In practice, formula_10 is estimated from the database of known protein structures, while formula_18 typically results from calculations or simulations. For example, formula_10 could be the conditional probability of finding the formula_20 atoms of a valine and a serine at a given distance formula_3 from each other, giving rise to the free energy difference formula_22. The total free energy difference of a protein, formula_23, is then claimed to be the sum of all the pairwise free energies:  where the sum runs over all amino acid pairs formula_25 (with formula_26) and formula_27 is their corresponding distance. In many studies formula_12 does not depend on the amino acid sequence. Intuitively, it is clear that a low value for formula_23 indicates that the set of distances in a structure is more likely in proteins than in the reference state. However, the physical meaning of these PMFs have been widely disputed since their introduction. The main issues are the interpretation of this "potential" as a true, physically valid potential of mean force, the nature of the reference state and its optimal formulation, and the validity of generalizations beyond pairwise distances. The first, qualitative justification of PMFs is due to Sippl, and based on an analogy with the statistical physics of liquids. For liquids, the potential of mean force is related to the radial distribution function formula_30, which is given by:  where formula_10 and formula_18 are the respective probabilities of finding two particles at a distance formula_3 from each other in the liquid and in the reference state. For liquids, the reference state is clearly defined; it corresponds to the ideal gas, consisting of non-interacting particles. The two-particle potential of mean force formula_35 is related to formula_30 by:  According to the reversible work theorem, the two-particle potential of mean force formula_35 is the reversible work required to bring two particles in the liquid from infinite separation to a distance formula_3 from each other. Sippl justified the use of PMFs - a few years after he introduced them for use in protein structure prediction - by appealing to the analogy with the reversible work theorem for liquids. For liquids, formula_30 can be experimentally measured using small angle X-ray scattering; for proteins, formula_10 is obtained from the set of known protein structures, as explained in the previous section. However, as Ben-Naim writes in a publication on the subject:  [...]the quantities, referred to as `statistical potentials,' `structure based potentials,' or `pair potentials of mean force', as derived from the protein data bank, are neither `potentials' nor `potentials of mean force,' in the ordinary sense as used in the literature on liquids and solutions.