TD_GENERALIZED_EXTREME_VALUE
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.

Generalized extreme value distribution (static).

Employ a target distribution given by a generalized extreme value distribution that is defined as

\[ p(s) = \frac{1}{\sigma} \, t(s)^{\xi+1} \, e^{-t(s)}, \]

where

\[ t(s) = \begin{cases} \left( 1 + \xi \left( \frac{s-\mu}{\sigma} \right) \right)^{-1/\xi} & \mathrm{if\ }\xi \neq 0 \\ \exp\left(- \frac{s-\mu}{\sigma} \right) & \mathrm{if\ } \xi = 0 \end{cases}, \]

and \(\mu\) is the location parameter which approximately determines the location of the maximum of the distribution, \(\sigma>0\) is the scale parameter that determines the broadness of the distribution, and \(\xi\) is the shape parameter that determines the tail behavior of the distribution. For \(\xi=0\), \(\xi>0\), and \(\xi<0\) the Gumbel, Frechet, and Weibull families of distributions are obtained, respectively.

The location parameter \(\mu\) is given using the LOCATION keyword, the scale parameter \(\sigma\) using the SCALE keyword, and the shape parameter \(\xi\) using the SHAPE keyword.

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

Compulsory keywords
LOCATION The \(\mu\) parameter of the generalized extreme value distribution.
SCALE The \(\sigma\) parameter for the generalized extreme value distribution given as a positive number.
SHAPE The \(\xi\) parameter for the generalized extreme value distribution.
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

Generalized extreme value distribution with \(\mu=0.0\), \(\sigma=2.0\), and \(\xi=0.0\) (Gumbel distribution)

td: TD_GENERALIZED_EXTREME_VALUE  LOCATION=0.0  SCALE=2.0 SHAPE=0.0

Generalized extreme value distribution with \(\mu=-5.0\), \(\sigma=1.0\), and \(\xi=0.5\) (Frechet distribution)

td: TD_GENERALIZED_EXTREME_VALUE  LOCATION=-5.0  SCALE=1.0 SHAPE=0.5

Generalized extreme value distribution with \(\mu=5.0\), \(\sigma=2.0\), and \(\xi=-0.5\) (Weibull distribution)

td: TD_GENERALIZED_EXTREME_VALUE  LOCATION=5.0  SCALE=1.0 SHAPE=-0.5

The generalized extreme value 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 Generalized extreme value distribution for argument 1 and uniform distribution for argument 2

td_gev: TD_GENERALIZED_EXTREME_VALUE  LOCATION=-5.0  SCALE=1.0 SHAPE=0.5

td_uni: TD_UNIFORM

td_pd: TD_PRODUCT_DISTRIBUTION DISTRIBUTIONS=td_gev,td_uni