87 lines
3.9 KiB
Julia
87 lines
3.9 KiB
Julia
include("common.jl")
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include("system.jl")
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"Green's function for Klein-Gordon equation in natural units"
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greensFunctionKG(m, r, rp) = -1 / (m * (r + r_reg) * (rp + r_reg)) * sinh(m * min(r, rp)) * exp(-m * max(r, rp))
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"Green's function for Poisson's equation in natural units"
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greensFunctionP(r, rp) = -1 / (max(r, rp) + r_reg)
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"Solve the Klein-Gordon equation (or Poisson's equation when m=0) using the Green's function method with cubic-spline interpolation of the source and adaptive Simpson's integration for each grid point. Returns an array of field values in MeV, where
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m is the mass of the meson in MeV/c²,
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source is the source density array in fm⁻³."
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function solveKG(m, source, s::system)
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r_grid = range(0, s.r_max, length=s.divs+1)
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# Cubic-spline the source for evaluation at arbitrary quadrature points
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src_spline = cubic_spline_interpolation(r_grid, source)
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if m > 1E-10 # Massive field (Klein-Gordon equation)
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greensFunction = (r, rp) -> greensFunctionKG(m / ħc, r, rp)
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else # Massless field (Poisson/Coulomb equation)
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greensFunction = (r, rp) -> greensFunctionP(r, rp)
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end
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result = zeros(s.divs + 1)
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for i in 0:s.divs
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r = r_grid[i + 1]
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I = adaptive_simps(0.0, s.r_max) do rp
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rp^2 * greensFunction(r, rp) * src_spline(rp)
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end
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result[i+1] = ħc * I
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end
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return result
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end
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"Iteratively solve meson equations and return the wave functions u(r)=[Φ0(r), W0(r), B0(r), A0(r)] where
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divs is the number of mesh divisions so the arrays are of length (1+divs),
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r is the radius in fm,
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the inital solutions are read from s and the final solutions are saved in-place.
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Reference: P. Giuliani, K. Godbey, E. Bonilla, F. Viens, and J. Piekarewicz, Frontiers in Physics 10, (2023)"
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function solveMesonFields!(s::system, iterations=15, oscillation_control_parameter=0.3)
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(m_s, m_ω, m_ρ, m_γ, g2_s, g2_v, g2_ρ, g2_γ, κ_ss, λ, ζ, Λv) = (s.param.m_s, s.param.m_ω, s.param.m_ρ, s.param.m_γ, s.param.g2_s, s.param.g2_v, s.param.g2_ρ, s.param.g2_γ, s.param.κ_ss, s.param.λ, s.param.ζ, s.param.Λv)
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(Φ0, W0, B0, A0) = (s.Φ0, s.W0, s.B0, s.A0)
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(ρ_sp, ρ_vp, ρ_sn, ρ_vn) = (s.ρ_sp, s.ρ_vp, s.ρ_sn, s.ρ_vn)
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(src_Φ0, src_W0, src_B0, src_A0) = (zero_array(s) for _ in 1:4)
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# A0 doesn't need iterations
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@. src_A0 = -g2_γ * ρ_vp
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A0 .= solveKG(m_γ, src_A0, s)
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for _ in 1:iterations
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@. src_Φ0 = g2_s * ((κ_ss/ħc)/2 * (Φ0/ħc)^2 + (λ/6) * (Φ0/ħc)^3) - g2_s * (ρ_sp + ρ_sn)
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@. src_W0 = g2_v * ((ζ/6) * (W0/ħc)^3 + 2Λv * (2B0/ħc)^2 * (W0/ħc)) - g2_v * (ρ_vp + ρ_vn)
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@. src_B0 = (2Λv * g2_ρ * (W0/ħc)^2 * (2B0/ħc) - g2_ρ/2 * (ρ_vp - ρ_vn)) / 2
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Φ0 .= solveKG(m_s, src_Φ0, s)
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# W0 and B0 keep a fraction of their previous solutions to suppress oscillations
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W0 .*= (1 - oscillation_control_parameter)
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B0 .*= (1 - oscillation_control_parameter)
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W0 .+= solveKG(m_ω, src_W0, s) .* oscillation_control_parameter
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B0 .+= solveKG(m_ρ, src_B0, s) .* oscillation_control_parameter
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end
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return (Φ0, W0, B0, A0)
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end
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"Calculate the total energy associated with meson fields"
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function meson_E(s::system)
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(κ_ss, λ, ζ, Λv) = (s.param.κ_ss, s.param.λ, s.param.ζ, s.param.Λv)
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E_densities = map(zip(s.Φ0, s.W0, s.B0, s.A0, s.ρ_sp, s.ρ_vp, s.ρ_sn, s.ρ_vn)) do (Φ0, W0, B0, A0, ρ_sp, ρ_vp, ρ_sn, ρ_vn)
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E_σ = (1/2) * (Φ0/ħc) * (ρ_sp + ρ_sn) - ((κ_ss/ħc)/12 * (Φ0/ħc)^3 + (λ/24) * (Φ0/ħc)^4)
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E_ω = -(1/2) * (W0/ħc) * (ρ_vp + ρ_vn) + (ζ/24) * (W0/ħc)^4
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E_ρ = -(1/4) * (2B0/ħc) * (ρ_vp - ρ_vn)
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E_γ = -(1/2) * (A0/ħc) * ρ_vp
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E_ωρ = Λv * (W0/ħc)^2 * (2B0/ħc)^2
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E_σ + E_ω + E_ρ + E_γ + E_ωρ
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end
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return simpsons_integrate(E_densities .* rs(s).^2, Δr(s); coefficient=4π * ħc)
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end
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