TD_EXPONENTIALLY_MODIFIED_GAUSSIAN

	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 exponentially modified Gaussian distributions (static).

Employ a target distribution that is given by a sum where each term is a product of one-dimensional exponentially modified Gaussian distributions,

\[ p(\mathbf{s}) = \sum_{i} \, w_{i} \prod_{k}^{d} \frac{\lambda_{k,i}}{2} \, \exp\left[ \frac{\lambda_{k,i}}{2} (2 \mu_{k,i} + \lambda_{k,i} \sigma_{k,i}^2 -2 s_{k}) \right] \, \mathrm{erfc}\left[ \frac{\mu_{k,i} + \lambda_{k,i} \sigma_{k,i}^2 - s_{k})}{\sqrt{2} \sigma_{k,i}} \right] \]

where \((\mu_{1,i},\mu_{2,i},\ldots,\mu_{d,i})\) are the centers of the Gaussian component, \((\sigma_{1,i},\sigma_{2,i},\ldots,\sigma_{d,i})\) are the standard deviations of the Gaussian component, \((\lambda_{1,i},\lambda_{2,i},\ldots,\lambda_{d,i})\) are the rate parameters of the exponential component, and \(\mathrm{erfc}(x)=1-\mathrm{erf}(x)\) is the complementary error function. The weights \(w_{i}\) are normalized to 1, \(\sum_{i}w_{i}=1\).

The centers \((\mu_{1,i},\mu_{2,i},\ldots,\mu_{d,i})\) are given using the numbered CENTER keywords, the standard deviations \((\sigma_{1,i},\sigma_{2,i},\ldots,\sigma_{d,i})\) using the the numbered SIGMA keywords, and the rate parameters \((\lambda_{1,i},\lambda_{2,i},\ldots,\lambda_{d,i})\) using the numbered LAMBDA keywords. The weights are given using the WEIGHTS keywords, if no weights are given are all terms weighted equally.

Examples

An exponentially modified Gaussian distribution in one-dimension

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

td1: TD_EXPONENTIALLY_MODIFIED_GAUSSIAN 
CENTER1
The center of each exponentially modified Gaussian distributions. 
=-10.0 
SIGMA1
The sigma parameters for each exponentially modified Gaussian distributions. 
=1.0 
LAMBDA1
The lambda parameters for each exponentially modified Gaussian distributions You
can use multiple instances of this keyword i.e. 
=0.25 The TD_EXPONENTIALLY_MODIFIED_GAUSSIAN action with label td1

A sum of two one-dimensional exponentially modified Gaussian distributions

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

td1: TD_EXPONENTIALLY_MODIFIED_GAUSSIAN ...
   
CENTER1
The center of each exponentially modified Gaussian distributions. 
=-10.0 
SIGMA1
The sigma parameters for each exponentially modified Gaussian distributions. 
=1.0 
LAMBDA1
The lambda parameters for each exponentially modified Gaussian distributions You
can use multiple instances of this keyword i.e. 
=0.5 
   
CENTER2
The center of each exponentially modified Gaussian distributions. 
=+10.0 
SIGMA2
The sigma parameters for each exponentially modified Gaussian distributions. 
=1.0 
LAMBDA2
The lambda parameters for each exponentially modified Gaussian distributions You
can use multiple instances of this keyword i.e. 
=1.0 
   
WEIGHTS
The weights of the distributions. 
=2.0,1.0 
   
...The TD_EXPONENTIALLY_MODIFIED_GAUSSIAN action with label td1

A sum of two two-dimensional exponentially modified Gaussian distributions

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

td1: TD_EXPONENTIALLY_MODIFIED_GAUSSIAN ...
   
CENTER1
The center of each exponentially modified Gaussian distributions. 
=-5.0,+5.0 
SIGMA1
The sigma parameters for each exponentially modified Gaussian distributions. 
=1.0,1.0 
LAMBDA1
The lambda parameters for each exponentially modified Gaussian distributions You
can use multiple instances of this keyword i.e. 
=0.5,0.5 
   
CENTER2
The center of each exponentially modified Gaussian distributions. 
=+5.0,+5.0 
SIGMA2
The sigma parameters for each exponentially modified Gaussian distributions. 
=1.0,1.0 
LAMBDA2
The lambda parameters for each exponentially modified Gaussian distributions You
can use multiple instances of this keyword i.e. 
=1.0,1.0 
   
WEIGHTS
The weights of the distributions. 
=1.0,1.0 
   
...The TD_EXPONENTIALLY_MODIFIED_GAUSSIAN action with label td1

Glossary of keywords and components

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 exponentially modified Gaussian distributions. You can use multiple instances of this keyword i.e. CENTER1, CENTER2, CENTER3...
SIGMA	The sigma parameters for each exponentially modified Gaussian distributions. You can use multiple instances of this keyword i.e. SIGMA1, SIGMA2, SIGMA3...
LAMBDA	The lambda parameters for each exponentially modified Gaussian distributions You can use multiple instances of this keyword i.e. LAMBDA1, LAMBDA2, LAMBDA3...
WEIGHTS	The weights of the distributions. 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.