We realize an experiment gathering two non-independent sub-experiments, named 1 and 2. For exemple, the measurment...

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We realize an experiment gathering two non-independent sub-experiments, named 1 and 2. For exemple, the measurment of two quantities in two interacting systems. We introduce the joint probability pij that the result is at the same time a; for the experiment 1 and b, for the experiment 2. Also, we define p the probability that the result of experiment 1 is ai (2) independently from the result of experiment 2, and reciprocally for p²). 1. Give the expression of p¹) and p2) in terms of probabilities pij. Verify the normalisation of p(¹) and p²). 2. Give the expression of the totale entropy S(12) associated with the (double)experiment. Also, give S(1) and S(2) the entropies associated with experiments 1 and 2. 3. Show that (1), (2) Pi Pj S(12) - (S(¹) + S(²)) = KB BΣPij In! Pij ij 4. Using In x < x-1, show that S(12) S(1) + S(2) We realize an experiment gathering two non-independent sub-experiments, named 1 and 2. For exemple, the measurment of two quantities in two interacting systems. We introduce the joint probability pij that the result is at the same time a; for the experiment 1 and b, for the experiment 2. Also, we define p the probability that the result of experiment 1 is ai (2) independently from the result of experiment 2, and reciprocally for p²). 1. Give the expression of p¹) and p2) in terms of probabilities pij. Verify the normalisation of p(¹) and p²). 2. Give the expression of the totale entropy S(12) associated with the (double)experiment. Also, give S(1) and S(2) the entropies associated with experiments 1 and 2. 3. Show that (1), (2) Pi Pj S(12) - (S(¹) + S(²)) = KB BΣPij In! Pij ij 4. Using In x < x-1, show that S(12) S(1) + S(2)

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