TD_CHISQUARED

	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-squared distribution (static).

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

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

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 2) 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.

Compulsory keywords

MINIMUM	The minimum of the chi-squared distribution.
SIGMA	The \(\sigma\) parameter of the chi-squared distribution given as a positive number.
KAPPA	The \(k\) parameter of the chi-squared distribution given as positive integer larger than 2.

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.

Examples

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

td: TD_CHISQUARED  MINIMUM=-10.0  SIGMA=2.0  KAPPA=2

The Chi-squared 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 Chi-squared distribution for argument 1 and uniform distribution for argument 2

td_chisq: TD_CHISQUARED  MINIMUM=10.0  SIGMA=2.0  KAPPA=2

td_uni: TD_UNIFORM

td_pd: TD_PRODUCT_DISTRIBUTION DISTRIBUTIONS=td_chisq,td_uni