Ideal gas in an energy landscape

problem

Consider N non-interacting point particles with mass m in a volume V.
The volume is divided into three distinct subvolumes V₁, V₂ and V₃ with V = V₁ + V₂ + V₃.
The particles can move freely between the subvolumes. 

The Hamiltonian of the system is

H = H₁(𝔯₁, 𝔭₁) + … + Hᵤ(𝔯ᵤ, 𝔭ᵤ) with Hᵢ = 𝔭ᵢ²/2m + U(𝔯ᵢ) (u = N, unicode lacks a subscript N)

where

U(𝔯ᵢ) = U₁ if 𝔯ᵢ ∈ V₁
        U₂ if 𝔯ᵢ ∈ V₂
        U₃ if 𝔯ᵢ ∈ V₃

with Uⱼ ∈ ℝ for j ∈ {1, 2, 3}.

The system is in contact with a heat reservoir of temperature T.
Let Nⱼ be the average number of particles in subvolume Vⱼ.
Determine the ratio N₁/N₂.