# 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
```