quantum mechanics - Is there an equivalent of probability current for the Wigner distribution?

I know that for a wavefunction, I can derive a probability current $\mathbf{J}$ that satisfies the continuity equation: $$\nabla \cdot \mathbf{J}=-\frac{\partial}{\partial t} \big|\psi\big|^2$$

Can a similar quantity be derived for the Wigner distribution? If so, what is it?

Answer

It's intuitively clear that this current must exist because the integral of the Wigner function is conserved by unitary evolution. This current is known as the Wigner flow, and it exists but it's not particularly pretty. For an example of the Wigner flow in use, see arXiv:1208.2970; in short, it is the current $$ J=\begin{pmatrix}J_x\\J_p\end{pmatrix} =\begin{pmatrix} \tfrac pm W(x,p,t) \\ -\sum_{l=0}^\infty\frac{(i\hbar/2)^{2l}}{(2l+1)!}\frac{\partial^{2l}W(x,p,t)}{\partial p^{2l}}\frac{\partial^{2l+1}V(x)}{\partial x^{2l+1}}\end{pmatrix} ,$$ where $V(x)$ is the system's potential energy, and it obeys the continuity equation $$ \frac{\partial W}{\partial t}+\frac{\partial J_x}{\partial x}+\frac{\partial J_p}{\partial p}=0. $$