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https://github.com/nsmith5/modela.jl


https://github.com/nsmith5/modela.jl

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   Model A Simulator

  


  


   Introduction

  


  


   

  


  


   Welcome to the Model A simulator! In the narrowest sense this is a place to see a particular stochastic partial differential equation solved that is of the following form:

  


  


   $$ \frac{\partial \phi (x,t)}{\partial t} = -M \left( W_o^2 \nabla^2 \phi - a_2 \phi - a_4 \phi^3 \right) + \zeta (x,t)   $$

  


  


   Where the driving force $\zeta(x,t)$ is a gaussian random variable with the following statistics:

  


  


   $$ \left< \zeta(x,t) \right> = 0, \,\,\,\,\,\,

   


   \left< \zeta(x_1, t_1) \zeta(x_2, t_2) \right> = 2 M k_b T \delta(x_2 - x_1) \delta(t_2 - t_1). $$

  


  


   The importance of Model A or

   

    time-dependent Ginzburg-Landau equation

   

   is that it is model of the dynamics of a non-conserved order parameter.

  


  


   Discrete models

  


  


   To understand what is meant by this, consider for a moment the Ising model.

  


  


   $$ \beta H = \sum_{\left< i, j \right>} J_{ij} \sigma_i \sigma_j + h \sum_i \sigma_i$$

  


  


   The variables $\sigma$ are the spin angular momenta of individual atoms in a discrete lattice. At this point if you'd like to simulate the dynamics of a lattice you can impose one of two different types of kinetics: Glauber or Kawasaki.

  


  


   In Glauber kinetics the total angular moment is not conserved. The free energy cost of a spin flip is calculated at any site and the probability of flipping is weighted accordingly.

  


  


   Alternatively, in Kawasaki dynamics, the total angular momentum is conserved. Specifically the angular momentum is conserved locally by implimenting a set of kinetics that local rearranges a small domain of spins instead of flipping them. Again, the cost of all rearrangements is considered and probabilities of those rearrangements are weighted accordingly.

  


  


   Continuum models

  


  


   In Ginzburg-Landau theory, continuous fields are considered at the

   thermodynamic variables of interest. These field are almost always coarse

   grained variables from discrete systems. If we take the Ising model again,

   for example, we can construct a coase grained field from our discrete spins

   through an averaging process. Consider of ball of radius $R$ centered at

   position $x$. We can define a field $S(x)$ such that it is the average of

   all spins, $\sigma$, inside the ball. As radius of the ball increases the

   field becomes continuous and we've constructed a coarsed-grained order

   parameter.  


  


   Ginzburg-Landau theory considers the thermodynamics of such fields and once

   again, if we'd like to simulate the dynamics of such fields we are left with

   a choice between two different types of kinetics: conserved and

   non-conserved. The continuum version of our discrete Glauber kinetics is

   called model A. As we stated before it simulated the dynamics of a

   non-conserved order parameter.  


  


   For a general order parameter $\phi$ the equation of motion is written as:

  


  


   $$ \frac{\partial \phi(x,t)}{\partial t}  = - M \frac{\delta H[\phi]}{\delta

   \phi(x,t)} + \zeta (x,t) $$ 


  


   Where the source term $\zeta$ follows the above mentioned statistics and

   $H[\phi]$ is the Ginzburg free energy functional of the system. In our case

   we'll use a simple $\phi^4$ model for the free energy functional: 


  


   $$ H_{\phi^4}[\phi] =  \int d^d x \lbrace  (W_o \nabla \phi)^2 + a_2

   (\phi)^2 + a_4 (\phi)^4  \rbrace $$ 


  


   Simulating

  


  


   All of the code is written in

   

    Julia

   

   (0.3 release should work) but also requires

   

    Python

   

   and a python package called

   

    Matplotlib

   

   .

  


  
    

       
  • 

        Julia info: http://julialang.org

       
    • 

       
  • 

        Matplotlib: http://matplotlib.org

       
    • 

      


  


   To enable Matplotlib calls from Julia just install the PyPlot package inside the Julia shell via:

  


  
julia> Pkg.add("PyPlots")



  


   Afterwards the simulation can be from the command line from the installation folder

  


  
> julia script.jl



  


   Parameters can be changed inside the script.jl where the relevent parameters are commented.

  


  


   Contact Info

  


  


   This model code repository and set of notes was made by me to learn and help teach topics in thermodynamics. The notes and code are free to use and distribute so long as they are properly attributed to me (eg. don't pretend you made these please).

  


  
    

       
  • 

        Nathan Smith

       
    • 

       
  • 

        nathan.smith5@mail.mcgill.ca

       
    • 

       
  • 

        http://www.nate-smith.net

       
    • 

      


  

  

  

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