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 λâÉáe^{μ/τ}, in
which case the answer can be written more simply as
Z_{G} = 1 + λe^{-/τ}.