Equation for Combined Site and Induced Heterogeneity

Derivation of Equation for Combined Site and Induced Heterogeneity 

We now make the usual approximation that at time t, all sites on patches having E are full and all with E Et are empty hence from Eq. (47), 

At a later time, all patches including the mth are partially filled and further increase of Et E q is associated solely with the effects of induced heterogeneity, then 

This latter situation requires that the increase of E due to induced heterogeneity on the lowest energy patch is larger than E q — Eoo, the range of E in the site heterogeneity. Substitution of Eq. (48) into (49) gives 

Since any distribution of site heterogeneity leads to the same conclusion, we may remove the restriction of patches of equal area and let be a function of x, such that n c — f fi oo + hx). Any distribution function for Wj and E o, with patches of varying area may be introduced. In such circumstances, for Et E q,  

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