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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 ($ε$) 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 $ε=4/5$. Here, the system remains homogeneous for small interaction strength ($ε<4/5$), and for anti-feromagnetic interactions ($ε<0$). 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 $ε= δ/n$, where phase separation occurs for $δ< -1$.