Note on the chemical potential of decoupled matter in the Universe
  Textbooks on cosmology exhibit a thermodynamic inconsistency for free streaming, decoupled matter. It is connected here to the chemical potential, which deviates from its equilibrium value $\mu=\alpha k_BT$, where $\alpha$ is the usual parameter of the Fermi-Dirac or Bose-Einstein distribution function. 
Theo M. Nieuwenhuizen and Claudia Pombo
astro-ph.CO cond-mat.stat-mech gr-qc