AVERAGE

This is part of the generic module |

Calculate the ensemble average of a collective variable

The ensemble average for a non-periodic, collective variable, \(s\) is given by the following expression:

\[ \langle s \rangle = \frac{ \sum_{t'=0}^t w(t') s(t') }{ \sum_{t'=0}^t w(t') } \]

Here the sum runs over a the trajectory and \(s(t')\) is used to denote the value of the collective variable at time \(t'\). The final quantity evaluated is a weighted average as the weights, \(w(t')\), allow us to negate the effect any bias might have on the region of phase space sampled by the system. This is discussed in the section of the manual on Analysis.

When the variable is periodic (e.g. TORSION) and has a value, \(s\), in \(a \le s \le b\) the ensemble average is evaluated using:

\[ \langle s \rangle = a + \frac{b - a}{2\pi} \arctan \left[ \frac{ \sum_{t'=0}^t w(t') \sin\left( \frac{2\pi [s(t')-a]}{b - a} \right) }{ \sum_{t'=0}^t w(t') \cos\left( \frac{2\pi [s(t')-a]}{b - a} \right) } \right] \]

- Examples

The following example calculates the ensemble average for the distance between atoms 1 and 2 and output this to a file called COLVAR. In this example it is assumed that no bias is acting on the system and that the weights, \(w(t')\) in the formulas above can thus all be set equal to one.

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The following example calculates the ensemble average for the torsional angle involving atoms 1, 2, 3 and 4. At variance with the previous example this quantity is periodic so the second formula in the above introduction is used to calculate the average. Furthermore, by using the CLEAR keyword we have specified that block averages are to be calculated. Consequently, after 100 steps all the information acquired thus far in the simulation is forgotten and the process of averaging is begun again. The quantities output in the colvar file are thus the block averages taken over the first 100 frames of the trajectory, the block average over the second 100 frames of trajectory and so on.

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This third example incorporates a bias. Notice that the effect the bias has on the ensemble average is removed by taking advantage of the REWEIGHT_BIAS method. The final ensemble averages output to the file are thus block ensemble averages for the unbiased canonical ensemble at a temperature of 300 K.

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