&=   f_b(m) {Z_{N-1,M-m}\over Z_{N,M}} = {1\over \zeta(b) m^b } {Z_{N-1,M-m}\over Z_{N,M}}.
\label{eq:pm}\end{aligned}$$ The particle-number distribution $p_\infty(m)$ behaves as $p_\infty(m)\simeq f_b(m)=m^{-b}/\zeta(b)$ for $m$ small while the contribution of $Z_{N-1,M-m}/Z_{N,M}$ in Eq. (\[eq:pm\]) may not be negligible for $m$ large. When the particle density $\rho$ is small, the latter contribution takes the form $e^{-\mu\,m}$ with $\mu$ depending on $\rho$ via the relation $\sum_{m=1}^\infty m \, p_\infty(m) = \rho$. $\mu$ corresponds to the negative chemical potential $\mu=(\partial/\partial M) \log Z_{N,M}$ [@Evans:2006uq], and decreases with increasing $\rho$. For $b>2$, there exists the critical density $\rho_c$ such that the relation $\sum_{m=1}^\infty m \, p_\infty(m) = \rho$ can be satisfied by $p_\infty(m) = f_b(m) e^{-\mu\, m}$ with $\mu\geq 0$ only for $\rho\leq \rho_c$: $$\rho_c = 
{\zeta(b-1)\over \zeta(b)}.
\label{eq:rhoc}$$ If $\rho>\rho_c$, $p_\infty(m)$ develops additionally a bump in the region $m\simeq M-N\rho_c$ indicating that most particles occupy a single site, which may be called condensation [@Evans:2006uq]. For $b\leq 2$ and $\rho$ finite, $p_\infty(m)$ is free from such a bump in the limit $N\to\infty$. Therefore two phases, fluid and condensate phase, can be defined depending on whether such condensation occurs or not in the stationary state, which leads to the phase diagram in the $(b,\rho)$ plane \[Fig. \[fig:model\] (b)\].

