What is the Gibbs sum for a system with two possible states and one particle?

[stunner5000pt]

a) Consider a system that may be unoccupied with energy zero or iccuped by one particle in either of two states, one of zer oeenrgy and one of energy epsilon. Show taht the Gibbs sum for this system is

$$z = 1 + \lambda / \lambda\exp(-\epsilon/\tau)$$

b) Show that the thermal average occupancy of the ssytem is
$$<N> = \frac{\lambda + \lambda\exp(-\epsilon/\tau)}{z}$$

c) show that the thermal average occupancy of the state at eneryg = epsilon is
$$<N(\epsilon)> = \lambda\exp(-\epsilon/\tau)/z$$

d) Find an expression for the theram laverage eneryg of the system
e) Allow the possibility tat the orbital at 0 and at epslon may be occupied each by one particle at the same time, show that
$$z = 1 + \lambda + \lambda\exp(-\epsilon/\tau) + \lambda^2 \exp(-\epsilon/\tau) = (1+ \lambda) [1 + \lambda \exp(\epsilon/\tau)]$$

I will post my attempted solutions in a seaparate post.

 

[stunner5000pt]

The Gibbs Sum or grand sum is given by

for this problem at least...
$$z = \sum_{N=0}^{1} \sum_{s(0)}^{s(1)}} \exp\left(\frac{N\mu-\epsilon_{s(N)}}{\tau}\right)$$

which comes out to

$$z = \sum_{N=0}^{1} \exp(\frac{N\mu}{\tau}) + \exp\left(\frac{N\mu-\epsilon_{s(N)}}{\tau}\right)$$

whic his

$$z = \exp(0) + \exp(\frac{\mu}{\tau}) + \exp(\frac{\mu-\epsilon}{\tau}$$
a term is missing because there is NO energy contribution by the s(0) state.

also we defined lambda like $\lambda = \exp(\mu/\tau}$

so
$$z = 1 + \lambda + \lambda \exp(\epsilon/\tau)$$

for b) similar concept applies

c) What is the diff between <N> and <N(epsilon)>?? Is N(epsilon)_ the number of particles whose energy actually contributes??

d) $$<\epsilon_{s(N)}> = \frac{\sum_{N=0}^{1} \sum_{s(0)}^{s(1)} \epsilon_{s(N)} \exp((N\mu-\epsilon)/\tau)) }{z}$$

$$<\epsilon_{s(N)}> = \sum_{N=0}^{1} \frac{\epsilon_{s(N)}\exp(\frac{N\mu-\epsilon}{\tau})}{z}$$

$$<\epsilon_{s(N)}> = \exp(\frac{\mu-\epsilon}{\tau})}{z} = \lambda\exp(-\epsilon/\tau)$$

e) i Dont quite understand

are they saying that we should accommodate N=1 only?? And N=0 is no more valid

 

[_Andreas]

c) <N> is the average number of particles in the system, all energies included. <N(epsilon)> is the average number of particles with the particular energy epsilon in the system.

 

[_Andreas]

d) The possible combinations are N=0, E=0; N=1, E=0; N=1, E=epsilon. You are on the right track in your first two attempts, so I think you should be able to figure it out from here.