By Stuart A. Rice
This sequence offers the chemical physics box with a discussion board for serious, authoritative reviews of advances in each region of the self-discipline.
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Additional resources for Advances in Chemical Physics Volume 140
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.
This does not affect the adiabatic dynamics. Hence provided that the ﬂux is maximal in the above sense, then this procedure ensures that both the structure and the dynamics of the subsystem are steady and unchanging in time. (See also the discussion of Fig. ) A corollary of this is that the ﬁrst entropy of the reservoirs increases at the greatest possible rate for any unconstrained ﬂux. This last point suggests an alternative interpretation of the transport coefﬁcient as the one corresponding to the correlation function evaluated at the point of maximum ﬂux.
Note that for large f the end corrections may be neglected. The change in entropy of the reservoir over the trajectory is ÁSr ½G ¼ Xr Á Áxr ¼ ÀXr Á Ár x ¼ ÀXr Á ðÁx À Á0 xÞ ¼ ÀXr Á ðxf À x0 À Á0 x½GÞ ð178Þ This is equal and opposite to the change on the reverse trajectory, ÁSr ½G ¼ ÀÁSr ½Gz . The unconditional probability of the trajectory is }ð½GjXr Þ ¼ f Y ½Ãr ðxi jx0iÀ1 ; Xr Þ}ss ðG0 jXr Þ i¼1 ¼ f Y 0 0 ½ÂÁ ðjGi À G0iÀ1 jÞeÀ½xi ÀxiÀ1 ÁXr =2kB e½xÁ;i ÀxÁ;iÀ1 ÁXr =2kB }ss ðG0 jXr Þ i¼1 ¼ f Y ½ÂÁ ðjGi ÀG0iÀ1 jÞeÀ½xi ÀxiÀ1 ÁXr =2kBe½xÁ;i ÀxÁ;iÀ1 ÁXr =2kBeÁt ½x_ iÀ1Àx_ Á;iÀ1 ÁXr =2kB}ss ðG0 jXrÞ i¼1 ¼ f Y ½ÂÁ ðjGi À G0iÀ1 jÞ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 0 }ss ðGf jXr Þ}ss ðG0 jXr ÞeÁ x½GÁXr =2kB eÀÁ xÁ ½GÁXr =2kB i¼1 0 0 0 0 Â eÀ½xf Àxf Àx0 þx0 ÁXr =4kB e½xÁ;f ÀxÁ;f ÀxÁ;0 þxÁ;0 ÁXr =4kB ð179Þ 50 phil attard This only depends on the probability of the termini, the total adiabatic works, and the total weight of stochastic transitions.