What is meant by "each occupied by one particle at the same time"

[JamesJames]

Gibbs sum DILEMNA

Consider a system that maybe unoccupied with energy zero or occupied by one particle in either of two states, one of energy zero and one of
energy $$\epsilon_{0}$$. Allow the possibility that the orbital at 0 and at $$\epsilon_{0}$$ may be each occupied by one particle at the same time. What is the Gibbs sum Z?

The Gibbs sum is given by

Z = $$\sum \lambda^{N}exp(-\epsilon_{S(N)}/\tau)$$

where N is the number of particles in each state.

Here is what I think:

Z = $$\lambda^{1}exp(-\epsilon_{0}/\tau) + \lambda^{1}exp(-\epsilon_{0}/\tau)$$

Reasoning: I think that N = 1 since it MUST BE OCCUPIED. I feel that the energy must always be $$\epsilon_{0}$$ since the orbital may be occupied by one particle at the same time...I am not at all convinced about this explanation and need clarification.

What is meant by "each occupied by one particle at the same time" in terms of its effect on the Gibbs sum?

 

[Nate810]

In this case, "each occupied by one particle at the same time" means that both states can be occupied simultaneously. Therefore, the Gibbs sum is given by: Z = \lambda^{2}exp(-\epsilon_{0}/\tau)

 

[wais]

In this context, "each occupied by one particle at the same time" means that both states, one with energy zero and one with energy \epsilon_{0}, can simultaneously have one particle occupying them. This is in contrast to a system where only one state can be occupied at a time.

In terms of its effect on the Gibbs sum, this means that the number of particles in each state is not fixed or restricted to only one. This allows for a greater range of possible states and configurations for the system, leading to a higher value for the Gibbs sum. In the specific case given, this would result in a doubling of the Gibbs sum compared to a system where only one state can be occupied at a time.

It is important to note that this concept is only applicable in systems where multiple particles can occupy the same state simultaneously, such as in quantum systems. In classical systems, each state can only be occupied by one particle at a time, so this concept does not apply.