Bounds on the Thermodynamic Behavior of Systems with Generalized Coulomb Interactions

Abstract

Various lower and upper bounds are derived for the energy, free energy, and pressure of particles interacting via an ${r}^{\ensuremath{-}n}$ pair potential in $\ensuremath{\nu}$ dimensions. The results include (i) lower bounds for mixtures with $n<2$, of the form $\mathrm{const}{\ensuremath{\rho}}^{\frac{1+2}{\ensuremath{\nu}}}$ as $\ensuremath{\rho}$ increases either isentropically or isothermally, complementing an upper bound of the same form in the isentropic case, recently derived by Kleban and Puff, and (ii) upper and lower bounds for the case $n>\ensuremath{\nu}$, $n>2$, which for the pressure are both of the form $\mathrm{const}{\ensuremath{\rho}}^{\frac{1+n}{\ensuremath{\nu}}}$ as $\ensuremath{\rho}$ increases isentropically.