Fermi and Bose gas in statistical mechanics

[Sang-Hyeon Han]

In statistical mehcanics(pathria, 3rd edition), I have some questions for ideal fermi and bose gases. The textbook handles the approximation for z(=e^βµ) and nλ^3 (n=N/V, λ : thermal de Broglie wavelength). It considers the cases that z<<1, z~1, nλ^3~1,<<1,→0 and so on. In here, I am confused that these approximation for z and nλ^3 are correlated with ecah other or not?? In some approximation , FD(or BE) function is almost equal to z and z is equal to nλ^3! I am really confused. So could you explain that for me??

 

[azntoon]

The approximation for z and nλ^3 are not necessarily correlated with each other. The Fermi-Dirac (or Bose-Einstein) function is equal to z only at low temperatures, when z is much smaller than 1. At higher temperatures, when z is close to 1, the FD (or BE) function is not equal to z, and it can be different from nλ^3 too. For example, in the ideal gas limit, when nλ^3→0, the FD (or BE) function tends to a constant value, while z→1.