Continuous vs Discrete Chains

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

sq(r,s)=b(s)2b2~2q(r,s)μp(r;s)q(r,s),

where ~2=Rg22, s[0,αp], q(x,0)=1, μ(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 Rg or Re will be slightly different depending on whether discrete or continuous chains are used. For continuous chains Rg2=b2N6 and Re2=b2N. For discrete chains Rg2=b2(N1)6 and Re2=b2(N1).

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)=eμK/NdrΦ(rr)q(r,s)

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

Φ(rr)=(32πb(s)2)3/2exp[3|rr|22]

The convolution can be solved using FFTs which requires the fourier transform of Φ(rr),

Φ^(kk)=exp[b(s)2(k2)6]

For Rg units

Φ^(kk)=exp[b(s)2b2(k2)1N1]

Density operator:

φK(rr)=p=1P(ϕpQpNps=DKNpqp(rr,s)eμKqp(rr,s))

the sum over s includes beads of species K.