Micro-canonical Ensemble of Ideal Bose Gas

[Ang Han Wei]

Hi,
can I know why the number of partitions separating different states have to be taken into account for the derivation of number of states in an ideal Bose Gas but not in the Fermi Gas?

What is the physical significance of this "partition"? In what ways can they vary?

 

[dauto]

That's just a math trick to easily calculate the total number of states. The partitions have no physical significance whatsoever.

 

[Ang Han Wei]

I see that there are some occasions when such partitions are called into use and they have to be taken into the total number of possible states in a factorial, while at other times, they do not appear.

How would one know when such partitions are to be taken into account?

 

[dauto]

The partitions should be used when they help solve the problem at hand. They have no physical significance and it is always possible to solve a problem without mentioning them. But sometimes they make it much easier to solve a problem. For instance, suppose you are calculating the number of ways that you can put two Fermions in three boxes, since each box can contain either zero or one fermion, than you can count them as follows
1: (1|1|0)
2: (1|0|1)
3: (0|1|1)
for a total of three states.
You can see that the first fermion can go in anyone of the three boxes, and second fermion can go on any of the remaining 2 boxes. 3*2=6= 3!/1!. At the ed you must divide by 2! because the fermions are identical so permutations among them won't create new states. You have then a total of 3!/(2!*1!)= 3 states. That can be generalized to

# of states = N!/[(N-n)!*n!] for n fermions on N boxes.

Now suppose you have two bosons that need to be placed in three boxes. since you can have more than one boson per boxe (no exclusion principle applies to bosons), there will be more states. Let's count them
1: (2|0|0)
2: (1|1|0)
3: (1|0|1)
4: (0|2|0)
5: (0|1|1)
6: (0|0|2)
for a total of six states. How can we generalize that? Doesn't seem obvious at first.
Here comes the math trick. Let's represent the particles by a '*'. the six states can now be represented as
1: (**||)
2: (*|*|)
3: (*||*)
4: (|**|)
5: (|*|*)
6: (||**)
Physically the '*' represents a particle while the '|' represents nothing whatsoever, it is just part of the notation. But that doesn't matter. Mathematically you have four symbols, Two '*'s and two '|'s. This symbols can be placed in 4!=24 different orders, but since the '*'s are identical we must divide by 2!, and the '|'s are also identical so we also divide by 2!. You have then a total of 4!/(2!*2!)= 6 states. That can be generalized to

# of states = (N+n-1)!/[(N-1)!*n!] for n bosons in N boxes.

Voila!

 

[PJC01]

The number of partitions separating different states must be taken into account for the derivation of the number of states in an ideal Bose gas because of the fundamental difference in the behavior of bosons and fermions. In an ideal Bose gas, multiple particles can occupy the same energy state, leading to a higher number of possible configurations or partitions. This is known as Bose-Einstein statistics. In contrast, fermions follow the Pauli exclusion principle, which states that no two particles can occupy the same energy state, leading to fewer possible configurations or partitions.

The physical significance of these partitions lies in their role in determining the thermodynamic properties of the gas. The number of partitions directly affects the density of states, which, in turn, determines the energy distribution and other thermodynamic quantities like entropy and specific heat. In an ideal Bose gas, the higher number of partitions leads to a larger density of states, resulting in a different energy distribution compared to an ideal Fermi gas.

The partitions can vary in several ways, including the number of particles, the volume of the system, and the energy levels available. As these parameters change, the number of partitions also changes, leading to different thermodynamic properties. For example, increasing the number of particles or decreasing the volume of the system will result in a higher number of partitions and a larger density of states, leading to a higher energy distribution and higher entropy.

In summary, the number of partitions plays a crucial role in understanding the behavior of an ideal Bose gas and is a fundamental concept in statistical mechanics. Its significance lies in its impact on the density of states and, ultimately, the thermodynamic properties of the gas.