The potential energy of a one-dimensional, anharmonic oscillator may be written as

\[
V(q)=c q^{2}-g q^{3}-f q^{4}
\]

where \(c, g\), and \(f\) are positive constants; quite generally, \(g\) and \(f\) may be assumed to be very small in value. Show that the leading contribution of anharmonic terms to the heat capacity of the

oscillator, assumed classical, is given by

\[
\frac{3}{2} k^{2}\left(\frac{f}{c^{2}}+\frac{5}{4} \frac{g^{2}}{c^{3}}ight) T
\]

and, to the same order, the mean value of the position coordinate \(q\) is given by

\[
\frac{3}{4} \frac{g k T}{c^{2}}
\]