.. _simplified_multi_state_steric_mass_action_model:

Simplified Multi-State Steric Mass Action
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The simplified multi-state steric mass action is the same as the multi-state SMA model described above (see Section :ref:`multi_state_steric_mass_action_model`), but with additional assumptions:

- Molecules are only exchanged between two adjacent states, that is, no transfer from state :math:`q_{i,1}` to state :math:`q_{i,3}` is allowed.

- Characteristic charge :math:`\nu_{i,j}` and shielding factor :math:`\sigma_{i,j}` only depend on the index of the state :math:`j`.

Thus, the exchange parameters :math:`k^{(i)}_{j\ell}`, the characteristic charge :math:`\nu_{i,j}`, and the shielding :math:`\sigma_{i,j}` can be parameterized with few degrees of freedom.
For all :math:`i = 1,\dots,N_{\text{comp}} - 1` and :math:`j,\ell = 0,\dots,M_i - 1` let

.. math::

    \begin{aligned}
        k^{(i)}_{j\ell} &= \begin{cases}
        0, & \text{for } \left\lvert j-\ell\right\rvert \neq 1 \\
        K^{(i)}_{ws} + j K^{(i)}_{ws,\text{lin}} - K^{(i)}_{ws,\text{quad}} j(j - M_i+2), & \text{for } \ell = j+1 \\
        K^{(i)}_{sw} + \ell K^{(i)}_{sw,\text{lin}} - K^{(i)}_{sw,\text{quad}} \ell(\ell - M_i+2), & \text{for } \ell = j-1, \end{cases}\\
        \nu_{i,j} &= \nu_{\text{min},i} + \frac{j}{M_i-1} \left( \nu_{\text{max},i} - \nu_{\text{min},i} \right) - \nu_{\text{quad},i} j (j-M_i+1), \\
        \sigma_{i,j} &= \sigma_{\text{min},i} + \frac{j}{M_i-1} \left( \sigma_{\text{max},i} - \sigma_{\text{min},i} \right) - \sigma_{\text{quad},i} j (j-M_i+1).
    \end{aligned}

Note that the characteristic charge :math:`\nu_{i,j}` has to be monotonically non-decreasing in the second index :math:`j` and all other rates and the steric factor :math:`\sigma_{i,j}` have to be non-negative.


For more information on model parameters required to define in CADET file format, see :ref:`simplified_multi_state_steric_mass_action_config`.