By Stuart A. Rice
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Extra resources for Advances in Chemical Physics Volume 140
Also, since only the leading term or terms in the small-t expansion have been retained above, the neglected terms must increasingly contribute as t increases and the above explicit results must become increasingly inapplicable. Hence these results hold for t in the < t < tlong . intermediate regime, tshort $ $ The matrix Lð^tÞ is called the transport matrix, and it satisﬁes Lð^tÞ ¼ E Lð^tÞT E ¼ LðÀ^tÞT ð55Þ This follows because, in the grouped representation, Qþ contains nonzero blocks only on the diagonal and is symmetric, and QÀ contains nonzero blocks only off the diagonal and is asymmetric.
In the expression for the second entropy 38 phil attard of the isolated system, Eq. (109), the isolated system ﬁrst entropy, Sð1Þ ðxÞ, must be replaced by the total ﬁrst entropy, Sð1Þ ðxÞ À x Á Xr . With this replacement and adding the stochastic second entropy, the total second entropy is ð2Þ Stotal ðÁ0 x; Ár x; xjXr ; tÞ ¼ 1 g ðxÞ : ½Á0 x2 þ fðx; ^tÞ Á ½Á0 x þ Sð1Þ ðxÞ À x Á Xr 2jtj 0 þ 12½Sð1Þ ðx00 Þ À Sð1Þ ðx0 Þ À Ár x Á Xr ð143Þ Setting the derivative with respect to Ár x to zero, one ﬁnds X00s ¼ Xr ð144Þ As in the linear case, in the steady state the subsystem force equals the reservoir force at the end of the transition.
B. Parity In addition to the coefﬁcients for the nonlinear second entropy expansion deﬁned earlier, Eqs. (83), (84), and (85) deﬁne qSð2Þ ðx; x0 jtÞ F ðx; tÞ 0 qx0 x ¼x q2 Sð2Þ ðx; x0 jtÞ y G ðx; tÞ qx0 qx0 x0 ¼x y and q2 Sð2Þ ðx0 ; xjtÞ G ðx; tÞ qx0 qx z ð89Þ ð90Þ ð91Þ x0 ¼x Under the parity operator, these behave as Eðx; tÞ ¼ EðEx; tÞ ¼ Eðx; ÀtÞ y y Fðx; tÞ ¼ EF ðEx; tÞ ¼ F ðx; ÀtÞ y y Gðx; tÞ ¼ E G ðEx; tÞE ¼ G ðx; ÀtÞ ð92Þ ð93Þ ð94Þ 29 the second law of nonequilibrium thermodynamics and Gz ðx; tÞ ¼ E Gz ðEx; tÞT E ¼ Gz ðx; ÀtÞT ð95Þ The matrices G and Gy are symmetric.
Advances in Chemical Physics Volume 140 by Stuart A. Rice