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**Sample text**

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. E. Exchange with a Reservoir Now the previously isolated subsystem is allowed to exchange x with an external reservoir that applies a thermodynamic force Xr .

In view of the previous results, in the nonlinear regime the second entropy for this may be written Sð2Þ ðx3 ; x2 ; x1 jt; tÞ ¼ Sð2Þ ðx3 ; x2 jtÞ þ Sð2Þ ðx2 ; x1 jtÞ À Sð1Þ ðx2 Þ ¼ 12 Gy ðx3 ; tÞ : ½x2 À x3 2 þ Fy ðx3 ; tÞ Á ½x2 À x3 þ Eðx3 ; tÞ þ 12 Gðx1 ; tÞ : ½x2 À x1 2 þ Fðx1 ; tÞ Á ½x2 À x1 þ Eðx1 ; tÞ À Sð1Þ ðx2 Þ ð96Þ The ﬁrst three terms arise from the expansion of Sð2Þ ðx3 ; x2 jtÞ about x3 , which accounts for the appearance of the daggers, and the second three terms arise from the expansion of Sð2Þ ðx2 ; x1 jtÞ about x1 .

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.