Model Edwards
===================

Arguments:
  * `type` (string) - Name of the model to use (`Edwards`)
  * `Nref` (float) - Reference degree of polymerization 
  * `bref` (float) - Reference statistical segment length
  * `initfields` (json object) - Describes how to initialize fields within model. See :ref:`label_init_model_fields`
 
  * `C` or `rho0` (float) - System density :math:`C = \rho_0 R_g^3 / N_{ref}`. Only one can be specified.
  * `B` or `u0` (float) - Excluded volume parameter :math:`B = u0 N_{ref}^2 / R_g^3`. Only one can be specified.

Example (python): ::
  
  fts.model(Nref=1.0,bref=1.0,C=4.0,B=2.0,type='Edwards')

Example (json): ::

  "model": {
    "Nref": 1.0,
    "bref": 1.0,
    "C": 4.0,
    "B": 2.0,
    "initfields": {
      "mu": {
        "type": "random",
        "mean": 1.0,
        "stdev": 0.1
      }
    },
    "type": "Edwards"
  },


  "model": {
    "type": "Edwards",
    "Nref": 1.0,
    "bref": 1.0,
    "nu0": 30, 
    "rho0": 10, 
    "initfields": {
        "mu": {
        "seed": 0,
        "type": "random"
      }   
    }   
  },


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

References: Delaney2016, Villet2014

The model

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

where 

.. math ::
  \beta U_{excl} = \frac{\nu}{2} \int d\pmb{r} \bar{\rho}(\pmb{r})^2 + self interaction

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

The field theory derived for this model is 

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

where 

.. math::
  H[w] = \frac{1}{2 \nu} \int d \pmb{r}  [w(\pmb{r})]^2 - \sum_p^P n_p \ln Q_p[i \Gamma * w]

By introducing the following variables :math:`B=\frac{\nu 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}`, and :math:`\mu = i 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] = \frac{1}{2 B} \int d \pmb{x}  [\mu(\pmb{x})]^2 - C \tilde{V} \sum_p^P \frac{\phi_p}{\alpha_p} \ln Q_p[\Gamma * \mu]

The density operator :math:`\varphi` is (for a continuous chain)

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

The force is 

.. math::
  \frac{\delta H[\mu]}{\delta \mu(\pmb{x})} = - \frac{1}{B} \mu(\pmb{x}) + \varphi(\pmb{x})

For a homogeneous saddle point :math:`\varphi(\pmb{x}) = C` and the saddle point is located at

.. math::
  \mu(\pmb{x}) = BC