One approach towards constructing the probability of a state in a system is by maximizing the constrained entropy

$$ S[p(n);\alpha,\beta] = -\sum_np(n)\log(p(n))\;+\; \alpha\big(\sum_nE_np(n) -\langle E\rangle\big)\;+\; \beta\big(\sum_np(n) - 1\big). $$ However, if there is a continuous symmetry in my Hamiltonian that preserves a charge $Q$, why not include this $$ S[p(n);\alpha,\beta,\gamma] = -\sum_np(n)\log(p(n))\;+\; \alpha\big(\sum_nE_np(n) -\langle E\rangle\big)\;+\; \beta\big(\sum_np(n) - 1\big) \\+\gamma\big(\sum_nq_np(n)+\langle Q\rangle \big)? $$ Does it make sense to do something like this or is there something clearly wrong with it.

Perhaps this makes more sense in for a quantum statistical system. Since the conserved charge commutes with the Hamiltonian, there will be many eigenstates $|n\rangle$ with the same charge $q_i$ for some $i$. Therefore, performing the sum in the second equation might make more sense.


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