First-Order Phase Transitions in Quantum-Coulomb Plasmas

A simplified model is suggested for the understanding (in principle) of the mechanism of a phase transition in a Coulomb system on a uniform neutralizing background. Quantum theory is taken into account only in so far as it provides discrete collective energy levels and, possibly, fermion statistics for the calculation of the collective modes. Other features stemming from the uncertainty relations are supposed to be irrelevant to the mechanism of the phase transition. (This is qualitatively justified in the text which goes with Fig. 1.) This procedure provides an ``effective quantum Hamiltonian'' H_qu, which incorporates the relevant quantum features and from which one can calculate the (approximate) quantum partition function using classical methods: $$z_{\mathrm{approx}}^{\mathrm{Qu}}=\frac{1}{{\mathrm{N!h}}^N}\mathrm{\int \cdots \int \hspace{0.17em}}\exp {\mathrm{\hspace{0.17em}(-\beta H}}_{\mathrm{qu}}{\mathrm{)d}}^3{x}_1{\mathrm{\cdots d}}^3{p}_N$$ . In the evaluation of this integral we use the same approximation in which the plasma modes are collective, viz., the RPA (random phase approximation). Because of the freezing‐out of the collective degrees of freedom at the relevant temperatures, and because the number of these degrees of freedom changes with temperature and density, H_qu describes a system with variable degrees of freedom, which seems to provide the mechanism for the phase transition. Preliminary numerical evaluations indicate a phase transition at T = 0 for an electron plasma at r_s = 7.6, i.e., just below the region of metallic densities. This is shown by finding a concave region in the free energy as a function of the density. The ground is then prepared for numerical evaluation of the transition at T ≠ 0. For white dwarfs we find a transition temperature of 10⁷ ∼ K°.