By Stuart A. Rice
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Additional resources for Advances in Chemical Physics, Vol. 140
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 ÁXr =2kB ½x00 þx0 ÁXr =2kB Át ½xÀ e e Á Á e _ x_ Á ÁXr =kB Zss ðXr Þ ð174Þ Now consider the forward transition, G ! G0 ! G00 , and its reverse, G00y ! G ! Gy (see Fig. 2). Note that G000 6¼ G0y , but that jG00 À G0 j ¼ jGy À G000 j. The 000 48 phil attard Figure 2.
G ! Gy (see Fig. 2). Note that G000 6¼ G0y , but that jG00 À G0 j ¼ jGy À G000 j. The 000 48 phil attard Figure 2. Forward and reverse transitions. The solid lines are the adiabatic trajectories over an inﬁnitesimal time step, and the dashed lines are the stochastic transitions. ratio of the forward to the reverse transition probabilities is }ðG00 }ðGy GjÁt ; Xr Þ G00y jÁt ; Xyr Þ ÂÁ ðjG00 À G0 jÞeÀ½x 00 ¼ y ÂÁ ðjG À G 00 ¼ eÀ½x e e =Zss ðXr Þ y þx000 ÁXy =2kB ½xy þx000 ÁXyr =2kB Át ½x_ 00y Àx_ 00y ÁXyr =kB À½x r Á jÞe e Á Á e =Zss ðXyr Þ _ x_ Á ÁXr =kB þxÁXr =2kB ½x00Á þxÁ ÁXr =2kB Át ½xÀ e e 00y y 00y y y 00y y 00y y eÀ½x þx ÁXr =2kB e½xÁ þxÁ ÁXr =2kB eÁt ½x_ Àx_ Á ÁXr =kB 00 ¼ 000 _ x_ Á ÁXr =kB þx0 ÁXr =2kB ½x00Á þx0Á ÁXr =2kB Át ½xÀ eÀ½x _ x_ Á ÁXr =kB þxÁXr =2kB ½x00Á þxÁ ÁXr =2kB Át ½xÀ e e 00 00 00 eÀ½xþx00 ÁXr =2kB eÀ½xÁ þxÁ ÁXr =2kB eÀÁt ½x_ þx_ Á ÁXr =kB 00 _ x_ ¼ e½xÁ þxÁ ÁXr =kB eÁt ½xþ 00 ÁXr =kB ð175Þ the even terms canceling.
The earlier analysis cannot decide the issue, since b3 can be zero or nonzero in the earlier results. But some insight can be gained into the possible behavior of the system from the following analysis. < t < tlong , the transport matrix is linear in t In the intermediate regime, tshort $ $ and it follows that Lð^tÞ ¼ ^t q _ þ tÞxðtÞi0 QðtÞ ¼ ^tkB hxðt qt ð56Þ or, equivalently, Lð^tÞ ¼ 1 ½QðtÞ þ SÀ1 ¼ ^tkB hxðt; tÞxðtÞi0 jtj ð57Þ That the time correlation function is the same using the terminal velocity or the coarse velocity in the intermediate regime is consistent with Eqs (53) and (54).