Navier-Stokes hydrodynamics near the critical point with out-of-equilibrium modes
We investigate how slow out-of-equilibrium modes (OEM), introduced to extend the regime of validity of hydrodynamics near the critical point, fundamentally reshape the power spectrum of dynamical density fluctuations. We have used the equation of motion of slow modes for the situation when the extensive nature of thermodynamics is not altered due to the introduction of OEM. We find that the extensivity condition puts an extra constraint on the coupling of OEM with the four divergence of velocity. In the absence of OEM, the dynamic structure factor exhibits three Lorentzian peaks, one at zero frequency ($ω$), called Rayleigh peak and the other two called Brillouin peaks located symmetrically about $ω=0$. When the system is away from the critical point the inclusion of the OEM preserves the symmetry of the distributions but all the peaks acquire reduced widths and the Brillouin peaks adopt unequal heights. Irrespective of the value of OEM (zero or nonzero) the Brillouin peaks vanish near the critical point. The width of the Rayleigh peak substantially reduced in presence of OEM. Such reductions of the widths in presence of the OEM indicates the reduction in the decay rate of the fluctuation which leads to slowing down of decay of the fluctuations, a distinctive feature of the critical point.