Boltzmann's Entropy and Large Deviation Lyapunov Functionals for Closed and Open Macroscopic Systems

I give a brief overview of the resolution of the apparent problem of reconciling time symmetric microscopic dynamic with time asymmetric equations describing the evolution of macroscopic variables. I then show how the large deviation function of the stationary state of the microscopic system can be used as a Lyapunov function for the macroscopic evolution equations.