# Phase Space Flow

Phase space flow shares characteristics with fluid flow such as incompressibility by Liouville's theorem. Extending the similarities one might be curious, does phase space flow have a characteristic number like the Reynold's number? Moreover, can phase space flow exhibit the characteristics of turbulence?

If so, can someone suggest papers or text concerning the aforementioned questions.

• Is the phase space in question refers to phase space of kinetic theory equations such as Boltzmann eq., or Vlasov eq.? If so, would phase space description of, say, plasma turbulence be an answer? Or are you talking about phase space of Hamiltonian mechanics? – user23660 Dec 3 '13 at 3:59
• I had phase space of Hamiltonian mechanics in mind, when I wrote the question. However, it would be interesting to hear about both. – sunspots Dec 3 '13 at 4:05

## 1 Answer

The problem with the phase space flow in Hamiltonian mechanics is that the flow itself is non-dynamical, that is, the flow is immediately defined for a given Hamiltonian, so there is no independent equation governing its evolution. Thus, Liouville equation is simply a transport of a scalar variable in a given flow.

So, dimensional analysis of the flow would be simply subset of dimensional analysis of underlying Hamiltonian structure.

Similarly, I do not think there is any sense of attempting to find turbulence in the phase space flows. Sure, time dependence of Hamiltonian can introduce changes in the phase space, including the type of changes associated with transitions to chaos: such as bifurcations, tori destruction ... but again, the flow itself is not the fundamental object in such transitions.

If we are talking about the phase space of kinetic equations, the same arguments apply. Even though the flow is 'more dynamical' especially if taken in the context of self-interacting system of equations such as Vlasov-Maxwell, in these equations the flow itself again is not a fundamental object, so rarely it is analyzed independently. However, most methods of (numerical) solutions for such equations like particle-in-cell method and its many variations do use approaches quite similar to that of hydrodynamics.