Phase Boundaries
================

.. module:: microscopic_gating.phase_boundaries

This module implements models for analyzing phase boundaries and capture islands
in the parameter space of the gating system.

Dome Phase Boundaries
---------------------

.. autoclass:: DomePhaseBoundaries
   :members:
   :member-order: bysource
   :show-inheritance:

Capture Island Window
---------------------

.. autoclass:: CaptureIslandWindow
   :members:
   :member-order: bysource
   :show-inheritance:

Theory Background
-----------------

Dome-Shaped Phase Boundaries
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The gating function :math:`G(\phi) = \theta(1-\theta)` produces a dome-shaped
dependence on concentration. Phase boundaries occur where:

.. math::

   G(\phi) = G_{\text{threshold}}

This yields two solutions (low and high concentration boundaries) that define
the "dome" region where gating is effective.

Capture Islands
~~~~~~~~~~~~~~~

Capture islands are regions in parameter space where the system exhibits
specific transport behavior. The window of concentrations supporting capture
depends on:

- Binding affinity :math:`K`
- Site pair counts :math:`N_{\text{pair}}`
- Geometric factors :math:`\chi`
- Chemical factors :math:`\kappa_B`