Nonextensive Statistical Mechanics

98 3 Generalizing What We Learntsteepest descent method [227, 334], the Khinchin large-numbers-law method [229,335], and the counting in the microcanonical ensemble [230, 231, 336].In the continuous (classic) limit, Eqs. (3.201), (3.202), and (3.203) take the formwithp(p, x) = e−β q [H(p,x)−U q ]q, (3.221)¯Z qβ q ≡β∫dp dx [p(p, x)]q , (3.222)and∫¯Z q ≡dp dx e −β q [H(p,x)−U q ]q , (3.223)H(p, x) being the Hamiltonian of the system.In the generic quantum case, Eqs. (3.201), (3.202), and (3.203) take the formwithandˆρ = e−β q (Ĥ−U q )q, (3.224)¯Z qβ q ≡Ĥ being the Hamiltonian of the system.βTrˆρ q , (3.225)¯Z q ≡ Tr e −β q (Ĥ−U q )q , (3.226)3.7 About the Escort Distribution and the q-Expectation ValuesWe have seen that the escort distributions play a central role in nonextensive statisticalmechanics. Let us start by analyzing their generic properties. We shall focus onthe discrete version, i.e.,P i ≡p q i∑ Wj=1 pq jW∑( p i = 1; q ∈ R) . (3.227)i=1We will note this transformation as follows: