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pimc.jl
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pimc.jl
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### A Pluto.jl notebook ###
# v0.19.40
using Markdown
using InteractiveUtils
# ╔═╡ 871e420a-94c3-4937-9df4-f1eef0ec936e
using LinearAlgebra, Random, Plots
# ╔═╡ 643c7ef8-e0d3-4c68-aa45-25a45f9604d5
# Define the KineticAction as an abstract type to allow various implementations
abstract type KineticAction end
# ╔═╡ fdf9fb5a-bd20-4390-8769-ba4a146505e4
# Primitive kinetic action
struct PrimitiveKineticAction <: KineticAction
lambda::Float64
end
# ╔═╡ f90b0f21-be82-4746-b304-91a96f2c81a0
function kinetic_action(action::PrimitiveKineticAction, R1, R2, tau)
return sum(norm(R1[i] - R2[i])^2 for i in 1:length(R1)) / (4 * action.lambda * tau)
end
# ╔═╡ 6a61102d-86c5-480a-ae40-d25f1f0a72cb
# Define the PotentialAction as an abstract type to allow various implementations
abstract type PotentialAction end
# ╔═╡ 77710cd1-0022-4d11-932c-7aca1dede122
# One-body potential action
struct OneBodyPotentialAction <: PotentialAction
V::Function
end
# ╔═╡ 8ae24fc3-fc8f-4d45-8ce8-cf651d3d72dc
# Two-body potential action
struct TwoBodyPotentialAction <: PotentialAction
V::Function
end
# ╔═╡ 30aad091-6425-4e05-a8a7-a708aad8d283
# Combine one-body and two-body potentials
struct CombinedPotentialAction <: PotentialAction
one_body::OneBodyPotentialAction
two_body::TwoBodyPotentialAction
end
# ╔═╡ 8340a1e4-54e2-4deb-bf0b-65cd33c3c336
begin
function potential_action(action::OneBodyPotentialAction, R)
return sum(action.V(r) for r in R)
end
function potential_action(action::TwoBodyPotentialAction, R)
return sum(action.V(R[i], R[j]) for i in 1:length(R) for j in i+1:length(R))
end
function potential_action(action::CombinedPotentialAction, R)
return potential_action(action.one_body, R) + potential_action(action.two_body, R)
end
end
# ╔═╡ d913c869-5ff7-454c-91b4-96e784a74439
# Path Integral Monte Carlo simulation parameters
mutable struct PIMCParams
n_particles::Int
n_beads::Int
n_dimensions::Int
tau::Float64
beta::Float64
n_steps::Int
step_size::Float64
kinetic_action::KineticAction
potential_action::PotentialAction
boxlength::Float64
periodic::Bool
bosonic::Bool
fermionic::Bool
function PIMCParams(; n_particles, n_beads, n_dimensions, tau, beta, n_steps, step_size, kinetic_action, potential_action, boxlength, periodic, bosonic, fermionic)
new(n_particles, n_beads, n_dimensions, tau, beta, n_steps, step_size, kinetic_action, potential_action, boxlength, periodic, bosonic, fermionic)
end
end
# ╔═╡ f68760e5-78b2-437f-bbc2-ae3ff9e17d1f
# Initialize particle positions
function initialize_positions(params::PIMCParams)
R = [rand(params.n_dimensions) for _ in 1:params.n_particles]
return [deepcopy(R) for _ in 1:params.n_beads]
end
# ╔═╡ 2d631856-dbe6-44e7-b850-c46cddae3dc7
# Apply periodic boundary conditions
function apply_pbc!(R, L)
for r in R
for i in 1:length(r)
r[i] -= L * floor(r[i] / L + 0.5)
end
end
end
# ╔═╡ 712d4d95-f946-4b2b-96e6-335637563d77
# Single-particle move
function single_particle_move!(path, params, b, accept_count)
i = rand(1:params.n_particles)
R_new = deepcopy(path[b])
R_new[i] .+= params.step_size * (rand(params.n_dimensions) .- 0.5)
if params.periodic
apply_pbc!(R_new, params.boxlength)
end
kinetic_old = kinetic_action(params.kinetic_action, path[mod(b-2, params.n_beads)+1], path[b], params.tau) +
kinetic_action(params.kinetic_action, path[b], path[mod(b, params.n_beads)+1], params.tau)
kinetic_new = kinetic_action(params.kinetic_action, path[mod(b-2, params.n_beads)+1], R_new, params.tau) +
kinetic_action(params.kinetic_action, R_new, path[mod(b, params.n_beads)+1], params.tau)
potential_old = potential_action(params.potential_action, path[b])
potential_new = potential_action(params.potential_action, R_new)
delta_action = kinetic_new - kinetic_old + params.tau * (potential_new - potential_old)
if rand() < exp(-delta_action)
path[b] = R_new
accept_count[] += 1
end
end
# ╔═╡ 4666e137-156e-4171-8c09-1cbb0ba69662
# Bisect move
function bisect_move!(path, params, b, accept_count)
bead1 = rand(1:params.n_beads)
bead2 = mod(bead1 + 2^rand(0:3), params.n_beads) + 1
mid_bead = div(bead1 + bead2, 2)
for k in bead1:bead2
for i in 1:params.n_particles
midpoint = 0.5 * (path[bead1][i] + path[bead2][i]) + params.step_size * (rand(params.n_dimensions) .- 0.5)
if params.periodic
midpoint = midpoint .- 1.0 .* floor.(midpoint ./ 1.0 .+ 0.5)
end
path[k][i] = midpoint
end
end
accept_count[] += 1 # Assume acceptance for simplicity
end
# ╔═╡ 686c063a-e1e1-4835-9c38-1bc17cc04cbd
# Multi-level, multi-bead move
function multi_level_move!(path, params, b, accept_count)
level = rand(1:3)
n_beads_move = 2^level
bead_start = rand(1:params.n_beads)
bead_end = mod(bead_start + n_beads_move, params.n_beads) + 1
for k in bead_start:bead_end
mid_bead = div(bead_start + bead_end, 2)
for i in 1:params.n_particles
midpoint = 0.5 * (path[bead_start][i] + path[bead_end][i]) + params.step_size * (rand(params.n_dimensions) .- 0.5)
if params.periodic
midpoint = midpoint .- 1.0 .* floor.(midpoint ./ 1.0 .+ 0.5)
end
path[k][i] = midpoint
end
end
accept_count[] += 1 # Assume acceptance for simplicity
end
# ╔═╡ bf84d32b-852a-4b06-a7bb-dbf482032a4a
# Estimators
function kinetic_energy_estimator(path, params)
n_beads = params.n_beads
tau = params.tau
lambda = (params.kinetic_action::PrimitiveKineticAction).lambda
kinetic_energy = 0.0
for b in 1:n_beads
kinetic_energy += sum(norm(path[b][i] - path[mod(b, n_beads) + 1][i])^2 for i in 1:params.n_particles)
end
return params.n_particles / (2.0 * lambda * tau) - kinetic_energy / (4.0 * lambda * tau^2 * n_beads)
end
# ╔═╡ 5db6c4e2-d111-446f-97f5-6093e212d2b5
function potential_energy_estimator(path, params)
n_beads = params.n_beads
potential_energy = 0.0
for b in 1:n_beads
potential_energy += potential_action(params.potential_action, path[b])
end
return potential_energy / n_beads
end
# ╔═╡ 4b9d51ef-6225-44f6-9dcf-6fa7d7aadb74
function virial_energy_estimator(path, params)
n_beads = params.n_beads
tau = params.tau
lambda = (params.kinetic_action::PrimitiveKineticAction).lambda
kinetic_energy = 0.0
potential_energy = 0.0
for b in 1:n_beads
kinetic_energy += sum(norm(path[b][i] - path[mod(b, n_beads) + 1][i])^2 for i in 1:params.n_particles)
potential_energy += potential_action(params.potential_action, path[b])
end
kinetic_energy = params.n_particles / (2.0 * lambda * tau) - kinetic_energy / (4.0 * lambda * tau^2 * n_beads)
potential_energy /= n_beads
return kinetic_energy + potential_energy
end
# ╔═╡ 6409a397-3dea-497c-9c24-df228a431fc0
# Visualization function to create a GIF of the path changes
function visualize_paths(all_paths, params, filename)
L = params.boxlength # Box length for periodic boundary conditions
n_particles = params.n_particles
n_beads = params.n_beads
n_dimensions = params.n_dimensions
if n_dimensions != 3
error("Visualization function currently supports only 3D paths.")
end
anim = @animate for step in 1:length(all_paths)
p = plot()
lower = minimum(minimum(all_paths[step]))
upper = maximum(maximum(all_paths[step]))
for i in 1:n_particles
x = [all_paths[step][bead][i][1] for bead in 1:n_beads]
y = [all_paths[step][bead][i][2] for bead in 1:n_beads]
z = [all_paths[step][bead][i][3] for bead in 1:n_beads]
plot!(x, y, z, marker=:circle, label="", lw=2, ms=4)
end
plot!(aspect_ratio=:equal, camera=(30, 30))
title!("Step $step")
end
gif(anim, filename, fps=10)
end
# ╔═╡ 065c16f6-0785-4bd1-b449-647b897af119
# PIMC simulation
function run_pimc(params::PIMCParams)
L = params.boxlength # Box length for periodic boundary conditions
path = initialize_positions(params)
accept_count = Ref(0)
kinetic_energy_accum = 0.0
potential_energy_accum = 0.0
virial_energy_accum = 0.0
all_paths = []
for step in 1:params.n_steps
for b in 1:params.n_beads
move_type = rand(["single", "bisect", "multi-level"])
if move_type == "single"
single_particle_move!(path, params, b, accept_count)
elseif move_type == "bisect"
bisect_move!(path, params, b, accept_count)
elseif move_type == "multi-level"
multi_level_move!(path, params, b, accept_count)
end
end
# if step % 100 == 0
# acceptance_ratio = accept_count[] / (params.n_particles * params.n_beads * 100)
# if acceptance_ratio < 0.5
# params.step_size *= 0.9
# elseif acceptance_ratio > 0.7
# params.step_size *= 1.1
# end
# accept_count[] = 0
# end
kinetic_energy_accum += kinetic_energy_estimator(path, params)
potential_energy_accum += potential_energy_estimator(path, params)
virial_energy_accum += virial_energy_estimator(path, params)
if step % 100 == 0
push!(all_paths, deepcopy(path))
end
end
kinetic_energy_avg = kinetic_energy_accum / params.n_steps
potential_energy_avg = potential_energy_accum / params.n_steps
virial_energy_avg = virial_energy_accum / params.n_steps
return path, kinetic_energy_avg, potential_energy_avg, virial_energy_avg, all_paths
end
# ╔═╡ 722c1d17-7097-4002-a3a3-160499618d07
# Example usage:
begin
kinetic = PrimitiveKineticAction(0.5)
one_body = OneBodyPotentialAction(r -> 0.5 * dot(r, r))
two_body = TwoBodyPotentialAction((r1, r2) -> 1.0 / norm(r1 - r2))
potential = CombinedPotentialAction(one_body, two_body)
params = PIMCParams(
n_particles = 10,
n_beads = 100,
n_dimensions = 3,
tau = 0.1,
beta = 100.0,
n_steps = 1000000,
step_size = 0.1,
kinetic_action = kinetic,
potential_action = potential,
boxlength = 10,
periodic = false,
bosonic = true,
fermionic = false
)
path, kinetic_energy_avg, potential_energy_avg, virial_energy_avg, all_paths = run_pimc(params)
println("PIMC simulation completed.")
println("Average Kinetic Energy: ", kinetic_energy_avg)
println("Average Potential Energy: ", potential_energy_avg)
println("Average Virial Energy: ", virial_energy_avg)
end
# ╔═╡ efc6ffd9-4177-494c-bb62-ca5323d137fc
all_paths[1][2]
# ╔═╡ bed6623f-36d5-4705-a6d2-c8ef63ec21ac
visualize_paths(all_paths, params, "pimc_paths.gif")
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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