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Thermodynamics of condensed matter with strong pressure-energy correlations

We show that for any liquid or solid with strong correlation between its $NVT$ virial and potential-energy equilibrium fluctuations, the temperature is a product of a function of excess entropy per particle and a function of density, $T=f(s)h(ρ)$. This implies that 1) the system's isomorphs (curves in the phase diagram of invariant structure and dynamics) are described by $h(ρ)/T={\rm Const.}$, 2) the density-scaling exponent is a function of density only, 3) a Gr{ü}neisen-type equation of state applies for the configurational degrees of freedom. For strongly correlating atomic systems one has $h(ρ)=\sum_nC_nρ^{n/3}$ in which the only non-zero terms are those appearing in the pair potential expanded as $v(r)=\sum_n v_n r^{-n}$. Molecular dynamics simulations of Lennard-Jones type systems confirm the theory.