The classical justification for the microcanonical ensemble relies on the fact that most many-body systems have just a 'small' (typically finite) number of conserved quantities (i.e. they violate ergodicity along just a few dimensions in the $n$-particle phase space). In addition to the energy, for example, both the total momentum and total angular momentum are conserved, and perhaps a few other functions too (e.g. the total number of certain types of particles, and quantities associated with interactions that might not be obvious a priori).

As long as the number of conserved quantities increases 'slowly' with the system size, classical statistical mechanics will describe the macroscopic behavior (to leading order) in the thermodynamic limit.

Are there examples of physical systems with distinct conserved quantities at every scale? If so, how would you compute the 'entropy' $S=k_B\log\Omega(E,C_1,C_2,\dots)$ for a given set of values $C_1,C_2,\dots$ for these conserved quantities? If you would not expect the existence of such systems, then why?


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