Notice that for large N the last term in the denominator ϕ ϕ / N ≪ ≪ 1 and can be neglected to give Δ Δ v ≈ ≈ v eff ∑ ∑ j = 1 j = N ϕ ϕ / N 1 − − j ϕ ϕ / N = v eff ∑ ∑ j = 1 j = N Δ Δ x 1 − − x j where Δ Δ x = ϕ ϕ N and x j = j ϕ ϕ N .