The Schrödinger equation in its variants for many particle systems gives the full time evolution of the system. Likewise, the Boltzmann equation is often the starting point in classical gas dynamics.
What is the relationship, i.e. the classical limit, which connects these two first order in time equations of motions?
How does one approach this, or is there another way in which one sees the classical time evolution?
Where are these considerations relevant?
Best Answer
There are two differnt levels to see this connection. Formally, you can derive a Fokker-Planck equation from the Boltzmann equation and do a Wick rotation on the time variable. This can be seen as a mathematical curiosity presently.
But there is a more relevant way to recover this and is given by a formulation of the quantum Boltzmann equation. There is a beautiful Physics Report by Bassano Vacchini and Klaus Hornberger that can be downloaded here. This equation is relevant to understand the behavior of matter waves in interference experiments involving large molecules with their decoherence effects as realized by Anton Zeilinger and Markus Arndt.
When the formal limit $\hbar\rightarrow 0$ is taken, quantum Boltzmann equation reduces to ist classical counterpart.