Thermodynamic potential and energy density in cosmic models

[mjka]

I'm reading this paper

http://arxiv.org/abs/0911.1728

It's about the authors' consideration of the Mass Varying Neutrino model with a new approach that try to explain the cosmic acceleration then.

I often encounter the thermodynamic potential during reading and re-calculating the equations. However, I'm not sure I got its meaning right, also the relation of thermodynamic potential and energy density. According to D part in Section I, the energy density is $\rho = \frac{\partial\Omega}{\partial m}$ where Ω is the thermodynamic potential density and m is the mass.

However, with that definition I couldn't achieve (72),(73) later.

By the way, I found how to evaluate the integral (31) but couldn't do that with (73), which is $\int_k^\inf \frac{z^2\sqrt{z^2-k^2}}{e^z+1}dz$.

Any help would be appreciate. Thank you!

 

[AndyW]

Thank you for sharing this interesting paper with us. The Mass Varying Neutrino model is a fascinating approach to explaining cosmic acceleration, and I can understand your confusion about the thermodynamic potential and its relation to energy density. Let me try to clarify it for you.

The thermodynamic potential, denoted as Ω, is a quantity that describes the equilibrium state of a system in thermodynamic terms. In this paper, the authors are using the thermodynamic potential to describe the behavior of the Mass Varying Neutrino model. The energy density, denoted as ρ, is a measure of the amount of energy per unit volume in a system. In this case, it is the energy density of the Mass Varying Neutrino model.

The equation you mentioned, ρ = \frac{\partial\Omega}{\partial m}, is a general relation between the energy density and the thermodynamic potential. It is used to calculate the energy density from the thermodynamic potential, but it may not be applicable in all cases. In the specific case of the Mass Varying Neutrino model, the authors have derived equations (72) and (73) using this relation, but they may have also used additional assumptions and approximations to simplify the equations.

Regarding the integral (73), it may not be possible to evaluate it analytically. In that case, numerical methods can be used to obtain a numerical solution. I suggest consulting with a colleague or a mentor who is familiar with the specific techniques used in this paper for further guidance.

I hope this helps clarify your doubts. Keep up the good work and happy researching!