Problems and Solutions on Thermodynamics and Statistical Mechanics

290 Problem8 €4 Solutiona on Thermodynamics €4 Statistical Mechanics

The reaction does not occur in free space, but one may think of it as
catalyzed by the walls of the box. Ignoring the walls except insofar as they
allow the reaction to occur, find

(a) The chemical potentials for the fermions.
(b) The average number of electron-positron pairs, in the two limits
kT >> mec2 and kT << mec2. (You may leave your answers in terms of
dimensionless definite integrals.)

(c) The neglect of the walls is not strictly permissible if they contain
a matter-antimatter imbalance. Supposing that this imbalance creates a
net chemical potential p # 0 for the electrons, what is then the chemical
potential of the positrons?

(d) Calculate the net charge of the system in the presence of this
imbalance in the limit kT >> p >> m,c2. (Again, your answer may be left
in terms of a dimensionless definite integral.)
(Chicago)
Solution:
(a) For a chemical reaction A +-+ B + C at equilibrium, p~ = pg +pc.
As the chemical potential of the photon gas p7 = 0, we obtain

pe+ + pe- = 0.

Considering the symmetry between particle and antiparticle, we have

Hence pe+ = pe- = 0.

(b) At the limit kT >> mec2, neglecting the electron mass and letting

E = cp, we obtain

V (kT)3 O0 x2dx

7r2 (h)3 ez + 1

• 
  
  
  
  • Ne+.
  
  
  
  

At the limit kT << mec2, the "1" in denominator of the Fermi factor

1

[exP(PJ(cP)2 + (meC2)2) + 11