# Relation between Wigner flows and entropy

After reading this question What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics? and some papers of Steuernagel’s group, like “The Quantum Liouville Equation is non-Liouvillian”

I started to think about the relation between Wigner flows and entropy.

I imagined an ensemble of Wigner quasi-probability distributions over the phase space of a closed system.

The ensemble has the property that the sum of W(x, p) over all its elements always equals some constant for each point [x, p] inside some shell of the phase space: the ensemble as a whole has constant density over the shell.

With “shell” I mean the set of all points that would imply total energy between some E1 and E2.

My question is: can you choose an ensemble and a Hamiltonian (keeping the system closed) in such a way that this property is respected at time 0 for some shell but is no longer true at some later time for that same shell?

This would be possible if divergence was not a function of the quasi-probability distribution, but it is not clear to me how changes in quasi-probability distribution affect divergence. For example: how does a rotation of the distribution around the centre of phase space affect the divergence field? Are there any systems for which the divergence field does not simply rotate by the same angle?

If you add the fact that for the ensemble as a whole probability on shells is roughly conserved over time (I am not completely sure about this, but it seems sensible, especially for a thick shell) I think an ensemble evolving in such a way would violate the second law of thermodynamics since it would move away from a uniform density distribution, that is the one with maximum entropy.

I don’t see how a non-Liouvillian flow can possibly not imply violation of the second law at least in some cases.

Do you think Wigner flows being non-Liouvillian could be somehow connected to this line of research?

https://www.nature.com/articles/s41598-019-40765-6

Here they claim that local violations of the second law are possible.

• The Wigner function is a special representation of a density matrix. A single density matrix of a mixed quantum state may describe an ensemble, so you almost certainly have no need of an ensemble of WFs. Any Hamiltonian with potential of order higher than quadratic would violate Liouville flow, so would deform flat regions (plateaus) in the WF. But I believe these plateaus have freakishly low not high entropy -- (Gaussians have high entropy). Jul 28 '19 at 22:43

For a closed system the entropy will remain constant in time. This is not particularly clear in the evolution equation for the Wigner function, but by remembering that the Wigner function contains the same information as a density matrix (as Cosmas points out) we can move to the density-operator picture:

$$$$W <=>\rho$$$$

$$$$\dot{\rho} = -\frac{i}{\hbar}[H, \rho]$$$$

Which is simply unitary evolution, and thus conserves information (entropy). However your "shells", if initiated at constant density, will not nessesarily remain at constant density.

For an open system things are more complicated, the evolution of the density operator (which maps to the Wigner function) is now determined by a Lindblad master equation (https://en.wikipedia.org/wiki/Lindbladian), while the time evolution of the Wigner function now contains diffusion terms (https://arxiv.org/abs/1703.04844). In this case the system entropy will in general change over time, exactly how is an active field of research (https://iopscience.iop.org/article/10.1088/1742-6596/258/1/012015/pdf#:~:text=In%20the%20quantum%20theory%20of,law%20of%20thermodynamics%20%5B1%5D.).

The specific paper you link has raised other questions on this forum, which may be interesting to you.

Have researchers managed to "reverse time"? If so, what does that mean for physics?