The notion of energy through multiple scales: From a molecular level to fluid flows and beyond

In the present paper, we review the consistent definition of macroscopic total energy in classical fluid mechanics, as a function of the microscopic canonical Hamiltonian field, based on a Lennard-Jones model with some spatially varying external field. The macroscopic total energy (sum of mechanical and internal energy) is proved to be equal to the equilibrium ensemble-averaged Hamiltonian. In particular, the conditions for including the effects of the external field both in the macroscopic potential energy and in the internal energy are discussed. {We present the notion of energy as defined in different scientific communities, starting from the standard macroscopic systems all the way down to small ones, which are gaining an increasing popularity.