The Kirkwood–Buff (KB) solution theory, due to John G. Kirkwood and Frank P. Buff, links macroscopic (bulk) properties to microscopic (molecular) details. Using statistical mechanics, the KB theory derives thermodynamic quantities from pair correlation functions between all molecules in a multi-component solution. The KB theory proves to be a valuable tool for validation of molecular simulations, as well as for the molecular-resolution elucidation of the mechanisms underlying various physical processes. For example, it has numerous applications in biologically relevant systems. The reverse process is also possible; the so-called reverse Kirkwood–Buff (reverse-KB) theory, due to Arieh Ben-Naim, derives molecular details from thermodynamic (bulk) measurements. This advancement allows the use of the KB formalism to formulate predictions regarding microscopic properties on the basis of macroscopic information.
The radial distribution function The radial distribution function (RDF), also termed the pair distribution function or the pair correlation function, is a measure of local structuring in a mixture. The RDF between components i {\displaystyle i} and j {\displaystyle j} positioned at r i {\displaystyle {\boldsymbol {r}}_{i}} and r j {\displaystyle {\boldsymbol {r}}_{j}} , respectively, is defined as:
g i j ( R ) = ρ i j ( R ) ρ i j bulk {\displaystyle g_{ij}({\boldsymbol {R}})={\frac {\rho _{ij}({\boldsymbol {R}})}{\rho _{ij}^{\text{bulk}}}}}
where ρ i j ( R ) {\displaystyle \rho _{ij}({\boldsymbol {R}})} is the local density of component j {\displaystyle j} relative to component i {\displaystyle i} , the quantity ρ i j bulk {\displaystyle \rho _{ij}^{\text{bulk}}} is the density of component j {\displaystyle j} in the bulk, and R = | r i − r j | {\displaystyle {\boldsymbol {R}}=|{\boldsymbol {r}}_{i}-{\boldsymbol {r}}_{j}|} is the inter-particle radius vector. Necessarily, it also follows that:
g i j ( R ) = g j i ( R ) {\displaystyle g_{ij}({\boldsymbol {R}})=g_{ji}({\boldsymbol {R}})}
Assuming spherical symmetry, the RDF reduces to:
g i j ( r ) = ρ i j ( r ) ρ i j bulk {\displaystyle g_{ij}(r)={\frac {\rho _{ij}(r)}{\rho _{ij}^{\text{bulk}}}}}
where r = | R | {\displaystyle r=|{\boldsymbol {R}}|} is the inter-particle distance. In certain cases, it is useful to quantify the intermolecular correlations in terms of free energy. Specifically, the RDF is related to the potential of mean force (PMF) between the two components by:
P M F i j ( r ) = − k T ln ( g i j ) {\displaystyle PMF_{ij}(r)=-kT\ln(g_{ij})}
where the PMF is essentially a measure of the effective interactions between the two components in the solution.
The Kirkwood–Buff integrals The Kirkwood–Buff integral (KBI) between components i {\displaystyle i} and j {\displaystyle j} is defined as the spatial integral over the pair correlation function:
G i j = ∫ V [ g i j ( R ) − 1 ] d R {\displaystyle G_{ij}=\int \limits _{V}[g_{ij}({\boldsymbol {R}})-1]\,d{\boldsymbol {R}}}
which in the case of spherical symmetry reduces to:
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