state $({{\tilde{t}}}\to\infty)$, the detailed balance condition $u(m_p+1, m_q-1) \, p_\infty(\{m\}_{pq}^{(+-)})  = u(m_q, m_p) \, p_\infty(\{m\})$ (and equivalently $u(m_q+1, m_p-1)  \, p_\infty(\{m\}_{pq}^{(-+)})  = u(m_p, m_q) \, p_\infty(\{m\})$) is satisfied if $u(m_p, m_q)$ is the multiplication of a function of $m_p$ and a function of $m_q$ [@0305-4470-38-19-R01], which is approximately true in the stationary state of our model; the denominator of Eq. (\[eq:u\]) saturates in the long time limit as confirmed in Fig. \[fig:density\] (b). Inserting Eq. (\[eq:u\]) in the detailed-balance condition, one finds that $p_\infty(\{m\})$ takes a factorized form $$\begin{aligned}
p_\infty(\{m\}) &= {1\over Z_{N,M}}\prod_{i=1}^N f_b(m_i), \nonumber\\
f_b(m)&={1\over \zeta(b) m^b},\nonumber\\
Z_{N,M}&=\prod_{i=1}^N \left(\sum_{m_i=1}^\infty f_b(m_i) \right) \delta\left(\sum_{j=1}^N m_j -M\right),
\label{eq:factor}\end{aligned}$$ with $\zeta(x)$ the Riemann zeta function and $Z_{N,M}$ the partition function. While many properties of the factorized states in Eq. (\[eq:factor\]) have been investigated [@PhysRevE.65.026102; @PhysRevLett.94.180601; @Evans:2006uq], we have seen that the case of $b=2$ is the model for the kinetics of product share in international trade, the properties of which are not fully understood.

Of main interest is the particle-number distribution at a single site, which is obtained from the whole particle configuration probability $p_\infty(\{m\})$ as $p_\infty (m) = \sum_{\{m_2, m_2, \ldots, \}} p_\infty (\{m,m_2,m_3,\ldots\})$ by using Eq. (\[eq:factor\]) [@0305-4470-38-19-R01] $$\begin{aligned}
p_\infty (m) 