Pion production from a classical source: Transport and hydrodynamical properties

We discuss how to determine the hydrodynamic and transport properties of a pion field produced in a high-energy collision. It is shown that a Boltzmann-type equation can be obtained for a relativistic generalization of the Wigner distribution function. In simple models, one can construct the expectation value of the normal-ordered energy-momentum tensor from the second moments of this distribution function. For a $\lambda {\phi }^4$ theory, treated in the Hartree approximation, we show that the transport equation for the pion field is similar, in the limit $\hslash \to 0$, to that for a single specie plasma, treated in the Vlasov approximation. We show that if pions are produced in a coherent state, the hydrodynamic properties of the field are those of an ideal fluid. In particular, pion production by a $c$-number source provides a realization in field theory of Landau's hydrodynamical model.