Model Edwards AB
===================

.. warning:
  
  This model is in beta. It isn't recommended that you use it unless you know what you're doing.

Arguments:

  * `nu0` (float) - Excluded volume between all species :math:`\nu_0`
  * `delta_nuAB` (float) - Extra excluded volume between A and B species :math:`\Delta \nu_{AB}`
  * `rho0` (float) - System density :math:`\rho_0`

Example (python) ::
  
  TODO

Example (json) ::

  "model": {
    "type": "EdwardsAB",
    "Nref": 1.0,
    "bref": 1.0,
    "nu0": 40, 
    "delta_nuAB": 10, 
    "rho0": 3,
    "initfields": {
      "mu_minus": {
        "center0": [ 0 ],
        "height": 1.0,
        "ngaussian": 1,
        "type": "gaussians",
        "width": 0.1 
      },  
      "mu_plus": {
        "seed": 0,
        "type": "random"
      }   
    }   
  }, 


Model Edwards AB Formalism
---------------------------

References: Park2020, Delaney2016

This is an extension of the Edwards model to include two species (A and B). This is very similar to the non-specific compressible model derived by Delaney2016 and the model derived by Park2020 in the absence of charges.

The model

.. math::
  \beta U(\pmb{r}) = \beta U_{bond}(\pmb{r}) + \beta U_{excl}

where 

.. math ::
  \beta U_{excl} = \frac{\nu_0}{2} \int d\pmb{r} (\bar{\rho}_A(\pmb{r}) + \bar{\rho}_B(\pmb{r}))^2 + \Delta \nu_{AB} \int d\pmb{r} \bar{\rho}_A(\pmb{r}) \bar{\rho}_B(\pmb{r}) + self interaction

where :math:`\bar{\rho}_K` is the smeared density of species :math:`K\in{A,B}` defined as :math:`\bar{\rho}_K = \Gamma_K * \hat{\rho}_K(\pmb{r}) = \int d\pmb{r} \Gamma_K (\pmb{r} - \pmb{r}') \hat{\rho}_K(\pmb{r}')`, where :math:`\Gamma_K(\pmb{r}) = \frac{1}{(2\pi a_K^2)^{3/2}} \exp{\left(-\frac{\lvert\pmb{r}\rvert}{2 a_K^2} \right)}` is the smearing function and :math:`\hat{\rho}_K` is the microscopic density.

To derive a field theory, we introduce the :math:`\bar{\rho}_+ = \bar{\rho}_A + \bar{\rho}_B` and :math:`\bar{\rho}_- = \bar{\rho}_A - \bar{\rho}_B`. Following two Hubbard-Stratonovich transforms, the field theory derived for this model is 

.. math:: 
  \mathcal{Z} = Z_0 \int \mathcal{D} w_+ \int \mathcal{D} w_- \exp{\left(-H[w_+,w_-]\right)}

where 

.. math::
  H[w_+, w_-] = \frac{1}{2 \nu_0 + \Delta \nu_{AB}} \int d \pmb{r}  [w_+(\pmb{r})]^2 + \frac{1}{\Delta \nu_{AB}} \int d \pmb{r}  [w_-(\pmb{r})]^2 - \sum_p^P n_p \ln Q_p[\Gamma_A * w_A, \Gamma_B * w_B]

with :math:`w_A = i w_+ - w_-` and :math:`w_B = i w_+ + w_-`.
By introducing the following variables :math:`B_0=\frac{\nu_0 N^2}{R_{g0}^3}`, :math:`B_{AB}=\frac{\Delta \nu_{AB} N^2}{R_{g0}^3}`, :math:`C=\frac{\rho_0 R_{g0}}{N}`, :math:`\tilde{V} = \frac{V}{R_{g0}^3}`, :math:`\pmb{x} = \frac{\pmb{r}}{R_{g0}}`, :math:`n_p = \frac{C \tilde{V} \phi_p}{\alpha_p}`, :math:`\mu_+ = i w_+(\pmb{r}) N`, :math:`\mu_- = w_-(\pmb{r}) N` (take a look at Duchs2014 if theres variables that were not defined), the field theory becomes

TODO: lots of these rescaling conventions are recycled from model to model. Can I combine into one place?

.. math::
  H[\mu_+, \mu_-] = - \frac{1}{2 B_0 + B_{AB}} \int d \pmb{x}  [\mu_+(\pmb{x})]^2 + \frac{1}{B_{AB}} \int d \pmb{x}  [\mu_-(\pmb{x})]^2  - C \tilde{V} \sum_p^P \frac{\phi_p}{\alpha_p} \ln Q_p[\Gamma_A * \mu_A, \Gamma_B * \mu_B]

The density operator of species K  :math:`\varphi_K` is (for a continuous chain)

.. math::
  \varphi_K (\pmb{x} ; [\mu]) = \sum_{p=1}^P \frac {\phi_p}{Q_p \alpha_p} \Gamma_K * \left( \int_{s \in K} ds q_p(x,s) q^\dagger_p(x,s) \right)

The forces are 

.. math::
  \frac{\delta H[\mu_+,\mu_-]}{\delta \mu_+(\pmb{x})} &= - \frac{2}{2B_0 + B_{AB}} \mu_+(\pmb{x}) + C \left(\varphi_A(\pmb{x}) + \varphi_B(\pmb{x})\right)\\
  \frac{\delta H[\mu_+,\mu_-]}{\delta \mu_-(\pmb{x})} &= \frac{2}{B_{AB}} \mu_-(\pmb{x}) + C \left(\varphi_B(\pmb{x}) - \varphi_A(\pmb{x})\right)