   TD_WELLTEMPERED
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Well-tempered target distribution (dynamic).

Use as a target distribution the well-tempered distribution  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_WELLTEMPERED BIASFACTORcompulsory keyword
The bias factor used for the well-tempered distribution. =10

Glossary of keywords and components
Compulsory keywords
 BIASFACTOR The bias factor used for the well-tempered distribution.