"""
Definition of the variance-constrained semi-grand canonical ensemble class.
"""
from ase import Atoms
from ase.data import atomic_numbers, chemical_symbols
from ase.units import kB
from typing import Dict, Union, List
from .. import DataContainer
from ..calculators.base_calculator import BaseCalculator
from .thermodynamic_base_ensemble import ThermodynamicBaseEnsemble
[docs]
class VCSGCEnsemble(ThermodynamicBaseEnsemble):
r"""Instances of this class allow one to simulate systems in the
variance-constrained semi-grand canonical (VCSGC) ensemble
(:math:`N\phi\kappa VT`), i.e. at constant temperature (:math:`T`), total
number of sites (:math:`N=\sum_i N_i`), and two additional dimensionless
parameters :math:`\phi` and :math:`\kappa`, which constrain average and
variance of the concentration, respectively.
The below examples treat the binary case, but the generalization of
to ternaries and higher-order systems is straight-forward.
The probability for a particular state in the VCSGC ensemble for a
:math:`2`-component system can be written
.. math::
\rho_{\text{VCSGC}} \propto \exp\Big[ - E / k_B T
+ \kappa N ( c_1 + \phi_1 / 2 )^2 \Big],
where :math:`c_1` represents the concentration of species 1, i.e.,
:math:`c_1=N_1/N`. (Please note that the quantities :math:`\kappa` and
:math:`\phi` correspond, respectively, to :math:`\bar{\kappa}` and
:math:`\bar{\phi}` in [SadErh12]_.) The :math:`\phi` may refer to any
of the two species. If :math:`\phi` is specified for species A, an
equivalent simulation can be carried out by specifying :math:`\phi_B` as
:math:`-2-\phi_A`. In general, simulations of :math:`N`-component
systems requires the specification of :math:`\phi` for :math:`N-1`
elements.
Just like the :ref:`semi-grand canonical ensemble <canonical_ensemble>`,
the VCSGC ensemble allows concentrations to change. A trial step consists
of changing the identity of a randomly chosen atom and accepting the change
with probability
.. math::
P = \min \{ 1, \, \exp [ - \Delta E / k_B T
+ \kappa N \Delta c_1 (\phi_1 + \Delta c_1 + 2 c_1 ) ] \}.
Note that for a sufficiently large value of :math:`\kappa`, say 200, the
probability density :math:`\rho_{\text{VCSGC}}` is sharply peaked around
:math:`c_1=-\phi_1 / 2`. In practice, this means that we can gradually
change :math:`\phi_1` from (using some margins) :math:`-2.1` to
:math:`0.1` and take the system continuously from :math:`c_1 = 0` to
:math:`1`. The parameter :math:`\kappa` constrains the fluctuations (or
the variance) of the concentration at each value of :math:`\phi_1`, with
higher values of :math:`\kappa` meaning less fluctuations. Unlike the
:ref:`semi-grand canonical ensemble <sgc_ensemble>`, one value of
:math:`\phi_1` maps to one and only one concentration also in multiphase
regions. Since the derivative of the canonical free energy can be expressed
in terms of parameters and observables of the VCSGC ensemble,
.. math::
k_B T \kappa ( \phi_1 + 2 \langle c_1 \rangle ) = - \frac{1}{N}
\frac{\partial F}{\partial c_1} (N, V, T, \langle c_1 \rangle ),
this ensemble allows for thermodynamic integration across multiphase
regions. This means that we can construct phase diagrams by directly
comparing the free energies of the different phases. This often makes the
VCSGC ensemble more convenient than the :ref:`semi-grand canonical ensemble
<sgc_ensemble>` when simulating materials with multiphase regions, such as
alloys with miscibility gaps.
When using the VCSGC ensemble, please cite
Phys. Rev. B **86**, 134204 (2012) [SadErh12]_.
Parameters
----------
structure
Atomic configuration to be used in the Monte Carlo simulation;
also defines the initial occupation vector.
calculator
Calculator to be used for calculating the potential changes
that enter the evaluation of the Metropolis criterion.
temperature
Temperature :math:`T` in appropriate units, commonly Kelvin.
phis
Average constraint parameters :math:`\phi_i`. The key denotes the
species. For a :math:`N`-component sublattice, there should be :math:`N - 1`
different :math:`\phi_i` (referred to as :math:`\bar{\phi}`
in [SadErh12]_).
kappa
parameter that constrains the variance of the concentration
(referred to as :math:`\bar{\kappa}` in [SadErh12]_)
boltzmann_constant
Boltzmann constant :math:`k_B` in appropriate
units, i.e., units that are consistent
with the underlying cluster expansion
and the temperature units. Default: eV/K.
user_tag
Human-readable tag for ensemble. Default: ``None``.
random_seed
Seed for the random number generator used in the Monte Carlo simulation.
dc_filename
Name of file the data container associated with the ensemble
will be written to. If the file exists it will be read, the
data container will be appended, and the file will be
updated/overwritten.
data_container_write_period
Period in units of seconds at which the data container is
written to file. Writing periodically to file provides both
a way to examine the progress of the simulation and to back up
the data. Default: 600 s.
ensemble_data_write_interval
Interval at which data is written to the data container. This
includes for example the current value of the calculator
(i.e., usually the energy) as well as ensembles specific fields
such as temperature or the number of atoms of different species.
Default: Number of sites in the :attr:`structure`.
trajectory_write_interval
Interval at which the current occupation vector of the atomic
configuration is written to the data container.
Default: Number of sites in the :attr:`structure`.
sublattice_probabilities
Probability for picking a sublattice when doing a random flip.
The list should be as long as the number of sublattices and should
sum up to 1.
Example
-------
The following snippet illustrate how to carry out a simple Monte Carlo
simulation in the variance-constrained semi-canonical ensemble. Here, the
parameters of the cluster expansion are set to emulate a simple Ising model
in order to obtain an example that can be run without modification. In
practice, one should of course use a proper cluster expansion::
>>> from ase.build import bulk
>>> from icet import ClusterExpansion, ClusterSpace
>>> from mchammer.calculators import ClusterExpansionCalculator
>>> # prepare cluster expansion
>>> # the setup emulates a second nearest-neighbor (NN) Ising model
>>> # (zerolet and singlet ECIs are zero; only first and second neighbor
>>> # pairs are included)
>>> prim = bulk('Au')
>>> cs = ClusterSpace(prim, cutoffs=[4.3], chemical_symbols=['Ag', 'Au'])
>>> ce = ClusterExpansion(cs, [0, 0, 0.1, -0.02])
>>> # set up and run MC simulation
>>> structure = prim.repeat(3)
>>> calc = ClusterExpansionCalculator(structure, ce)
>>> phi = 0.6
>>> mc = VCSGCEnsemble(structure=structure, calculator=calc,
... temperature=600,
... dc_filename='myrun_vcsgc.dc',
... phis={'Au': phi},
... kappa=200)
>>> mc.run(100) # carry out 100 trial swaps
"""
def __init__(self, structure: Atoms,
calculator: BaseCalculator,
temperature: float,
phis: Dict[str, float],
kappa: float,
boltzmann_constant: float = kB,
user_tag: str = None,
random_seed: int = None,
dc_filename: str = None,
data_container: str = None,
data_container_write_period: float = 600,
ensemble_data_write_interval: int = None,
trajectory_write_interval: int = None,
sublattice_probabilities: List[float] = None) -> None:
self._ensemble_parameters = dict(temperature=temperature,
kappa=kappa)
self._boltzmann_constant = boltzmann_constant
# Save ensemble parameters
for sym, phi in phis.items():
if isinstance(sym, str):
chemical_symbol = sym
else:
chemical_symbol = chemical_symbols[sym]
phi_sym = 'phi_{}'.format(chemical_symbol)
self._ensemble_parameters[phi_sym] = phi
super().__init__(
structure=structure,
calculator=calculator,
user_tag=user_tag,
random_seed=random_seed,
dc_filename=dc_filename,
data_container=data_container,
data_container_class=DataContainer,
data_container_write_period=data_container_write_period,
ensemble_data_write_interval=ensemble_data_write_interval,
trajectory_write_interval=trajectory_write_interval,
boltzmann_constant=boltzmann_constant
)
# Save phis (need self.configuration to check sublattices so
# we do it last)
self._phis = get_phis(phis)
# Check that each sublattice has N - 1 phis
for sl in self.sublattices.active_sublattices:
count_specified_elements = 0
for number in sl.atomic_numbers:
if number in self._phis.keys():
count_specified_elements += 1
if count_specified_elements != len(sl.atomic_numbers) - 1:
raise ValueError('phis must be set for N - 1 elements on a '
'sublattice with N species')
if sublattice_probabilities is None:
self._flip_sublattice_probabilities = self._get_flip_sublattice_probabilities()
else:
self._flip_sublattice_probabilities = sublattice_probabilities
def _do_trial_step(self):
""" Carries out one Monte Carlo trial step. """
sublattice_index = self.get_random_sublattice_index(
probability_distribution=self._flip_sublattice_probabilities)
return self.do_vcsgc_flip(
phis=self.phis, kappa=self.kappa, sublattice_index=sublattice_index)
@property
def temperature(self) -> float:
""" Temperature :math:`T` (see parameters section above). """
return self.ensemble_parameters['temperature']
@property
def phis(self) -> Dict[int, float]:
r"""
Average constraint parameters :math:`\phi_i` for all species but one
(referred to as :math:`\bar{\phi}` in [SadErh12]_).
"""
return self._phis
@property
def kappa(self) -> float:
r"""
Variance constraint parameter :math:`\bar{\kappa}` (see parameters section above).
"""
return self.ensemble_parameters['kappa']
def _get_ensemble_data(self) -> Dict:
"""
Returns a dict with the default data of the ensemble. This includes
atom counts and free energy derivative.
"""
data = super()._get_ensemble_data()
# free energy derivative
data.update(self._get_vcsgc_free_energy_derivatives(self.phis, self.kappa))
# species counts
data.update(self._get_species_counts())
return data
def get_phis(phis: Union[Dict[int, float], Dict[str, float]]) -> Dict[int, float]:
"""Get phis as used in the VCSGC ensemble.
Parameters
----------
phis
phis that will be transformed to the format used by the ensemble.
"""
if not isinstance(phis, dict):
raise TypeError(f'phis has the wrong type: {type(phis)}')
# Translate to atomic numbers if necessary
phis_ret = {}
for key, phi in phis.items():
if isinstance(key, str):
atomic_number = atomic_numbers[key]
phis_ret[atomic_number] = phi
elif isinstance(key, int):
phis_ret[key] = phi
return phis_ret