Write $A(k,T)=\gamma k^2+\mu^2(T)$ and use units with $k_B=1$. The <Gaussian functional integral> gives the fluctuation free-energy density, up to terms linear in $T$ that do not affect the heat capacity,
$$
f(T)=\frac T2\int\frac{d^dk}{(2\pi)^d}
\log\frac{A(k,T)}T.
$$
Since the heat capacity per volume is $c=-T\,d^2f/dT^2$,
$$
c=\frac12\int\frac{d^dk}{(2\pi)^d}
\left[1-\frac{2T\dot\mu^2+T^2\ddot\mu^2}{\gamma k^2+\mu^2}
+\frac{T^2(\dot\mu^2)^2}{(\gamma k^2+\mu^2)^2}
\right],
$$
where a dot denotes $d/dT$. Thus
$$
g(T)=2T\frac{d\mu^2}{dT}+T^2\frac{d^2\mu^2}{dT^2},
\qquad
h(T)=T^2\left(\frac{d\mu^2}{dT}\right)^2.
$$
For $\mu^2=a(T-T_c)$, these reduce to $g(T)=2aT$ and $h(T)=a^2T^2$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 303 / 1 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-303/1/b/ii/solution