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We develop a rigorous theory of hard-sphere dynamics in the kinetic regime,away from thermal equilibrium. In the low density limit, the empirical densityobeys a law of large numbers and the dynamics is governed by the Boltzmannequation. Deviations from this behaviour are described by dynamicalcorrelations, which can be fully characterized for short times. This providesboth a fluctuating Boltzmann equation and large deviation asymptotics.

Fluctuation theory in the Boltzmann--Grad limit