Totally Asymmetric Simple Exclusion Process: particles hop right if the next site is empty. Exactly solvable via Bethe ansatz. Watch phase transitions between low-density, high-density, and maximal current phases.
TASEP (Spitzer 1970, MacDonald-Gibbs-Pipkin 1968): 1D lattice with hard-core exclusion, unidirectional hopping. Open boundary TASEP phase diagram: LD phase (ρ=α<1/2, J=α(1-α)) for α<β and α<1/2; HD phase (ρ=1-β, J=β(1-β)) for β<α and β<1/2; MC phase (ρ=1/2, J=1/4) otherwise. Exact solution: matrix product ansatz (Derrida 1992). Fluctuations at MC: Kardar-Parisi-Zhang universality class (height fluctuations ~ t^{1/3}).