- #1

snatchingthepi

- 148

- 38

- Homework Statement:
- See post

- Relevant Equations:
- See post

I'm given the following density of states

$$ \Omega(E) = \delta(E) + N\delta(E-\Delta) + \theta(E-\Delta)\left(\frac{1}{\Delta}\right)\left(\frac{E}{N\Delta}\right)^N $$

where $ \Delta $ is a positive constant. From here I have to "calculate the canonical partition function as a function of $$ x=\beta\Delta $$ using the incomplex gamma function

$$ \Gamma(n,x) = \int_x^\infty dt e^{-t} t^{n-1} $$

I know this can be solved for a partition function by taking a Laplace transform of the density of states. I can do the first two term very easily, but for the third term

$$ z_{can} = \int_0^\infty \theta(E-\Delta)\left(\frac{1}{\Delta}\right)\left(\frac{E}{N\Delta}\right)^N exp[-\beta E] dE $$

I'm not sure how to go forward from here. I've never seen an integral like this. I am thinking the step function changes the integral lower bound, but I'm kinda strung out so near the end of term, and am not seeing where to go now. Can anyone please help out?

$$ \Omega(E) = \delta(E) + N\delta(E-\Delta) + \theta(E-\Delta)\left(\frac{1}{\Delta}\right)\left(\frac{E}{N\Delta}\right)^N $$

where $ \Delta $ is a positive constant. From here I have to "calculate the canonical partition function as a function of $$ x=\beta\Delta $$ using the incomplex gamma function

$$ \Gamma(n,x) = \int_x^\infty dt e^{-t} t^{n-1} $$

I know this can be solved for a partition function by taking a Laplace transform of the density of states. I can do the first two term very easily, but for the third term

$$ z_{can} = \int_0^\infty \theta(E-\Delta)\left(\frac{1}{\Delta}\right)\left(\frac{E}{N\Delta}\right)^N exp[-\beta E] dE $$

I'm not sure how to go forward from here. I've never seen an integral like this. I am thinking the step function changes the integral lower bound, but I'm kinda strung out so near the end of term, and am not seeing where to go now. Can anyone please help out?