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source site 1. Optimum Intermediate Point The optimum intermediate point of the sequential transition may be obtained by maximizing the corresponding second entropy. À1 ! 2 qf 1 ð~x2 Þ g ð~ x2 Þ ½x1 À x3 jtj 0 q~x2 t À1 z ¼ g ð~ x2 Þ g ð~ x2 Þ½x3 À x1 3 2 0 ð135Þ 0 ¼ ^tf 1 ðx1 Þ À ^tf 1 ðx3 Þ À Sð~ x2 Þ½x2 À ~ x2 þ which has solution x2 À ~ x2 ¼ ^t Sð~ x2 Þ À using Eq. (119). This shows that the departure of the optimum point from the midpoint is of second order (linear in t and in x3 À x1 ), and that l is of linear order in t.
see 23), and the correlation function, Eq. (17), the second entropy may be written at all times as Sð2Þ ðx0 ; xjtÞ ¼ Sð1Þ ðxÞ þ 12½SÀ1 À QðtÞSQðtÞT À1 : ðx0 þ QðtÞSxÞ2 ð64Þ It is evident from this that the most likely terminal position is xðx; tÞ ¼ ÀQðtÞSx, as expected from the deﬁnition of the correlation function, and the fact that for a Gaussian probability means equal modes. This last point also ensures that the reduction condition is automatically satisﬁed, and that the maximum value of the second entropy is just the ﬁrst entropy, Sð2Þ ðx; tÞ Sð2Þ ðx0 ; xjtÞ ¼ Sð1Þ ðxÞ ð65Þ This holds for all time intervals t, and so in the optimum state the rate of production of second entropy vanishes.
see url X00 is determined by one-half of the external change in the total ﬁrst entropy. The factor of 12 occurs for the conditional transition probability with no speciﬁc correlation between the terminal states, as this preserves the singlet probability during the reservoir induced transition [4, 8, 80]. The implicit assumption underlying this is that the conductivity of the reservoirs is much greater than that of the subsystem. The second entropy for the stochastic transition is the same as in the linear case, Eq.