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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2017

Paper 2

Replace the sum by the <integral> against the <density of states> and set $x=\epsilon/(kT)$, $y=\mu/T$:
$$
\log\mathcal Z=AV(kT)^{a+1}\int_0^\infty x^a\log(1+e^{y/k-x})\,dx.
$$
Therefore
$$
\boxed{b=a+1,\qquad f(y)=Ak^{a+1}\int_0^\infty x^a\log(1+e^{y/k-x})\,dx,\qquad\log\mathcal Z=VT^bf(\mu/T).}
$$
For the given $a>0$, the <integral> converges at zero and infinity for every finite $y$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 34D / b / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 34D / b / ii / Solution

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Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 34D / b / ii / Solution

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