Polymers.jl

A Julia implementation of the pivot algorithm: a Markov chain Monte carlo (MCMC) sampler for the self-avoiding walk (SAW) model of a linear polymer chain.

Interactive examples

Click on any of the (static) images below to view interactive examples (generated with Plotly).

2D SAW with 100 000 steps	Pivot algorithm in 2D for a walk with 10 000 steps
3D SAW with 100 000 steps	Pivot algorithm in 3D for a walk with 10 000 steps

Usage

Setup

Requirements

Julia 1.1-1.3 is required.

Specifically, this project was developed using Julia 1.1.0 and appears to work up to Julia 1.3. Later versions either don't produce walks correctly or don't plot them correctly.

Quick start

Start by activating the project environment in the Julia REPL:

julia> ]
(v1.1) pkg> activate .
(Polymers) pkg> instantiate

Initialize a 3-dimensional polymer as a straight line with 1000 steps:

using Polymers

poly3d = Polymer(1000, 3)

Initialize a 2-dimensional polymer and run the pivot algorithm for 10000 iterations:

poly = Polymer(1000)
mix!(poly, 10000)

Plot a polymer:

poly_plot(poly)

Make an interactive plot:

poly_plot(poly, true)

Produce an animation of the pivot algorithm run for 100 successful steps and save it to "pivot_anim.gif":

anim(100, Polymer(100), "pivot_anim.gif")

Other initializers

The stairs function initializes a polymer in the shape of a staircase.

The bridge function initializes a polymeer in the shape of a self-avoiding bridge, which is Markovian and does not take long to generate. This initialization appears to greatly improve mixing, as apparent from the following examples, both of which have 100000 steps and are the result of running the pivot algorithm for 10000 iterations.

2D SAW initialized from a line	2D SAW initialized from a bridge

To do

• It should be possible to parallelize the pivot operation since every point on the tail (pivoted) segment of a walk is pivoted independently. This should lead to a roughly linear (in the number of available CPU cores) speedup. However, note this requires some Julia implementation of (parallel) map-reduce (see Folds.jl).
• Hashing lattice sites in order to update the reverse index associated with a Polymer struct currently appears to be a bottleneck.
• Interactive animations generated with Plotly.