Renormalization group flow equations and the phase transition in O(N)-models

We derive and solve numerically self-consistent flow equations for a general O(N)-symmetric effective potential without any polynomial truncation. The flow equations combined with a sort of a heat-kernel regularization are approximated in next-to-leading order of the derivative expansion. We investigate the method at finite temperature and study the nature of the phase transition in detail. Several beta functions, the WilsonFisher fixed point in three dimensions for various N are analyzed and various critical exponents beta, nu, delta and eta are independently calculated in order to emphasize the reliability of this novel approach.