# Evaluating partition function (stuck for weeks)

• dumbperson
In summary, the conversation discusses the calculation of a partition function for a particular system. The individual is stuck on an expression for the partition function and is seeking guidance on how to transform it into an integral over a function. A method using a "Hubbard stratonovich" transformation is suggested, which leads to a large and complex partition function. The individual is unsure of how to proceed and is seeking further assistance.
dumbperson
Hi, I'm trying to calculate the partition function for a certain system and I arrived at an expression for the partition function $Z$, and have been stuck here for two weeks at the least. This is not a homework problem. If this is the wrong place to post a question like this, could you please direct me as to where I could ask this? My goal is to express this partition function into an integral over some function, as long as I get the sigmas/spins out.

I will leave out the details of the system (it's not exactly physics), but I'll gladly provide more info if you want any, but it looks like the Ising model.

$$Z = \sum_{\vec{G}}\exp{\left[\sum_{i} \left(\frac{J(M-1)}{2}\sum_{j}\sum_{\alpha}\sigma_{ij}^\alpha - \sum_{\alpha}\sum_{j}\sigma_{ij}^\alpha \frac{\theta_i + \theta_j}{4} + \frac{J}{4} \sum_{j}\left(\sum_{\alpha}\sigma_{ij}^\alpha \right)^2 - \frac{1}{2}\sum_{j}h_{ij}^0 - \frac{JMN}{4} \right) \right]}$$

where $$\sigma_{ij}^\alpha \in \{-1,1\}$$
and the summation over G means to sum over all possible configurations of $$\{\sigma_{ij}^\alpha\}$$
where $i<j$, and $j = 1,...,N$ and $\alpha = 1,...,M$
$$\sum_{\vec{G}} = \sum_{\{\sigma_{ij}^\alpha \} = \pm 1} = \sum_{\sigma_{11}^1 = \pm 1 }\sum_{\sigma_{11}^2 = \pm 1 } ... \sum_{\sigma_{12}^1 = \pm 1 }\sum_{\sigma_{12}^2 = \pm 1 }...$$

$M$ are the amount of different values of $\alpha$ (so alpha goes from 1 to $M$), $N$ are the amount of different values for$i$ and $j, J$ and $\theta_i$ are (unknown) constants, and $h_{ij}^0$ is a function of $J, \theta_i, \theta_j$.Normally what is done with partition functions such as these is that the term inside the exponent is linearized by using a "Hubbard stratonovich" transformation:

$$e^{-\frac{1}{2}Ks^2} = \left(\frac{K}{2\pi} \right)^{1/2}\int e^{-\frac{1}{2}Kx^2 - iKsx} dx$$
$$e^{\frac{1}{2}Ks^2} = \left(\frac{K}{2\pi} \right)^{1/2}\int e^{-\frac{1}{2}Kx^2 + Ksx} dx$$
or

$$e^{\frac{1}{2}\sum_{ij}K_{ij}s_is_j} =\left(\frac{\det{K}}{(2\pi)^N} \right)^{1/2} \int^{\infty}_{-\infty}...\int^{\infty}_{\infty}\prod_{k=1}^Nd\phi_k \exp{\left[-\frac{1}{2}\sum_{ij}\phi_i K_{ij}\phi_j + \sum_{ij} s_iK_{ij}\phi_j\right]}$$

I have tried to use this here, first I tried to write the partition function as

$$Z = \sum_{G}\exp{\left[\sum_i \left(\frac{J(M-1)}{2}m_i - \frac{\theta_i}{2} m_i + \frac{J}{4}m_i^2 - \frac{J}{2}\sum_{j<k}m_{ij}m_{ik} - \frac{1}{2}\sum_j h_{ij}^0 - \frac{JMN}{4} \right) \right]}$$

where
$$m_{ij} = \sum_{\alpha} \sigma_{ij}^\alpha , \qquad m_{i} = \sum_{j}m_{ij}$$

The parts of the exponent with $m_i , m_i^2$ i can write as an integral over some integration variable with subscript $i$, and then I'm left with the " cross terms" . I used that
$$m_{ij}m_{ik} = \frac{1}{2}m_{ij}^2 + \frac{1}{2}m_{ik}^2 + \frac{1}{2}(m_{ij}+m_{ik})^2$$

and this transformation :

$$e^{-\frac{1}{2}Ks^2} = \left(\frac{K}{2\pi} \right)^{1/2}\int e^{-\frac{1}{2}Kx^2 - iKsx} dx$$

which leads to a final partition function that looks like (if I've done everything correctly...) that is enormous and I have no idea what to do with it

$$Z = \exp{\left[- \frac{N^2K}{4}-\frac{1}{2}\sum_i \sum_j h_{ij}^0 \right]}\sum_{G}\left(\prod_i \left(\frac{K}{4\pi M}\right)^{1/2} \int \exp{\left[-\frac{K}{4M}x_i^2 + \frac{K}{2M} m_i x_i + \frac{K(M-1)}{2M}m_i - \frac{\theta_i}{2}m_i\right]}dx_i \right) \\ \prod_{i}\prod_{k<j}\left( \left(\frac{K}{4\pi M} \right)^{1/2}\int \exp{\left[-\frac{K}{4M}x_{ij}^2 + \frac{K}{2M}m_{ij}x_{ij} \right]} dx_{ij} \left(\frac{K}{4\pi M} \right)^{1/2} \int \exp{\left[ -\frac{K}{4M}x_{ik}^2 + \frac{K}{2M}m_{ik}x_{ik} \right]} dx_{ik} \right) \\ \left( \left(\frac{1}{2\pi} \right)^{1/2} \int \exp{\left[-\frac{1}{2}x_{ijk}^2 - i \sqrt{\frac{K}{M}}(m_{ij}+m_{ik})x_{ijk} \right]}dx_{ijk}\right)$$This last expression is really long and in the preview it looked separated over several lines (I used \\) but it doesn't seem to be separated now. How could I do that?

Does anyone have any ideas on how to proceed, or how I could try this differently? I'd greatly appreciate it! :)

I'm sorry. How do I put math symbols within a sentence? I think it used to be within single dollar signs.

No one? :-(

dumbperson said:
I'm sorry. How do I put math symbols within a sentence? I think it used to be within single dollar signs.
Double \$ (as you did now) or double # (for inline math).
We don't have a forum for statistical physics, I moved it to general physics - that is beyond homework level.

## 1. What is a partition function and why is it important in scientific research?

A partition function is a mathematical concept used to calculate the thermodynamic properties of a system. It represents the sum of all possible energy states of a system. It is important in scientific research because it allows us to understand and predict the behavior of complex systems, such as chemical reactions and phase transitions.

## 2. How do you evaluate a partition function?

To evaluate a partition function, you need to know the energy levels of the system, as well as the degeneracy (number of states with the same energy) of each level. Then, you can use the equation Z = Σg_ie^(-E_i/kT), where Z is the partition function, g_i is the degeneracy of the i-th energy level, E_i is the energy of the i-th level, k is the Boltzmann constant, and T is the temperature in Kelvin.

## 3. What are some common challenges when evaluating a partition function?

One common challenge is determining the correct energy levels and their corresponding degeneracies. This can be especially difficult for complex systems. Another challenge is choosing the appropriate temperature range for the system, as the partition function is highly sensitive to changes in temperature.

## 4. How can I check if I have correctly evaluated the partition function?

One way to check the accuracy of your partition function is to compare it with experimental data. If the calculated values match or closely match the experimental values, it is a good indication that the partition function has been correctly evaluated. Additionally, you can check for physical consistency, such as ensuring that the partition function is always positive and increases with increasing temperature.

## 5. Can a partition function be used for any type of system?

In theory, a partition function can be used for any system that has well-defined energy levels. However, it is most commonly used in thermodynamics and statistical mechanics to describe the behavior of physical and chemical systems. It may not be applicable to systems that are highly complex or do not have well-defined energy levels.

Replies
26
Views
3K
Replies
1
Views
1K
Replies
1
Views
1K
Replies
3
Views
651
Replies
1
Views
1K
Replies
9
Views
1K
Replies
1
Views
2K
Replies
2
Views
4K
Replies
0
Views
503
Replies
0
Views
822