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

• $n_p=\frac{C\tilde{V}{\overline{\varphi }}_p}{{\alpha }_p}$ - 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

• ${\varphi }_p(rr)={\rho }_p/{\rho }_0$ - spatially dependent volume fraction of p th molecule type

• ${\overline{\varphi }}_p=\frac{n_p{N}_p}{\sum _{i=1}^P{n}_i{N}_i}$ - 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

• $R_g=\sqrt{\frac{b^{2}N}{6}}$, 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

• $\phi$, 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