Generalization of Shannon’s theorem for Tsallis entropy

Abstract

By using the assumptions that the entropy must (i) be a continuous function of the probabilities {pi}(pi∈(0,1)∀i), only; (ii) be a monotonic increasing function of the number of states W, in the case of equiprobability; (iii) satisfy the pseudoadditivity relation Sq(A+B)/k=Sq(A)/k+Sq(B)/k+(1−q)Sq(A)Sq(B)/k2 (A and B being two independent systems, q∈R and k a positive constant), and (iv) satisfy the relation Sq({pi})=Sq(pL,pM)+pL qSq({pi/pL})+pM qSq({pi/pM}), where pL+pM=1(pL=∑i=1WLpi and pM=∑i=WL+1Wpi), we prove, along Shannon’s lines, that the unique function that satisfies all these properties is the generalized Tsallis entropy Sq=k(1−∑i=1Wpiq)/(q−1).