Time reversal invariant evolution equation in dilute quantum gas

For a dilute gas interacting through a short range potential, we have obtai ned, by starting from the quantum mechanical Hamiltonian, a closed set of e volution equations for the particle distribution and for the expectation va lues of the four-field-operator products. The set of equations is invariant under time reversal. The evolution equation for the particle distribution depends on the initial conditions, and reduces to the Boltzmann transport e quation for almost all initial states.