Action: TD_MULTICANONICAL

Module	ves
Description	Usage
Multicanonical target distribution (dynamic).

Details and examples

Multicanonical target distribution (dynamic).

Use the target distribution to sample the multicanonical ensemble that is introduced in the first paper cited below. 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: 1. the energy or, 2. the energy and an order parameter.

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 (for details see the second paper cited below.

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 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.

When only the potential energy is used as CV the method is equivalent to the Wang-Landau algorithm that is discussed in the last paper cited below. 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:

$p(E)= \left\{\begin{array}{ll} \frac{1}{E_2-E_1} & \mathrm{if} \quad E_1<E<E_2 \\ 0 & \mathrm{otherwise} \end{array}\right.$

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.

Examples

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.

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

# Use energy and volume as CVs
energyThe ENERGY action with label energy calculates the following quantities: Quantity    Type    Description  
energy scalar the internal energy: ENERGYCalculate the total potential energy of the simulation box. More details

# Basis functions
bf1: BF_LEGENDRELegendre polynomials basis functions. More details ORDERThe order of the basis function expansion=20 MINIMUMThe minimum of the interval on which the basis functions are defined=-25000 MAXIMUMThe maximum of the interval on which the basis functions are defined=-23500

# Target distributions
The BF_LEGENDRE action with label bf1 calculates somethingTD_MULTICANONICALMulticanonical target distribution (dynamic). More details ...
 LABELa label for the action so that its output can be referenced in the input to other actions=td_multi
 MIN_TEMPMinimum temperature=400
 MAX_TEMPMaximum temperature=600
... TD_MULTICANONICAL

# Expansion
The TD_MULTICANONICAL action with label td_multi calculates somethingVES_LINEAR_EXPANSIONLinear basis set expansion bias. This action has hidden defaults. More details ...
 ARGthe labels of the scalars on which the bias will act=energy
 BASIS_FUNCTIONSthe label of the one dimensional basis functions that should be used=bf1
 TEMPthe system temperature - this is needed if the MD code does not pass the temperature to PLUMED=500.0
 GRID_BINSthe number of bins used for the grid=1000
 TARGET_DISTRIBUTIONthe label of the target distribution to be used=td_multi
 LABELa label for the action so that its output can be referenced in the input to other actions=b1The VES_LINEAR_EXPANSION action with label b1 calculates the following quantities: Quantity    Type    Description  
b1.bias scalar the instantaneous value of the bias potential
b1.force2 scalar the instantaneous value of the squared force due to this bias potential.
... VES_LINEAR_EXPANSION
VES_LINEAR_EXPANSIONLinear basis set expansion bias. This action uses the defaults shown here. More details ...
 ARGthe labels of the scalars on which the bias will act=energy
 BASIS_FUNCTIONSthe label of the one dimensional basis functions that should be used=bf1
 TEMPthe system temperature - this is needed if the MD code does not pass the temperature to PLUMED=500.0
 GRID_BINSthe number of bins used for the grid=1000
 TARGET_DISTRIBUTIONthe label of the target distribution to be used=td_multi
 LABELa label for the action so that its output can be referenced in the input to other actions=b1
 GRID_FMT the format to use when outputting the numbers in the grids=%14.9f
... VES_LINEAR_EXPANSION

# Optimization algorithm
OPT_AVERAGED_SGDAveraged stochastic gradient decent with fixed step size. This action has hidden defaults. More details ...
  BIASthe label of the VES bias to be optimized=b1
  STRIDEthe frequency of updating the coefficients given in the number of MD steps=500
  LABELa label for the action so that its output can be referenced in the input to other actions=o1
  STEPSIZEthe step size used for the optimization=1.0
  FES_OUTPUThow often the FES(s) should be written out to file=500
  BIAS_OUTPUThow often the bias(es) should be written out to file=500
  TARGETDIST_OUTPUThow often the dynamic target distribution(s) should be written out to file=500
  COEFFS_OUTPUT how often the coefficients should be written to file=10
  TARGETDIST_STRIDEstride for updating a target distribution that is iteratively updated during the optimization=100
... OPT_AVERAGED_SGD
The OPT_AVERAGED_SGD action with label o1 calculates the following quantities: Quantity    Description  
o1.value a scalarOPT_AVERAGED_SGDAveraged stochastic gradient decent with fixed step size. This action uses the defaults shown here. More details ...
  BIASthe label of the VES bias to be optimized=b1
  STRIDEthe frequency of updating the coefficients given in the number of MD steps=500
  LABELa label for the action so that its output can be referenced in the input to other actions=o1
  STEPSIZEthe step size used for the optimization=1.0
  FES_OUTPUThow often the FES(s) should be written out to file=500
  BIAS_OUTPUThow often the bias(es) should be written out to file=500
  TARGETDIST_OUTPUThow often the dynamic target distribution(s) should be written out to file=500
  COEFFS_OUTPUT how often the coefficients should be written to file=10
  TARGETDIST_STRIDEstride for updating a target distribution that is iteratively updated during the optimization=100
 COEFFS_FILE the name of output file for the coefficients=coeffs.data
... 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.

Full list of keywords

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

Keyword	Type	Default	Description
THRESHOLDThis keyword do not have examples	compulsory	5	Maximum exploration free energy in kT
EPSILONThis keyword do not have examples	compulsory	10	The zeros of the target distribution are changed to e^-EPSILON
MIN_TEMP	compulsory	none	Minimum temperature
MAX_TEMP	compulsory	none	Maximum temperature
STEPS_TEMPThis keyword do not have examples	optional	not used	Number of temperature steps
SIGMAThis keyword do not have examples	optional	not used	The standard deviation parameters of the Gaussian kernels used for smoothing the target distribution

References

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

Quantity	Type	Description
energy	scalar	the internal energy