I was wondering if there are what I would like to call "higher order" conservation laws. We have systems of equations like Euler and Navier-Stokes-Fourier, which can be used to describe the conversation of mass, momentum and energy, under some assumptions.
From another perspective, one can take moments of the Boltzmann equation in order to find these conversation laws, by Chapsman-Enskog expansion one can even show that under some linearization, the Euler equations or Navier-Stokes-Fourier equations are found again.
Looking from the last perspective, the zeroth moment gives mass conversation, the first gives momentum and the second energy. However, we can take as high order moments as we want, but these will not lead to conversation laws from classical physics.
Why are there no conversation laws expect for the ones above? Is it because there is no need for "higher order" quantities to be conserved in order to describe what we see in nature, or is there a more fundamental reason, making these higher order laws obsolete, or even something else then conservation laws?