# Talk:Capillary waves

## Thermal capillary waves

Hello, now I'm writing the same article for 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 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 $k_{B}T$ rather than ${\frac {1}{2}}k_{B}T$. That's because each mode has to degrees of freedom $A_{{mn}}$ and $B_{{mn}}$, since each wave is $A_{{mn}}\cos({\frac {2\pi }{L}}mx+{\frac {2\pi }{L}}ny)+B_{{mn}}\sin({\frac {2\pi }{L}}mx+{\frac {2\pi }{L}}ny)$, with the energy of each mode proportional to $A_{{mn}}^{2}+B_{{mn}}^{2}$. This obviously lead to the mean energy of each mode to be $k_{B}T$. That was the real notation and now lets turn to the complex notation.

Each mode with the fixed wave vector is presented as $h_{{\mathbf {k}}}\exp(i\;{\mathbf {k}}\cdot {\boldsymbol {\tau }})$, ${\mathbf {k}}=({\frac {2\pi }{L}}m,{\frac {2\pi }{L}}n),\;m,n\in {\mathbb {Z}}$ — wave vector, ${\boldsymbol {\tau }}$$(x,y)$ vector. The energy is proportional to $h_{{\mathbf {k}}}^{*}h_{{\mathbf {k}}}$ (indeed it is $E_{{\mathbf {k}}}={\frac {\sigma L^{2}}{2}}\left({\frac {2}{a_{c}^{2}}}+{\mathbf {k}}^{2}\right)h_{{\mathbf {k}}}^{*}h_{{\mathbf {k}}}$). According to equipartition:

$\left\langle x_{i}{\frac {\partial H}{\partial x_{j}}}\right\rangle =\delta _{{ij}}k_{B}T,$

we obtain:

$\left\langle h_{{\mathbf {k}}}\,{\frac {\partial }{\partial h_{{\mathbf {k}}}}}\left[{\frac {\sigma L^{2}}{2}}\left({\frac {2}{a_{c}^{2}}}+{\mathbf {k}}^{2}\right)h_{{\mathbf {k}}}^{*}h_{{\mathbf {k}}}\right]\right\rangle =\left\langle h_{{\mathbf {k}}}\left[{\frac {\sigma L^{2}}{2}}\left({\frac {2}{a_{c}^{2}}}+{\mathbf {k}}^{2}\right)h_{{\mathbf {k}}}^{*}\right]\right\rangle =\left\langle E_{{\mathbf {k}}}\right\rangle =k_{B}T.$

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. Grigory Sarnitskiy. 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 $\sin$ terms should appear in the modes, not $\cos$. If, on the other hand, periodic boundary conditions are applied, the opposite applies: only $\cos$, not $\sin$. This may explain the factor of $2$ that's missing... but I still have to think more carefully about this. --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 ${\frac {\partial h(x,y)}{\partial {\mathbf {n}}}}{\Big |}_{{wall}}=0$, with ${\mathbf {n}}$ normal to the wall. I suppose it is valid at least for more or less long waves (several minimal wavelengths), which contribute most to $\langle h\rangle$. 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 ${\frac {1}{2}}k_{B}T$ for each mode. Still I'll think it over again and will wait for your reply. 91.76.179.246 12:53, 3 February 2009 (CET) Well now I'm not sure at all in the variant above. Have to dig the question. 91.76.179.246 15:29, 3 February 2009 (CET)