# Simplifying Partition Sum

#### zemozamster

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.

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#### Pmb

PHF Hall of Fame
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.

#### studiot

@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?

#### Pmb

PHF Hall of Fame
@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?

#### studiot

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

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#### Pmb

PHF Hall of Fame
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)!

#### zemozamster

@studiot
Thank you for posting the output. My current workstation loads the equations instantly and thus, I was not aware of any issue.

I take it you are dividing the quantity (normalised?) x between a reservoir, represented by q and some excitation levels, represented by p?
Quite similar to this idea.
You can interpret this as a problem of connectivity between two disjoint sets {a}s and {b}s. The system energy is defined by the maximum cost of existing links.