Model Edwards

Arguments:

• type (string) - Name of the model to use (Edwards)

• Nref (float) - Reference degree of polymerization

• bref (float) - Reference statistical segment length

• initfields (json object) - Describes how to initialize fields within model. See Initialize Model Fields

• C or rho0 (float) - System density $C={\rho }_0{R}_g^3/N_{ref}$. Only one can be specified.

• B or u0 (float) - Excluded volume parameter $B=u0N_{ref}^2/R_g^3$. Only one can be specified.

Example (python):

fts.model(Nref=1.0,bref=1.0,C=4.0,B=2.0,type='Edwards')

Example (json):

"model": {
  "Nref": 1.0,
  "bref": 1.0,
  "C": 4.0,
  "B": 2.0,
  "initfields": {
    "mu": {
      "type": "random",
      "mean": 1.0,
      "stdev": 0.1
    }
  },
  "type": "Edwards"
},


"model": {
  "type": "Edwards",
  "Nref": 1.0,
  "bref": 1.0,
  "nu0": 30,
  "rho0": 10,
  "initfields": {
      "mu": {
      "seed": 0,
      "type": "random"
    }
  }
},

Model Edwards Formalism

References: Delaney2016, Villet2014

The model

$$\beta U(rr)=\beta U_{bond}(rr)+\beta U_{excl}$$

where

$$\beta U_{excl}=\frac{\nu }{2}\int drr\overline{\rho }(rr{)}^2+selfinteraction$$

where $\overline{\rho }$ is the smeared density defined as $\overline{\rho }=\mathrm{\Gamma }\ast \hat{\rho }(rr)=\int drr\mathrm{\Gamma }(rr-r{r}^{\prime })\hat{\rho }(r{r}^{\prime })$, where $\mathrm{\Gamma }(rr)=\frac{1}{(2\pi a^2{)}^{3/2}}\exp (-\frac{|rr|}{2a^2})$ is the smearing function and $\hat{\rho }$ is the microscopic density.

The field theory derived for this model is

$$\mathcal{Z}=Z_0\int \mathcal{D}w\exp (-H[w])$$

where

$$H[w]=\frac{1}{2\nu }\int drr[w(rr){]}^2-\sum _p^P{n}_p\ln Q_p[i\mathrm{\Gamma }\ast w]$$

By introducing the following variables $B=\frac{\nu N^2}{R_{g0}^3}$, $C=\frac{{\rho }_0{R}_{g0}}{N}$, $\tilde{V}=\frac{V}{R_{g0}^3}$, $xx=\frac{rr}{R_{g0}}$, $n_p=\frac{C\tilde{V}{\varphi }_p}{{\alpha }_p}$, and $\mu =iw(rr)N$ (take a look at Duchs2014 if theres variables that were not defined), the field theory becomes

TODO: lots of these rescaling conventions are recycled from model to model. Can I combine into one place?

$$H[\mu ]=\frac{1}{2B}\int dxx[\mu (xx){]}^2-C\tilde{V}\sum _p^P\frac{{\varphi }_p}{{\alpha }_p}\ln Q_p[\mathrm{\Gamma }\ast \mu ]$$

The density operator $\phi$ is (for a continuous chain)

$$\phi (xx;[\mu ])=\sum _{p=1}^P\frac{{\varphi }_p}{Q_p{\alpha }_p}\mathrm{\Gamma }\ast ({\int }_{s}ds{q}_p(x,s)q_p^{†}(x,s))$$

The force is

$$\frac{\delta H[\mu ]}{\delta \mu (xx)}=-\frac{1}{B}\mu (xx)+\phi (xx)$$

For a homogeneous saddle point $\phi (xx)=C$ and the saddle point is located at

$$\mu (xx)=BC$$