User talk:Dduque

2009-01-30T18:49:24Z

91.76.179.101:

Hi there Daniel. Many thanks for re-ordering the [[conferences]] section. I think the years must have slipped my attention.
All the best --[[User:Carl McBride | Carl McBride]] 13:37, 10 May 2007 (CEST)

==Capillary waves==
Hello, could you please look at [[Talk:Capillary waves]]? [[Special:Contributions/91.76.179.101|91.76.179.101]] 19:47, 30 January 2009 (CET)

User talk:Dduque

2009-01-30T18:47:19Z

91.76.179.101:

Hi there Daniel. Many thanks for re-ordering the [[conferences]] section. I think the years must have slipped my attention.
All the best --[[User:Carl McBride | Carl McBride]] 13:37, 10 May 2007 (CEST)

==Capillary waves==
Hello, could you please look at [Talk:Capillary waves]?[[Special:Contributions/91.76.179.101|91.76.179.101]] 19:47, 30 January 2009 (CET)

Talk:Capillary waves

2009-01-30T18:45:24Z

91.76.179.101: New page: ==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%...

==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)

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User talk:Dduque

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Talk:Capillary waves