Phase space representation of non-Hermitian system with $\mathcal{P}\mathcal{T}$-symmetry

We present a phase space study of non-Hermitian Hamiltonian with $\mathcal{P}\mathcal{T}$-symmetry based on the Wigner distribution function. For an arbitrary complex potential, we derive a generalized continuity equation for the Wigner function flow and calculate the related circulation values. Studying vicinity of an exceptional point, we show that a $\mathcal{P}\mathcal{T}$-symmetric phase transition from an unbroken $\mathcal{P}\mathcal{T}$-symmetry phase to a broken one is a second-order phase transition.