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Nuclear collisions from 0.3 to 2 GeVInucleon are studied in a microscopic theory based on Vlasov's self-consistent mean field and Uehling-Uhlenbeck's two-body collision term which respects the Pauli principle. T h e theory explains simultaneously the observed collective flow and the pion multiplicity and gives their dependence o n the nuclear equation of state.PACS numbers: 25.70.-z One of the most intriguing motivations for studying relativistic nucleus-nucleus collisions is the unique opportunity to explore compressed and excited nuclear matter in the laboratory. A signature of compression is the collective sidewards flow predicted theoretically by nuclear fluid dynamics' and classical microscopic many-body s i m u~a t i o n s .~ Recently, the predicted collective sidewards flow has been observed in highmultiplicity selected collisions of heavy n~c l e i .~,~ Another observable compression effect predicted by fluid dynamics is the dependence of the pion multiplicity o n the nuclear compression energy at high densit i e~.~ The pion multiplicities have been measured for near-central collisions of Ar (0.3-1.8 GeV/nucleon)Both data sets present a challenge to microscopic theories: The flow calculationsL4 done with the standard intranuclear-cascade p r~~r a m s ' .~ result in forward-peaked angular distributions, in contrast to the data. and the calculated pion m~l t i~l i c i t i e s~-~ drastical-I ly overestimate the experimental yields. These large discrepancies are surprising in view of the success of the cascade model in describing inclusive data.'j8 It has been conjectured6 that the difference between measured pion yields and cascade predictions is due to the neglect of compression energy in the cascade approach and thus may be used to extract the nuclear equation of state at high densities.In this Letter we present a microscopic theory which explains for the first time simultaneously both the observed collective flow and the pion multiplicity and gives their dependence o n the nuclear equation of state. Our approach is based on Vlasov's equation for the evolution of the single-particle distribution function f of a collisionless plasma in a self-consistent mean potential field supplemented by UehlingUhlenbeck's quantum mechanical extension of Boltzmann's two-body collision term which respects the Pauli principle. This extended Boltzmann equation can be ~r i t t e n~. '~ The Vlasov equation is solved by simultaneous numerical integration of the classical equations of motion of fifteen parallel ensembles of A p + A T test particles, which are initially assigned Fermi momenta and random positions in a sphere of nuclear radiiis. The ensemble-averaged phase-space density is computed at each synchronization time step in a six-dimensional sphere around each test particle. This ensemble a~e r a g i n g~, '~ ensures a reasonably smooth singleparticle distribution function f (r,p,r) which is used to determine the mean field U ( n) and the Pauli-blocking probability ( 1 -,f) ( 1 -, f ) .