RingPolymerArrays.jl provides the RingPolymerArray{T}, an AbstractArray{T,3}, useful for representing ring polymer systems using the imaginary-time path integral formalism.
A RingPolymerArray can be used to store positions or momenta for ring polymer systems, where the three dimensions represent each degree of freedom per atom, each atom and each ring polymer bead.
For example, this RingPolymerArray represents a 3D system with 2 atoms, and 4 ring polymer beads:
julia> RingPolymerArray(reshape(1:24, 3, 2, 4))
3×2×4 RingPolymerArray{Int64}:
[:, :, 1] =
1 4
2 5
3 6
[:, :, 2] =
7 10
8 11
9 12
[:, :, 3] =
13 16
14 17
15 18
[:, :, 4] =
19 22
20 23
21 24In a system with disparate particle masses, it is desirable to treat heavier atoms classically, without any replicas.
This is made possible by the classical keyword argument in the constructor that allows us to specify which atoms should be treated classically.
Here, the first atom will have a fixed value for all beads:
julia> a = RingPolymerArray{Float64}(reshape(1:24, 3, 2, 4); classical=[1])
3×2×4 RingPolymerArray{Float64}:
[:, :, 1] =
1.0 4.0
2.0 5.0
3.0 6.0
[:, :, 2] =
1.0 10.0
2.0 11.0
3.0 12.0
[:, :, 3] =
1.0 16.0
2.0 17.0
3.0 18.0
[:, :, 4] =
1.0 22.0
2.0 23.0
3.0 24.0These values will remain fixed, even after attempted modification:
julia> a .+= rand(3, 2, 4)
3×2×4 RingPolymerArray{Float64}:
[:, :, 1] =
1.78368 4.85881
2.90528 5.81841
3.57008 6.31728
[:, :, 2] =
1.78368 10.6239
2.90528 11.9604
3.57008 12.4336
[:, :, 3] =
1.78368 16.1968
2.90528 17.8464
3.57008 18.8612
[:, :, 4] =
1.78368 22.7994
2.90528 23.1735
3.57008 24.7504This works by internally storing only a single copy of the classical positions and allowing setindex! to operate only on the first bead for atoms labelled as classical.
Alongside the RingPolymerArray we also provide the NormalModeTransformation{T} which can be used to convert any AbstractArray{T,3} of ring polymer coordinates to and from normal mode coordinates:
julia> transform = NormalModeTransformation{Float64}(4);
julia> transform_to_normal_modes!(fill(1.0, (2,3,4)), transform)
2×3×4 Array{Float64, 3}:
[:, :, 1] =
2.0 2.0 2.0
2.0 2.0 2.0
[:, :, 2] =
-8.65956e-17 -8.65956e-17 -8.65956e-17
-8.65956e-17 -8.65956e-17 -8.65956e-17
[:, :, 3] =
0.0 0.0 0.0
0.0 0.0 0.0
[:, :, 4] =
2.59787e-16 2.59787e-16 2.59787e-16
2.59787e-16 2.59787e-16 2.59787e-16
julia> transform_from_normal_modes!(fill(1.0, (2,3,4)), transform)
2×3×4 Array{Float64, 3}:
[:, :, 1] =
1.70711 1.70711 1.70711
1.70711 1.70711 1.70711
[:, :, 2] =
-0.707107 -0.707107 -0.707107
-0.707107 -0.707107 -0.707107
[:, :, 3] =
0.292893 0.292893 0.292893
0.292893 0.292893 0.292893
[:, :, 4] =
0.707107 0.707107 0.707107
0.707107 0.707107 0.707107When using a RingPolymerArray, the normal mode transformation is applied to both classical
and quantum atoms, where the classical atoms will have the value of the scaled centroid.
In normal mode coordinates, the centroid is given by the first bead coordinate divided by sqrt(nbeads).
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