Computational Statistical Mechanics

In this part of the chapter, our aim is to describe a few of the highlights of both the molecular dynamics and Monte Carlo approaches to computational statistical mechanics. We begin with a review of the notion of microstate, this time with the idea of identifying how one might treat such states from the perspective of the computer. Having identified the nature of such states, we will take up in turn the alternatives offered by the molecular dynamics and Monte Carlo schemes for effecting the averages over these microstates. 

As noted in the preface, the availability of computers has in many cases changed the conduct of theoretical analysis. One of the fields to benefit from the emergence 

In the simplest classical terms, carrying out a molecular dynamics simulation involves a few key steps. First, it is necessary to identify some force law that governs the interactions between the particles which make up the system. The microscopic origins of the force laws themselves will be taken up in chap. 4, and for the moment we merely presuppose their existence. It is then imagined that these particles evolve under their mutual interactions according to Newton s second law of motion. If we adopt the most naive picture in which all of the atoms are placed in a closed box at fixed energy, we find a set of 3A coupled second-order ordinary 

A host of clever schemes have been devised for integrating these equations of motion in a manner that is at once efficient and stable. One well-known integration scheme is that of Verlet (though we note that more accurate methods are generally used in practice). One notes that for a sufficiently small time step, the position at times t 8t may be expressed as 

By adding the two resulting expressions and rearranging terms, we find 

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W. H. Hoover, Computational Statistical Mechanics, Elsevier, Amsterdam, 1991. 

F. F. Abraham, Adv. Phys., 35, 1,1986. Computational Statistical Mechanics Methodology, Applications and Supercomputing. 

Abraham. F. E. Computational statistical mechanics Methodology, applications, and super-computing. Adv. Phys. 3)5, 1-111 (1986). 

Hoover WG (1991) Computational statistical mechanics, studies in modem thermodynamics 11. Elsevier Science, Amsterdam/Oxford/New York/Tokyo Ecawa N, Donaldson RR, Komanduii R, Koenig W, McKeown PA, MoriwaM T, Stowers IF... 

W. G. Hoover, Computational Statistical Mechanics, Elsevier, New York, 1991. [An alternative derivation of the canonical distribution is possible by incorporating temperature as a constraint in the equations of motion of the molecules. The resulting constrained equations are known as Nose-Hoover mechanics. For more details, see this reference and the references cited therein.]... 

Monte Carlo methods are perhaps the most frequently used in computational statistical mechanics. In particular, the Metropolis Monte Carlo technique has been used extensively in simulation of liquids. Monte Carlo methods are probabilistic, rather than deterministic, procedures atoms are moved more or less randomly during the course of the simulation. In a Metropolis Monte Carlo simulation of a molecular system, the following steps would be followed ... 

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... 

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