Shortcut: HISTOGRAM

Module	gridtools
Description	Usage
Accumulate the average probability density along a few CVs from a trajectory.
output value	type
the estimate of the histogram as a function of the argument that was obtained	grid

Details and examples

Accumulate the average probability density along a few CVs from a trajectory.

When using this shortcut it is supposed that you have some collective variable $\zeta$ that gives a reasonable description of some physical or chemical phenomenon. As an example of what we mean by this suppose you wish to examine the following SN2 reaction:

$\textrm{OH}^- + \textrm{CH}_3Cl \rightarrow \textrm{CH}_3OH + \textrm{Cl}^-$

The distance between the chlorine atom and the carbon is an excellent collective variable, $\zeta$ , in this case because this distance is short for the reactant, $\textrm{CH}_3Cl$ , because the carbon and chlorine are chemically bonded, and because it is long for the product state when these two atoms are not chemically bonded. We thus might want to accumulate the probability density, $P(\zeta)$ , as a function of this distance as this will provide us with information about the overall likelihood of the reaction. Furthermore, the free energy, $F(\zeta)$ , is related to this probability density via:

$F(\zeta) = - k_B T \ln P(\zeta)$

Accumulating these probability densities is precisely what this shortcut can be used to do. Furthermore, the conversion of the histogram to the free energy can be achieved by using the method CONVERT_TO_FES.

We calculate histograms within PLUMED using a method known as kernel density estimation. This shortcut action thus uses the KDE and ACCUMULATE actions to build up the time average of the histogram.

In PLUMED the value of $\zeta$ at each discrete instant in time in the trajectory is accumulated. A kernel, $K(\zeta-\zeta(t'),\sigma)$ , centered at the current value, $\zeta(t)$ , of this quantity is generated with a bandwidth $\sigma$ , which is set by the user. These kernels are then used to accumulate the ensemble average for the probability density:

$\langle P(\zeta) \rangle = \frac{ \sum_{t'=0}^t w(t') K(\zeta-\zeta(t'),\sigma) }{ \sum_{t'=0}^t w(t') }$

Here the sums run over a portion of the trajectory specified by the user. 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.

A discrete analogue of kernel density estimation can also be used. In this analogue the kernels in the above formula are replaced by Dirac delta functions. When this method is used the final function calculated is no longer a probability density - it is instead a probability mass function as each element of the function tells you the value of an integral between two points on your grid rather than the value of a (continuous) function on a grid.

Additional material and examples can be also found in this tutorial, which also introduces the technique known as block averaging.

Examples

The following input monitors two torsional angles during a simulation and outputs a continuous histogram as a function of them at the end of the simulation.