By Stuart A. Rice
This sequence presents the chemical physics box with a discussion board for serious, authoritative reviews of advances in each region of the self-discipline.
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Extra resources for Advances in Chemical Physics, Vol. 140
In the ﬁnal equality, one can see that, due to the orthogonality condition, the even and odd parts of the regression contribute separately to the maximum value of the second entropy. Note that in the expression for the second entropy a term independent of x0 has been neglected in this section. Hence the second entropy does not here reduce to the ﬁrst entropy as it does in Section 2. 1. Optimum Intermediate Point The optimum intermediate point of the sequential transition may be obtained by maximizing the corresponding second entropy.
It is negative and represents the cost of the order that is the constrained static state x. The second term is also negative and is quadratic in the coarse velocity. It represents the cost of maintaining the dynamic order that is induced in the system for a nonzero ﬂux x. The third and fourth terms sum to a positive number, at least in the optimum state, where they cancel with the second term. As will become clearer shortly, they represent the production of ﬁrst entropy as the system returns to equilibrium, and it is these terms that drive the ﬂux.
Explicitly 2 2 then, in the intermediate regime it follows that 1 1 g ðxÞ þ SðxÞ jtj 0 2 1 1 Gy ðx; tÞ ¼ g ðxÞ þ SðxÞ jtj 0 2 Gðx; tÞ ¼ ð116Þ ð117Þ and Gz ðx; tÞ ¼ À1 g ðxÞ þ ^tgz ðxÞ 3 jtj 0 ð118Þ with gz ðxÞ ¼ qf 1 ðxÞ=qx 3 ð119Þ the second law of nonequilibrium thermodynamics 33 which will be used later. These further expansions justify the second entropy given earlier, Eq. (109). In view of the reduction condition, Eq. (87), and the earlier scaling of G and F, the reduction condition (108) can be written to higher order: jtj À1 g ðxÞ : fðx; ^tÞ2 þ Ot2 2 0 Eðx; tÞ $ Sð1Þ ðxÞ þ ð120Þ Since Eðx; tÞ is an even function of t, this shows that f 1 ðxÞ Á gÀ1 ðxÞXðxÞ ¼ 0 0 ð121Þ This says that f 1 is orthogonal to the usual thermodynamic force X (using the inner product with metric g ).
Advances in Chemical Physics, Vol. 140 by Stuart A. Rice