6805: 1094 Activities 8

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

Thermal number distribution functions

Goal: Explore the number distributions for bosons and fermions.

1. 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:
   
   
   
   
   
   
2. For a pion, what role will the chemical potential play?
   
   
   
   
   
   
3. 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 without calculating the integral explicitly. [Hint: make an appropriate change of variables.]
   
   
   
   
   
   
   
   
4. 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?
   
   
   
   
   
   
   
   
5. 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.
   
   
   
   
   
   
   
   

Thermodynamics of massless bosons

Goal: Calculate the energy density of massless pions.

1. 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?
   
   
   
   
   
   
   
   
2. Your result should be π²/10 T⁴. Did you get this?
   
   
   
   
3. 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?
   
   
   
   
4. Did you verify that pressure = 1/3 energy density for a massless boson?
   
   
   
   

Deconfinement transition temperature by equating pressure

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

1. At a phase boundary, what quantities must be the same for each phase that is in equilibrium? [Choices: temperature, chemical potential, energy density, pressure.]
   
   
   
   
2. 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.
3. 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:
   
   
   
   
   
   
4. 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?
   
   
   
   
   
   
5. Equate these pressures and solve for the temperature at which they are equal in terms of B. [You should find Tc = (45B/17π²)^1/4.]
   
   
   
   
   
   
   
   
6. From fits to hadron masses, it is found that B^1/4 is about 200 MeV. What is your prediction for the transition temperature?