Kinetic-Energy Corrections to Klein's Theorem for the Thermodynamic Functions of a Plasma

Abstract

It is shown that the average thermal energy of a system of $N$ free electrons and $N$ free protons with density $n=\frac{N}{V}$ and temperature $T$ has the form $E=2N[\frac{3}{2}kT+Tf(\ensuremath{\eta})+{f}_{1}(n, T)+{{f}_{2}}^{\ensuremath{'}}(n, T)],$ where $f$, the leading term in the electrostatic energy, is an arbitrary function of $\ensuremath{\eta}=T{n}^{\ensuremath{-}\frac{1}{3}}$ in agreement with the original Klein theorem. ${f}_{1}$ is a kinetic-energy correction which is related to the electrostatic energy ${{f}_{2}}^{\ensuremath{'}}$ by the differential equation $\ensuremath{-}\frac{n}{T}\frac{\ensuremath{\partial}}{\ensuremath{\partial}n}{({f}_{1}+{{f}_{2}}^{\ensuremath{'}})}_{T}=\frac{T}{3}\frac{\ensuremath{\partial}}{\ensuremath{\partial}T}{\left(\frac{2{f}_{1}+{{f}_{2}}^{\ensuremath{'}}}{T}\right)}_{n}.$ Some consequences of this result are discussed.