  We study the effects of hydrodynamic fluctuations in non-boost-invariant
longitudinal expansion of matter formed in relativistic heavy ion collisions.
We formulate the theory of thermal noise within second-order viscous
hydrodynamics treating noise as a perturbation on top of the
non-boost-invariant flow. We develop a numerical simulation model to treat the
(1+1)-dimension hydrodynamic evolution. The code is tested to reproduce the
analytic results for the Riemann solver for expansion of matter in vacuum. For
viscous hydrodynamic expansion, the initial energy density distribution are
obtained by reproducing the measured charged hadron rapidity distribution at
the RHIC energies. We show that the longitudinal rapidity correlations arising
from space-time dependent thermal noise and from an induced thermal
perturbation have distinct structures. In general, the rapidity correlations
are found to be dominated by temperature fluctuations at small rapidity
separation and velocity fluctuations at large rapidities. We demonstrate that
thermal noise produce ridge-like two-particle rapidity correlations which
persist at moderately large rapidities. The magnitude and pattern of the
correlations are quite sensitive to various second-order dissipative formalisms
and to the underlying equations of state, especially at large rapidities. The
short-range part of the rapidity correlation is found to be somewhat enhanced
as compared to that in boost-invariant flow of matter.
