Transport Models
================

.. module:: microscopic_gating.transport

This module implements transport models that connect bridge dynamics to
effective diffusion coefficients.

Jump Diffusion
--------------

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

Poisson Escape Averaging
------------------------

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

Concentration Dependent Diffusion
---------------------------------

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

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

Jump Diffusion Mapping
~~~~~~~~~~~~~~~~~~~~~~

The residence time and effective diffusion are related to the escape rate:

.. math::

   \tau_{\text{res}} = \frac{1}{k_{\text{esc}}}

   D_{\text{eff}}(n_b) = \frac{\ell^2}{2d} k_{\text{esc}}(n_b)

where :math:`\ell` is the jump length and :math:`d` is the spatial dimension.

Poisson Averaging
~~~~~~~~~~~~~~~~~

When the escape rate depends exponentially on bridge count:

.. math::

   k_{\text{esc}}(n_b) = \tilde{k}_0 e^{-\alpha n_b}

The Poisson-averaged rate has a closed form:

.. math::

   k_{\text{esc}}(\phi) = \tilde{k}_0 \exp[-\lambda(1-e^{-\alpha})]

Concentration-Dependent Diffusion
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The full expression for effective diffusion:

.. math::

   D_{\text{eff}}(\phi) = D_{\text{free}} \exp[-\lambda(\phi)(1-e^{-\beta\epsilon_{\text{eff}}})]