Fluctuation-dissipation theorem indicates the presence of hydrodynamic (local thermal) fluctuations in otherwise deterministic theory of viscous hydrodynamics. We have formulated a general theory of thermal fluctuations within causal second-order viscous hydrodynamic evolution of matter formed in relativistic heavy ion collisions. The fluctuation is treated perturbatively on top of boost-invariant longitudinal expansion [(0+1)-dim] as well as for realistic (1+1)-dim expansion. A numerical model simulation was developed for hydrodynamic evolution in (1+1)-dim which was tested to reproduce the analytic results for the Riemann solutions of expansion of matter in vacuum at early times, and the Landau-Khalatnikov wave propagation inside the medium for later times. Numerical simulation of thermal noise is performed for various second-order dissipative evolution equations using lattice QCD equation of state. Phenomenological effects of thermal fluctuations for the two-particle rapidity correlations showed that viscous damping of the correlation is at most ~20%.Further, significant damping was found at small rapidity separations when second-order dissipative hydrodynamics was employed instead of first-order Navier-Stokes theory. As compared to the conformal equation of state, the softer lattice QCD EOS causes reduced propagation of the fluctuations and leads to a pronounced peak in
the rapidity correlations. In contrast to Bjorken flow, enhanced velocity gradients at large rapidities induce quite large fluctuations and thereby larger rapidity correlations.
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