Model Lorentz-like equation with continuous spectrum

•Non-analytical dependence of hydrodynamic eigenvalues is demonstrated.•The problems with extended hydrodynamics for soft interactions are shown.•The limits if validity of Grad’s methods is explicitly stated for such models.

We present a new model of the Lorentz gas kinetic equation for a system where the integral collision operator has a spectrum consisting of a continuous and discrete part. The spectral gap between the two kinds of the spectrum is an adjustable parameter of the model. This allows for the analysis of the existence and property of the hydrodynamical eigenstates and the meaning of the Grad’s method of moments for the transition between hard and soft interactions.