By Stuart A. Rice
This sequence presents the chemical physics box with a discussion board for severe, authoritative reviews of advances in each sector of the self-discipline.
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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 ! 1 ð67Þ the second law of nonequilibrium thermodynamics 23 This follows directly by setting Q to zero in either of the two previous expressions. In the intermediate regime, ½SÀ1 À QðtÞ SQ ðtÞT À1 $ ðQþ ÞÀ1 =2jtj ¼ a2 =jtj; and ðI þ QðtÞSÞ $ tQÀ S þ jtj Qþ S. Hence the second entropy goes like Sð2Þ ðx0 ; xjtÞ $ Sð1Þ ðxÞ þ þ jtj þ À1 2 jtj ðQ Þ : x þ x ÁðQþ ÞÀ1 ½QÀ þ ^tQþ Sx 4 2 jtj þ À1 ðQ Þ : ð½QÀ þ ^tQþ SxÞ2 4 ð68Þ The terms that are linear in the time interval must add up to a negative number, and so as the ﬂux spontaneously develops these terms approach zero from below.
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.
Forward and reverse transitions. The solid lines are the adiabatic trajectories over an inﬁnitesimal time step, and the dashed lines are the stochastic transitions. ratio of the forward to the reverse transition probabilities is }ðG00 }ðGy GjÁt ; Xr Þ G00y jÁt ; Xyr Þ ÂÁ ðjG00 À G0 jÞeÀ½x 00 ¼ y ÂÁ ðjG À G 00 ¼ eÀ½x e e =Zss ðXr Þ y þx000 ÁXy =2kB ½xy þx000 ÁXyr =2kB Át ½x_ 00y Àx_ 00y ÁXyr =kB À½x r Á jÞe e Á Á e =Zss ðXyr Þ _ x_ Á ÁXr =kB þxÁXr =2kB ½x00Á þxÁ ÁXr =2kB Át ½xÀ e e 00y y 00y y y 00y y 00y y eÀ½x þx ÁXr =2kB e½xÁ þxÁ ÁXr =2kB eÁt ½x_ Àx_ Á ÁXr =kB 00 ¼ 000 _ x_ Á ÁXr =kB þx0 ÁXr =2kB ½x00Á þx0Á ÁXr =2kB Át ½xÀ eÀ½x _ x_ Á ÁXr =kB þxÁXr =2kB ½x00Á þxÁ ÁXr =2kB Át ½xÀ e e 00 00 00 eÀ½xþx00 ÁXr =2kB eÀ½xÁ þxÁ ÁXr =2kB eÀÁt ½x_ þx_ Á ÁXr =kB 00 _ x_ ¼ e½xÁ þxÁ ÁXr =kB eÁt ½xþ 00 ÁXr =kB ð175Þ the even terms canceling.
Advances in Chemical Physics, Vol. 140 by Stuart A. Rice