*Write your name and answers on this sheet and hand it in at the
end.*

Work with others at your table on these activities. Argue about the answers but work efficiently!

Goal: Explore the number distributions for bosons and fermions.

- In the Mathematica notebook
`thermal_distributions.nb`, the distribution functions for relativistic bosons and fermions are defined as a function of momentum*p*(and other variables).*Write down the formulas here:*

*For a pion, what role will the chemical potential play?*

*Write down the formula for the total number of particles for a massless gas of pions and derive the dependence of the total number (integration of the distribution function) on the temperature*[Hint: make an appropriate change of variables.]*without*calculating the integral explicitly.

- Now evaluate it exactly in Mathematica.
*Why do we multiply the formula by 3 for pions? What is your result for Ntotal? Did you get the same dependence on temperature as you predicted?*

- Finally, let's explore the low-temperature limit of fermion distribution
(as it would be along the x-axis of the QCD phase diagram). In the notebook,
there is an expression to plot the distribution at high temperature (everything
is measured in GeV here).
*Add plots to the same graph for 1/10 and 1/100 of this temperature. What happens? Does it agree with your expectations? Describe the occupation numbers in the zero-temperature limit and why it is consistent with fermions.*

Goal: Calculate the energy density of massless pions.

- Continuing in the Mathematica notebook
`thermal_distributions.nb`, let's find the energy density of a gas of massless pions. We need to add something to the integral for the total number to get the expectation value of the energy.*What do you add?*

- Your result should be π
^{2}/10*T*^{4}.*Did you get this?*

- Follow the notebook to compare to the Maxwell-Boltzmann limit.
*What is the percentage difference for the two results in the constant that multiplies the kBT dependence?*

*Did you verify that pressure = 1/3 energy density for a massless boson?*

Goal: Estimate the transition temperature at zero baryon chemical potential based on the simple bag model of confinement.

*At a phase boundary, what quantities must be the same for each phase that is in equilibrium?*[Choices: temperature, chemical potential, energy density, pressure.]

- In the MIT bag model, the true ground state of the vacuum has a lower energy density (-B) than the "perturbative" vacuum in the deconfined phase. This creates the pressure that confines quarks in a hadron. The pressure is the same value but positive: +B. We'll find the critical temperature for the deconfinement phase transition by equating pressures.
- For the hadron gas phase, we'll add B to the result for the pion gas
pressure from the last section.
*Write this expression for P*_{π}here:

- For the QGP plasmas, we have a massless gas still (we'll treat both gluons
and quarks as bosons at a first pass), but we have to change the factor in
front from 3 for the pion to an appropriate value for quarks and gluons.
*What is this factor? There are 8 gluons, each of which can have 2 helicities (spin along or opposite the momentum). There are 2 light quarks and 2 light anti-quarks, each with 3 colors and 2 spins (spin-up and spin-down). Add the factors for gluons and quarks. What do you get for the pressure in this phase?*

*Equate these pressures and solve for the temperature at which they are equal in terms of B.*[You should find Tc = (45B/17π^{2})^{1/4}.]

- From fits to hadron masses, it is found that B
^{1/4}is about 200 MeV.*What is your prediction for the transition temperature?*

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