Perturbed distribution function, calculation

In practice it is helpful to know the order of magnitude of the sample size N needed to reach a reasonably accurate free energy. The inaccuracy model described above presents an effective way to relate the sample size N and the finite sampling error through perturbation distribution functions. Alternatively, one can develop a heuristic that does not involve distribution functions and is determined by exploring the common behavior of free energy calculations for different systems [25]. Although only FEP calculations are considered in this section, the analysis extends to NEW calculations. 

It would be valuable if one could proceed with a reliable free energy calculation without having to be too concerned about the important phase space and entropy of the systems of interest, and to analyze the perturbation distribution functions. The OS technique [35, 43, 44, 54] has been developed for this purpose. Since this is developed from Bennett s acceptance ratio method, this will also be reviewed in this section. That is, we focus on the situation in which the two systems of interest (or intermediates in between) have partial overlap in their important phase space regions. The partial overlap relationship should represent the situation found in a wide range of real problems. 

This procedure of calculating perturbed distribution functions is of use if the molecules of the medium are under the effect of a static electric field only and if the system is in well-defined conditions of thermodynamic equilibrium. If, however, the system is acted on by an alternating field... 

Calculation of a perturbed distribution function can be approached in various ways (1) direct solution of the Boltzmann equation for the distribution function in the perturbed system, (2) distribution-difference methods, (3) local calculations, and (4) normal-mode expansion methods. 

The perturbation process can be quantified using distribution functions of the potential energy change AU (for FEP) or work W (for NEW). The procedure is the same for both the FEP and NEW calculations, thus we use the term perturbation and the notation x to unify both. 

The main lines of the Prigogine theory14-16-17 are presented in this section. A perturbation calculation is employed to study the IV-body problem. We are interested in the asymptotic solution of the Liouville equation in the limit of a large system. The resolvent method is used (the resolvent is the Laplace transform of the evolution operator of the N particles). We recall the equation of evolution for the distribution function of the velocities. It contains, first, a part which describes the destruction of the initial correlations this process is achieved after a finite time if the correlations have a finite range. The other part is a collision term which expresses the variation of the distribution function at time t in terms of the value of this function at time t, where t > t t—Tc. This expresses the fact that the system has a memory because of the finite duration of the collisions which renders the equations non-instantaneous. 

Note that the binary HMSA [60] scheme gives the solute-solvent radial distribution function only in a limited range of solute-solvent size ratio. It fails to provide a proper description for such a large variation in size. Thus, here the solute-solvent radial distribution function has been calculated by employing the well-known Weeks-Chandler-Anderson (WCA) perturbation scheme [118], which requires the solution of the Percus-Yevick equation for the binary mixtures [119]. 

The ASEP/MD method, acronym for Averaged Solvent Electrostatic Potential from Molecular Dynamics, is a theoretical method addressed at the study of solvent effects that is half-way between continuum and quantum mechanics/molecular mechanics (QM/MM) methods. As in continuum or Langevin dipole methods, the solvent perturbation is introduced into the molecular Hamiltonian through a continuous distribution function, i.e. the method uses the mean field approximation (MFA). However, this distribution function is obtained from simulations, i.e., as in QM/MM methods, ASEP/MD combines quantum mechanics (QM) in the description of the solute with molecular dynamics (MD) calculations in the description of the solvent. 

The MFA [1] introduces the perturbation due to the solvent effect in an averaged way. Specifically, the quantity that is introduced into the solute molecular Hamiltonian is the averaged value of the potential generated by the solvent in the volume occupied by the solute. In the past, this approximation has mainly been used with very simplified descriptions of the solvent, such as those provided by the dielectric continuum [2] or Langevin dipole models [3], A more detailed description of the solvent has been used by Ten-no et al. [4], who describe the solvent through atom-atom radial distribution functions obtained via an extended version of the interaction site method. Less attention has been paid, however, to the use of the MFA in conjunction with simulation calculations of liquids, although its theoretical bases are well known [5]. In this respect, we would refer to the papers of Sese and co-workers [6], where the solvent radial distribution functions obtained from MD [7] calculations and its perturbation are introduced a posteriori into the molecular Hamiltonian. 

The evaluation of the free energy is essential to quantitatively treat a chemical process in condensed phase. In this section, we review methods of free-energy calculation within the context of classical statistical mechanics. We start with the standard free-energy perturbation and thermodynamic integration methods. We then introduce the method of distribution functions in solution. The method of energy representation is described in its classical form in this section, and is combined with the QM/MM methodology in the next section. 

Monte Carlo techniques were first applied to colloidal dispersions by van Megen and Snook (1975). Included in their analysis was Brownian motion as well as van der Waals and double-layer forces, although hydrodynamic interactions were not incorporated in this first study. Order-disorder transitions, arising from the existence of these forces, were calculated. Approximate methods, such as first-order perturbation theory for the disordered state and the so-called cell model for the ordered state, were used to calculate the latter transition, exhibiting relatively good agreement with the exact Monte Carlo computations. Other quantities of interest, such as the radial distribution function and the excess pressure, were also calculated. This type of approach appears attractive for future studies of suspension properties. 

Electric Saturation of the Kerr Effect. When calculating the dectric reorientation function of molecules (172) for molecular gases, it was justifiable to use the approximation (169a) for the perturbed statistical distribution function. In the general case, calcidations have to be carried out with a distribution function of the form (169) where, for axially... 

The calculation of the perturbed radial distribution function g(2) for steady-state viscous flow is carried out using Eq. 34 for continuity in molecular pair phase space. The development is by no means simple but it is carried out without the introduction of additional assumptions beyond the formulation of appropriate boundary conditions. Those used by Kirkwood and co-workers27 38 make use of the following ... 

The traditional role of perturbation theory in reactor physics has been to estimate, with a first-order accuracy, the effect of an alteration in the reactor on its reactivity. Lately, application of perturbation theory techniques has increased significantly in both scope and volume. Two general trends characterize these developments (1) improvement of the accuracy of reactivity calculation, and (2) extension of the use of second-order perturbation theory formulations for estimating the effect of a perturbation on integral parameters other than reactivity, and to nuclear systems other than reactors. These trends reflect two special features of perturbation theory. First, it provides exact expressions for the effect of an alteration in the reactor on its reactivity. For small, and especially local alterations, these perturbation expressions are easier and cheaper to apply than other approaches. Second, second-order perturbation theory formulations can be applied with distribution functions pertaining only to the unperturbed system. 

If the flux distribution in the perturbed reactor were known, it could have been used in Eq. (8) to give the exact reactivity worth associated with the perturbation. The flux and other distribution-function perturbations are also required for many applications other than reactivity calculations. A few of these applications, in homogeneous and inhomogeneous systems, will be mentioned in the sections to follow. 

The formulations presented in this section are for the calculation of perturbed fluxes. Analogous formulations can be derived and applied for the calculation of perturbed adjoints, kernels for the integral transport equations (see Section IV), generalized functions (see Section V), and other distribution-function perturbations. The presentation is restricted to exposition of the general formulation of the methods, without considering the technical details of the solution. 

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