Let conserved momentum remember a molecular impulse

Collisions relax nonconserved velocity distortions rapidly, but they do not destroy total momentum. At long wavelengths a transverse momentum disturbance spreads diffusively through viscosity. Its characteristic radius grows as the square root of elapsed time, so the affected volume grows as time to half the spatial dimension. A fixed total impulse distributed through that increasing volume gives a local amplitude with an algebraic, rather than exponential, decay. Coupling a tagged velocity to this field creates persistent correlation.

Hydrodynamic fluctuations add fluctuating stress and heat flux to deterministic balance laws, with strengths linked to dissipation at thermal equilibrium. These terms represent unresolved microscopic fluctuations after coarse graining; they are not arbitrary external forcing. Averaging the noise to zero does not erase its effects on correlations, because products of fluctuations can survive averaging. Nonlinear couplings among slow modes are central to the tail mechanism and are absent from a purely independent single-particle relaxation ansatz.

The diffusion coefficient is the time integral of a velocity correlation. An algebraic tail can therefore contribute noticeably even after the correlation becomes visually small. In three dimensions a t⁻³ᐟ² tail is integrable and leaves a slowly decaying cutoff error. In a simple two-dimensional t⁻¹ approximation, the integral grows logarithmically with the upper cutoff. Refined self-consistent theories can introduce logarithmic changes, so the simple power law should not be presented as a universal final theorem for every two-dimensional fluid.

Finite simulations introduce their own cutoffs: periodic images return sound and momentum disturbances, and small systems restrict the longest wavelengths. A late-time feature that moves when box size changes can be a finite-size artifact. The original Alder–Wainwright calculations explicitly compared molecular dynamics with collective hydrodynamic flow and considered image interference. Reporting a diffusion coefficient from a finite trace therefore requires specifying the correlation integration window, boundary conditions, and evidence that the relevant tail or cutoff is controlled.

A positive long-time tail is a decaying equilibrium memory, not proof of a kinetic instability. Instabilities require a specified nonequilibrium state and a growing perturbation under its dynamics. Similarly, an unstable discretization can create artificial growth even when every physical mode is damped. A convincing calculation separates physical growth rates, conservation errors, numerical resolution, and finite-size changes. Basic BGK or deterministic diffusion alone cannot reproduce all correlated fluctuation spectra merely by fitting one coefficient.

The sampling lab illustrates another limit of independent models. Reset and set success probability to 0.5 and sample size to 25; predict sample-mean variance 0.01. Reset and set probability to 0.5 and size to 100; predict 0.0025. Its exact binomial law assumes independent samples. Correlated fluid observations can have a larger variance and a different effective sample size. Playback only highlights exact outcomes, so it cannot generate correlated hydrodynamic time traces or discover a long-time tail.