Path integral relations numerical methods

We have obtained a broad range of detailed quantitative results relating to universality in the absence of the electron-phonon interaction by various theoretical techniques. The methods include perturbation theory, the coherent potential approximation (CPA), field theory, path integral methods, numerical calculations and the potential well analogy. The results include the density of states, the nature of the wave functions, the mean free path, the energy dependent... 

As we have aheady said, the grand canonical Monte Carlo provides a mean to determining the chemical potential, and hence, die free energy of the system. In other MC and MD calculations a numerical value for the free energy can always be obtained by means of an integration of thermodynamic relations along a path which links the state of interest to one for which the free energy is already known, for example, the dilute gas or the low-temperature solid. Such a procedure requires considerable computational effort, and it has alow numerical stability. Several methods have been proposed and tested. 

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