Phase separation transition in anti-ferromagnetically interacting particle systems

One dimensional non-equilibrium systems with short-range interaction can undergo phase transitions from homogeneous states to phase separated states as interaction ($\epsilon$) among particles is increased. One of the model systems where such transition has been observed is the extended Katz-Lebowitz-Spohn (KLS) model with ferro-magnetically interacting particles at $\epsilon=4/5$. Here, the system remains homogeneous for small interaction strength ($\epsilon<4/5$), and for anti-feromagnetic interactions ($\epsilon<0$). We show that the phase separation transitions can also occur in anti-ferromagnetic systems if interaction among particles depends explicitly on the size of the block ($n$) they belong to. We study this transition in detail for a specific case $\epsilon = \delta/n$, where phase separation occurs for $\delta < -1$.