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?
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.
You are probably asking if there is a limit where the Schrodinger equation for many particles interacting with a potential reproduces the Boltzmann equation for many classical particles colliding in a potential. The answer is no, because the Boltzmann equation is irreversible in time, while the Schrodinger equation is reversible. The first order BE does not have a symmetry between forward and backward in time evolution, it's not an equation for complex amplitudes. It has an entropy which constantly increases. The Schrodinger equation is completely reversible.
It should be added that this is also true of classical particle dynamics--- as Loschmidt noted, it is impossible for the reversible classical particle dynamics to ever produce exact irreversible Boltzmann evolution. But Boltzmann understood that the equation was only approximate, valid only when multiparticle correlations could be ignored. But Boltzmann also understood from physical intuition that this was the case most of the time in real gasses. So there is a sense in which it is possible to arrive at the Boltzmann equation from a statistical description of a classical gas. But it requires a truncation of the statistical description to only the function f(x,p) which describes the expected number of particles at position x and momentum p. This truncation is lossy, and it is the reason for the emergence of irreversibility.
So the more nuanced answer is that you can find a Boltzmann equation when you can truncate the statistical description into a low dimensional projection, and get the best approximate statistical evolution in this truncation.
Your intuition was probably that the SE should reduce to a BE because both describe the behavior of a bunch of particles in a statistical way. This is incorrect, because the SE is not a statistical description. Absent a measurement, which is not described by SE anyway, the SE gives you the time evolution of the complete state. There is nothing statistical going on without measurement.
The SE is also describing waves in a humongous 3N dimensional space, so that it describes all the entanglements between all the particles. To get to the BE, you need to truncate the space to just the expected number of particles at position x with momentum p. This truncation doesn't work with probability amplitudes--- only probabilities can be truncated like this. The reason is that if you have a truncated state X which internally can be one of two possible micro-states A or B, if you have probability, you can just say "the probability of X is the sum of the probability of A and the probability of B", but you can't say "the amplitude of X is the sum of the amplitude of A and the amplitude of B", because that's just wrong for quantum evolution.
So truncated partial descriptions are only for classical probabilities. So to reproduce classical Boltzmann dynamics you need to pass to a statistical description, which means density matrices, then take the classical limit of SE for density matrices, then project this probabilistic description from the 6N dimensional phase space to the 6 dimensional Boltzmann function.
The first step is the classical limit of QM, the second step is the original derivation of the Boltzmann equation from the full description of the stochastic gas in 6N dimensional phase space. You can't relate the equations in any other meaningful way.
Irreversible equations such as the Boltzmann equation can be obtained rigorously as scaling limits of reversible microscopic equations such as a multiparticle Schroedinger equation.
A good entry point for studied about your question is the survey paper by
You can follow up with reading one of the many papers citing it obtained with author:spohn kinetic in http://scholar.google.com
Less rigorous versions of the same technique are ubiquitous in nonequilibrium statistical mechanics. I recommend two nice books: the book by
