Download Advances in Chemical Physics Volume 140 by Stuart A. Rice PDF

By Stuart A. Rice

This sequence offers the chemical physics box with a discussion board for severe, authoritative reviews of advances in each zone of the self-discipline.

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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. Neglecting it in the previous analysis yields the leading contributions. D. Linear Limit of Nonlinear Coefficients In the linear limit, differentiation of the second entropy, Eq. (19), gives the relationship between the two sets of coefficients. One obtains Eðx; tÞ ¼ 12½A þ A0 þ B þ BT Š : x2 0 Fðx; tÞ ¼ ½A þ B Š : x T ð136Þ ð137Þ Fy ðx; tÞ ¼ ½A þ BŠ : x ð138Þ Gðx; tÞ ¼ A0 ð139Þ Gy ðx; tÞ ¼ A ð140Þ and Gz ðx; tÞ ¼ BT ð141Þ These expressions can be used to confirm the consistency between the linear and the nonlinear results given earlier.

Sð1Þ ðxÞ þ Sð1Þ ðx0 Þ; jtj ! 1 ð67Þ the second law of nonequilibrium thermodynamics 23 This follows directly by setting Q to zero in either of the two previous expressions. In the intermediate regime, ½SÀ1 À QðtÞ SQ ðtÞT ŠÀ1 $ ðQþ ÞÀ1 =2jtj ¼ a2 =jtj; and ðI þ QðtÞSÞ $ tQÀ S þ jtj Qþ S. Hence the second entropy goes like Sð2Þ ðx0 ; xjtÞ $ Sð1Þ ðxÞ þ þ jtj þ À1  2 jtj  ðQ Þ : x þ x ÁðQþ ÞÀ1 ½QÀ þ ^tQþ ŠSx 4 2 jtj þ À1 ðQ Þ : ð½QÀ þ ^tQþ ŠSxÞ2 4 ð68Þ The terms that are linear in the time interval must add up to a negative number, and so as the flux spontaneously develops these terms approach zero from below.

The first equality in expression (86) for the second entropy contains two terms involving x0 . The quadratic term is negative and represents the entropy cost of ordering the system dynamically; whether the departure from zero is positive or negative, any fluctuation in the flux represents order and is unfavorable. The linear term can be positive or negative; it is this term that encourages a nonzero flux that drives the system back toward equilibrium in the future, which obviously increases the first entropy.

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