The authors study the effect of the maximum entropy principle (MEP) on the thermodynamic behaviour of gases. The MEP relies on the kinetic theory of gases, and yields local constitutive equations of extended thermodynamics. There are two extreme cases in the kinetic theory: dominance of particle interactions, and free flight. In its current form the MEP gives the phase density that maximizes the entropy at each instant of time. This is appropriate in the case of dominant particle interaction, but it is not adequate for free flight. Here the authors introduce a modified MEP capable of linking both extreme cases. To illustrate how the modified MEP works, the authors consider an example which leads in the case of dominant particle interactions to Euler equations. In addition, the authors prove a representation theorem which relates global solutions of Euler equations with shock interactions for arbitrary large variations of initial data.