Nonextensive Statistical Mechanics

70 3 Generalizing What We LearntTable 3.3 Three binary subsystems: joint probabilities p A+B+Cijk(i, j, k = 1, 2). The quantitieswithout (within) square brackets [ ] correspond to state 1 (state 2) of subsystem C. Themarginalprobabilities where we have summed over the states of B are defined as indicated in the Table.The marginal probabilities where we have summed over the states of A are defined as follows:p (A)+B+C11≡ p A+B+C111+ p A+B+C211, p (A)+B+C21≡ p A+B+C121+ p A+B+C221, p (A)+B+C12≡ p A+B+C112+ p A+B+C212and p (A)+B+C22≡ p A+B+C122+ p A+B+C222.Themarginal probabilities where we have summed over thestates of both A and B are defined as follows: p (A)+(B)+C1≡ p A+B+C111+ p A+B+C121+ p A+B+C211+1+p A+B+C221and p (A)+(B)+C2≡ p A+B+C112+ p A+B+C122+ p A+B+C212+ p A+B+C222. Of course, p (A)+(B)+Cp (A)+(B)+C2= 1A\ B 1 212p A+B+C111p A+B+C121p A+(B)+C11≡ p A+B+C111+ p A+B+C121[p A+B+C112] [p A+B+C122] [p A+(B)+C12≡ p A+B+C112+ p A+B+C122]p A+B+C211p A+B+C221p A+(B)+C21≡ p A+B+C211+ p A+B+C221[p A+B+C212] [p A+B+C222] [p A+(B)+C22≡ p A+B+C212+ p A+B+C222]p (A)+B+C11p (A)+B+C21p (A)+(B)+C1[p (A)+B+C12] [p (A)+B+C22] [p (A)+(B)+C2]If N = 3, we shall note p A+B+C111, p A+B+C112, p A+B+C121,...,and p A+B+C222the correspondingjoint probabilities. Of course, they satisfy p A+B+C111+ ...+ p A+B+C222= 1(see Table 3.3).The joint probabilities corresponding to the general case are noted p A 1+A 2 +...+A Np A 1+A 2 +...+A N11...1,11...2,...,and p A 1+A 2 +...+A N22...2. They satisfy∑p A 1+A 2 +...+A Ni 1 i 2 ...i N= 1 (N = 1, 2, 3,...) , (3.121)i 1 ,i 2 ,...,i N =1,2and can be represented as a 2 × 2 × ...× 2 hypercube, which is associated with 2 Nstates. There are N sets of marginal probabilities where we have summed over onesubsystem. They are noted p (A 1)+A 2 +...+A Ni 2 ...i N, p A 1+(A 2 )+...+A Ni 1 i 3 ...i N, ..., and p A 1+A 2 +...+A Ni 1 i 2 ...i N−1.There are N(N − 1)/2 sets of marginal probabilities where we have summed overtwo subsystems and so on.For future use, let us right away introduce the notation corresponding to the mostgeneral case of N distinguishable discrete subsystems. Subsystem A r is assumed tohave W r states (r = 1, 2,...,N). The joint probabilities for the whole system are}, such that{p A 1+A 2 +...+A Ni 1 i 2 ...i N∑W 1 W 2∑...i 1 =1 i 2 =1∑W Ni N =1p A 1+A 2 +...+A Ni 1 i 2 ...i N= 1 (N = 1, 2, 3,...) . (3.122)These probabilities can be represented in a W 1 × W 2 × ... × W N hypercube.The marginal probabilities are obtained by summing over the states of at least one