Beta equilibrium for free and interacting nucelar models

[Leonardo Machado]

Good evening.
I'm currently studing dense matter and nuclear matter above 10^8 g/cm^3, and i know well how to insert a beta equilibirum condition in a free Fermi gás at T=0.
$$\sqrt{K^2_{F,n}+m^2_n}=\sqrt{K^2_{F,p}+m^2_p}+\sqrt{K^2_{F,e}+m^2_e}$$
But how do i insert the same condition of radioactive equilibrium for an interacting model ?

 

[LightHero]

In an interacting model, the beta equilibrium condition would need to include the effects of interactions on the free Fermi gas. This could be done by introducing additional terms to account for interactions between nucleons and electrons. For example, one could include a term that represents the interaction energy between neutrons and protons, or a term that represents the electrostatic interaction between electrons and protons. Additionally, one would need to consider the effect of temperature on the beta equilibrium. At finite temperatures, the energies of the particles will be affected by the thermal motion of particles in the system, and this needs to be taken into account when calculating the beta equilibrium condition.