ratio of the population of any state to the population of any other state

# Ratio of the population of any state to the

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ratio of the population of any state to the population of any other state having a lower energy equals 0.99? In other words, what is the minimum temperature needed to guarantee that all states will be equally populated to within 1%? A numerical answer is required. I notice that the lowest ratio will be: ( ) ( ) βε β ε β ε 7 1 5 , , = e P P . Then, I want the temperature such that : βε 7 01 . 0 = e B B B k T k T T k ε ε ε 10 ln 2 7 1 10 ln 7 2 1 7 10 ln 2 = = = (c) Derive a formula for the internal energy U of the entire system of particles. For a very high temperature, say B k T ε 1000 > , approximately what is the value of U? A numerical answer in units of ε N is required. The expected energy of a single particle is given by βε βε βε βε βε βε βε βε βε βε βε βε βε βε βε βε βε βε ε ε ε ε ε ε ε ε β ε 7 4 3 0 7 4 3 7 4 3 0 7 4 3 7 4 3 7 4 3 1 + + + + + + + = = + + + + + + + = = = e e e e e e e e e N E N U e e e e e e e e e Z Z e e E i i i i i At a very high temperature, the states are all equally populated for maximum entropy.

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