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| 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. |
Well-tempered target distribution (dynamic).
Use as a target distribution the well-tempered distribution [8] 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 [111] 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.
| BIASFACTOR | The bias factor used for the well-tempered distribution. |
Employ a well-tempered target distribution with a bias factor of 10
td_welltemp: TD_WELLTEMPERED BIASFACTOR=10
1.8.14