Editing Talk:Capillary waves

==Thermal capillary waves==
Hello, now I'm writing the same article for [http://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BF%D0%BB%D0%BE%D0%B2%D1%8B%D0%B5_%D0%BA%D0%B0%D0%BF%D0%B8%D0%BB%D0%BB%D1%8F%D1%80%D0%BD%D1%8B%D0%B5_%D0%B2%D0%BE%D0%BB%D0%BD%D1%8B Russian wikipedia]. While I was deducing the expression for mean square amplitude I found my result to be two times less than common one (that is in [http://books.google.com/books?id=_ydSF_XUVeEC&printsec=frontcover&hl=ru#PPA115,M1 Molecular Theory of Capillarity] and refered to in many articles). Could you please tell me weather I am right or not.

I claim that the mean energy of each mode is <math>k_B T</math> rather than <math>\frac{1}{2} k_B T</math>. That's because each mode has to degrees of freedom <math>A_{mn}</math> and <math>B_{mn}</math>, since each wave is <math>A_{mn} \cos(\frac{2\pi}{L}mx+\frac{2\pi}{L}ny) + B_{mn} \sin(\frac{2\pi}{L}mx+\frac{2\pi}{L}ny)</math>, with the energy of each mode proportional to <math>A_{mn}^2+B_{mn}^2</math>. This obviously lead to the mean energy of each mode to be <math>k_B T</math>. That was the real notation and now lets turn to the complex notation.

Each mode with the fixed wave vector is presented as <math>h_\mathbf{k} \exp(i \; \mathbf{k} \cdot \boldsymbol{\tau})</math>, <math>\mathbf{k}=(\frac{2 \pi}{L}m, \frac{2 \pi}{L}n), \; m,n \in \mathbb{Z}</math> — wave vector, <math>\boldsymbol{\tau}</math> — <math>(x,y)</math> vector. The energy is proportional to <math>h_\mathbf{k}^*h_\mathbf{k}</math> (indeed it is <math>E_\mathbf{k}=\frac{\sigma L^2}{2} \left( \frac{2}{a_c^2} + \mathbf{k}^2 \right) h^*_\mathbf{k} h_\mathbf{k}</math>). According to [http://en.wikipedia.org/wiki/Equipartition equipartition]:
:<math>
\left \langle x_i \frac{\partial H}{\partial x_j} \right \rangle = \delta_{ij}  k_B T,
</math>
we obtain:
:<math>
\left \langle h_\mathbf{k} \, \frac{\partial}{\partial h_\mathbf{k}} \left[ \frac{\sigma L^2}{2} \left( \frac{2}{a_c^2} + \mathbf{k}^2 \right) h^*_\mathbf{k} h_\mathbf{k} \right] \right \rangle = \left \langle h_\mathbf{k} \left[ \frac{\sigma L^2}{2} \left( \frac{2}{a_c^2} + \mathbf{k}^2 \right) h^*_\mathbf{k} \right] \right \rangle = \left \langle E_\mathbf{k} \right \rangle = k_B T.
</math>
And again we get the same result. What do you think of it? Is there a mistake? Please help, I'm really stuck with it. [http://en.wikipedia.org/wiki/User:Yrogirg Grigory Sarnitskiy]. [[Special:Contributions/91.76.179.101|91.76.179.101]] 19:45, 30 January 2009 (CET)

=== A quick comment ===

I may be wrong, but looking at your derivation it seems the boundary conditions are not correctly
described. If the system is fixed to some immobile frame, only <math>\sin</math> terms should appear in the modes, not <math>\cos</math>. If, on the other hand, periodic boundary conditions are applied, the opposite applies: only <math>\cos</math>, not <math>\sin</math>. This may explain the factor of <math>2</math> that's missing... but I still have to think more carefully about this. --[[User:Dduque|Dduque]] 09:58, 3 February 2009 (CET)
:Thank you. Yes, I didn't appreciate the importance of boundary conditions. I think it is quite reasonable to take something like <math>\frac{\partial h(x,y)}{\partial \mathbf{n}} \Big|_{wall}=0</math>, with <math>\mathbf{n}</math> normal to the wall. I suppose it is valid at least for more or less long waves (several minimal wavelengths), which contribute most to <math>\langle h \rangle</math>. It is likely there is no need to study molecular interaction between liquid and solid surface in this case — boundary conditions will not be affected by the material of the walls at least for real-life systems. So we get <math>\frac{1}{2} k_B T</math> for each mode. Still I'll think it  over again and will wait for your reply. [[Special:Contributions/91.76.179.246|91.76.179.246]] 12:53, 3 February 2009 (CET)