  Field theory at nonvanishing temperature beyond perturbation theory is
discussed for the $N$-component $O(N)$-symmetric scalar theory. We compute the
effective potential directly in three dimensions using an exact evolution
equation for an effective action with an infrared cutoff. A suitable truncation
is solved numerically. We obtain a detailed quantitative picture of the scaling
form of the equation of state in the vicinity of the critical temperature of a
second order phase transition.
