Boltzmann distribution derivation.

AI Thread Summary
The discussion focuses on the derivation of the Boltzmann distribution, specifically addressing the calculation of partial derivatives with respect to discrete states like n1, n2, and on. Participants question the applicability of derivatives to discrete variables, noting that derivatives are typically defined for continuous functions. It is clarified that the transition from discrete to continuous variables is valid due to the large number of particles involved, allowing for the use of Stirling's approximation. This approximation facilitates the mathematical treatment of these states as continuous. The conversation emphasizes the importance of understanding this transition in the context of statistical mechanics.
kidsasd987
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please check the video at 5:33.

how can we find the partial derivative w.r.t n1 n2 and on? isn't each state (n1, n2 and on) one discrete state not a continuous variable? is it because we can have multiple particles in the given energy state?

However its a finite discrete number. as far as I know, derivative is defined on continuous(complete) functions.
 
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kidsasd987 said:
However its a finite discrete number. as far as I know, derivative is defined on continuous(complete) functions
Correct. But between 5:16 and 5:20 the transition from discrete to continuous is being made. The reason this can be done is that n is always large (and Stirling's formula is a good approximation)
 
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