Source code for mchammer.ensembles.semi_grand_canonical_ensemble

"""
Definition of the 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 collections import OrderedDict
from typing import Dict, Union, List

from .. import DataContainer
from ..calculators.base_calculator import BaseCalculator
from .thermodynamic_base_ensemble import ThermodynamicBaseEnsemble


[docs]class SemiGrandCanonicalEnsemble(ThermodynamicBaseEnsemble): """Instances of this class allow one to simulate systems in the semi-grand canonical (SGC) ensemble (:math:`N\\Delta\\mu_i VT`), i.e. at constant temperature (:math:`T`), total number of sites (:math:`N=\\sum_i N_i`), relative chemical potentials (:math:`\\Delta\\mu_i=\\mu_i - \\mu_1`, where :math:`i` denotes the species), and volume (:math:`V`). The probability for a particular state in the SGC ensemble for a :math:`m`-component system can be written .. math:: \\rho_{\\text{SGC}} \\propto \\exp\\Big[ - \\big( E + \\sum_{i>1}^m \\Delta\\mu_i N_i \\big) \\big / k_B T \\Big] with the *relative* chemical potentials :math:`\\Delta\\mu_i = \\mu_i - \\mu_1` and species counts :math:`N_i`. Unlike the :ref:`canonical ensemble <canonical_ensemble>`, the number of the respective species (or, equivalently, the concentrations) are allowed to vary in the SGC ensemble. A trial step thus consists of randomly picking an atom and changing its identity with probability .. math:: P = \\min \\Big\\{ 1, \\, \\exp \\big[ - \\big( \\Delta E + \\sum_i \\Delta \\mu_i \\Delta N_i \\big) \\big / k_B T \\big] \\Big\\}, where :math:`\\Delta E` is the change in potential energy caused by the swap. There exists a simple relation between the differences in chemical potential and the canonical free energy :math:`F`. In a binary system, this relationship reads .. math:: \\Delta \\mu = - \\frac{1}{N} \\frac{\\partial F}{\\partial c} ( N, V, T, \\langle c \\rangle). Here :math:`c` denotes concentration (:math:`c=N_i/N`) and :math:`\\langle c \\rangle` the average concentration observed in the simulation. By recording :math:`\\langle c \\rangle` while gradually changing :math:`\\Delta \\mu`, one can thus in principle calculate the difference in canonical free energy between the pure phases (:math:`c=0` or :math:`1`) and any concentration by integrating :math:`\\Delta \\mu` over that concentration range. In practice this requires that the average recorded concentration :math:`\\langle c \\rangle` varies continuously with :math:`\\Delta \\mu`. This is not the case for materials with multiphase regions (such as miscibility gaps), because in such regions :math:`\\Delta \\mu` maps to multiple concentrations. In a Monte Carlo simulation, this is typically manifested by discontinuous jumps in concentration. Such jumps mark the phase boundaries of a multiphase region and can thus be used to construct the phase diagram. To recover the free energy, however, such systems require sampling in other ensembles, such as the :ref:`variance-constrained semi-grand canonical ensemble <sgc_ensemble>`. Parameters ---------- structure : :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] chemical_potentials : Dict[str, float] chemical potential for each species :math:`\\mu_i`; the key denotes the species, the value specifies the chemical potential in units that are consistent with the underlying cluster expansion 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. This 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 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 SemiGrandCanonicalEnsemble # 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 (T=600 K, delta_mu=0.8 eV/atom) structure = prim.repeat(3) calc = ClusterExpansionCalculator(structure, ce) mc = SemiGrandCanonicalEnsemble(structure=structure, calculator=calc, temperature=600, data_container='myrun_sgc.dc', chemical_potentials={'Ag': 0, 'Au': 0.8}) mc.run(100) # carry out 100 trial swaps TODO ---- * add check that chemical symbols in chemical potentials are allowed """ def __init__(self, structure: Atoms, calculator: BaseCalculator, temperature: float, chemical_potentials: Dict[str, float], 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, boltzmann_constant: float = kB, sublattice_probabilities: List[float] = None) -> None: self._ensemble_parameters = dict(temperature=temperature) # add chemical potentials to ensemble parameters self._chemical_potentials = get_chemical_potentials(chemical_potentials) for atnum, chempot in self.chemical_potentials.items(): mu_sym = 'mu_{}'.format(chemical_symbols[atnum]) self._ensemble_parameters[mu_sym] = chempot self._boltzmann_constant = boltzmann_constant super().__init__( structure=structure, 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) if sublattice_probabilities is None: self._flip_sublattice_probabilities = self._get_flip_sublattice_probabilities() else: self._flip_sublattice_probabilities = sublattice_probabilities @property def temperature(self) -> float: """ temperature :math:`T` (see parameters section above) """ return self.ensemble_parameters['temperature'] 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_sgc_flip(self.chemical_potentials, sublattice_index=sublattice_index) @property def chemical_potentials(self) -> Dict[int, float]: """ chemical potentials :math:`\\mu_i` (see parameters section above) """ return self._chemical_potentials def _get_ensemble_data(self) -> Dict: """Returns the data associated with the ensemble. For the SGC ensemble this specifically includes the species counts. """ # generic data data = super()._get_ensemble_data() # species counts data.update(self._get_species_counts()) return data
def get_chemical_potentials(chemical_potentials: Dict[Union[int, str], float]): """ Gets values of chemical potentials.""" if not isinstance(chemical_potentials, dict): raise TypeError('chemical_potentials has the wrong type: {}' .format(type(chemical_potentials))) cps = OrderedDict([(key, val) if isinstance(key, int) else (atomic_numbers[key], val) for key, val in chemical_potentials.items()]) return cps