CLSolvers.jl

This is the code generating the simulations done in: https://arxiv.org/abs/2105.02735

To run the code you need a version of Julia installed, then you can make separate scripts or follow the Main.jl file which can be run line by line using the Julia vscode extension.

Instantiate

To initialize the project run these comments inside the Julia REPL (From inside the project directory)

    import Pkg
    Pkg.activate(".")
    Pkg.instantiate()

For more information see: https://docs.julialang.org/en/v1/stdlib/Pkg/

Now all dependencies should be downloaded and the code is ready to be run.

Example

Under is an example simulating the Anharmonic oscillator on the canonical Schwinger-Keldysh contour and plotting the two-point function compared to the true solution.

using CLSolvers
using DifferentialEquations, StochasticDiffEq
using Random
rng = MersenneTwister(12345);

# In this file the true solutions used in the paper are stored as
# dictionares
include("Scro_Data.jl")

# Setup the contour
t_steps = 64
contour = SchwingerKeldyshContour(1.0,0.5,0.0,0.0)
discContour = discretizeContour(contour,t_steps/4,t_steps/4,2*t_steps/4)

# Get out the needed information from the contour
a = getContourDistances(discContour, contour) # Points separation
CC = getContour(discContour, contour)         # Contour points (Complex numbers)
tp = CLSolvers.spread_timepoints(discContour,contour) # Parameterization points

# initialize the field
y0 = vcat(randn(rng, Float64, t_steps),zeros(Float64,t_steps))

# Setup parameters for the simulation
args = (N_Tr = 50, # Number of trajectories 
          θ = 0.6, # Impliciteness of solver
          dt = 1e-4, tol = 5e-2, dtmax = 1e-3, #Langevin stepsize paramater
          tspan=(0.0,10.0), # Langevin time span
          adaptive=true) # Select adaptive step-size or fixed step-size

# Get parameters and functions on a form that the StochasticDiffEq can read
params = getParamArray(p,a)
sde_a, sde_b, _ = CLSolvers.get_sde_funcs(p)

ff = SDEFunction(sde_a,sde_b)
prob_sde2 = SDEProblem(ff,sde_b,y0,args.tspan,params)

# Funciton that is run before each trajectory (Initialize the trajectory)
function prob_func(prob,i,repeat)
    println("Starting ensamble: ",i)
    remake(prob,u0=vcat(randn(rng, Float64, t_steps),zeros(Float64,t_steps)))
end

# Setup a ensamble simulation (See https://diffeq.sciml.ai/stable/tutorials/sde_example/#Ensemble-Simulations)
ensemble_prob = EnsembleProblem(prob_sde2,prob_func=prob_func)

# Run the simulation
@time sol = solve(ensemble_prob,ImplicitEM(theta=args.θ, symplectic=false),
            EnsembleThreads(), trajectories=args.N_Tr,
            progress = true, saveat = 0.01, save_start = false, 
            dtmax=args.dtmax, dt=args.dt,
            adaptive=args.adaptive,
            abstol=args.tol,reltol=args.tol,maxiter=10^6)

# Compute the observables
avg_Re, avg_Im, avg2_Re, avg2_Im, corr0t_Re, corr0t_Im, corr0t_2_Re, corr0t_2_Im, corrt0_Re, corrt0_Im = get_statistics(sol)

# Plot the two-point function
plot_realtime_corr0t(corr0t_Re, corr0t_Im, CC; true_solution=dfScroSolMinkBeta1)