5. FTS Glossary¶
There is quite a bit of math involved in performing field theoretic simulations. If you don’t understand what a symbol means elsewhere in the documentation, hopefully it will be defined here.
\(V\) - cell volume
\(d\) - dimentionality of system
\(P\) - number of different molecule types
\(n = \sum_{p=1}^P n_p\) - total number of molecules
\(n_p = \frac{C \tilde{V} \bar{\phi}_p}{\alpha_p}\) - number of copies of p th molecule type
\(N_p\) - degree of polymerization of p th molecule type
\(N\) - reference degree of polymerization
\(\tilde{V} = V/R_g^d\) - scaled cell volume .
\(\alpha_p = N_p / N\) - scaled length of p th molecule type
\(\phi_p (\pmb{r}) = \rho_p / \rho_0\) - spatially dependent volume fraction of p th molecule type
\(\bar{\phi}_p = \frac{n_p N_p}{\sum_{i=1}^P n_i N_i}\) - overall volume fraction of p th molecule type
\(C = \frac{\rho_0 R_g^d}{N}\) - dimentionless chain concentration, where d is dimensionality of system
\(\rho_0 = \sum_{p=1}^P \frac{n_p N_p}{V}\) - overall monomer number density
\(v_0 = 1/\rho_0\) - reference volume, average volume per monomer
\(w_i\) - species or exchange field. Typically if i is a letter than \(w_i\) is a species field, while if i is a number or \(\pm\) then \(w_i\) is an exchange field.
\(\mu_i = \gamma_i w_i N\) - rescaled (and possibly Wick rotated) exchange field
\(\gamma_i = {1,\sqrt{-1}}\) - depending on whether the field is Wick rotated
\(R_g = \sqrt{\frac{b^2N}{6}}\), the radius of gyration of reference polymer chain consisting of N+1 beads
\(B =\beta u_0 N^2 / R_g^d\), the dimensionless excluded volume, where d is dimensionality of system.
\(E = 4 \pi l_B \sigma^2 N^2 / R_g\), the dimensionless electrostatic interaction strength
\(\varphi\), fluctuating electrostatic potential field
\(a_i\), smearing length of species i
\(\tilde{a}_i\), smearing length of species i in units of \(R_g\). \(\tilde{a}_i = a_i / R_g\)
\(z_i\), charge on species i
\(l_B\), the Bjerrum length