Consider the one-dimensional nucleation and growth process.
(a) Let $h(x, t)$ be the density of holes of length $x$ at time $t$. Show that this hole length distribution evolves according to
$$
\left(\frac{\partial}{\partial t}-2 v \frac{\partial}{\partial x}\right) h(x, t)=2 \gamma(t) \int_{x}^{\infty} h(y, t) d y-\gamma(t) x h(x, t)
$$(b) Show that the density of holes (or islands) $n(t)$ and the fraction $1-\rho(t)$ of uncovered space are given by
$$
n(t)=\int_{0}^{\infty} h(x, t) d x, \quad 1-\rho(t)=\int_{0}^{\infty} x h(x, t) d x
$$
(c) Show that for instantaneous nucleation, $\gamma(t)=\sigma \delta(t)$, the governing equation turns into the wave equation $h_{t}-2 v h_{x}=0$ whose general solution is $h(x, t)=$ $h_{0}(x+2 v t) .$ Show that the appropriate initial condition is $h_{0}(x)=\sigma^{2} e^{-\sigma x}$ and therefore $h(x, t)=\sigma^{2} e^{-\sigma x} e^{-2 \sigma v t}$
(d) Show that for homogeneous nucleation, $\gamma(t)=\gamma$ when $t>0$, the hole length distribution becomes
$$
h(x, t)=(\gamma t)^{2} e^{-\gamma x t} e^{-\gamma v t^{2}}
$$
(e) Let $g(x, t)$ be the density of holes of length $x$ at time $t$. Show that this distribution obeys
$$
\begin{aligned}
\left(\frac{\partial}{\partial t}+2 v \frac{\partial}{\partial x}\right) g(x, t)=& \gamma(t)[1-\rho(t)] \delta(x)+2 v \frac{h(0, t)}{[n(t)]^{2}} \\
& \times\left[\int_{0}^{x} g(y, t) g(x-y, t) d y-2 n(t) g(x, t)\right]
\end{aligned}
$$