For the first time we are able to construct a sampling method for the transition from relativistic hydrodynamics to particle transport, which, in every sample, preserves the local conservation of energy, momentum, baryon number, strangeness, and electric charge microcanonically. The proposed method is essential for studying fluctuations and correlations by means of stochastic hydrodynamics. It is also useful for studying small systems. The method is based on Metropolis sampling applied to particles within distinct patches of the switching space-time surface, where hydrodynamic and kinetic evolutions are matched.

Figure:  Demonstration of the sampling with conservation laws over the patch, where total baryon number, strangeness, and charge are enforced to be 0, while total energy and momentum are fixed. The patch consists of 3 cells with arbitrarily selected normals dσ_μ1=(500.0,50.0,20.0,30.0), dσ_μ2=(500.0,40.0,80.0,30.0), and dσ_μ3=(500.0,20.0,20.0,20.0) fm³; collective velocities v₁=(0.2,0.3,0.4), v₂=(0.1,0.5,0.5), v₃=(0.3,0.4,0.2); and temperatures T₁=0.155, T₂=0.165, and T₃=0.175 GeV. Mean multiplicities of selected hadrons in the cells are shown in panel (a): they are unchanged compared to standard grand-canonical Cooper-Frye sampling. However, the scaled variances of multiplicities in the whole patch, shown in panel (b), differ from the standard Cooper-Frye result and coincide within 0.5% with the microcanonical expectation in the thermodynamic limit, computed using analytic formulas from EPJC58, 83. In panel (c), the nontrivial correlations, generated by conservation laws, are shown in contrast to no correlations in the standard Cooper-Frye sampling. Correlations are defined as (A,B)≡⟨(A−⟨A⟩)(B−⟨B⟩)⟩, where ⟨⟩ denotes the average over samples; σ_2A≡(A,A).