The convergence of the totally asymmetric simple exclusion process to thesolution of the Burgers equation is a classical result. In his seminal 1981paper, Herman Rost proved the convergence of the density fields and localequilibrium when the limiting solution of the equation is a rarefaction fan. Animportant tool of his proof is the subadditive ergodic theorem. We prove hisresults by showing how second class particles transport the rarefaction-fansolution, as characteristics do for the Burgers equation, avoidingsubadditivity. In the way we show laws of large numbers for tagged particles,fluxes and second class particles, and simplify existing proofs in the shockcases. The presentation is self contained.
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