Estimate the partition function by analyzing a graphic

In summary, the conversation discusses the partition function Z and how it is derived by summing all Boltzmann factors. The author presents a different method for computing the partition function by measuring the y values directly from a graph. However, the author's method only provides ratios of values and does not take into account the estimative of the exponents.
  • #1
LCSphysicist
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Homework Statement
All below.
Relevant Equations
All below
1596796127472.png

I am not sure, but since the partition function Z is just the sum of all Boltzmann Factor
1596796184138.png

We can just add:
1596797180297.png

(some terms don't appear in the image, by the way, the estimative is nice, the result is above ANS)

But i didn't understand what the author did:
1596797273294.png


While i didn't even care about the image of the functions (P(s), what he call by height), his resolution is based on it, not only, he talk nothing about the estimative of the exponents (the values on the abscissa), while my resolution is based on it.

What method he used?? Do you understand?
 

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  • #2
I was confused at first because the first two terms of your sum are missing. What I see starts at ##+e^{-0.7}##.
The author is doing the same as you, but instead of measuring the x coordinates off the graph and computing each y coordinate, he is measuring the y values directly. This gives him cm rather than the actual Boltzmann factor, but he is only interested in ratios of values since it is going to be normalized to a total of 1 anyway.
 
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1. What is the partition function in statistical mechanics?

The partition function is a mathematical concept used in statistical mechanics to describe the distribution of particles in a system. It is a sum over all possible states of a system and is a key component in calculating thermodynamic properties such as energy and entropy.

2. How is the partition function related to the probability of a state?

The partition function is directly related to the probability of a state in a system. The probability of a state is equal to the Boltzmann factor, which is derived from the partition function. The higher the partition function, the higher the probability of a state occurring.

3. How can a graphic be used to estimate the partition function?

A graphic can be used to estimate the partition function by plotting the logarithm of the partition function against the energy levels of a system. The slope of the resulting line is equal to the inverse of the temperature, which can then be used to calculate the partition function.

4. What information is needed to estimate the partition function from a graphic?

To estimate the partition function from a graphic, you will need to know the energy levels of a system and the corresponding degeneracies (number of states with the same energy). This information can then be used to plot a graph and calculate the partition function.

5. Why is it important to accurately estimate the partition function?

The partition function is a fundamental concept in statistical mechanics and is used to calculate various thermodynamic properties. Accurately estimating the partition function allows for more accurate predictions of the behavior of a system, which is crucial in many scientific fields such as chemistry, physics, and biology.

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