... state.1
Recall from the simulations in class that the particle spends most of its time in the well and most of the rest of its time near the saddle points where the net velocity is small.
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... state:2
This is certainly important for the case for migrating particles that have a large compliance (low stiffness) such as vacancies.
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... independently:3
Note that $ \int_{-\infty}^\infty e^{-\frac{p_1^2}{2 m_1 kT}} dp_1 = \sqrt{2 \pi m_1 kT}$.
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... parts4
The boundary conditions may be chosen to make the " $ uv\ensuremath{\left.\mbox{\rule{0pt}{16pt}}\right\vert}_{BC}$" part vanish, or the surface term may be considered to be small compared to the volume.
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