Applications

In this chapter we will consider some applications of the general ideas presented in previous chapters. In particular, we will consider how to obtain the thermodynamic properties of a solid at temperature greater than zero. We have seen that the thermodynamics of a system can be described from the knowledge of a relevant thermodynamic potential and therefore, ultimately, from its partition function. In Sec.  we obtained with Eq.  the partition function of a system of classical particles interacting via a potential energy $U$ at constant volume $V$ and constant temperature $T$, and in particular that $Q$, and so $Z$, is completely determined by $U$. In many cases an exact analytical form for $U$ is not available and as a result the Boltzmann factor $\exp\{-\beta U\}$ cannot be integrated. However, under certain circumstances it may be possible to approximate $U$ and obtain an analytic expression for $Q$. If an analytical approximation to $U$ is not available, it may still be possible to obtain the value of $U$ for an ensemble of configurations, using computer simulation. We shall show how this can then be used to obtain stochastic approximations of the partition function.

If the system is not in the classical limit and quantum effects are important, then it is not possible to factorise the kinetic and the potential components of the energy and the partition function cannot be written as in , i.e. as the product of a kinetic (perfect gas) term $Z_P$ and a term $Q$ that only depends on the potential energy. In this case the terms entering the partition function would be the Boltzmann factors of the total energies, which are the eigenvalues of the Hamiltonian.