Statistical mechanics molecular simulation

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. 

Exploiting the principles of statistical mechanics, atomistic simulations allow for the calculation of macroscopically measurable properties from microscopic interactions. Structural quantities (such as intra- and intermolecular distances) as well as thermodynamic quantities (such as heat capacities) can be obtained. If the statistical sampling is carried out using the technique of molecular dynamics, then dynamic quantities (such as transport coefficients) can be calculated. Since electronic properties are beyond the scope of the method, the atomistic simulation approach is primarily applicable to the thermodynamics half of the standard physical chemistry curriculum. 

Molecular dynamics assigns numerical values to states, thereby making states observable, at least for model substances. With numerical values assigned to states, theoretical relations from kinetic theories to compute values by simulation for experimentally accessible observables. Thus, molecular dynamics is more closely related to kinetic theory compared to statistical mechanics. Molecular dynamics, in a sense, is less sophisticated but more direct than statistical mechanics. 

Atomic and molecular simulation methods can generally be categorized as either equilibrated or dynamic. Static simulations attempt to determine the structural and thermodynamic properties such as crystal structure, sorption isotherms, and sorbate binding. Structural simulations are often carried out using energy minimization schemes that are similar to molecular mechanics. Elquihbrium prop>erties, on the other hand, are based on thermodynamics and thus rely on statistical mechanics and simulating the system state function. Monte Carlo methods are then used to simulate these systems stochastically. 

In recent years, the development of cubic equations of state has benefited by parallel developments in applied statistical mechanics, molecular theory and simulation, primarily with respect to intermolecular interactions. Furthermore, the accumulated experience with cubic equations of state, the large databases with optimum pure component and binary interaction parameters and the familiarity of applied scientists, chemical and process engineers in industry guarantee that these models will retain their leading position in applied research and development in the years to come. 

A new opportunity, which creates good prospects for avoiding many problems eoimected with the theoretical description of adsorption on heterogeneous surfaces, has appeared as a result of the introduction of computer simulation methods [17,18]. Over the last three deeades, computer simulations have grown into a third fundamental discipline of research in addition to experiment and theory. The study of adsorption on heterogeneous solid surfaces has especially benefited fi om the molecular simulation method, first of aU, because of the complexity of interactions of adsorbate molecules with differently distributed active centers that are not easily described by the methods of statistical mechanics. Computer simulation can, in principle at least, provide an exact solution of the assumed model. 

Theoretical prediction and experimental measurement of the thermodynamic stability of a-helix has been one of the central issues in biophysical research. Pioneering work was done by Zimm and Bragg [38] which was then improved by Lifson and Roig [39]. Much work has since been and is currently being done with vigorous use of statistical mechanics, molecular dynamics, and other types of computer simulation. Only a few examples are cited here [40-45]. Since the statistical mechanical treatment of helix stability has been developed primarily on a few... 

The simulation of the macroscopic properties and of the molecular organisation obtained for a system of N model molecules at a certain temperature and pressure (T, P) typically proceeds through one of the two current mainstream methods of computational statistical mechanics molecular namics or Monte Carlo [1,2]. MD sets up and solves step by step the equations of motion for all the particles in the system and calculates properties as time averages from the trajectories obtained. MC calculates instead average properties fi-om equilibrium configurations of the system obtained with an algorithm designed... 

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. 

This chapter concentrates on describing molecular simulation methods which have a counectiou with the statistical mechanical description of condensed matter, and hence relate to theoretical approaches to understanding phenomena such as phase equilibria, rare events, and quantum mechanical effects. 

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. 

Berendsen. H.J.C., Van Gunsteren, W.F. Practical algorithms for dynamic simulations, in Molecular Dynamics Simulations of Statistical Mechanical Systems, G. Ciccotti, ed., Soc. Italiana di Fisica, Bologna (1987) 43-65. 

An important though deman ding book. Topics include statistical mechanics, Monte Carlo sim illation s. et uilibrium and non -ec iiilibrium molecular dynamics, an aly sis of calculation al results, and applications of methods to problems in liquid dynamics. The authors also discuss and compare many algorithms used in force field simulations. Includes a microfiche containing dozens of Fortran-77 subroutines relevant to molecular dynamics and liquid simulations. 

In Chapter 2, a brief discussion of statistical mechanics was presented. Statistical mechanics provides, in theory, a means for determining physical properties that are associated with not one molecule at one geometry, but rather, a macroscopic sample of the bulk liquid, solid, and so on. This is the net result of the properties of many molecules in many conformations, energy states, and the like. In practice, the difficult part of this process is not the statistical mechanics, but obtaining all the information about possible energy levels, conformations, and so on. Molecular dynamics (MD) and Monte Carlo (MC) simulations are two methods for obtaining this information... 

A very important aspect of both these methods is the means to obtain radial distribution functions. Radial distribution functions are the best description of liquid structure at the molecular level. This is because they reflect the statistical nature of liquids. Radial distribution functions also provide the interface between these simulations and statistical mechanics. 

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. 

Eds Ciccotti G., Frenkel D., McDonald I. R.) Simulation of Liquids and Solids Molecular Dynamics and Monte Carlo Methods in Statistical Mechanics (North-Holland Physics Publishing, Amsterdam) (1987). 

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