By Stuart A. Rice
This sequence offers the chemical physics box with a discussion board for severe, authoritative reviews of advances in each quarter of the self-discipline.
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Extra info for Advances in Chemical Physics Volume 140
In the optimum or steady state, the third term on the right-hand side is minus twice the fourth term, and so the sum of these two terms is jtj t x Á ðQþ ÞÀ1 ½QÀ þ ^tQþ Sx % x Á XðxÞ 4 4 ð69Þ where the asymmetric coupling has been neglected. For t > 0, this is one-quarter of the ﬁrst entropy production and is positive, which justiﬁes the above physical interpretation of these two terms. In summary, following a ﬂuctuation the system is initially dynamically disordered, and the ﬂux is zero. If the ﬂux were constrained to be zero for an intermediate time, the second entropy would be less than the ﬁrst entropy of the ﬂuctuation, and it would have decreased at a constant rate, Sð2Þ ðx ¼ 0; xjtÞ ¼ Sð1Þ ðxÞ þ jtj þ Q : ðSxÞ2 4 ð70Þ ignoring the asymmetric term.
Since the most likely subsystem force equals that imposed by the reservoir, Xs % Xr , with relatively negligible error, the asymptote can be written _ 0 ðtjG1 ÞÞ ! ÇXr Á Lðx1 ; Æ1ÞX1 % ÇLðx1 ; Æ1Þ : X2r ; Xr Á xðG t ! Æt ð159Þ This is equal and opposite to the term added to the integrand and so cancels with it leaving a short-ranged integrand. Since this asymptote gives zero contribution to the integral, because of this cancelation or, equivalently, because it is odd, the integral is insensitive to the value of t provided that it is in the intermediate regime.
From the quadratic term it follows that 1 8½Gðx1 ; tÞ 1 þ Gy ðx3 ; tÞ þ 16 ½Sðx1 Þ þ Sðx3 Þ ¼ 14½Gðx1 ; 2tÞ þ Gy ðx3 ; 2tÞ ð100Þ To satisfy this, G must contain terms that scale inversely with the time interval, and terms that are independent of the time interval. As in the linear case, the expansion is nonanalytic, and it follows that Gðx; tÞ ¼ 1 1 g ðxÞ þ g ðxÞ þ g ðxÞ þ ^tg ðxÞ 2 3 jtj 0 t 1 ð101Þ and that, in view of the parity rule (94), Gy ðx; tÞ ¼ 1 1 g ðxÞ À g ðxÞ þ g ðxÞ À ^tg ðxÞ 2 3 jtj 0 t 1 ð102Þ From Eq.
Advances in Chemical Physics Volume 140 by Stuart A. Rice