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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?

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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 $$ \frac{\partial W}{\partial t}+\frac{\partial J_x}{\partial x}+\frac{\partial J_p}{\partial p}=0. $$

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  • $\begingroup$ Impeccable. You may rewrite the transport (continuity) phase-space flux, for $\mathrm{sinc}(z)\equiv \sin z / z$, as $J_x=pW/m$ and $J_p= -W\mathrm{sinc} ({\hbar \over 2} \overleftarrow{\partial_p} \overrightarrow{\partial_x} )~~ \partial_x V(x)$. Observe how the Wigner flow deformation modifies the incompressible Liouville flow by total derivative corrections of $O(\hbar^2)$. $\endgroup$
    – Cosmas Zachos
    Commented Mar 20, 2017 at 21:31
  • $\begingroup$ @Cosmas that expression is horrifying. $\endgroup$
    – Emilio Pisanty
    Commented Mar 21, 2017 at 7:10
  • $\begingroup$ I guess... but we always dance around our nightmares... At the origin, sinc is 1, and it keeps cropping up and winking at us... $\endgroup$
    – Cosmas Zachos
    Commented Mar 21, 2017 at 13:57
  • $\begingroup$ Yeah, fair enough, but what does the "origin" mean in $\mathrm{sinc}(z)$ when $z=\partial_p\partial_x$? Or what does the sinc function evaluated on derivatives mean other than its series expansion? Does it connect to anything else? (Don't get me wrong, though, it's a neat identity.) $\endgroup$
    – Emilio Pisanty
    Commented Mar 21, 2017 at 14:45
  • $\begingroup$ Of course, nothing but a series expansion. It just reminds you of the mnemonics of manipulations you'd be brave enough to perform, etc. Yes, the origin means hbar smaller than the inverse gradients of the functions acted upon... the usual painful "classical" limit.... $\endgroup$
    – Cosmas Zachos
    Commented Mar 21, 2017 at 15:03

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