	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 Von Mises distributions (static).

Employ a target distribution that is given by a sum where each term is a product of one-dimensional Von Mises distributions,

\[ p(\mathbf{s}) = \sum_{i} \, w_{i} \prod_{k}^{d} \frac{\exp\left(\kappa_{k,i} \, \cos (s_{k}-\mu_{k,i}) \right)} {2\pi I_{0}(\kappa_{k,i})} \]

where \((\mu_{1,i},\mu_{2,i},\ldots,\mu_{d,i})\) are the centers of the distributions, \((\kappa_{1,i},\kappa_{2,i},\ldots,\kappa_{d,i})\) are parameters that determine the extend of each distribution, and \(I_{0}(x)\) is the modified Bessel function of order 0. The weights \(w_{i}\) are normalized to 1, \(\sum_{i}w_{i}=1\).

The Von Mises distribution is defined for periodic variables with a periodicity of \(2\pi\) and is analogous to the Gaussian distribution. The parameter \( \sqrt{1/\kappa}\) is comparable to the standard deviation \(\sigma\) for the Gaussian distribution.

To use this target distribution you need to give the centers \((\mu_{1,i},\mu_{2,i},\ldots,\mu_{d,i})\) by using the numbered CENTER keywords and the "standard deviations" \((\sqrt{1/\kappa_{1,i}},\sqrt{1/\kappa_{2,i}},\ldots,\sqrt{1/\kappa_{d,i}})\) using the numbered SIGMA keywords.

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.
CENTER	The centers of the Von Mises distributions. You can use multiple instances of this keyword i.e. CENTER1, CENTER2, CENTER3...
SIGMA	The "standard deviations" of the Von Mises distributions. You can use multiple instances of this keyword i.e. SIGMA1, SIGMA2, SIGMA3...
WEIGHTS	The weights of the Von Mises distributions. Have to be as many as the number of centers given with the numbered CENTER keywords. If no weights are given the distributions are weighted equally. The weights are automatically normalized to 1.
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

Sum of two Von Mises distribution in one dimension that have equal weights as no weights are given.

TD_VONMISES ...
 CENTER1=+2.0 SIGMA1=0.6
 CENTER2=-2.0 SIGMA2=0.7
 LABEL=td
... TD_VONMISES

Sum of two Von Mises distribution in two dimensions that have different weights. Note that the weights are automatically normalized to 1 such that specifying WEIGHTS=1.0,2.0 is equal to specifying WEIGHTS=0.33333,0.66667.

TD_VONMISES ...
 CENTER1=+2.0,+2.0 SIGMA1=0.6,0.7
 CENTER2=-2.0,+2.0 SIGMA2=0.7,0.6
 WEIGHTS=1.0,2.0
 LABEL=td
... TD_VONMISES