TD_GENERALIZED_NORMAL
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.

Target distribution given by a sum of generalized normal distributions (static).

Employ a target distribution that is given by a sum where each term is a product of one-dimensional generalized normal distributions (version 1, also know as an exponential power distribution), defined as

\[ p(\mathbf{s}) = \sum_{i} \, w_{i} \prod_{k}^{d} \frac{\beta_{k,i}}{2 \, \alpha_{k,i} \, \Gamma(1/\beta_{k,i})} \exp\left( -\left\vert \frac{s_{k}-\mu_{k,i}}{\alpha_{k,i}} \right\vert^{\beta_{k,i}} \right) \]

where \((\mu_{1,i},\mu_{2,i},\ldots,\mu_{d,i})\) are the centers of the distributions, \((\alpha_{1,i},\alpha_{2,i},\ldots,\alpha_{d,i})\) are the scale parameters of the distributions, \((\beta_{1,i},\beta_{2,i},\ldots,\beta_{d,i})\) are the shape parameters of the distributions, and \(\Gamma(x)\) is the gaamma function. The weights \(w_{i}\) are normalized to 1, \(\sum_{i}w_{i}=1\).

Employing \(\beta=2\) results in a Gaussian (normal) distributions with mean \(\mu\) and variance \(\alpha^2/2\), \(\beta=1\) gives the Laplace distribution, and the limit \(\beta \to \infty\) results in a uniform distribution on the interval \([\mu-\alpha,\mu+\alpha]\).

The centers \((\mu_{1,i},\mu_{2,i},\ldots,\mu_{d,i})\) are given using the numbered CENTER keywords, the scale parameters \((\alpha_{1,i},\alpha_{2,i},\ldots,\alpha_{d,i})\) using the numbered SCALE keywords, and the shape parameters \((\beta_{1,i},\beta_{2,i},\ldots,\beta_{d,i})\) using the numbered SHAPE keywords. The weights are given using the WEIGHTS keywords, if no weights are given are all terms weighted equally.

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.

CENTER The center of each generalized normal distribution. You can use multiple instances of this keyword i.e. CENTER1, CENTER2, CENTER3...
ALPHA The alpha parameters for each generalized normal distribution. You can use multiple instances of this keyword i.e. ALPHA1, ALPHA2, ALPHA3...
BETA The beta parameters for each generalized normal distribution. You can use multiple instances of this keyword i.e. BETA1, BETA2, BETA3...
WEIGHTS The weights of the generalized normal distribution. By default all are weighted equally.
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

A generalized normal distribution in one-dimensional

td1: TD_GENERALIZED_NORMAL CENTER1=+20.0  ALPHA1=5.0  BETA1=4.0

A sum of two one-dimensional generalized normal distributions

TD_GENERALIZED_NORMAL ...
 CENTER1=+20.0  ALPHA1=5.0  BETA1=4.0
 CENTER2=-20.0  ALPHA2=5.0  BETA2=3.0
 LABEL=td1
... TD_GENERALIZED_NORMAL

A sum of two two-dimensional generalized normal distributions

TD_GENERALIZED_NORMAL ...
 CENTER1=-20.0,-20.0 ALPHA1=5.0,3.0 BETA1=2.0,4.0
 CENTER2=-20.0,+20.0 ALPHA2=3.0,5.0 BETA2=4.0,2.0
 WEIGHTS=2.0,1.0
 LABEL=td1
... TD_GENERALIZED_NORMAL