  We propose a one-dimensional (1D) diffusion equation (heat equation) for
systems in which the diffusion constant (thermal diffusivity) varies
alternately with a spatial period $a$. We solve the time evolution of the field
(temperature) profile from a given initial distribution, by diagonalising the
Hamiltonian, i.e., the Laplacian with alternating diffusion constants, and
expanding the temperature profile by its eigenstates. We show that there are
basically phases with or without edge states. The edge states affect the heat
conduction around heat baths. In particular, rapid heat transfer to heat baths
would be observed in a short time regime, which is estimated to be $t<10^{-2}$s
for $a\sim 10^{-3}$m system and $t< 1$s for $a\sim 10^{-2}$m system composed of
two kinds of familiar metals such as titanium, zirconium and aluminium, gold,
etc. We also discuss the effective lattice model which simplifies the
calculation of edge states up to high energy. It is suggested that these high
energy edge states also contribute to very rapid heat conduction in a very
short time regime.
