Continuous vs Discrete Chains
=================================

In :ref:`Model Melt Chi AB`, we discussed how to derive:

.. math::
  \frac{\partial}{\partial s} q(r,s) = \frac{b(s)^2}{b^2} \tilde{\nabla}^2 q(r,s) - \mu_p (r; s) q(r,s),

where :math:`\tilde{\nabla}^2 = R_g^2 \nabla^2`, :math:`s \in [0,\alpha_p]`, :math:`q(x,0) = 1`, :math:`\mu(r,s)` is the field corresponding to the species at contour position `s` along molecule `p`.

Discuss operator splitting and how this leads to a form that is similar to the discrete chains

Note that Nref means different things for continuous vs discrete chains. For continuous chains, it is the contour length (with Nref + 1 contour steps). For discrete chains, `N` is the number of beads. This means that the reference lengths :math:`R_g` or :math:`R_e` will be slightly different depending on whether discrete or continuous chains are used. For continuous chains :math:`R_g^2 = \frac{b^2 N}{6}` and :math:`R_e^2 = b^2 N`. For discrete chains :math:`R_g^2 = \frac{b^2 (N-1)}{6}` and :math:`R_e^2 = b^2 (N-1)`. 

The different meanings of Nref for different chain types is necessary so that when a user specifies Nref = 100 and nbeads = 100, the lengths in the simulation don't need to be correted by a factor of N/(N-1) (TODO: is this correct?).

Discrete chains
-----------------

References: GHF, Koski2013, Matsen2012 (Macromol 45:8502)

.. math::
  q(r,s+1) = e^{-\mu_K / N} \int d r' \Phi (r-r') q(r',s) 

where :math:`K` is the identity of the :math:`s+1` bead. :math:`\Phi(r)` is the bond probability which for a discrete Gaussian chain is:

.. math::
  \Phi(\pmb{r}) = \left(\frac{3}{2\pi b(s)^2} \right)^{3/2} \exp \left[ \frac{-3 \vert \pmb{r} \vert^2 }{2} \right]

The convolution can be solved using FFTs which requires the fourier transform of :math:`\Phi(\pmb{r})`,

.. math::
  \hat{\Phi}(\pmb{k}) = \exp \left[ \frac{b(s)^2 (-k^2)}{6} \right]

For Rg units

.. math::
  \hat{\Phi}(\pmb{k}) = \exp \left[ \frac{b(s)^2}{b^2} (-k^2) \frac{1}{N-1} \right]

Density operator:

.. math::
  \varphi_K (\pmb{r}) =  \sum_{p=1}^P \left( \frac{\phi_p}{Q_p N_p}  \sum_{s = D_K}^{N_p} q_p(\pmb{r},s) e^{\mu_K} q_p^\dagger (\pmb{r},s) \right)

the sum over `s` includes beads of species `K`.