Partition function from the density of states

  • #1
snatchingthepi
148
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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?
 

Answers and Replies

  • #2
Abhishek11235
175
39
Please tell us exact step where you are stuck. The integral is doable using incomplete gamma functions
 
  • #3
snatchingthepi
148
38
I am unsure *how* to do this integrla with the incomplete gamma function. My thought hit a dead-end at

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

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

let $$ x=\beta \Delta $$

and for the incomplete gamma function let

$$ t = \frac{-xE}{\Delta}, dt = \frac{-x}{\Delta} dE $$

so

$$ z_{can} = \left(\frac{1}{N \Delta}\right)^N \int_x^\infty dt t^N exp[-t] $$

$$ z_{can} = \left(\frac{1}{N \Delta}\right)^N \Gamma(N+1, x) $$

I'm not convinced of my math in these last few steps.
 
Last edited:
  • #4
nrqed
Science Advisor
Homework Helper
Gold Member
3,765
295
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?
You could also replace the factor ##E^N## by ##(-1)^N## times the N-th derivative of the exponential with respect to ##\beta##. The integral will be trivial and then you can apply the N-th derivative wrt ##\beta## on the result to get the final answer.
 
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