Many models of heavy ion collisions employ relativistic hydrodynamics to describe the system evolution at high densities. The Cooper-Frye formula is applied in most of these models to turn the hydrodynamical fields into particles. However, the number
of particles obtained from the Cooper-Frye formula is not always positive-definite. Physically negative contributions of the Cooper-Frye formula are particles that stream backwards into the hydrodynamical region. We quantify the Cooper-Frye negative contributions in a coarse-grained transport approach, which allows to compare them to the actual number of underlying particles crossing the transition hypersurface. It is found that the number of underlying inward crossings is much smaller than the one the Cooper-Frye formula gives under the assumption of equilibrium distribution functions. The magnitude of Cooper-Frye negative contributions is also investigated as a function of hadron mass, collision energy in the range $E_{rm lab} = 5-160A$ GeV, and collision centrality. The largest negative contributions we find are around 13% for the pion yield at midrapidity at $E_{rm lab} = 20A$ GeV collisions.
In most heavy ion collision simulations involving relativistic hydrodynamics, the Cooper-Frye formula is applied to transform the hydrodynamical fields to particles. In this article the so-called negative contributions in the Cooper-Frye formula are
studied using a coarse-grained transport approach. The magnitude of negative contributions is investigated as a function of hadron mass, collision energy in the range of $E_{rm lab} = 5$--$160A$ GeV, collision centrality and the energy density transition criterion defining the hypersurface. The microscopic results are compared to negative contributions expected from hydrodynamical treatment assuming local thermal equilibrium. The main conclusion is that the number of actual microscopic particles flying inward is smaller than the negative contribution one would expect in an equilibrated scenario. The largest impact of negative contributions is found to be on the pion rapidity distribution at midrapidity in central collisions. For this case negative contributions in equilibrium constitute 8--13% of positive contributions depending on collision energy, but only 0.5--4% in cascade calculation. The dependence on the collision energy itself is found to be non-monotonous with a maximum at 10-20$A$ GeV.
