Download Advances in Chemical Physics, Vol. 140 by Stuart A. Rice PDF

By Stuart A. Rice

This sequence offers the chemical physics box with a discussion board for severe, authoritative reviews of advances in each region of the self-discipline.

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Hence it follows that ð1Þ ð1Þ Stotal ðGjXr Þ ¼ Sð1Þ s ðGÞ þ Sr ðxðGÞjXr Þ ¼ const: À xðGÞ Á Xr ð151Þ The final term is the subsystem-dependent part of the reservoir entropy, which arises from exchanging x between the two. Using this the so-called static probability distribution is }st ðGjXr Þ ¼ 1 eÀxðGÞÁXr =kB Zst ðXr Þ ð152Þ where kB is Boltzmann’s constant. This is the analogue of Boltzmann’s distribution and hence will yield the usual equilibrium results. However, it is dynamically disordered; under time reversal, xðGÞ ) xðGy Þ ¼ ExðGÞ, and Xr ) Xyr  EXr , it remains unchanged: }st ðGjXr Þ ¼ }st ðGy jXyr Þ ð153Þ It is clear that the true nonequilibrium probability distribution requires an additional factor of odd parity.

Writing twice the left-hand side as the expansion about the first argument plus the expansion about the second argument, it follows that 2Sð2Þ ðx3 ; x1 j2tÞ ¼ 12 Gy ðx3 ; 2tÞ : ½x1 À x3 Š2 þ Fy ðx3 ; 2tÞ Á ½x1 À x3 Š þ Eðx3 ; 2tÞ þ 12 Gðx1 ; 2tÞ : ½x3 À x1 Š2 þ Fðx1 ; 2tÞ Á ½x3 À x1 Š þ Eðx1 ; 2tÞ ð98Þ As in the linear case, the optimum point is approximated by the midpoint, and it is shown later that the shift is of second order. Hence the right-hand side of 30 phil attard Eq. (97) is given by Eq.

This is the required result, which shows the stationarity of the steady-state probability under the present transition probability. This result invokes the preservation of the steady-state probability during adiabatic evolution over intermediate time scales. 4. Forward and Reverse Transitions The previous results for the transition probability held over intermediate time scales. On infinitesimal time scales the adiabatic evolution of the steady-state probability has to be accounted for. The unconditional transition probability over an infinitesimal time step is given by }ðG00 GjÁt ; Xr Þ ¼ Ãr ðG00 jG0 ; Xr Þ}ss ðGjXr Þ _ r =kB ÀÁt x_ Á ÁXr =kB ¼ Ãr ðG00 jG0 ; Xr Þ}ss ðG0 jXr ÞeÁt xÁX e ¼ ÂÁ ðjG00 À G0 jÞ À½x00 þx0 ŠÁXr =2kB ½x00 þx0 ŠÁXr =2kB Át ½xÀ e e Á Á e _ x_ Á ŠÁXr =kB Zss ðXr Þ ð174Þ Now consider the forward transition, G !

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