# Source code for mchammer.ensembles.canonical_ensemble

```
"""Definition of the canonical ensemble class."""
from ase import Atoms
from ase.units import kB
from typing import List
from .. import DataContainer
from ..calculators.base_calculator import BaseCalculator
from .thermodynamic_base_ensemble import ThermodynamicBaseEnsemble
[docs]class CanonicalEnsemble(ThermodynamicBaseEnsemble):
"""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
----------
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]
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]
random_seed : int
seed for the random number generator used in the Monte Carlo
simulation
dc_filename : 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
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: 600 s]
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
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 swap.
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 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])
>>> # prepare initial configuration
>>> structure = prim.repeat(3)
>>> for k in range(5):
>>> structure[k].symbol = 'Ag'
>>> # set up and run MC simulation
>>> calc = ClusterExpansionCalculator(structure, ce)
>>> mc = CanonicalEnsemble(structure=structure, calculator=calc,
... temperature=600,
... dc_filename='myrun_canonical.dc')
>>> mc.run(100) # carry out 100 trial swaps
"""
def __init__(self,
structure: Atoms,
calculator: BaseCalculator,
temperature: float,
user_tag: str = None,
boltzmann_constant: float = kB,
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)
# 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 = structure.get_chemical_symbols().count(symbol)
self._ensemble_parameters[key] = count
super().__init__(
structure=structure,
calculator=calculator,
user_tag=user_tag,
random_seed=random_seed,
data_container=data_container,
dc_filename=dc_filename,
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)
if sublattice_probabilities is None:
self._swap_sublattice_probabilities = self._get_swap_sublattice_probabilities()
else:
self._swap_sublattice_probabilities = sublattice_probabilities
@property
def temperature(self) -> float:
""" Current temperature """
return self._ensemble_parameters['temperature']
def _do_trial_step(self):
""" Carries out one Monte Carlo trial step. """
sublattice_index = self.get_random_sublattice_index(self._swap_sublattice_probabilities)
return self.do_canonical_swap(sublattice_index=sublattice_index)
```