Exact Boltzmann Factor - Comparing y1 & y2 w/ 7 & 8 Particles

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However, for a small number of particles, such as 7 in this example, the "exact" function can provide an accurate approximation.
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The image is showing functions:
y1=e^(ln(n0)-x)
y2=invDigamma(digamma(n0+1)-x)-1

where y2 is the "exact" version. How exactly is it exact?

For this example (with 7 particles):
https://bouman.chem.georgetown.edu/S02/lect21/lect21.htm

if I use
y2 = Round( invDigamma(digamma(n0+1)-0.4*x)-1 )
for x=0, 1, 2, 3
i get (n0, n1, n2, n3) = (3, 2, 1, 1) which agrees with the example.

But for 8 particles I think the correct answer should be (3, 3, 1, 1) which I cannot get with the "exact" function.

Am I thinking about this incorrectly?
 
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The "exact" version of the function is designed to accurately approximate the number of particles for a given value of x. However, it is not always exact because the actual number of particles may not be an integer and thus cannot be represented exactly. In this example, the exact answer for 8 particles is (3, 3, 1, 1), which cannot be represented precisely with the "exact" function. Therefore, you will not get the exact answer using the "exact" version of the function.
 

FAQ: Exact Boltzmann Factor - Comparing y1 & y2 w/ 7 & 8 Particles

1. What is the Exact Boltzmann Factor?

The Exact Boltzmann Factor is a mathematical formula used in statistical mechanics to determine the probability of a particle being in a specific energy state. It takes into account the energy of the particle, the temperature, and the number of particles in the system.

2. How do you calculate the Exact Boltzmann Factor?

The Exact Boltzmann Factor can be calculated using the formula e^(-E/kT), where E is the energy of the particle, k is the Boltzmann constant, and T is the temperature in Kelvin.

3. What is the significance of comparing y1 & y2 w/ 7 & 8 particles?

Comparing y1 and y2 with 7 and 8 particles allows us to see how the Exact Boltzmann Factor changes with a difference of only one particle. This can help us understand how the probability of a particle being in a specific energy state is affected by the number of particles in the system.

4. How does the Exact Boltzmann Factor relate to thermodynamics?

The Exact Boltzmann Factor is a fundamental concept in thermodynamics, as it helps us understand the relationship between energy, temperature, and entropy in a system. It is used to calculate the thermodynamic properties of a system, such as the internal energy and entropy.

5. Can the Exact Boltzmann Factor be applied to all types of particles?

Yes, the Exact Boltzmann Factor can be applied to all types of particles, including atoms, molecules, and even subatomic particles. It is a universal formula that is used to describe the behavior of particles in a wide range of systems.

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