Understanding Molecular Simulation (PDF)

Chapter 2 Statistical Mechanics (p. 10-11)

The topic of this book is computer simulation. Computer simulation allows us to study properties of many-particle systems. However, not all properties can be directly measured in a simulation. Conversely, most of the quantities that can be measured in a simulation do not correspond to properties that aremeasured in real experiments. To give a specific example: in aMolecular Dynamics simulation of liquid water, we could measure the instantaneous positions and velocities of all molecules in the liquid. However, this kind of information cannot be compared to experimental data, because no real experiment provides us with such detailed information. Rather, a typical experiment measures an average property, averaged over a large number of particles and, usually, also averaged over the time of the measurement. If we wish to use computer simulation as the numerical counterpart of experiments, we must know what kind of averages we should aim to compute. In order to explain this, we need to introduce the language of statistical mechanics. This we shall do here. We provide the reader with a quick (and slightly dirty) derivation of the basic expressions of statistical mechanics. The aimof these derivations is only to show that there is nothing mysterious about concepts such as phase space, temperature and entropy and many of the other statistical mechanical objects that will appear time and again in the remainder of this book.

2.1 Entropy and Temperature

Most of the computer simulations that we discuss are based on the assumption that classical mechanics can be used to describe the motions of atoms and molecules. This assumption leads to a great simplification in almost all calculations, and it is therefore most fortunate that it is justified in many cases of practical interest. Surprisingly, it turns out to be easier to derive the basic laws of statistical mechanics using the language of

Daan Frenkel is based at the FOM Institute for Atomic and Molecular Physics and at the Department of Chemistry, University of Amsterdam. His research has three central themes: prediction of phase behavior of complex liquids, modeling the (hydro) dynamics of colloids and microporous structures, and predicting the rate of activated processes. He was awarded the prestigious Spinoza Prize from the Dutch Research Council in 2000.

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"... this book brilliantly lays down the scientific foundations of the simulational approach ..." Prof. Kurt Binder in Physics World, 1997

"... a treasure. The book is a marvellous mix of just enough formalism with an informal and readable style, sufficient detail to understand methodological advances, appropriate mathematics ..." Prof. Mark A. Ratner in Physics Today, 1997