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Independent symmetric three-level system

For

N

noninteracting units with energies

- ϵ, 0, ϵ

, put

x = β ϵ

. Then

Z = (1 + 2 cosh x)^{N}, E = - \frac{2 N ϵ s i n h x}{1 + 2 c o s h x},

and

S = N k_{B} [lo g (1 + 2 cosh x) - \frac{2 x s i n h x}{1 + 2 c o s h x}] .

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