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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The analysis will be carried out using the nonlinear results for the subsystem; in the linear regime the correspondence given in the preceding subsection can be applied. The second entropy then is a function of three constrained variables, Sð2Þ ðÁ0 x; Ár x; xjt; Xr Þ. The internal part, which accounts for the resistance to the ﬂux, is as given earlier; it characterizes the transition x ! x0 . The external part is entirely inﬂuenced by the reservoir, and it consists of two parts: the equilibration with the reservoir for the initial state x, and the transition x0 !
42 phil attard Accordingly, the change in the entropy of the reservoir associated with the current point in phase space is ÁSr ðG1 ; tÞ ¼ Xr Á xÁ ðG1 Þ Z 1 t _ 0 ðtjG1 ÞÞ þ ^t Xr Á Ls ðx1 ÞXr dt ½Xr Á xðG ¼ 2 Àt Z 1 t _ 0 ðtjG1 ÞÞ þ ^t Xr Á Lðx1 ; ^tÞXr ¼ dt ½Xr Á xðG 2 Àt ð158Þ where ^t ¼ signðtÞ. In the second equality, the symmetric part of the transport matrix appears, Ls ðxÞ ¼ ½Lðx; þ1Þ þ Lðx; À1Þ=2, which is independent of ^t. This means that the second term in the integrand is odd in time and it gives zero contribution to the integral.
The scaling of Gðx; tÞ motivates writing Gz ðx; tÞ $ 1 z 1 g ðxÞ þ gz ðxÞ þ gz ðxÞ þ ^tgz ðxÞ 2 3 jtj 0 t 1 ð113Þ In view of the parity rule (95), gz and gz are symmetric matrices and gz and gz 0 2 1 3 are antisymmetric matrices. The derivative of the second entropy force is qFðx; tÞ ¼ Gðx; tÞ þ Gz ðx; tÞ qx ð114Þ which in the intermediate regime becomes qf 0 ðxÞ qf 1 ðxÞ 1 1 þ^t ¼ ½g ðxÞþ gz ðxÞ þ ½g ðxÞ þ gz ðxÞ þ g ðxÞ þ gz ðxÞ þ^tgz ðxÞ 0 1 2 3 2 qx qx jtj 0 t 1 ð115Þ Clearly, the ﬁrst two bracketed terms have to individually vanish.