Extensive Nature of Long Range Interactions: Role of Disorder

The omnipresent disorder in physical systems makes it imperative to investigate its effect on the spatial range of interactions for which system remains thermodynamically extensive. Previously known bounds on the statistical free energy for clean systems \cite{fish} indicate it to be extensive only for the spatially short range interactions (decaying faster than $r^{-d}$ at large distance $r$ with $d$ as system dimension). We analyze the bounds for quantum systems with different types of disordered many body potentials e.g annealed, quenched, Gaussian or power law distributed. Our results indicate the dependence of the bounds on the multiple distribution parameters representing the potential which in turn permits, in contrast to clean potentials, more freedom to achieve the extensive limits even for arbitrary spatial ranges of the interactions.