Computer simulation statistical mechanics

Despite writing the chapters quite independently, the authors wanted to give a true unity to the book. Thus, throughout the work, they aimed at using coherent notation and reasonable designations. Consequently, logic sometimes forced them to distance themselves somewhat from awkward traditions. Nevertheless, this problem of notation has not always been easy to solve, due to the large number of disciplines concerned by the study of polymers namely, computer simulation, statistical mechanics and theory of liquids, description of the... 

The interfacial pair correlation functions are difficult to compute using statistical mechanical theories, and what is usually done is to assume that they are equal to the bulk correlation function times the singlet densities (the Kirkwood superposition approximation). This can be then used to determine the singlet densities (the density and the orientational profile). Molecular dynamics computer simulations can in... 

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

Progress in the theoretical description of reaction rates in solution of course correlates strongly with that in other theoretical disciplines, in particular those which have profited most from the enonnous advances in computing power such as quantum chemistry and equilibrium as well as non-equilibrium statistical mechanics of liquid solutions where Monte Carlo and molecular dynamics simulations in many cases have taken on the traditional role of experunents, as they allow the detailed investigation of the influence of intra- and intemiolecular potential parameters on the microscopic dynamics not accessible to measurements in the laboratory. No attempt, however, will be made here to address these areas in more than a cursory way, and the interested reader is referred to the corresponding chapters of the encyclopedia. 

Flere we consider various aspects of statistical mechanics (see also chapter A2.3 and [2, 3]) that have a direct bearing on computer simulation metiiodology. 

Tuckerman M E and Hughes A 1998 Path integral molecular dynamics a computational approach to quantum statistical mechanics Classical and Quantum Dynamics In Condensed Phase Simulations ed B J Berne, G Ciccotti and D F Coker (Singapore World Scientific) pp 311-57... 

Prenkel, D. Pree energy computation and first order phase transitions. In Molecular Dynamic Simulation of Statistical Mechanical Systems, Enrico Fermi Summer School, Varenna 1985, G. Ciccotti and W. Hoover, eds. North Holland, Amsterdam (1986) 43-65. 

Due to the noncrystalline, nonequilibrium nature of polymers, a statistical mechanical description is rigorously most correct. Thus, simply hnding a minimum-energy conformation and computing properties is not generally suf-hcient. It is usually necessary to compute ensemble averages, even of molecular properties. The additional work needed on the part of both the researcher to set up the simulation and the computer to run the simulation must be considered. When possible, it is advisable to use group additivity or analytic estimation methods. 

In this chapter we provide an introductory overview of the imphcit solvent models commonly used in biomolecular simulations. A number of questions concerning the formulation and development of imphcit solvent models are addressed. In Section II, we begin by providing a rigorous fonmilation of imphcit solvent from statistical mechanics. In addition, the fundamental concept of the potential of mean force (PMF) is introduced. In Section III, a decomposition of the PMF in terms of nonpolar and electrostatic contributions is elaborated. Owing to its importance in biophysics. Section IV is devoted entirely to classical continuum electrostatics. For the sake of completeness, other computational... 

Essentially, the RISM and extended RISM theories can provide infonnation equivalent to that obtained from simulation techniques, namely, thermodynamic properties, microscopic liquid structure, and so on. But it is noteworthy that the computational cost is dramatically reduced by this analytical treatment, which can be combined with the computationally expensive ab initio MO theory. Another aspect of such treatment is the transparent logic that enables phenomena to be understood in terms of statistical mechanics. Many applications have been based on the RISM and extended RISM theories [10,11]. 

The partition function Z is given in the large-P limit, Z = limp co Zp, and expectation values of an observable are given as averages of corresponding estimators with the canonical measure in Eq. (19). The variables and R ( ) can be used as classical variables and classical Monte Carlo simulation techniques can be applied for the computation of averages. Note that if we formally put P = 1 in Eq. (19) we recover classical statistical mechanics, of course. 

D. Nicholson, N. G. Parsonage. Computer Simulations and Statistical Mechanics of Adsorption. London Academic Press, 1982. 

A complete set of intermolecular potential functions has been developed for use in computer simulations of proteins in their native environment. Parameters have been reported for 25 peptide residues as well as the common neutral and charged terminal groups. The potential functions have the simple Coulomb plus Lennard-Jones form and are compatible with the widely used models for water, TIP4P, TIP3P and SPC. The parameters were obtained and tested primarily in conjunction with Monte Carlo statistical mechanics simulations of 36 pure organic liquids and numerous aqueous solutions of organic ions representative of subunits in the side chains and backbones of proteins... 

The linkage of microscopic and macroscopic properties is not without challenges, both theoretical and experimental. Statistical mechanics and thermodynamics provide the connection between molecular properties and the behavior of macroscopic matter. Coupled with statistical mechanics, computer simulation of the structure, properties, and dynamics of mesoscale models is now feasible and can handle the increase in length and time scales. 

In the case of molten salts, no obvious model based on statistical mechanics is available because the absence of solvent results in very strong pair correlation effects. It will be shown that the fundamental properties of these liquids can be described by quasi-chemical models or, alternatively, by computer simulation of molecular dynamics (MD). 

The lattice gas has been used as a model for a variety of physical and chemical systems. Its application to simple mixtures is routinely treated in textbooks on statistical mechanics, so it is natural to use it as a starting point for the modeling of liquid-liquid interfaces. In the simplest case the system contains two kinds of solvent particles that occupy positions on a lattice, and with an appropriate choice of the interaction parameters it separates into two phases. This simple version is mainly of didactical value [1], since molecular dynamics allows the study of much more realistic models of the interface between two pure liquids [2,3]. However, even with the fastest computers available today, molecular dynamics is limited to comparatively small ensembles, too small to contain more than a few ions, so that the space-charge regions cannot be included. In contrast, Monte Carlo simulations for the lattice gas can be performed with 10 to 10 particles, so that modeling of the space charge poses no problem. In addition, analytical methods such as the quasichemical approximation allow the treatment of infinite ensembles. 

Equation (4-5) can be directly utilized in statistical mechanical Monte Carlo and molecular dynamics simulations by choosing an appropriate QM model, balancing computational efficiency and accuracy, and MM force fields for biomacromolecules and the solvent water. Our group has extensively explored various QM/MM methods using different quantum models, ranging from semiempirical methods to ab initio molecular orbital and valence bond theories to density functional theory, applied to a wide range of applications in chemistry and biology. Some of these studies have been discussed before and they are not emphasized in this article. We focus on developments that have not been often discussed. 

Andricioaei, I. Straub, J.E., Computational methods for the simulation of classical and quantum many body systems sprung from the nonextensive thermostatistics. In Nonextensive Statistical Mechanics and Its Application, Abe, S. Okamoto, Y., Eds., Lecture Notes in Physics. Springer Berlin, Heidelberg, New York, 2001, ch. IV, pp. 195-235... 

On several occasions, the reader will notice a direct connection between the topics covered in the book and other, related areas of statistical mechanics, such as the methodology of computer simulations, nonequilibrium dynamics or chemical kinetics. This is hardly a surprise because free energy calculations are at the nexus of statistical mechanics of condensed phases. 

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