Particle creations in a variational formulation of relativistic thermodynamics with dissipation

In modeling relativistic thermodynamics, we frequently regard the particle number as a conserved quantity. The number conservation law, which comes from the requirement that the pull-back construction from fluid-matter 3-space has the same degrees of freedom as the number of independent components of the four-dimensional Lagrangian variations, supports the assumption. On the other hand, Andersson and Comer[1] argue that their variational model is compatible with non-vanishing particle creation rates. We support the result and prove that a proper variational formula considering the effect of the creation rate enables us to treat the non-vanishing creation rate without any additional requirement.