TD_MULTITHERMAL_MULTIBARIC
 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.

Multithermal-multibaric target distribution (dynamic).

Use the target distribution to sample the multithermal-multibaric ensemble [90] [85]. In this way, in a single molecular dynamics simulation one can obtain information about the simulated system in a range of temperatures and pressures. This range is determined through the keywords MIN_TEMP, MAX_TEMP, MIN_PRESSURE, and MAX_PRESSURE. One should also specified the target pressure of the barostat with the keyword PRESSURE.

The collective variables (CVs) used to construct the bias potential must be:

1. the potential energy and the volume or,
2. the potential energy, the volume, and an order parameter.

Other choices of CVs or a different order of the above mentioned CVs are nonsensical. The third CV, the order parameter, must be used when the region of the phase diagram under study is crossed by a first order phase transition [91] .

The algorithm will explore the free energy at each temperature and pressure up to a predefined free energy threshold $$\epsilon$$ specified through the keyword THRESHOLD (in kT units). If only the energy and the volume are being biased, i.e. no phase transition is considered, then THRESHOLD can be set to around 5. 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.

It is also important to specify the number of intermediate temperatures and pressures to consider. This is done through the keywords STEPS_TEMP and STEPS_PRESSURE. If the number of intermediate temperature and pressures is too small, then holes might appear in the target distribution. If it is too large, the performance will deteriorate with no additional advantage.

We now describe the algorithm more rigorously. The target distribution is given by

$p(E,\mathcal{V},s)= \begin{cases} 1/\Omega_{E,\mathcal{V},s} & \text{if there is at least one } \beta',P' \text{ such} \\ & \text{that } \beta' F_{\beta',P'}(E,\mathcal{V},s)<\epsilon \text{ with} \\ & \beta_1>\beta'>\beta_2 \text{ and } P_1<P'<P_2 \\ 0 & \text{otherwise} \end{cases}$

with $$F_{\beta',P'}(E,\mathcal{V},s)$$ the free energy as a function of energy $$E$$ and volume $$\mathcal{V}$$ (and optionally the order parameter $$s$$) at temperature $$\beta'$$ and pressure $$P'$$, $$\Omega_{E,\mathcal{V},s}$$ is a normalization constant, and $$\epsilon$$ is the THRESHOLD. In practice the condition $$\beta' F_{\beta',P'}(E,\mathcal{V},s)<\epsilon$$ is checked in equally spaced points in each dimension $$\beta'$$ and $$P'$$. The number of points is determined with the keywords STEPS_TEMP and STEPS_PRESSURE. In practice the target distribution is never set to zero but rather to a small value controlled by the keyword EPSILON. The small value is e^-EPSILON.

Much like in the Wang-Landau algorithm [120] or in the multicanonical ensemble [11] , a flat histogram is targeted. The idea behind this choice of target distribution is that all regions of potential energy and volume (and optionally order parameter) that are relevant at all temperatures $$\beta_1<\beta'<\beta_2$$ and pressure $$P_1<P'<P_2$$ are included in the distribution.

The free energy at temperature $$\beta'$$ and pressure $$P'$$ is calculated from the free energy at $$\beta$$ and $$P$$ using:

$\beta' F_{\beta',P'}(E,\mathcal{V},s) = \beta F_{\beta,P}(E,\mathcal{V},s) + (\beta' - \beta) E + (\beta' P' - \beta P ) \mathcal{V} + C$

with $$C$$ such that $$F_{\beta',P'}(E_m,\mathcal{V}_m,s_m)=0$$ with $$E_{m},\mathcal{V}_m,s_m$$ the position of the free energy minimum. $$\beta F_{\beta,P}(E,\mathcal{V},s)$$ is not know from the start and is instead found during the simulation. Therefore $$p(E,\mathcal{V},s)$$ is determined iteratively as done in the well tempered target distribution [118].

The output of these simulations can be reweighted in order to obtain information at all temperatures and pressures in the targeted region of Temperature-Pressure plane. The reweighting can be performed using the action REWEIGHT_TEMP_PRESS.

The multicanonical ensemble (fixed volume) can be targeted using TD_MULTICANONICAL.

Examples

The following input can be used to run a simulation in the multithermal-multibaric ensemble. The region of the temperature-pressure plane that will be explored is 260-350 K and 1 bar- 300 MPa. The energy and the volume are used as collective variables. Legendre polynomials are used to construct the two dimensional 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.

Click on the labels of the actions for more information on what each action computes
# Use energy and volume as CVs
energy: ENERGY
vol: VOLUME
# Basis functions
bf1: BF_LEGENDRE =10 =-14750 =-12250
bf2: BF_LEGENDRE =10 =6.5 =8.25
# Target distribution - 1 bar = 0.06022140857 and 300 MPa = 180.66422571
td_multi: TD_MULTITHERMAL_MULTIBARIC ...
=260
=350
=180.66422571
=0.06022140857
=0.06022140857

...
# Bias expansion
b1: VES_LINEAR_EXPANSION ...
=energy,vol
=bf1,bf2
=300.0
=200,200
=td_multi

...
# Optimization algorithm
o1: OPT_AVERAGED_SGD ...
=b1
=500

=1.0
=500
=500
=500
=100
=100

... OPT_AVERAGED_SGD

The multithermal-multibaric target distribution can also be used to explore regions of the phase diagram crossed by first order phase transitions. Consider a system of 250 atoms that crystallizes in the FCC crystal structure. The region of the temperature-pressure plane that will be explored is 350-450 K and 1bar-1GPa. We assume that inside this region we can find the liquid-FCC coexistence line that we would like to obtain. In this case in addition to the energy and volume, an order parameter must also be biased. The energy, volume, and an order parameter are used as collective variables to construct the bias potential. We choose as order parameter the FCCUBIC. Legendre polynomials are used to construct the three dimensional 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.

Click on the labels of the actions for more information on what each action computes
# Use energy, volume and FCCUBIC as CVs
energy: ENERGY
vol: VOLUME
fcc: FCCUBIC =1-256 ={CUBIC D_0=0.4 D_MAX=0.5}  ={RATIONAL R_0=0.45 NN=12 MM=24}
# Basis functions
bf1: BF_LEGENDRE =8 =-26500 =-23500
bf2: BF_LEGENDRE =8 =8.0 =11.5
bf3: BF_LEGENDRE =8 =0.0 =256.0
# Target distribution
td_multitp: TD_MULTITHERMAL_MULTIBARIC ...

=350.0
=450.0
=0.06022140857
=602.2140857
=301.10704285
=250.0,0.1,10.0
=15
=20
=20
...
# Expansion
b1: VES_LINEAR_EXPANSION ...
=energy,vol,fcc.morethan
=bf1,bf2,bf3
=400.0
=40,40,40
=td_multitp

...
# Optimization algorithm
o1: OPT_AVERAGED_SGD ...
=b1
=500

=1.0
=500
=500
=500
=100
=500

... OPT_AVERAGED_SGD
Glossary of keywords and components
Compulsory keywords
 THRESHOLD ( default=5 ) Maximum exploration free energy in kT. EPSILON ( default=10 ) The zeros of the target distribution are changed to e^-EPSILON. MIN_TEMP Minimum energy. MAX_TEMP Maximum energy. MIN_PRESSURE Minimum pressure. MAX_PRESSURE Maximum pressure. PRESSURE Target pressure of the barostat used in the MD engine. STEPS_TEMP ( default=20 ) Number of temperature steps. STEPS_PRESSURE ( default=20 ) Number of pressure steps.
Options
 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.