**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.
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Last edited by zemozamster; Jul 4th 2017 at 01:04 AM.
Reason: Missing 't' inside exp() of modified Hamiltonian under method 2
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