Continuous vs Discrete Chains

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

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

where \(\tilde{\nabla}^2 = R_g^2 \nabla^2\), \(s \in [0,\alpha_p]\), \(q(x,0) = 1\), \(\mu(r,s)\) 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 \(R_g\) or \(R_e\) will be slightly different depending on whether discrete or continuous chains are used. For continuous chains \(R_g^2 = \frac{b^2 N}{6}\) and \(R_e^2 = b^2 N\). For discrete chains \(R_g^2 = \frac{b^2 (N-1)}{6}\) and \(R_e^2 = b^2 (N-1)\).

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)

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

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

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

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

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

For Rg units

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

Density operator:

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

the sum over s includes beads of species K.