Classical Systems with Interactions

Abstract

In the ideal systems considered in the previous chapters the Hamiltonian H_n(q, p; α) and the Hamiltonian operator $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{H}_{N} \left( \alpha \right)$ are, respectively, a sum of one-particle dynamical functions and of one-particle operators (kinetic energy and harmonic oscillators). Real systems are characterized by the fact that, besides the kinetic energy, H_n(q, p; α) and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{H}_{N} \left( \alpha \right)$ also include the potential energy, which describes how the particles, atoms, or molecules, interact with each other. As has been shown throughout the text, ideal quantum systems are more complex than classical ideal systems. This complexity is also greater in systems with interaction and so from here onward only classical systems will be considered. Since, in general, the interaction potential in real systems is not known exactly, it is rather common to introduce the so-called reference systems in which the potential energy of interaction is relatively simple, while their thermodynamic properties are nevertheless similar to those of real systems, and so they provide a qualitative description of the latter. Along this chapter some approximate methods for the determination of the free energy of an interacting system are considered. In the last section, a brief summary of the so-called numerical simulation methods is provided.