===================
Potts model context
===================

* Digitize space (lattice)

  * Lattice sites represent discrete bits of material
  * Each color represents membership in a phase/grain

    .. image:: images/potts_grains.jpg
       :scale: 75%

* Potts algorithm seeks to minimize the total energy

* Total grain boundary energy: :math:`E`

  .. math::
  
       E=\sum_i^N \; \sum_j^nJ(1-\delta_{ij}) \\

  :math:`N` -- number of sites; :math:`n` -- number of neighboring sites; :math:`J` -- bond energy


* Local energy consideration: :math:`\Delta E=-2,\;\; r=1.0` (see below)

   .. figure:: images/potts_grain_growth.jpg
      :scale: 75%
      :alt: Missing figure demonstrates :math:`\Delta E=-2`

* Rates :math:`r` computed to mimic atomic processes -- Boltzmann distribution

  .. math::
        
        r=\begin{cases}
          1                        & \Delta E\leq 0\\
          e^{-\frac{\Delta E}{kT}} & \Delta E > 0
        \end{cases}

  * Events that decrease energy have higher rates

  .. image:: images/boltzmann.jpg
     :width: 75%