is not uniquely determined because of the freedom to add a linear combination of $\tilde{G}_{\mbox{\tiny(k)}}$ to $\tilde{H}$.

The $3n$ variables $x_{i\mbox{\tiny(k)}}$ satisfy the relations $$\begin{aligned}
\{
x_{i_1{\mbox{\tiny($\rm k_1$)}}}, 
x_{i_2{\mbox{\tiny($\rm k_2$)}}}, 
x_{i_3{\mbox{\tiny($\rm k_3$)}}}
\}_{\mbox{\tiny{NB}}} 
&=& \varepsilon_{i_1 i_2 i_3}~~~~\mbox{for ~$\rm k_1=k_2=k_3$}~,
\label{NB-n-rel1}\\
\{
x_{i_1{\mbox{\tiny($\rm k_1$)}}}, 
x_{i_2{\mbox{\tiny($\rm k_2$)}}}, 
x_{i_3{\mbox{\tiny($\rm k_3$)}}}
\}_{\mbox{\tiny{NB}}} 
&=& 0~~~~~~~~~~\mbox{otherwise}.
\label{NB-n-rel2}\end{aligned}$$ The first type of relation (Eq. (\[NB-n-rel1\])) is invariant under the time evolution (Eq. (\[N-eqf-n\])) irrespective of the form of $\tilde{H}$. To be more specific, for infinitesimal transformations $x_{i\mbox{\tiny(k)}} \to x'_{i\mbox{\tiny(k)}}=
x_{i\mbox{\tiny(k)}}+(dx_{i\mbox{\tiny(k)}}/dt) dt$, $$\begin{aligned}
\{x'_{\mbox{\tiny(k)}}, y'_{\mbox{\tiny(k)}}, z'_{\mbox{\tiny(k)}}\}
_{\mbox{\tiny{NB}}} 
= 1~\label{NB-n-rel'}\end{aligned}$$ hold. We can also show an important relation, $$\begin{aligned}
\frac{\partial(x'_{\mbox{\tiny(1)}}, y'_{\mbox{\tiny(1)}}, z'_{\mbox{\tiny(1)}}, 
\cdots, 
x'_{\mbox{\tiny({\it n})}}, y'_{\mbox{\tiny({\it n})}}, z'_{\mbox{\tiny({\it n})}})}
{\partial(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}}, z_{\mbox{\tiny(1)}}, \cdots, 
x_{\mbox{\tiny({\it n})}}, y_{\mbox{\tiny({\it n})}}, z_{\mbox{\tiny({\it n})}})}
= 1~,\label{LT}\end{aligned}$$ which guarantees the Liouville theorem, the conservation law of the phase space volume under time development. On the other hand, the second type of relation (Eq. (\[NB-n-rel2\])) does not always hold, unless there is no interaction between the $n$ subsystems, i.e., $\tilde{H}$ has a form such as $\tilde{H}(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}}, 
\cdots, z_{\mbox{\tiny({\it n})}})
= \sum_{\rm k} \tilde{H}_{\mbox{\tiny(k)}}
(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})$.

### Partition functions {#Partition functions}

It is well known that the partition function $Z_{\rm H}$ for a canonical ensemble 