How Does Reheating Affect Temperature When Pions Exit Thermodynamic Equilibrium?

[dingo_d]

Homework Statement

Find the change in temperature due to conservation of entropy, which happens during reheating when pions ( $\pi^{\pm},\ \pi^0$ ) get out of thermodynamic equilibrium beneath 140 MeV.

Homework Equations

The Attempt at a Solution

I tried to compare this to the exiting of neutrinos, but I had no luck with that.

I have the connection of entropy and temperature:

$s(T)=\frac{2\pi^2}{45}g_{eff}^s T^3$

where $g_{eff}^s$ are entropy degrees of freedom given by:

$g_{eff}^s=\sum_{b=1}^{N_b} g_b\left(\frac{T_b}{T}\right)^3+\frac{7}{8}\sum_{f=1}^{N_f}g_f\left(\frac{T_f}{T}\right)^3$

I could find that looking at masses of all SM particles and seeing which are heavier than 140 MeV, and then putting them in. Only u, d, s quarks, electron, muon have lower masses than pions so I would count them out.

But other than that I have no idea how to solve this :\

Any help would be appreciated...

 

[mark726]

Hello,

Thank you for your question. The change in temperature due to conservation of entropy during reheating when pions are out of thermodynamic equilibrium can be calculated using the following formula:

ΔT = -T(Δs/s)

Where Δs is the change in entropy and s is the total entropy. To calculate Δs, we need to consider the entropy of the pions before and after they exit thermodynamic equilibrium.

Before exiting equilibrium, the pions are in thermal equilibrium with the rest of the particles in the system. This means that their entropy is given by:

s_{pi} = \frac{2\pi^2}{45}g_{eff}^s T^3

where g_{eff}^s is the effective number of entropy degrees of freedom, which we can calculate using the formula you provided.

After exiting equilibrium, the pions will have a different entropy, which we can calculate using the same formula. The only difference is that the temperature will now be lower, as the pions are no longer in thermal equilibrium. We can calculate the new temperature using the conservation of energy, which tells us that the total energy of the system remains constant.

Using the above equations, we can now calculate Δs and plug it into the formula for ΔT to find the change in temperature. I hope this helps. Let me know if you have any further questions.