Entropy production and pressure anisotropy across fast shocks in the ten-moment Gaussian closure

We analyze shock thermodynamics of the source-free ten-moment Gaussian closure in one-dimensional variation, focusing on fast shocks. The longitudinal subsystem (ρ,u,Pxx) is algebraically identical to the Euler equations with γ=3, while the transverse degrees of freedom enter through a Schur complement that is advected in smooth flow and transmitted unchanged through fast shocks. For transversely isotropic shear-free states, this structure yields a closed entropy–anisotropy theory: the post-shock anisotropy decomposes exactly into a reversible (isentropic) component and an irreversible enhancement, ΔA(r,Δs;R1)=9R1r2 (e2Δs−1)(R1r2e2Δs+2)(R1r2+2), where r=ρ2/ρ1 is the compression ratio, Δs is the Gaussian entropy jump, and R1=Pxx,1/Pyy,1 is the upstream pressure ratio. The weak-shock conversion factor is κ(R1)=18R1/(R1+2)2, and the strong-shock saturation is ΔA→9/(4R1+2). For the full three-dimensional pressure tensor, we prove that fast shocks transmit the normalized shear vector p/Pxx and the normalized Schur complement C=ρ−1(Π−ppT/Pxx) unchanged, implying detP2=β(r) r2 detP1 and Δs=12 ln (β/r3) independent of shear. Inversion formulas allow Δs to be inferred directly from second moments, providing analytical benchmarks for numerical methods and diagnostics for kinetic simulation data. Numerical verification confirms all predictions.