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Example text
As if it were isolated), and let x00 ¼ x0 þ Ár x be the state after the externally induced changes. By conservation the change in the reservoir is Áxr ¼ ÀÁr x, and the change in the reservoir first entropy is Áxr Á Xr ¼ ÀÁr x Á Xr . Even when the subsystem force is nonlinear, the reservoir force remains independent of x. The analysis will be carried out using the nonlinear results for the subsystem; in the linear regime the correspondence given in the preceding subsection can be applied. The second entropy then is a function of three constrained variables, Sð2Þ ðÁ0 x; Ár x; xjt; Xr Þ.
Hence whether the time step is infinitesimal or intermediate, the adiabatic change in the odd work is ½xÁ ðG0 Þ À xÁ ðGÞ Á Xr ¼ ÀÁt Xr Á Ls ðxðG0 ÞÞXs ðG0 Þ % ÀÁt Ls ðxðGÞÞ : X2r ð168Þ The final approximation is valid if the adiabatic change in the macrostate is relatively negligible, jx0 À xj(jxj, and if the departure of the internal force from the reservoir force is relatively negligible, jX0s À Xr j(jXr j. Again denoting the adiabatic evolution over the intermediate time Át by a prime, G0 ¼ GðÁt jGÞ, the adiabatic change in the even exponent that appears in the steady-state probability distribution is À½xðG0 Þ À xðGÞ Á Xr % Át Xr Á LðxðGÞ; þ1ÞXs ðGÞ % Át Ls ðxðGÞÞ : X2r ð169Þ This is equal and opposite to the adiabatic change in the odd exponent.
Onsager, in his first two papers, combined the general expression, Eq. (61), with the restricted one, Eq. (63), to make a variational principle that is almost the same as the present second entropy and that is generally valid. However, in the later paper with Machlup [32], he proposes variational principles based solely on the two restricted expressions and applies these to the constrained states where they have no physical meaning. Prigogine [11, 83] uses both restricted expressions in his work and interprets them as the rate of entropy production, which is not correct for the constrained states to which he applies them.