Go Back   Physics Help Forum > College/University Physics Help > Advanced Mechanics

Advanced Mechanics Advanced Mechanics Physics Help Forum

Reply
 
LinkBack Thread Tools Display Modes
Old Jul 3rd 2017, 05:21 AM   #1
Junior Member
 
Join Date: Jul 2017
Posts: 1
Simplifying Partition Sum

I have a Hamiltonian of a system as $\displaystyle H(x\in X) = \max\limits_{a,b} \left( p_{ab} + q\frac{ n_b}{ n_a} \right) x_{ab}$
where $\displaystyle n_a=\sum_b x_{ab}, n_b=\sum_a x_{ab}$, and $\displaystyle X = \{ x = [x_{ab}]_{a\in A}^{b\in B} |x_{ab}\in\{0,1\}, n_a\geq 1 \}$. Here, $\displaystyle p_{ab}$ is a random variable with gamma distribution and $\displaystyle q$ is a constant.
I need to simplify/find close-form expression for Partition sum $\displaystyle Z = \sum_{x\in X} e^{-\beta H(x) }$.

My attempts:

Method 1:
Modify Hamiltonian as $\displaystyle H(x,t) = t$ with additional constraints $\displaystyle \left( p_{ab} + q\frac{ n_b}{ n_a} \right) x_{ab} \leq t$ for all $\displaystyle a,b$.
Then I think the partition sum should be $\displaystyle Z = \sum_{x\in X} \int_{0}^{\infty} e^{-\beta H(x,t) } dt$.
I have no clue how to simplify this due to having mixed integer and linear parameters.


Method 2:
Modify Hamiltonian as $\displaystyle H(x) = \frac{1}{t} \ln \frac{1}{AB} \sum_{a,b} \exp \left( t \left( p_{ab} + q\frac{ n_b}{ n_a} \right) x_{ab} \right)$.
As $\displaystyle t\to\infty$, I can obtain the original Hamiltonian.

Then,
$\displaystyle Z = \sum_{x\in X} e^{-\beta H(x) } = \prod\limits_{a,b}\sum\limits_{x_{a,b}=0,1} \sum\limits_{n_a\geq1} \sum\limits_{n_b\geq 0} e^{-\beta H(x) }$
$\displaystyle Z = \sum\limits_{x_{a,b}=0,1} \sum\limits_{n_a\geq1} \sum\limits_{n_b\geq 0} e^{-\beta H(x) } \prod\limits_{a}\delta_{n_a,\sum_b x_{ab}} \prod\limits_{b}\delta_{n_b,\sum_a x_{ab}}$

Substitute $\displaystyle \delta_{n_a,\sum_b x_{ab}} = \int_{0}^{2\pi}\frac{d\lambda}{2\pi}e^{\imath \lambda (n_a-\sum_b x_{ab})}$ to decouple $\displaystyle x_{ab}$ variables.

I have used this method for a simpler form of Hamiltonian such as $\displaystyle H=p_{ab}x_{ab} - q n_a - r n_b$ in which after above step I could separate the variables $\displaystyle x_{ab}$ to a product term where I could substitute $\displaystyle x_{ab}=0,1$ and carryout the integrals.
For above choice of Hamiltonian, I cannot decouple $\displaystyle x_{ab}$ variables.

I would really appreciate if any of you could guide me/provide alternate method to simplify the partition sum.
Thanks.

Last edited by zemozamster; Jul 4th 2017 at 12:04 AM. Reason: Missing 't' inside exp() of modified Hamiltonian under method 2
zemozamster is offline   Reply With Quote
Old Jul 18th 2017, 10:41 AM   #2
Pmb
Physics Team
 
Join Date: Apr 2009
Location: Boston's North Shore
Posts: 912
I'd love to help but all I see is a bunch of strange symbols which looks like Latex code. I don't read Latex code so as such I can't help.

Is there a way for you to post it so I can see math symbols instead of code? Thanks.
Pmb is online now   Reply With Quote
Old Jul 18th 2017, 12:40 PM   #3
Senior Member
 
Join Date: Apr 2015
Location: Somerset, England
Posts: 508
@PMB
The Tex looks OK to me although I had to wait 23 seconds for mathjax to appear and another 20 for it to parse the Tex and display.

Thanks Mash for the update.

@zemozamster
I take it you are dividing the quantity (normalised?) x between a reservoir, represented by q and some excitation levels, represented by p?
studiot is offline   Reply With Quote
Old Jul 18th 2017, 12:55 PM   #4
Pmb
Physics Team
 
Join Date: Apr 2009
Location: Boston's North Shore
Posts: 912
Originally Posted by studiot View Post
@PMB
The Tex looks OK to me although I had to wait 23 seconds for mathjax to appear and another 20 for it to parse the Tex and display.
What is mathjax? Could the problem be that I don't have something installed that I need to in order that I see it?
Pmb is online now   Reply With Quote
Old Jul 18th 2017, 01:10 PM   #5
Senior Member
 
Join Date: Apr 2015
Location: Somerset, England
Posts: 508
Hi, PMB

I am not an expert in mathjax.
It is something used by the forum (and many others), whereby the tex or mathml script is sent to another site for translation into something that can be displayed and then sent back to this forum and displayed.

It stopped working a week or so back and I complained so Mash fixed it.
But it is very slow, I have noticed this slowness on other forums too.

If it genuinely does not come up for you try refreshing the page and then waintin one minute.

Meanwhile I have taken a screenshot of the OP. It should blow up well enough to be readable.

Hope this helps
Attached Thumbnails
Simplifying Partition Sum-partition1.jpg  
studiot is offline   Reply With Quote
Old Jul 18th 2017, 01:13 PM   #6
Pmb
Physics Team
 
Join Date: Apr 2009
Location: Boston's North Shore
Posts: 912
Ah ha! I see it now. You were right. All I had to do was wait a bit. Muchas gracias (and no, I don't speak Spanish)!
Pmb is online now   Reply With Quote
Reply

  Physics Help Forum > College/University Physics Help > Advanced Mechanics

Tags
partition, simplifying, sum


« Help is needed | - »

Thread Tools
Display Modes


Similar Physics Forum Discussions
Thread Thread Starter Forum Replies Last Post
Question about simplifying physics equations!? muon321 Theoretical Physics 11 Oct 7th 2013 04:45 PM


Facebook Twitter Google+ RSS Feed