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

By Stuart A. Rice

This sequence presents the chemical physics box with a discussion board for serious, authoritative reviews of advances in each quarter of the self-discipline.

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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 ! 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 ŠÁXy =2kB ½xy þx000 ŠÁXyr =2kB Át ½x_ 00y Àx_ 00y ŠÁXyr =kB À½x r Á jÞe e Á Á e =Zss ðXyr Þ _ x_ Á ŠÁXr =kB þxŠÁXr =2kB ½x00Á þxÁ ŠÁXr =2kB Át ½xÀ e e 00y y 00y y y 00y y 00y y eÀ½x þx ŠÁXr =2kB e½xÁ þxÁ ŠÁXr =2kB eÁt ½x_ Àx_ Á ŠÁXr =kB 00 ¼ 000 _ x_ Á ŠÁXr =kB þx0 ŠÁXr =2kB ½x00Á þx0Á ŠÁXr =2kB Át ½xÀ eÀ½x _ x_ Á ŠÁXr =kB þxŠÁXr =2kB ½x00Á þxÁ ŠÁXr =2kB Át ½xÀ e e 00 00 00 eÀ½xþx00 ŠÁXr =2kB eÀ½xÁ þxÁ ŠÁXr =2kB eÀÁt ½x_ þx_ Á ŠÁXr =kB 00 _ x_ ¼ e½xÁ þxÁ ŠÁXr =kB eÁt ½xþ 00 ŠÁXr =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 Š 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).

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