Bounds, Successive Approximations, and Thermodynamic Limits for Distribution Functions, and the Question of Phase Transitions for Classical Systems with Non-negative Interactions

It is shown rigorously that, for systems with non-negative interactions and integrable Mayer $f$ bonds (e.g., hard spheres), distribution functions and the thermodynamic ratio $\frac{\rho }{z}$ exist in the limit $V\to \infty$, and are analytic functions of the activity $z$, for $z<\mathrm{}\frac{1}{f}$ in the right-hand half-plane, with $f$ the absolute value of the integral of the $f$ bond. Heuristic arguments then indicate that these functions are continuous in $z$ for all $z<\mathrm{}\infty$. This would mean that no Ehrenfest phase transitions are possible for such systems.