We will discuss ideas from statistical mechanics applied to the analysis of the periodic generalized KdV equation (gKdV). We first discuss invariance of the Gibbs measure for gKdV. Previously, Bourgain proved the invariance for KdV and mKdV.  We show that the same invariance holds for the quartic gKdV.  One of the consequences of invariance of the Gibbs measure is the recurrence property for rough solutions.  For KdV and mKdV, dynamics of smooth solutions are also recurrent since these PDEs are completely integrable, but much less is understood with a higher order nonlinearity.  In the second part of the talk, we will study the concentration properties of certain micro-canonical measures, and discuss the insight these properties may provide into the long-time behavior of smooth solutions to the quartic gKdV.  The latter results are part of a joint work in progress with Vladimir Sverak and Ofer Zeitouni (University of Minnesota).