• This lecture should give a brief description about how the Fairness Function relates to entropy
• This lecture may be moved around if students ask about it early on in the discussion of Fairness

Given two uncorrelated systems, AA and BB, we can show that the fairness of the combined system is equal to the sum of the fairnesses of the two separate systems. This means that FF is extensive. $$\mathcal{F}_A = -k \sum_i P_i \ln P_i$$ $$\mathcal{F}_B = -k \sum_i P_i \ln P_i$$ $$\mathcal{F}_{AB} = -k \sum_{ij} P_{ij} \ln\left( P_{ij} \right)$$ $$= -k \sum_{ij} P_iP_j \ln\left( P_iP_j \right)$$ $$= -k \sum_{ij} P_i P_j \left(\ln P_i + \ln P_j\right) $$ $$= -k \left(\sum_{ij} P_i P_j \ln P_i\right) -k \left(\sum_{ij} P_i P_j \ln P_j\right)$$ $$= -k \left(\sum_i P_i \ln P_i\right)\left(\sum_j P_j \right) -k \left(\sum_j P_j \ln P_j\right)\left(\sum_i P_i\right)$$ $$= -k \left(\sum_i P_i \ln P_i\right) -k \left(\sum_j P_j \ln P_j\right)$$ $$= \mathcal{F}_A + \mathcal{F}_B$$