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The scaling of Gðx; tÞ motivates writing Gz ðx; tÞ $ 1 z 1 g ðxÞ þ gz ðxÞ þ gz ðxÞ þ ^tgz ðxÞ 2 3 jtj 0 t 1 ð113Þ In view of the parity rule (95), gz and gz are symmetric matrices and gz and gz 0 2 1 3 are antisymmetric matrices. The derivative of the second entropy force is qFðx; tÞ ¼ Gðx; tÞ þ Gz ðx; tÞ qx ð114Þ which in the intermediate regime becomes qf 0 ðxÞ qf 1 ðxÞ 1 1 þ^t ¼ ½g ðxÞþ gz ðxÞ þ ½g ðxÞ þ gz ðxÞ þ g ðxÞ þ gz ðxÞ þ^tgz ðxÞ 0 1 2 3 2 qx qx jtj 0 t 1 ð115Þ Clearly, the ﬁrst two bracketed terms have to individually vanish.
G00 due to the perturbations by the reservoir. The stochastic transition probability due to the reservoir can be taken to be Ãr ðG00 jG0 ; Xr Þ ¼ ÂÁ ðjG00 À G0 jÞeÀ½x 00 Àx0 ÁXr =2kB ½x00Á Àx0Á ÁXr =2kB ð171Þ e The ﬁrst factor is an appropriately normalized short-ranged function, such as a Gaussian or Heaviside step function, that represents the strength of the reservoir perturbations over the time interval. To leading order the normalization factor does not depend on the reservoir forces or the points in phase space.
The third and fourth terms sum to a positive number, at least in the optimum state, where they cancel with the second term. As will become clearer shortly, they represent the production of ﬁrst entropy as the system returns to equilibrium, and it is these terms that drive the ﬂux. 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 !