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# Quiz2Sol - Quiz 2 Gibbs Distribution 1 Consider two...

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i T B B H 2 2 =0 2 1 0 Bonus question Hint Hint Solution T t ± t ² ² t t ± t ² t ² ² ² ² ² ² Quiz 2: Gibbs Distribution T ε T ε T ε x x x H. t H t H ± ∂ F/∂H T ε T ε x x / x P ε ε/ ± ± ε P ε dε T Z ε/T ε/T F T Z T ε/T ε/T 1. Consider two electrons in a two-level ”atom” in equilibrium at temperature . The level separation is . a) Evaluate the partition function of this system and its free energy. b) Find the entropy and the energy of the system in the limits and . : log (1 + ) , when 1. Suppose the system is now subject to a magnetic ±eld The energy of a spin-up electron is and of a spin-down electron is . c) Evaluate the partition function of this system and its free energy. d) Evaluate the linear magnetic susceptibility = ( ) in the limits and . : cosh 1 + 2, when 1. Suppose that we are now dealing with a large number of two-level systems and the level spacing is distributed with the probability distribution function ( ) = ² exp ( ²). e) Evaluate approximately the mean value of the magnetic susceptibility = ( ) ( ) in the limit ². a) = 1 + 4 exp ( ) + exp ( 2 ) = log = log [1 + exp ( 2 ) + 4 exp ( )] 1

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T T T T T T T T T B B B B B B B B B H B B T T B b B ± ² B ± b B ±² B ± B ± T t i I I 2 2 2 2 2 2 2 =0
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