TD_CHI

	This is part of the ves module
	It is only available if you configure PLUMED with ./configure –enable-modules=ves . Furthermore, this feature is still being developed so take care when using it and report any problems on the mailing list.

Chi distribution (static).

Employ a target distribution given by a chi distribution that is defined as

\[ p(s) = \frac {2^{1-\frac{k}{2}}} {\sigma \, \Gamma\left(\frac{k}{2}\right) } \, \left(\frac{s-a}{\sigma}\right)^{k-1} \, \exp\left(- \frac{1}{2} \left(\frac{s-a}{\sigma}\right)^2\right), \]

where \(a\) is the minimum of the distribution that is defined on the interval \([a,\infty)\), the parameter \(k\) (given as a positive integer larger than 1) determines how far the peak of the distribution is from the minimum (known as the "degrees of freedom"), and the parameter \(\sigma>0\) determines the broadness of the distribution.

The minimum \(a\) is given using the MINIMUM keyword, the parameter \(k\) is given using the KAPPA keyword, and the parameter \(\sigma\) is given using the SIGMA keyword.

This target distribution action is only defined for one dimension, for multiple dimensions it should be used in combination with the TD_PRODUCT_DISTRIBUTION action.

Examples

Chi distribution with \(a=10.0\), \(\sigma=2.0\), and \(k=2\)

Click on the labels of the actions for more information on what each action computes

td: TD_CHI 
MINIMUM
compulsory keyword 
The minimum of the chi distribution. 
=10.0 
SIGMA
compulsory keyword 
The sigma parameter of the chi distribution given as a positive number. 
=2.0 
KAPPA
compulsory keyword 
The k parameter of the chi distribution given as positive integer larger than 1.
=2 The TD_CHI action with label td

The Chi distribution is only defined for one dimension so for multiple dimensions we have to use it in combination with the TD_PRODUCT_DISTRIBUTION action as shown in the following example where we have a uniform distribution for argument 1 and a Chi distribution for argument 1

Click on the labels of the actions for more information on what each action computes

td_uni: TD_UNIFORM The TD_UNIFORM action with label td_uni
td_chi: TD_CHI 
MINIMUM
compulsory keyword 
The minimum of the chi distribution. 
=-10.0 
SIGMA
compulsory keyword 
The sigma parameter of the chi distribution given as a positive number. 
=2.0 
KAPPA
compulsory keyword 
The k parameter of the chi distribution given as positive integer larger than 1.
=2 The TD_CHI action with label td_chi
td_pd: TD_PRODUCT_DISTRIBUTION 
DISTRIBUTIONS
compulsory keyword 
Labels of the one-dimensional target distribution actions for each argument to be
used in the product distribution. 
=td_uni,td_chi The TD_PRODUCT_DISTRIBUTION action with label td_pd

Glossary of keywords and components

Compulsory keywords

MINIMUM	The minimum of the chi distribution.
SIGMA	The sigma parameter of the chi distribution given as a positive number.
KAPPA	The k parameter of the chi distribution given as positive integer larger than 1.

Options

SHIFT_TO_ZERO	( default=off ) Shift the minimum value of the target distribution to zero. This can for example be used to avoid negative values in the target distribution. If this option is active the distribution will be automatically normalized.
NORMALIZE	( default=off ) Renormalized the target distribution over the intervals on which it is defined to make sure that it is properly normalized to 1. In most cases this should not be needed as the target distributions should be normalized. The code will issue a warning (but still run) if this is needed for some reason.
WELLTEMPERED_FACTOR	Broaden the target distribution such that it is taken as [p(s)]^(1/gamma) where gamma is the well tempered factor given here. If this option is active the distribution will be automatically normalized.