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# Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

This lecture had a large amount of spoken content not captured in these notes. Reference to section 4 [1] was made for additional details.

# Grand canonical ensemble

Fig 1.1: Ensemble pictures

We are now going to allow particles to move to and from the system and the reservoir. The total number of states in the system is

so for , and , we have

where the chemical potential \index{chemical potential} and temperature \index{temperature} are defined respectively as

With as the set of all possible configuration pairs , we define the grand partition function

So that the probability of finding a given state with energy and particle numbers is

For a classical system we have

whereas in a quantum content we have

We want to do this because the calculation of the number of states

can quickly become intractable. We want to go to the canonical ensemble was because the partition function

yields the same results, but can be much easier to compute. We have a similar reason to go to the grand canonical ensemble, because this computation, once we allow the number of particles to vary also becomes very hard.

We are now going to define a notion of equilibrium so that it includes

- All forces are equal (mechanical equilibrium)
- Temperatures are equal (no net heat flow)
- Chemical potentials are equal (no net particle flow)

We’ll isolate a subsystem, containing a large number of particles fig. 1.2.

Fig 1.2: A subsystem to and from which particle motion is allowed

When we think about Fermions we have to respect the “Pauli exclusion” principle \index{Pauli exclusion principle}.

Suppose we have just a one dimensional Fermion system for some potential as in fig. 1.3.

Fig 1.3: Energy level filling in a quantum system

For every momentum there are two possible occupation numbers

our partition function is

We’d find that this calculation with this constraint becomes essentially impossible.

We’ll see that relaxing this constraint will allow this calculation to become tractable.

# References

[1] RK Pathria. *Statistical mechanics*. Butterworth Heinemann, Oxford, UK, 1996.