Source code for mchammer.ensembles.vcsgc_ensemble

"""
Definition of the variance-constrained semi-grand canonical ensemble class.
"""

import numpy as np

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): """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 <vcsgc_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 Sadigh, B. and Erhart, P., Phys. Rev. B **86**, 134204 (2012) [SadErh12]_. Parameters ---------- atoms : :class:`Atoms <ase.Atoms>` atomic configuration to be used in the Monte Carlo simulation; also defines the initial occupation vector calculator : :class:`BaseCalculator <mchammer.calculators.ClusterExpansionCalculator>` calculator to be used for calculating the potential changes that enter the evaluation of the Metropolis criterion temperature : float temperature :math:`T` in appropriate units [commonly Kelvin] phis : Dict[str, float] average constraint parameters :math:`\\phi_i`; the key denotes the species; for a N-component sublattice, there should be N - 1 different `\\phi_i` (referred to as :math:`\\bar{\\phi}` in [SadErh12]_) kappa : float parameter that constrains the variance of the concentration (referred to as :math:`\\bar{\\kappa}` in [SadErh12]_) boltzmann_constant : float 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 : str human-readable tag for ensemble [default: None] data_container : str 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 random_seed : int seed for the random number generator used in the Monte Carlo simulation ensemble_data_write_interval : int 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 data_container_write_period : float 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: np.inf] trajectory_write_interval : int interval at which the current occupation vector of the atomic configuration is written to the data container. sublattice_probabilities : List[float] 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 from mchammer.ensembles import VCSGCEnsemble # 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 atoms = prim.repeat(3) calc = ClusterExpansionCalculator(atoms, ce) phi = 0.6 mc = VCSGCEnsemble(atoms=atoms, calculator=calc, temperature=600, data_container='myrun_vcsgc.dc', phis={'Au': phi}, kappa=200) mc.run(100) # carry out 100 trial swaps """ def __init__(self, atoms: Atoms, calculator: BaseCalculator, temperature: float, phis: Dict[str, float], kappa: float, boltzmann_constant: float = kB, user_tag: str = None, data_container: DataContainer = None, random_seed: int = None, data_container_write_period: float = np.inf, 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__( atoms=atoms, calculator=calculator, user_tag=user_tag, data_container=data_container, random_seed=random_seed, 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 elements') 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) 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]: """ phis :math:`\\phi_i`, one for each species but their sum must be :math:`-2.0` (referred to as :math:`\\bar{\\phi}` in [SadErh12]_) """ return self._phis @property def kappa(self) -> float: """ kappa :math:`\\bar{\\kappa}` constrain parameter (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: Dict[Union[int, str], float]) -> Dict[int, float]: """Get phis as used in the vcsgc ensemble. Parameters ---------- phis the phis that will be transformed to the format the ensemble use. """ if not isinstance(phis, dict): raise TypeError('phis has the wrong type: {}'.format(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