Tractable exploration of phase space

Statistical mechanics is a powerful and elegant theory, at whose core lie the concepts of phase space, probability distributions, and partition functions. With these concepts, statistical mechanics explains physical properties, such as entropy, and physical phenomena, such as irreversibility, all in terms of microscopic properties.

The stimulating nature of statistical mechanical theories notwithstanding, calculation of partition functions requires increasingly severe approximations and idealizations as the density of systems increases and particles begin interacting. Even with unrealistic approximations, the partition function is not calculable for systems at high densities. The difficulty lies with the functional dependence of the partition function on the system volume. This dependence cannot be analytically determined, because of the impossible calculation of the configurational integral when the potential energy is not zero. For five decades then, after Gibbs posited the foundation of statistical mechanics, scientists were limited to solving only problems that accepted analytical solutions.

The invention of digital computers ushered in a new era of computational methods that numerically determine the partition function, rendering feasible the connection between microscopic Hamiltonians and thermodynamic behavior for complex systems.

Two large classes of computer simulation method were created in the 1940s and 1950s:

1. Monte Carlo simulations, and

2. Molecular dynamics simulations.

Both classes generate ensembles of points in the phase space of a well defined system. Computer simulations start with amicroscopicmodel of the Hamiltonian.