By Stuart A. Rice
This sequence offers the chemical physics box with a discussion board for serious, authoritative reviews of advances in each sector of the self-discipline.
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III. NONLINEAR THERMODYNAMICS A. Quadratic Expansion In the nonlinear regime, the thermodynamic force remains formally deﬁned as the ﬁrst derivative of the ﬁrst entropy, XðxÞ ¼ qSð1Þ ðxÞ qx ð81Þ However, it is no longer a linear function of the displacement. In other words, the second derivative of the ﬁrst entropy, which is the ﬁrst entropy matrix, is no longer constant: SðxÞ ¼ q2 Sð1Þ ðxÞ qx qx ð82Þ For the second entropy, without assuming linearity, one can nevertheless take x0 to be close to x, which will be applicable for t not too large.
Hence the right-hand side of 30 phil attard Eq. (97) is given by Eq. (96) evaluated at the midpoint x2 ¼ ~x2 : Sð2Þ ðx3 ; ~ x2 ; x1 jt; tÞ ¼ 18½Gðx1 ; tÞ þ Gy ðx3 ; tÞ : ½x3 À x1 2 þ 12½Fðx1 ; tÞ À Fy ðx3 ; tÞ Á ½x3 À x1 þ Eðx1 ; tÞ þ Eðx3 ; tÞ À 12½Sð1Þ ðx1 Þ þ Sð1Þ ðx3 Þ 1 ½x3 À x1 2 : ½Sðx1 Þ þ Sðx3 Þ þ 16 ð99Þ Here the ﬁrst entropy Sð1Þ ð~ x2 Þ has been expanded symmetrically about the terminal points to quadratic order. Each term of this may be equated to half the corresponding one on the righthand side of Eq.
Theories that invoke the Principle of Minimum Dissipation, [10–12, 32] or the Principle of 22 phil attard Maximum Dissipation, [13–18] are fundamentally ﬂawed from this point of view. These two principles are diametrically opposed, and it is a little surprising that they have both been advocated simultaneously. Using the quadratic expression for the second entropy, Eq. (22), the reduction condition, Eq. (23), and the correlation function, Eq. (17), the second entropy may be written at all times as Sð2Þ ðx0 ; xjtÞ ¼ Sð1Þ ðxÞ þ 12½SÀ1 À QðtÞSQðtÞT À1 : ðx0 þ QðtÞSxÞ2 ð64Þ It is evident from this that the most likely terminal position is xðx; tÞ ¼ ÀQðtÞSx, as expected from the deﬁnition of the correlation function, and the fact that for a Gaussian probability means equal modes.