Source code for mchammer.ensembles.canonical_ensemble

"""Definition of the canonical ensemble class."""

import numpy as np

from ase import Atoms
from ase.units import kB

from .. import DataContainer
from .base_ensemble import BaseEnsemble
from ..calculators.base_calculator import BaseCalculator


[docs]class CanonicalEnsemble(BaseEnsemble): """Instances of this class allow one to simulate systems in the canonical ensemble (:math:`N_iVT`), i.e. at constant temperature (:math:`T`), number of atoms of each species (:math:`N_i`), and volume (:math:`V`). The probability for a particular state in the canonical ensemble is proportional to the well-known Boltzmann factor, .. math:: \\rho_{\\text{C}} \\propto \\exp [ - E / k_B T ]. Since the concentrations or equivalently the number of atoms of each species is held fixed in the canonical ensemble, a trial step must conserve the concentrations. This is accomplished by randomly picking two unlike atoms and swapping their identities. The swap is accepted with probability .. math:: P = \\min \\{ 1, \\, \\exp [ - \\Delta E / k_B T ] \\}, where :math:`\\Delta E` is the change in potential energy caused by the swap. The canonical ensemble provides an ideal framework for studying the properties of a system at a specific concentration. Properties such as potential energy or phenomena such as chemical ordering at a specific temperature can conveniently be studied by simulating at that temperature. The canonical ensemble is also a convenient tool for "optimizing" a system, i.e., finding its lowest energy chemical ordering. In practice, this is usually achieved by simulated annealing, i.e. the system is equilibrated at a high temperature, after which the temperature is continuously lowered until the acceptance probability is almost zero. In a well-behaved system, the chemical ordering at that point corresponds to a low-energy structure, possibly the global minimum at that particular concentration. 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] 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. """ def __init__(self, atoms: Atoms, calculator: BaseCalculator, temperature: float, user_tag: str = None, boltzmann_constant: float = kB, 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) -> None: self._ensemble_parameters = dict(temperature=temperature) self._boltzmann_constant = boltzmann_constant # add species count to ensemble parameters symbols = set([symbol for sub in calculator.sublattices for symbol in sub.chemical_symbols]) for symbol in symbols: key = 'n_atoms_{}'.format(symbol) count = atoms.get_chemical_symbols().count(symbol) self._ensemble_parameters[key] = count 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) # setup sublattice probabilities self.sublattice_probabilities = get_swap_sublattice_probabilities(self.configuration) @property def temperature(self) -> float: """ temperature :math:`T` (see parameters section above) """ return self.ensemble_parameters['temperature'] @property def boltzmann_constant(self) -> float: """ Boltzmann constant :math:`k_B` (see parameters section above) """ return self._boltzmann_constant def _do_trial_step(self): """ Carries out one Monte Carlo trial step. """ self._total_trials += 1 sublattice_index = self.get_random_sublattice_index() sites, species = self.configuration.get_swapped_state(sublattice_index) potential_diff = self._get_property_change(sites, species) if self._acceptance_condition(potential_diff): self._accepted_trials += 1 self.update_occupations(sites, species) def _acceptance_condition(self, potential_diff: float) -> bool: """ Evaluates Metropolis acceptance criterion. Parameters ---------- potential_diff change in the thermodynamic potential associated with the trial step """ if potential_diff < 0: return True else: return np.exp(-potential_diff / (self.boltzmann_constant * self.temperature)) > \ self._next_random_number()
[docs] def get_random_sublattice_index(self) -> int: """Returns a random sublattice index based on the weights of the sublattice. Todo ---- * add unit test """ pick = np.random.choice(range(0, len(self.sublattices)), p=self.sublattice_probabilities) return pick
def get_swap_sublattice_probabilities(cm): """ Returns the probabilities of picking a sublattice in a ConfigurationManager for a canonical swap. """ sublattice_probabilities = [] for i, sl in enumerate(cm.sublattices): if cm.is_swap_possible(i): sublattice_probabilities.append(len(sl.indices)) else: sublattice_probabilities.append(0) norm = sum(sublattice_probabilities) if norm == 0: raise ValueError('No canonical swaps are possible on any of the active sublattices.') sublattice_probabilities = [p / norm for p in sublattice_probabilities] return sublattice_probabilities