of those attractions between all pairs of particles at site $p$ and $q$, we find that individual particles attract one another with equal strength for $b=1$. The quadratic scaling $b=2$, reproducing the linear scaling in Eq. (\[eq:DApAp\]), implies inhomogeneous interaction between particles; the interaction between a particle at site $p$ and another at $q$ is proportional to $m_p m_q$. It should be mentioned that a site having only one particle is not allowed to send a particle elsewhere, i.e., $u_{pq}=0$ if $m_p=1$, since our study is restricted to the products maintaining non-zero share in the studied period.

Stationary-state distribution and phase diagram {#sec:pd}
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To see how particles are distributed under Eq. (\[eq:u\]), let us consider the evolution of the probability of finding a particle configuration $\{m\}=\{m_p|p=1,2,\ldots, N\}$ with time ${{\tilde{t}}}$ [@0305-4470-38-19-R01]: $$\begin{aligned}
& {\partial  p_{{{\tilde{t}}}} (\{m\})  \over \partial{{\tilde{t}}}} = 
\sum_{p<q} 
\left[u(m_p+1, m_q-1) \, p_{{\tilde{t}}}(\{m\}_{pq}^{(+-)}) \right. \nonumber\\
& - u(m_q, m_p) \, p_{{\tilde{t}}}(\{m\}) +  u(m_q+1, m_p-1)  \, p_{{\tilde{t}}}(\{m\}_{pq}^{(-+)}) \nonumber\\
 &\left.  - u (m_p, m_q) \, p_{{\tilde{t}}}(\{m\}) \right],\end{aligned}$$ where $\{m\}_{pq}^{(\pm, \mp)}$ is identical to $\{m\}$ except at sites $p$ and $q$ such that $\{m\}_{pq}^{(\pm, \mp)}= \{m_1, m_2, \ldots, m_{p-1}, m_p\pm 1, m_{p+1}, \ldots, m_{q-1}, m_q\mp 1, m_{q+1}, \ldots\}$. In the stationary 