**Problem : **

Write down the Gibbs sum. Be sure to get all of the indices correct.

*Z*_{G}(

*μ*,

*τ*) =

*e*^{(Nμ-)/τ}
**Problem : **

Give the expression for the absolute probability that a system will be
found in the state with *N*_{1} particles and energy .

*P*(

*N*_{1},

) =

**Problem : **

Give an expression for the average value of a property *A* for a system
in diffusive and thermal contact with a "reservoir". A "reservoir" is a
huge system next to our smaller system with large energy and number of
particles.

<

*A* > =

**Problem : **

Give an expression for the average number of particles in a system that
is in thermal and diffusive contact with a reservoir.

We are looking for < *N* >, which we can calculate using the formula we
just derived.

<

*N* > =

**Problem : **

Suppose that we have a system that can be unoccupied or can have one
particle in a state with energy . Write the Gibbs sum
for this system.

One possible state has *N* = 0, for which we say that the energy
is also zero. So the first term in the sum is 1.
The second possible state has *N* = 1, and energy . We can
write the total sum as:

*Z*_{G} = 1 + *e*^{μ-/τ}

We sometimes simplify this by defining *λ*âÉá*e*^{μ/τ}, in
which case the answer can be written more simply as
*Z*_{G} = 1 + *λe*^{-/τ}.