Finite-site canonical law and microcanonical shell

Fix a finite alphabet $A=\{1,\dots ,q\}$, $q\ge 1$, and real single-site energies $e_1,\dots ,e_q$ in a fixed energy unit. At inverse parameter $\beta \in \mathbb{R}$, define $Z_1(\beta ):={\sum }_{j=1}^q{e}^{-\beta e_j},p_j(\beta ):=e^{-\beta e_j}\mathrm{/}{Z}_1(\beta ),E(\beta ):={\sum }_{j=1}^q{p}_j(\beta )e_j\mathrm{.}$ For $N\ge 1$, a configuration $i=(i_1,\dots ,i_N)\in A^N$ has additive energy $H_N(i):={\sum }_{r=1}^N{e}_{i_r}$. Given a width ${\delta }_N\ge 0$, the shell is ${\mathrm{\Gamma }}_N(\beta ,{\delta }_N):=\{i\in A^N:\mathrm{\mid }{H}_N(i)-NE(\beta )\mathrm{\mid }\le {\delta }_N\}\mathrm{.}$ When this finite set is nonempty, its microcanonical law means the uniform probability on ${\mathrm{\Gamma }}_N$, and its dimensionless entropy is $\log \mathrm{\mid }{\mathrm{\Gamma }}_N\mathrm{\mid }$. A type is a vector $t=(n_1\mathrm{/}N,\dots ,n_q\mathrm{/}N)$, where the nonnegative integer counts sum to $N$; its multiplicity is $N!\mathrm{/}{\prod }_j{n}_j!$.

The convention fixes both the shell center and the law within the shell. It makes no physical claim that an apparatus has resolution ${\delta }_N$.