  Josephson junctions have broad applications in metrology, quantum information
processing, and remote sensing. For these applications, the electronic noise is
a limiting factor. In this work we study the thermal noise in narrow Josephson
junctions using a tight-binding Hamiltonian. For a junction longer than the
superconducting coherence length, several self-consistent gap profiles appear
close to a phase difference $\pi$. They correspond to two stable solutions with
an approximately constant phase gradient over the thin superconductor connected
by a $2\pi$ phase slip, and a solitonic branch. The current noise power
spectrum has pronounced peaks at the transition frequencies between the
different states in each branch. We find that the noise is reduced in the
gradient branches in comparison to the zero-length junction limit. In contrast,
the solitonic branch exhibits an enhanced noise and a reduced current due to
the pinning of the lowest excitation energy to close to zero energy.
