  We explore the implications of the averaged null energy condition for thermal
states of relativistic quantum field theories. A key property of such thermal
states is the thermalization length. This lengthscale generalizes the notion of
a mean free path beyond weak coupling, and allows finite size regions to
independently thermalize. Using the eigenstate thermalization hypothesis, we
show that thermal fluctuations in finite size `fireballs' can produce states
that violate the averaged null energy condition if the thermalization length is
too short or if the shear viscosity is too large. These bounds become very weak
with a large number N of degrees of freedom but can constrain real-world
systems, such as the quark-gluon plasma.
