Trend to equilibrium of the Wigner function describing a quantum particle subject to decoherence

This paper studies the long-time behavior of a spatially homogeneous Wigner equation describing a quantum particle subject to decoherence. The model extends the classical Wigner–Fokker–Planck equation by including a general decoherence term derived from microscopic collision mechanisms. An explicit representation of the equilibrium state is obtained, and a necessary and sufficient condition ensuring that the equilibrium state belongs to L^2 is established, highlighting the balance between friction and decoherence. Under suitable assumptions on the initial data, exponential convergence in L^2 toward equilibrium is rigorously proved.