Microscopic limits of PDEs modeling macroscopic heat conduction

We show that it is possible to construct microscopic-level discrete equations from macroscopic modeling PDEs for heat conduction in one space dimension. The significance of this result is that, in general, one starts from microscopic theories and then take their continuum limits to obtain the corresponding macroscopic PDEs, whereas here it is demonstrated that the reverse procedure is also possible. While our focus is on heat conduction, we discuss the applicability of our methodology to other physical systems.