In this talk, I will review some ideas and tools that allow to bridge from a particle system described through equations of motion (e.g. Newton’s 2nd law), to a fluid description ruled by a hydrodynamic partial differential equation (e.g. the Navier-Stokes equations) through the lens of large deviation theory. From the point of view of a physicist, I will explain why in some contexts we should take into account finite particle-number effects in this process and how to do so. To illustrate these principles, I will focus on the specific case of a gas composed of active particles. If time allows, I will also address the limitations of this approach when the hydrodynamic limit leads to a scalar conservation law partial differential equation (PDE).