By Stuart A. Rice
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17). Explicitly, QðtÞ ¼ E AðtÞÀ1 BðtÞE SÀ1 ð44Þ This invokes the result, hxðx; tÞxi0 ¼ hxðt þ tÞxðtÞi0 , which is valid since the mode is equal to the mean for a Gaussian conditional probability. À1 ! 1 À1 QðtÞS $ E a þ ^ta3 þ a4 a þ ^tb3 þ b4 E jtj 2 jtj 2 ð45Þ $ E ½ I À taÀ1 a3 À jtjaÀ1 a4 ½ÀI þ taÀ1 b3 þ jtjaÀ1 b4 E þ Ot2 2 2 2 2 $ ÀI À taÀ1 ½ a3 þ b3 þ jtjaÀ1 ½ a4 þ b4 þ Ot2 2 2 Here we have used the symmetry and commuting properties of the matrices to obtain the ﬁnal line. This shows that the correlation matrix goes like QðtÞ $ ÀSÀ1 þ tQÀ þ jtjQþ þ Ot2 ð46Þ where QÀ is block-adiagonal and Qþ ¼ aÀ1 =2 is block-diagonal.
Imagining that the transitions occur sequentially, 24 phil attard x ! x0 ! x00 , with x0 ¼ x þ Á0 x and x00 ¼ x0 þ Ár x, the second entropy for the stochastic transition is half the difference between the total ﬁrst entropies of the initial and ﬁnal states, r 1 ð1Þ r r ð1Þ Sð2Þ r ðÁ xjx; Xr Þ ¼ 2 ½S ðÁ x þ xÞ À ðÁ x þ xÞ Á Xr À S ðxÞ þ x Á Xr ð71Þ The factor of 12 arises from the reversibility of such stochastic transitions. One can debate whether x or x0 should appear here, but it does not affect the following results to leading order.
The total adiabatic change in state over the trajectory is Á0 x½G ¼ f X ½x0i À xi À 12 ½x0f À xf þ x00 À x0 ð176Þ i¼0 and the total adiabatic change in the odd work parameter is Á0 xÁ ½G ¼ f X ½x0Á;i À xÁ;i À 12 ½x0Á;f À xÁ;f þ x0Á;0 À xÁ;0 ð177Þ i¼0 Clearly, the adiabatic work done on the forward trajectory is equal and opposite to that done on the reverse trajectory, Á0 x½Gz Á Xyr ¼ ÀÁ0 x½G Á Xr , and the adiabatic odd work is the same on both trajectories, Á0 xÁ ½Gz Á Xyr ¼ Á0 xÁ ½G Á Xr .