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Additional resources for Advances in Chemical Physics Volume 140
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 : x2 0 Fðx; tÞ ¼ ½A þ B : x T ð136Þ ð137Þ Fy ðx; tÞ ¼ ½A þ B : 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 À1 $ 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 Sx 4 2 jtj þ À1 ðQ Þ : ð½QÀ þ ^tQþ SxÞ2 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.