Ornstein-Zernike equation

In statistical mechanics the Ornstein–Zernike equation (named after Leonard Ornstein and Frits Zernike) is an integral equation for defining the direct correlation function. It basically describes how the correlation between two molecules can be calculated. Its applications can mainly be found in fluid theory.

The derivation below is heuristic in nature: rigorous derivations require extensive graph analysis or functional techniques. The interested reader is referred to the text book for the full derivation.

It is convenient to define the total correlation function:

which is a measure for the "influence" of molecule 1 on molecule 2 at a distance ${\displaystyle r_{12}}$ away with ${\displaystyle g(r_{12})}$ as the radial distribution function. In 1914 Ornstein and Zernike proposed to split this influence into two contributions, a direct and indirect part. The direct contribution is defined to be given by the direct correlation function, denoted ${\displaystyle c(r_{12})}$. The indirect part is due to the influence of molecule 1 on a third molecule, labeled 3, which in turn affects molecule 2, directly and indirectly. This indirect effect is weighted by the density and averaged over all the possible positions of particle 3. This decomposition can be written down mathematically as

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