Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 36A / a

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- Solution a

Solution

For each one-particle state

i

, a boson occupation number can be any

n_{i} \geq 0

. Its contribution to the grand canonical partition function is the geometric series

\sum_{n_{i} = 0}^{\infty} e^{- β n_{i} (ϵ_{i} - μ)} = \frac{1}{1 - e ^{- β (ϵ_{i} - μ)}} .

The noninteracting modes factorize, so

Z = \prod_{i} \frac{1}{1 - e ^{- β (ϵ_{i} - μ)}}, lo g Z = - i \sum lo g (1 - e^{- β (ϵ_{i} - μ)}) .

Convergence requires

μ < min_{i} ϵ_{i}

. Since the lowest energy is zero, the limiting maximum chemical potential is

μ_{m a x} = 0.

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