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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. |
Multicanonical target distribution (dynamic).
Use the target distribution to sample the multicanonical ensemble [11] [86]. In this way, in a single molecular dynamics simulation one can obtain information about the system in a range of temperatures. This range is determined through the keywords MIN_TEMP and MAX_TEMP.
The collective variables (CVs) used to construct the bias potential must be:
Other choices of CVs or a different order of the above mentioned CVs are nonsensical. The second CV, the order parameter, must be used when one aims at studying a first order phase transition in the chosen temperature interval [87].
The algorithm will explore the free energy at each temperature up to a predefined free energy threshold \(\epsilon\) specified through the keyword THRESHOLD (in kT units). If only the energy is biased, i.e. no phase transition is considered, then THRESHOLD can be set to 1. If also an order parameter is used then the THRESHOLD should be greater than the barrier for the transformation in kT. For small systems undergoing a freezing transition THRESHOLD is typically between 20 and 50.
When only the potential energy is used as CV the method is equivalent to the Wang-Landau algorithm [113]. The advantage with respect to Wang-Landau is that instead of sampling the potential energy indiscriminately, an interval is chosen on the fly based on the minimum and maximum targeted temperatures.
The algorithm works as follows. The target distribution for the potential energy is chosen to be:
MISSING EQUATION TO BE FIXED
where the energy limits \(E_1\) and \(E_2\) are yet to be determined. Clearly the interval \(E_1–E_2\) chosen is related to the interval of temperatures \(T_1-T_2\). To link these two intervals we make use of the following relation:
\[ \beta' F_{\beta'}(E) = \beta F_{\beta}(E) + (\beta' - \beta) E + C, \]
where \(F_{\beta}(E)\) is determined during the optimization and we shall choose \(C\) such that \(F_{\beta'}(E_{m})=0\) with \(E_{m}\) the position of the free energy minimum. Using this relation we employ an iterative procedure to find the energy interval. At iteration \(k\) we have the estimates \(E_1^k\) and \(E_2^k\) for \(E_1\) and \(E_2\), and the target distribution is:
\[ p^k(E)=\frac{1}{E_2^k-E_1^k} \quad \mathrm{for} \quad E_1^k<E<E_2^k. \]
\(E_1^k\) and \(E_2^k\) are obtained from the leftmost solution of \(\beta_2 F_{\beta_2}^{k-1}(E_1^k)=\epsilon\) and the rightmost solution of \(\beta_1 F_{\beta_1}^{k-1}(E_2^k)=\epsilon\). The procedure is repeated until convergence. This iterative approach is similar to that in TD_WELLTEMPERED.
The version of this algorithm in which the energy and an order parameter are biased is similar to the one described in TD_MULTITHERMAL_MULTIBARIC.
The output of these simulations can be reweighted in order to obtain information at all temperatures in the targeted temperature interval. The reweighting can be performed using the action REWEIGHT_TEMP_PRESS.
| THRESHOLD | ( default=1 ) Maximum exploration free energy in kT. |
| MIN_TEMP | Minimum temperature. |
| MAX_TEMP | Maximum temperature. |
| STEPS_TEMP | Number of temperature steps. Only for the 2D version, i.e. energy and order parameter. |
| SIGMA | The standard deviation parameters of the Gaussian kernels used for smoothing the target distribution. One value must be specified for each argument, i.e. one value per CV. A value of 0.0 means that no smoothing is performed, this is the default behavior. |
The following input can be used to run a simulation in the multicanonical ensemble. The temperature interval to be explored is 400-600 K. The energy is used as collective variable. Legendre polynomials are used to construct the bias potential. The averaged stochastic gradient descent algorithm is chosen to optimize the VES functional. The target distribution is updated every 100 optimization steps (200 ps here) using the last estimation of the free energy.
# Use energy and volume as CVs energy: ENERGY # Basis functions bf1: BF_LEGENDRE ORDER=20 MINIMUM=-25000 MAXIMUM=-23500 # Target distributions TD_MULTICANONICAL ... LABEL=td_multi SIGMA=50.0 MIN_TEMP=400 MAX_TEMP=600 THRESHOLD=1 ... TD_MULTICANONICAL # Expansion VES_LINEAR_EXPANSION ... ARG=energy BASIS_FUNCTIONS=bf1 TEMP=500.0 GRID_BINS=1000 TARGET_DISTRIBUTION=td_multi LABEL=b1 ... VES_LINEAR_EXPANSION # Optimization algorithm OPT_AVERAGED_SGD ... BIAS=b1 STRIDE=500 LABEL=o1 STEPSIZE=1.0 FES_OUTPUT=500 BIAS_OUTPUT=500 TARGETDIST_OUTPUT=500 COEFFS_OUTPUT=10 TARGETDIST_STRIDE=100 ... OPT_AVERAGED_SGD
The multicanonical target distribution can also be used to explore a temperature interval in which a first order phase transitions is observed.
1.8.14