Use the Takahashi method of Section 13.1 for a system of point masses and harmonic springs of length \(a\). Allow the particles to pass through each other, so that the partition function can be evaluated in a closed form. Show that the system is stable at zero pressure. Determine the average distance between particles that are far apart on the chain and the variance of that distance. Determine the structure factor and plot it for several values of the parameter \(m \omega^{2} a^{2} / k T\), where \(m\) is the mass and \(m \omega^{2}\) is the spring constant. Show that the specific heat of this system is independent of temperature, as given by the equipartition theorem.