the measurement of NO gas [243, 244]. During the... 

The reader of this volume may possibly draw the conclusion that progress to date in our application of nonequilibrium approaches has been dismally small. There is considerable justification for such a conclusion, but progress is being made, nonetheless. Although much can still be learned by application of equilibrium models, one must be wary of oversimplification of the complexities of the real world. It will be necessary eventually to accept the existence of the complications and prepare to deal with them. These papers should help put into proper levels of importance such factors as the slow and unknown rates of many important chemical reactions and the influence of water movement, mixing, and circulation on the composition of water bodies. Instances of progress in study of details of the chemistry of certain specific simple systems which are included in some of the papers may stimulate wider use of nonequilibrium approaches. In future symposia like this one, one may hope real progress will be documented. 

A difficulty with the nonequilibrium approach is that one must estimate the time constant or time constants for solvent equilibration with the solvent. This may be estimated from solvent viscosities, from diffusion constants, or from classical trajectory calculations with explicit solvent. Estimating the time constant for solvation dynamics presents new issues because there is more than one relevant time scale [69, 80]. Fortunately, though, the solvation relaxation time seems to depend mostly on the solvent, not the solute. Thus it is very reasonable to assume it is a constant along the reaction path. 

As a last note we underline that, while it is easy to see that nonequilibrium effects have to be taken into aceount in the presence of fast changes in the electronie distribution of the solute, or of an oseillating external field (as that exploited to measure molecular optical properties), nonequilibrium approaches for nuclear vibrational analyses are still open. 

Nilsson KGI (1987) A comparison of the enzyme-catalysed formation of peptides and oligosaccharides in various hydroorganic solutions using the nonequilibrium approach. 

Other expressions are available in Ref [57]. An alternative to the diffusion-like treatment of dispersion, in terms of a nonequilibrium approach, was given by BCronberg and Westerterp [58]. 

Note that the dependence of the shear viscosity on shear rate has been explicitly noted. Alternatively, a non-equilibrium MD simulation can be conducted in which the response of the system to an external perturbation have been considered. The most widely used nonequilibrium approach for viscosity calculations is called "SLLOD" algorithm[146] in which a shear rate is imposed on the system and the resulting stress is computed. The shear viscosity is found at a given shear rate from where Pij is an off-diagonal component of the stress tensor, i is the direction of flow caused by the imposed shear rate, and j is the direction normal to the flow. For the SLLOD algorithm, special "sliding brick" boundary conditions are also typically used, which require modification of the Ewald sum if charged systems are simulated. [147]... 

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