Processor sharing is a mathematical idealization of round-robin scheduling algorithms commonly used in computer time-sharing. It is a fundamental example of a non-head-of-the-line service discipline. For such disciplines, it is typical that any Markov description of the system state is infinite dimensional. Due to this, measure-valued stochastic processes are becoming a key tool used in the modeling and analysis of stochastic network models operating under various non-head-of-the-line service disciplines. In this talk, we discuss a new approach to studying the asymptotic behavior of fluid model solutions (formal functional law of large numbers limits) for critically loaded processor sharing queues. For this, we introduce a notion of relative entropy associated with measure-valued fluid model solutions. This approach is developed with idea that similar notions involving relative entropy may be helpful for understanding the asymptotic behavior of critical fluid model solutions for stochastic networks operating under protocols naturally described by measure-valued processes.