chapter 1 section 1.1

STATISTICAL MECHANICS AND COMPUTER SIMULATION

The state of a liter of water can be described using only a few macroscopic equilibrium properties. This is remarkable, considering that the fluid consists of some mutually interacting atoms. The secret lies in the word macroscopic. All atomic processes and fluctuations take place extremely fast on a very small distance scale and will be averaged out while we perform our measurement. Only those particular combinations of atomic coordinates that are essentially time independent can be macroscopic observable. To compute the equilibrium properties of a bulk substance, there is little need to know the exact trajectory of every atom in the system. Rather, we like to calculate the average properties starting from a proper atomistic model. Thermodynamics relates macroscopic equilibrium properties with each other and statistical mechanics provides the link between these quantities and the atomistic description on a microscopic level [1,2].

The fundamental quantity in equilibrium statistical mechanics is the partition function, the sum of the Boltzmann factors over all possible states s of a canonical system of N atoms in a volume V at a temperature T

Here is Boltzmann 's constant and is the total energy of the system in state s. The Boltzmann factors are proportional to the probability of finding the system in state s. One therefore can also calculate the average of any dynamical property A as

The ergodic theorem states that this ensemble average is equal to the time average of A that would be obtained by following the natural dynamics of the system for a sufficiently long time [2]. The canonical partition function Z is connected to the thermodynamic Helmholtz free energy F by

The average value of most equilibrium properties is related in one way or another to (derivatives of) the Helmholtz free energy F, and can therefore be expressed in terms of the partition function Z. For instance, the energy E can be calculated as follows,

In view of the central role played by the partition function, the aim of many theoretical studies in equilibrium statistical mechanics is to find (approximate) expressions for the partition function, for specific model systems. An alternative route to extract the macroscopic information from a microscopic model, is to perform computer simulations [3]. There are two basic styles of computer simulations: Monte Carlo (MC) and molecular dynamics (MD). In an MD simulation one solves Newton's equation of motion numerically for a system containing several hundreds up to several millions of molecules, in discrete time steps. The desired macroscopic observables are computed simply by time averaging. Hence, the MD simulation follows the natural time evolution of the system, which includes dynamical information. With the use of periodic boundaries, a small number of atoms (usually 100-10000) is usually already enough to approximate the behavior of an infinite system.

The Monte Carlo method can be considered a numerical implementation of Gibbs' concept of averaging over an ensemble of equilibrium states: in MC simulations, macroscopic observables are obtained by directly computing properly weighted averages over the accessible states of a many-particle system (cf. eqn. 1.2). Because the order in which this averaging is carried out has no physical meaning, the MC method contains no dynamical information. In practice, the averaging over the most probable states of this system is achieved by using a random sampling procedure. Starting from a configuration, one or more particles are moved randomly. Each move is accepted or rejected with a certain probability, such that the probability of finding the system in a state s is proportional to the Boltzmann factor (in practice, only the potential energy needs to be considered in this sampling).

As computer simulations of classical many-body systems are, in principle, exact (i.e. can be carried out to any desired accuracy), they can be used to test the assumptions made in approximate theories of many-body systems. Simulations are also of great practical use in predicting the macroscopic properties of new materials or novel substances, before embarking on the expensive task of actually making these materials. Moreover, computer simulations can be used to asses the behavior of materials under conditions that are not easily achieved in experiments. Finally, computer simulations sometimes allow us to predict qualitatively new behavior, or new phases of known substances, and thereby act as a guide to both theoretical and experimental work.

chapter 1 section 1.1