Maximum entropy theory

A maximum entropy method (MEM) map of electron density is obtained by maximizing S = -Σ p log p subject to whatever data are available. MEM is derived as the only reconstruction technique that is consistent with simple and general requirements. The method is very widely applicable, but, in this paper, attention is focused on the problem of producing electron density maps in crystallography. The entropy formula can also be derived by analogy with a thermodynamic system of quanta, but it is shown that this model can be misleading, and can break down in practice. MEM applied to a different problem related to quantum fluctuations in the thermodynamic model is shown to lead to formulae equivalent to the maximum determinant method. It is argued that direct MEM will produce superior maps.