By Stuart A. Rice

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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.

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 ). 0 It is always possible to write f 1 ðxÞ ¼ g ðxÞÈðxÞXðxÞ 0 ð122Þ The matrix È is underdetermined by this equation.

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. Beyond the intermediate regime, in the long time limit the correlation function vanishes, QðtÞ ! 0. In this regime the second entropy is just the sum of the two ﬁrst entropies, as is expected, Sð2Þ ðx0 ; xjtÞ ! Sð1Þ ðxÞ þ Sð1Þ ðx0 Þ; jtj !