  We present a three-lane exclusion process that exhibits the same universal
fluctuation pattern as generic one-dimensional Hamiltonian dynamics with
short-range interactions, viz., with two sound modes in the Kardar-Parisi-Zhang
(KPZ) universality class (with dynamical exponent $z=3/2$ and symmetric
Pr\"ahofer-Spohn scaling function) and a superdiffusive heat mode with
dynamical exponent $z=5/3$ and symmetric L\'evy scaling function. The lattice
gas model is amenable to efficient numerical simulation. Our main findings,
obtained from dynamical Monte-Carlo simulation, are: (i) The frequently
observed numerical asymmetry of the sound modes is a finite time effect. (ii)
The mode-coupling calculation of the scale factor for the $5/3$-L\'evy-mode
gives at least the right order of magnitude. (iii) There are significant
diffusive corrections which are non-universal.
