Equilibrium state proof with lagrange

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SUMMARY

The forum discussion centers on the derivation of the equilibrium state probability \( p_j = e^{1-\lambda} \) transitioning to \( p_j = \frac{1}{N} \) as presented in the lecture notes from the University of Texas. The key insight is that since the probability \( p_i \) is uniform across all microstates, the sum of probabilities \( \sum p_i = 1 \) leads to the conclusion that \( N p_i = 1 \), thereby establishing \( p_i = \frac{1}{N} \). This demonstrates the fundamental principle of equal probability in statistical mechanics.

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Guffie
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Hello,

I have been trying to follow the start of these lecture notes I found online and I have having trouble understanding what is happening between two steps.

The notes I am looking at are located:
http://pillowlab.cps.utexas.edu/teaching/CompNeuro10/slides/slides16_EntropyMethods.pdf

On the bottom of slide 4 they get p_j = e^{1-λ} which is clear to me.

The next line says = 1/N.

I am having trouble understanding how they get to 1/N, from my understanding the N should represent the number of micro states however I don't understand how they jump from e^{1-λ} to 1/N.

Would anyone be able to clarify this for me?

Thank you,
 
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The probability is the same for all j, so therefore you have [itex]\sum p_i = 1 = N p_i[/itex].
 

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