Energy Density Distribution

The Energy Density Distribution is the function that gives how much field energy sits at each point in space, combining the field's magnitude with its spatial variation. It describes where the collective field built from discrete pulses holds its energy: local concentrations depend both on the field amplitude and on how sharply it changes from place to place. The field is physically realizable only if its total energy is finite, which the book expresses as ∫|∂Ψ/∂t|² dx < ∞ and ∫|∇Ψ|² dx < ∞, with amplitudes scaling as A_n ∝ (n+1)^(−α) for α > 1/2. Normalized by the total field energy, ρ_I(x) = ρ_E(x)/E⟐, it becomes the density from which the field's information content is measured. The formal statement is ρ_E(x) = (ħ²/(2m)) |∇Ψ(x)|² + V(x) |Ψ(x)|², the standard Schrödinger and Gross–Pitaevskii energy density, where ρ_E(x) is the energy density at position x, ħ the reduced Planck constant, m the mass of the field quantum, V(x) the potential energy at x, Ψ(x) the collective field function, ∇Ψ(x) its spatial gradient and x the spatial coordinate; the factor ħ²/(2m) and V(x) turn |∇Ψ|² and |Ψ|² into energy per volume. Dimensions as the book gives them: ρ_E [J/m³], ħ [𝕄·𝕃²·𝕋⁻¹], m [𝕄], V(x) [𝕄·𝕃²·𝕋⁻²], |Ψ|² [𝕃⁻³], Ψ [𝕃⁻³/²], |∇Ψ|² [𝕃⁻⁵], ∇Ψ [𝕃⁻⁵/²], x [𝕃]. It connects to the Collective Field Function, Energy Concentration, Field Complexity Measure, Amplitude Scaling and Energy Conservation.