Continuous vs Discrete Chains

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

$$\frac{\partial }{\partial s}q(r,s)=\frac{b(s{)}^2}{b^2}{\tilde{\mathrm{\nabla }}}^{2}q(r,s)-{\mu }_p(r;s)q(r,s),$$

where ${\tilde{\mathrm{\nabla }}}^2=R_g^2{\mathrm{\nabla }}^2$, $s\in [0,{\alpha }_p]$, $q(x,0)=1$, $\mu (r,s)$ 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)

$$q(r,s+1)=e^{-{\mu }_K/N}\int d{r}^{\prime }\mathrm{\Phi }(r-r^{\prime })q(r^{\prime },s)$$

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

$$\mathrm{\Phi }(rr)={\left(\frac{3}{2\pi b(s{)}^2}\right)}^{3/2}\exp \left[\frac{-3|rr{|}^2}{2}\right]$$

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

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

For Rg units

$$\hat{\mathrm{\Phi }}(kk)=\exp [\frac{b(s{)}^2}{b^2}(-k^2)\frac{1}{N-1}]$$

Density operator:

$${\phi }_K(rr)=\sum _{p=1}^P(\frac{{\varphi }_p}{Q_p{N}_p}\sum _{s=D_K}^{N_p}{q}_p(rr,s)e^{{\mu }_K}{q}_p^{†}(rr,s))$$

the sum over s includes beads of species K.