  The emergent fluctuating hydrodynamics of a viscoelastic fluid modeled by the
multiparticle collision dynamics (MPC) approach is studied. The fluid is
composed of flexible, Gaussian phantom polymers, which interact by local
momentum-conserving stochastic MPC collisions. For comparison, the analytical
solution of the linearized Navier-Stokes equation is calculated, where
viscoelasticity is taken into account by a time-dependent shear relaxation
modulus. The fluid properties are characterized by the transverse velocity
autocorrelation function in Fourier space as well as in real space. Various
polymer lengths are considered---from dumbbells to (near-)continuous polymers.
Viscoelasticity affects the fluid properties and leads to strong correlations,
which overall decay exponentially in Fourier space. In real space, the
center-of-mass velocity autocorrelation function of individual polymers
exhibits a long-time tail independent of polymer length, which decays as
$t^{-3/2}$, similar to a Newtonian fluid, in the asymptotic limit $t \to
\infty$. Moreover, for long polymers an additional power-law decay appears at
time scales shorter than the longest polymer relaxation time with the same time
dependence, but negative correlations, and the polymer length dependence
$L^{-1/2}$. Good agreement is found between the analytical and simulation
results.
