Nested sampling method
==================================

:Authors: Noam, Rob, Gabor, Livia
:Date: 2015


Overview
-------------------------------------

The nested sampling method was first introduced by John Skilling [#f1]_
in the field of applied probability and inference to efficiently sample
probability densities in high-dimensional spaces where the regions contributing
most of the probability mass are exponentially localized.
It is an annealing algorithm that creates a sequence of probability distributions,
each more concentrated than the previous one near the high-likelihood region of the
parameter space - i.e. in the context of materials, the low-energy region of configuration space.


Since its original inception, nested sampling has also been applied to atomistic systems,
and its several advantages mean it became a powerful method to sample atomic
configuration spaces.[#f2]_
* calculate the partition function, hence all thermodynamic properties become accessible

    - calculate the heat capacity and locate the phase transitions
    - with an order parameter calculate the free energy

* the sampling process itself is independent of temperature, thus the calculation of thermodynamic quantities is a simple post-processing step
* no prior knowledge is needed of the phases or phase transitions
* can be used with both constant volume and constant pressure calculations
* calculate the entire phase diagram in an automated way
* considerable computational gain over parallel tempering

.. [#f1] J. Skilling, in Nested Sampling, edited by Rainer Fischer, Roland Preuss, and Udo von Toussaint, AIP Conf. Proc. No. 735 (AIP, New York, 2004), p. 395
.. [#f2] L.B. Partay, G. Csanyi and N. Bernstein Nested sampling for materials, Eur. Phys. J. B (2021) 94:159

Publications
-------------------------------------
.. include:: ../README.rst
   :start-after: (references are available in bibtex format in the ``NS_publications.bib`` file)
   :end-before: ----

Algorithm
-------------------------------------

Nested sampling is an iterative, "top-down" method, it samples the Potential Energy Landscape (PES)
through a series of nested energy levels, starting from the high energy region (gas phase) and going towards the
global minimum. The sampling is done using a set of walkers - these are randomly generated configurations at the initial step of the sampling,
and the number of these determine the "resolution" of which the PES is sampled.
Then, at every iteration step, the walker with the highest energy (enthalpy in case of constant pressure sampling)
is removed, and a new one is generated to keep the number of walkers constant.
To generate the new configuration, one of the existing walkers is randomly selected and cloned, and this cloned configuration is
modified (through an MCMC, a total-enthalpy Hamiltonian MC or a Galilean MC walk), such that its energy (enthalpy) does not exceed
the that of the last removed walker.
The iteration is continued until a satisfactory low energy configuration is reached.

.. image:: NS_PES_animation.gif

``pymatnest``
-------------------------------------

The ``pymatnest`` package is a software
library written in Fortran 95/python for the purpose of carrying out
nested sampling calculations with a variety of options suitable for different systems.
It can be used with the supplied fortran potential models and it also has interfaces with the following packages:

   - ``LAMMPS``

MC and MD step algorithms
+++++++++++++++++++++++++++++++++++++

.. include:: MC_MD_steps.txt