“…Explicitly, ϵ l = 1 + 4 πl Bo p 2 ρ b /3, where l Bo = e 2 /4 πϵ o k B T is the Bjerrum length in the vacuum characterized by its permittivity ϵ o , electronic charge, e , and the Boltzmann constant, k B . In eq , the term proportional to the volume V is related to self-energy of dipoles and interactions among the dipoles. , In particular, free energy density , f b becomes directly proportional to the self-energy of dipoles in the limit when the number density of dipoles (ρ b ) vanishes i.e., ρ b → 0 e.g., for a dipolar gas. The logarithmic term appearing in the expression for f b results from multibody interactions among the dipoles and is plotted in Figure (a).…”