Consider a composite system consisting of two simple systems connected thermally and allowing the exchange of one type of entity (N₁), but with an impenetrable barrier to all other entities and interaction.  The virtual change in entropy in the appropriate virtual process is

The closure conditions demand

therefore

Since dS must vanish for arbitrary dE⁽¹⁾ and dN₁⁽¹⁾, the conditions for equilibrium are

Therefore also

Thus, just as temperature differentials can be looked upon as a sort of "potential" for heat flux and the relationship of pressure differentials can be looked upon as a sort of "potential" for relationship volume changes, so the entity potential can be looked upon as a sort of "potential" for matter flux.  A difference in entity potential provides a "generalized force" for matter flow.  If the fundamental relationship is known, this differential can be turned into an explicit result for the rate of entity diffusion.  The entities can be individuals or organizations and their cultures, i.e.  their meme pool.  This is particularly crucial step for a predictive model. 

The ecological approach such as used by Cavalli-Sforza and Feldman [1981] is in good agreement with observations but it cannot be considered predictive because the meme infection or acquisition rates could not be calculated from some underlying principles.  Here the situation is significantly different.  The final equilibrium condition is predictable from underlying principles.  There is an uncertainty in the details of the path, but the final state is determinable.  Before I can take this step I need some additional mathematical tools that I shall develop in the following pages.

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