Varying and extremizing L with respect to ρ ρ i leads to 0 ≡ ≡ δ δ L = δ δ ( − − ∑ ∑ i k B ρ ρ i ln ⁡ ⁡ ρ ρ i ) + δ δ ( λ λ 1 − − ∑ ∑ i λ λ 1 ρ ρ i ) + δ δ ( λ λ 2 U − − ∑ ∑ i λ λ 2 ρ ρ i E i ) = ∑ ∑ i [ δ δ ( − − k B ρ ρ i ln ⁡ ⁡ ρ ρ i ) + δ δ ( λ λ 1 ρ ρ i ) + δ δ ( λ λ 2 E i ρ ρ i ) ] = ∑ ∑ i [ ∂ ∂ ∂ ∂ ρ ρ i ( − − k B ρ ρ i ln ⁡ ⁡ ρ ρ i ) δ δ ( ρ ρ i ) + ∂ ∂ ∂ ∂ ρ ρ i ( λ λ 1 ρ ρ i ) δ δ ( ρ ρ i ) + ∂ ∂ ∂ ∂ ρ ρ i ( λ λ 2 E i ρ ρ i ) δ δ ( ρ ρ i ) ] = ∑ ∑ i [ − − k B ln ⁡ ⁡ ρ ρ i − − k B + λ λ 1 + λ λ 2 E i ] δ δ ( ρ ρ i ) .