Nonequilibrium simulations

VER in liquid O 2 is far too slow to be studied directly by nonequilibrium simulations. The force-correlation function, equation (C3.5.2), was computed from an equilibrium simulation of rigid O2. The VER rate constant given in equation (C3.5.3) is proportional to the Fourier transfonn of the force-correlation function at the Oj frequency. Fiowever, there are two significant practical difficulties. First, the Fourier transfonn, denoted [Pg.3041]

Multiphase and nonequilibrium simulations are extremely difficult. These usually entail both a large amount of computing resources and a lot of technical expertise on the part of the researcher. Readers of this book are urged to refer such projects to specialists in this area. 

It is also possible to simulate nonequilibrium systems. For example, a bulk liquid can be simulated with periodic boundary conditions that have shifting boundaries. This results in simulating a flowing liquid with laminar flow. This makes it possible to compute properties not measurable in a static fluid, such as the viscosity. Nonequilibrium simulations give rise to additional technical difficulties. Readers of this book are advised to leave nonequilibrium simulations to researchers specializing in this type of work. 

As we will see further in the book, almost all methods for calculating free energies in chemical and biological problems by means of computer simulations of equilibrium systems rely on one of the three approaches that we have just outlined, or on their possible combination. These methods can be applied not only in the context of the canonical ensemble, but also in other ensembles. As will be discussed in Chap. 5, AA can be also estimated from nonequilibrium simulations, to such extent that FEP and TI methods can be considered as limiting cases of this approach. 

This relation is called the Jarzynski equality (hereafter referred to as JE) and can be used to recover free energies from nonequilibrium simulations or experiments (see Section IV.B.2). The FT in Eq. (27) becomes the Crooks fluctuation theorem (hereafter referred to as GET) [45, 46] ... 

Figure 43. Solvation dynamics from MD simulations for isomer 2. (a) The linear-response calculated time-resolved Stokes shifts for indole-protein, indole-water, and their sum. (b) Direct nonequilibrium simulations of the time-resolved Stokes shifts for indole-water, indole-protein, and their sum. Note the lack of slow component in indole-water relaxation in both (a) and (b), which is opposite to isomer 1 in Fig. 42. Also shown is the indole-water (within 5 A of indole) with coupled long-time negative solvation, (c) Relaxation between indole-lys79 and indole-glu4. The interaction energy changes from these two residues nearly cancel each other, (d) The distance changes between the indole and two charged residues, but both residues move away from the indole ring.

Maginn EJ, Bell AT, Theodorou DN (1993) Transport diffusivity of methane in silicalite from equilibrium and nonequilibrium simulations. J. Phys. Chem. 97 4173—4181... 

At all events, the role of Coulombic forces for VET in solution was first examined in a molecular dynamics simulation for the 680 cm-1 C-Cl vibration of the CH3CI molecule (modeled as a diatomic) in water solvent by Whitnell et al. (19). The (classical) relaxation time Ti was determined both by nonequilibrium simulations and by use of the classical Landau-Teller (LT) formula (1,3,19,20). 

Whereas it is clear that nonequilibrium simulations are important in some circumstances, it is not clear that one needs to develop any special NEMD techniques. For example, imagine chemical engineers who wish to understand the behavior of a prototypical lubricant under shear. The most straightforward simulation would mimic the real experimental setup and would require only standard equilibrium MD methods. The fluid could be placed between two surfaces, and these surfaces could be given equal and opposite velocities, as shown in Figure 1. Standard Newtonian equations of motions would be sufficient to evolve the system in time. 

Finally we note several future directions which should be studied (a) Our final results for the VER rate depend on a width parameter y. Unfortunately we do not know which value is the most appropriate for y. Nonequilibrium simulations (with some quantum corrections [39]) might help this situation, and they are useful to investigate energy pathways or sequential IVR (intramolecular vibrational energy redistribution) [40] in a protein, (b) This work is motivated by pioneering spectroscopic experiments by Romesberg s group. The calculation of the VER rate and the linear or nonlinear response functions, related to absorption or 2D-IR (or 2D-Raman) spectra [41—44], is desirable, (c)... 

Most of the quantities normally specified in an equilibrium stage simulation (number of stages, feed stage location, feed flow rates and composition, reflux ratio, distillate or bottoms rate, and so on) must be specified for a comparable nonequilibrium simulation. By including the hydraulic or pressure drop equations in the model it is not necessary to specify the pressure of each stage for a nonequilibrium simulation only the pressure of the top stage and of the condenser need be specified. 

Discuss how the fundamental models of mass transfer in Sections 12.1.7 (binary systems) and 12.2.4 (multicomponent systems) may be used to estimate mass transfer rates for use in a nonequilibrium simulation of an existing distillation column. Your essay should address the important question of how the model parameters are to be estimated. 

A number of manipulations are possible, once this formalism has been established. There are useful analogies both with the Eulerian and Lagrangian pictures of incompressible fluid flow, and with the Heisenberg and Schrodinger pictures of quantum mechanics T, chapter 7], [M, chapter 11]. These analogies are particularly useful in formulating the equations of classical response theory [39], linking transport coefficients with both equilibrium and nonequilibrium simulations [35]. 

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]... 

Figure 4.3.2. The linear response relaxation function C(t) (dashed and dotted lines] and the non-equilibrium solvation function S(t) (solid line) computed for the Stockmayer-CH3Cl model deseribed in Section 4. In the nonequilibrium simulation the ion charge is switehed on at t = 0. The dotted and dashed lines rq>re-sent C(t) obtained from equihbiium simulations with uneharged and charged i

E. J. Maginn, A. T. Bell, and D. N. Theodorou, /. Phys. Chem., 97, 4173 (1993). Transport Diffusivity of Methane in Silicalite from Equilibrium and Nonequilibrium Simulations. 

In a simulation system, we investigate the equilibrium selective adsorption and nonequilibrium transport and separation of gas mixture in the nanoporous carbon membrane are modeled as slits from the layer structure of graphite. A schematic representation of the system used in our simulations is shown in Fig. 11.21(a) and (b), in which the origin of the coordinates is at the center of simulation box and transport takes place along the x-direction in the nonequilibrium simulations. In the equilibrium simulations, the box as shown in Fig. 11.21(a) is employed, whose size is set as 85.20 nm x 4.92 nm x (1.675 + JV) nm in x-, y-, and z-directions, respectively, where JV is the pore width, i.e. the separation distance between the centers of carbon atoms on the two layers forming a slit pore (Fig. 11.21). is the separation distance between two centers of adjacent carbon atom L is the pore length JV is the pore width, A... 

FIGU RE 11.21 Schematic representation of the simulation boxes. The H-, L-and M-areas correspond to the high and low chemical potential control volumes, and membrane, respectively. Transport takes place along the x-direction in the nonequilibrium simulations, (a) Equilibrium adsorption simulations and (b) nonequilibrium transport simulations. L is the membrane thickness and W is the pore width. 

Rotational relaxation can also be investigated by nonequilibrium simulations. The solute molecule is subjected to an instantaneous jump in its angular momentum, and the energy and orientations dynamics are followed. This is repeated for an ensemble of initial solute positions and velocities, from which time-dependent averages are computed. 

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