Reverse Monte Carlo methods
Most modelling techniques in the solid-state sciences are what would be called forward modelling. The starting point is an exact expression for a Hamiltonian or free energy, together with a way of using this to generate numerical quantities that can be compared with experiment. The starting expression could be a representation of the Schrödinger equation, or a numerical approximation to the forces between atoms. The models can also be based on differential equations to incorporate time-dependence (e.g. the Lagrangian of classical mechanics), or the partition function to obtain thermodynamic averages. The solutions to the models can be exact in some cases (particularly when the models are used to describe a system at zero temperature), but may frequently have some degree of statistical uncertainty. For example, in molecular dynamics, the dynamic equations are solved using discrete time steps over a limited time span, and in Monte Carlo simulations only a portion of the total phase space available to a system is sampled. Whatever approach is taken, the end point of forward modelling may be to provide new insights, to predict behaviour, or to interpret observations. A key point towards this end is to be able to reproduce some experimental data in order to check that the simulations are representative of reality, even though the true value of a simulation is to provide information, insight or understanding that cannot be extracted from experimental data.
In this chapter we will consider a second type of modelling technique, which is called inverse modelling. This is best described by comparison with forward modelling.