The Connection between Classical and Statistical thermodynamics

13.4 Fermi-Dirac Distribution

• Fermions are particles that are identical and indistinguishable.
• Fermions include particles such as electrons, positrons, protons, neutrons, etc. They all have half-integer spin.
• Fermions obey the Pauli exclusion principle, i.e. each quantum state can only accept one particle.
• Therefore, for fermions Nj cannot be larger than gj.
• FD statistic is useful in characterizing free electrons in semi-conductors and metals.

For FD statistics, the quantum states of each energy level can be classified into two groups: occupied Nj and unoccupied (gj-Nj), similar to head and tail situation (Note, quantum states are distinguishable!)

• For FD statistics, the quantum states of each energy level can be classified into two groups: occupied Nj and unoccupied (gj-Nj), similar to head and tail situation (Note, quantum states are distinguishable!)
• The thermodynamic probability for the jth energy level is calculated as
• where gj is N in the coin-tossing experiments.
• The total thermodynamic probability is

W and ln(W) have a monotonic relationship, the configuration which gives the maximum W value also generates the largest ln(W) value.

• W and ln(W) have a monotonic relationship, the configuration which gives the maximum W value also generates the largest ln(W) value.
• The Stirling approximation can thus be employed to find maximum W

• There are two constrains
• Using the Lagrange multiplier

See white board for details

13.5 Bose-Einstein distribution

• Bosons have zero-spin (spin factor is 1).
• Bosons are indistinguishable particles.
• Each quantum state can hold any number of bosons.
• The thermodynamic probability for level j is
• The thermodynamic probability of the system is

Finding the distribution function

13.6 Diluted gas and Maxwell-Boltzman distribution

• Dilute: the occupation number Nj is significantly smaller than the available quantum states, gj >> Nj.
• The above condition is valid for real gases except at very low temperature.
• As a result, there is very unlikely that more than one particle occupies a quantum state. Therefore, the FD and BE statistics should merge there.

The above two slides show that FD and BE merged.

• The above two slides show that FD and BE merged.
• The above “classic limit” is called Maxwell-Boltzman distribution.
• Notice the difference
• They difference is a constant. Because the distribution is established through differentiation, the distribution is not affected by such a constant.

Summary

• Boltzman statistics:
• Fermi-Dirac statistics:
• Bose-Einstein statistics:
• Problem 13-4: Show that for a system of N particles obeying Maxwell-Boltzmann statistics, the occupation number for the jth energy level is given by