next up previous
Next: References Up: Theory of the Depletion Previous: Summary

Appendix: Pressure Sum Rule

Consider a rod, at an angle tex2html_wrap_inline1164 to the normal of a wall, with its centre-of-mass velocity tex2html_wrap_inline1166 along the normal direction, of mass m and angular velocity tex2html_wrap_inline1170 about an axis parallel to the wall but perpendicular to the rod. Suppose one end of the rod, with normal component of velocity


is infinitesimally close to the wall and undergoes an instantaneous energy conserving collision; the momentum transfer from the rod to the wall can be easily found to be:


Since both tex2html_wrap_inline1166 and tex2html_wrap_inline1170 independently obey Gaussian distribution with their variances given by the equipartition of the thermal energy,


it follows, from Eq.(40), that tex2html_wrap_inline1176 also obeys a Gaussian distribution:


with a variance tex2html_wrap_inline1178 . The pressure can then be integrated to give:


which is the required result. Although derived by kinetic theory, this result must of course correspond to that found from an appropriate free energy derivative. Accordingly it remains applicable to rods in solution (rather than in vacuo), so long as p is interpreted as an osmotic pressure.

Yong Mao
Tue Sep 17 16:40:38 BST 1996