By William Geraint V. Rosser

Advent to Statistical Physics (Mathematics and Its functions) [Paperback

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If P (A | B) > P (A), we say that A and B are positively correlated, while P (A | B) < P (A) corresponds to a negative correlation between A and B. 2 Joint Probability Let us now consider an event that is an element of the set A as well as of the set B. Then, the event is contained also in A ∩ B. The probability P (A ∩ B) is called the joint probability that the event is contained in both classes. 3 Combined Probabilities P (A ∩ B) = P (A | B)P (B) = P (B | A)P (A). 55) This representation allows a natural definition of statistically independent events.

33). 109) of the equation of motion and insert the ˆ after the propagator exp{L(t ˆ − t0 )}. We obtain identity operator Pˆ + Q dGα (t) ˆ − t0 )} Pˆ + Q ˆ LG ˆ 0 = exp{L(t α dt 0 ˆ − t0 )}Pˆ LG ˆ ˆ ˆˆ 0 = exp{L(t α + exp{L(t − t0 )}QLGα . 114) βγ where we have introduced the M × M frequency matrix Ωαγ , ˆ 0 ). 116) t0 ˆ =L ˆ1 + L ˆ 2 . 41) with L ˆ 1 by Q ˆL ˆ and L ˆ 2 by by a derivative with respect to the time. If we replace L ˆ we arrive at Pˆ L, ˆ ˆˆ ˆ LG ˆ 0α = eQL(t−t0 ) Q ˆ LG ˆ 0α eL(t−t0 ) Q t ˆ QL(t −t0 ) Q ˆ LG ˆ 0α .

This equation is the differential form of the Chapman–Kolmogorov equation, which is denoted in the literature as the forward differential Chapman– Kolmogorov equation. The right-hand side of this equation defines the general ˆ Markov . structure of the Markovian L 32 2. Evolution and Probabilistic Concepts If we want to specify the differential Chapman–Kolmogorov equation, we must consider that by definition the components Bαβ (X, t) must form a positive-definite matrix and that W (X | Z; t) must be a nonnegative function.

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