Action: TD_WELLTEMPERED

Module	ves
Description	Usage
Well-tempered target distribution (dynamic).

Details and examples

Well-tempered target distribution (dynamic).

Use as a target distribution the well-tempered distribution discussed in the first paper cited below, which is given by

$p(\mathbf{s}) = \frac{e^{-(\beta/\gamma) F(\mathbf{s})}} {\int d\mathbf{s}\, e^{-(\beta/\gamma) F(\mathbf{s})}} = \frac{[P_{0}(\mathbf{s})]^{1/\gamma}} {\int d\mathbf{s}\, [P_{0}(\mathbf{s})]^{1/\gamma}}$

where $\gamma$ is a so-called bias factor and $P_{0}(\mathbf{s})$ is the unbiased canonical distribution of the CVs. This target distribution thus corresponds to a biased ensemble where, as compared to the unbiased one, the probability peaks have been broaden and the fluctuations of the CVs are enhanced. The value of the bias factor $\gamma$ determines by how much the fluctuations are enhanced.

The well-tempered distribution can be view as sampling on an effective free energy surface $\tilde{F}(\mathbf{s}) = (1/\gamma) F(\mathbf{s})$ which has largely the same metastable states as the original $F(\mathbf{s})$ but with barriers that have been reduced by a factor of $\gamma$ . Generally one should use a value of $\gamma$ that results in effective barriers on the order of few $k_{\mathrm{B}}T$ such that thermal fluctuations can easily induce transitions between different metastable states.

At convergence the relationship between the bias potential and the free energy surface is given by

$F(\mathbf{s}) = - \left(\frac{1}{1-\gamma^{-1}} \right) V(\mathbf{s})$

This target distribution depends directly on the free energy surface $F(\mathbf{s})$ which is quantity that we do not know a-priori and want to obtain. Therefore, this target distribution is iteratively updated according to

$p^{(m+1)}(\mathbf{s}) = \frac{e^{-(\beta/\gamma) F^{(m+1)}(\mathbf{s})}} {\int d\mathbf{s}\, e^{-(\beta/\gamma) F^{(m+1)}(\mathbf{s})}}$

where $F^{(m+1)}(\mathbf{s})$ is the current best estimate of the free energy surface obtained according to

$F^{(m+1)}(\mathbf{s}) = - V^{(m+1)}(\mathbf{s}) - \frac{1}{\beta} \log p^{(m)}(\mathbf{s}) = - V^{(m+1)}(\mathbf{s}) + \frac{1}{\gamma} F^{(m)}(\mathbf{s})$

The frequency of performing this update needs to be set in the optimizer used in the calculation. Normally it is sufficient to do it every 100-1000 bias update iterations.

Examples

Employ a well-tempered target distribution with a bias factor of 10

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

td_welltemp: TD_WELLTEMPEREDWell-tempered target distribution (dynamic). More details BIASFACTORThe bias factor used for the well-tempered distribution=10
The TD_WELLTEMPERED action with label td_welltemp calculates something

Full list of keywords

The following table describes the keywords and options that can be used with this action

Keyword	Type	Default	Description
BIASFACTOR	compulsory	none	The bias factor used for the well-tempered distribution

References

More information about how this action can be used is available in the following articles: