Interactions from diffraction data: historical and comprehensive overview of simulation assisted methods

A large part of statistical mechanics is concerned with the determination of condensed matter structure on the basis of known microscopic interactions. An increasing emphasis has been put on the opposite situation in the last decades as well, where structural data, e.g. pair-distance statistics, are known from diffraction experiments, and one looks for the corresponding interaction functions. The solution of this inverse problem was searched for within the integral equation theories of condensed matter in the early investigations, but before long computer simulation assisted methods were suggested. The interest in this field showed an increasing trend after some attempts appeared in the late 1980s. Several methods were published in the 1990s, and one-two methods appear annually nowadays. In this paper a comprehensive and historical overview is given on the solution of the inverse problem with simulation assisted methods. Emphasis is put on the theoretical grounds of the methods, on the choice of possible input structural functions, on the numerically local or global schemes of the potential modifications, on some advantages and limits of the different methods and on the scientific impact of the methods