The average energy per particle comes out to be (3/2)kT for the classical distribution as given by Equation (1.158). The average energy obtained by Equation (1.159) which applies to a system of distinguishable particles is practically the same except at very low temperatures 10 -11 Kelvin which is of no practical consequence in macroscopic systems.
But a system of distinguishable particles follows Fermi-Dirac Statistics as shown in Chapter 1_Part 8_Section1.8.2.3. At low temperatures it has the form shown in Figure (1.83). At 0 Kelvin there are 2 electrons or 2 fermions of opposite spins per quantum state up to E F ( Fermi Energy) and above E F the permissible states are empty. This rectangular distribution becomes more and more skewed with rise in temperature until above degeneracy temperature the system of distinguishable particles becomes non- degenerate and behaves classically as shown in Figure (1.82).
Figure 1.83. The average number of particles per permissible quantum state for particles obeying Fermi-Dirac Statistics at Temperature (a) 0 Kelvin (b) T 1 Kelvin (c) T 2 Kelvin.
A very illustrative example of these two conditions are the conducting electrons in a metal which is a degenerate system following the distribution in Figure (1.83) and conducting electrons in semiconductors which is non-degenerate following Figure(1.82).
Here it is noteworthy that in Figure (1.82) the number of electrons is 1 or less. Here 1 means 1 pair of opposite spin. Less than 1 means the state has less than 100% occupancy.
For Bosonic system consisting of indistinguishable particles things are very different. As shown in Figure (1.84), The number of particles at lower energies is much higher than the classical case If we have a fixed number of particles then at low temperature we have Bose Condensate which we have talked about in Chapter 1_Part 8_1.8.2,2. At higher temperature the distribution of Bosons cannot be distinguished from that of Fermions.
Figure 1.84. The average number of particles per state at a given temperature for particles obeying Bose-Einstein Statistics.
This discussion implies that at low temperatures the distribution is going to be critically effected by the fact that the system is bosonic or fermionic.
In solids we do not see any difference in the distribution in Bosonic System say Diamond isotope 12 ( 6 p’s, 6 n’s and 6 e’s) and Fermionic System say Diamond isotope 13 ( 6 p’s, 7 n’s and 6 e’s). Whatever difference is seen can be explained in terms of atomic masses. Here Diamond isotope 12 though indistinguishable are still distinguishable since the individual atoms are tied to their respective positions. The difference in the two distribution shows up in liquids and gases.
In gases the density is very low hence interatomic distances is very large and so the degeneracy temperature is too low for observing the difference.
So to see the difference in a bosonic system and fermionic system we need a quantum liquid. It is sufficiently dense so that we have an achievable degeneracy temperature . At the same time it is sufficiently mobile so that an indistinguishable system really becomes indistinguishable. This liquid should be at a temperature below Degeneracy Temperature. Then it is a proper quantum liquid.