[MMTK] NMA and mean square fluctuations

[Konrad Hinsen]

On 14.08.2008, at 10:49, Matias Saavedra wrote:

> Which is the theoretical basis for the calculation of atomic mean  
> square fluctuations with normal modes? In particular the relation  
> of mean square fluctuations with boltzmann constant, temperature  
> and modes frequencies (eigenvalues). The typical relation in order  
> to calculate B-factors. I cannot find a reference wich explain this  
> issue with detail.

The B factors are the diagonal elements of the position  
autocorrelation matrix. This can be written as a thermal average,  
i.e. an integral over the Boltzmann distribution times terms of the  
form <(r_i-<r_i>)(r_j-<r_j>)>. For a harmonic potential, this  
Boltzmann distribution is a Gaussian distribution which makes it  
possible to calculate the integral exactly. The result is that the  
position autocorrelation matrix is kT times the inverse of the force  
constant matrix.

> Why the calculation of the fluctuations suposes a division of the  
> squared eigenvectors by the squared mode frequencies ?

That is just the spectral representation of the inverse force  
constant matrix. Any symmetric matrix can be written as

  	M = sum_i[e_i u_i u_i],

where e_i are its eigenvectors and u_i are its eigenvectors. The  
eigenvectors of the (mass weighted) force constant matrix are the  
squared frequencies.

> And finally, why the criterion for the applicability of classical  
> mechanics is hv<<KbT ? and how this affects the interpretation of NMA?

The separation of energy levels in a quantum harmonic oscillation is  
hv. Classical mechanics yields a continuum of energy levels, which  
becomes a good approximation when the energy (kT) is high enough to  
be made up of many quantum excitations (hv).

Konrad.
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Konrad Hinsen
Centre de Biophysique Moléculaire, CNRS Orléans
Synchrotron Soleil - Division Expériences
Saint Aubin - BP 48
91192 Gif sur Yvette Cedex, France
Tel. +33-1 69 35 97 15
E-Mail: hinsen at cnrs-orleans.fr
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