This paper discusses the evaluation of hydrodynamic variables in the presence of spontaneous fluctuations, such as in molecular simulations of fluid flows. The principal point is that hydrodynamic variables such as fluid velocity and temperature must be defined in terms of mechanical variables such as momentum and energy density). Because these relations are nonlinear and because fluctuations of mechanical variables are correlated, care must be taken to avoid introducing a bias when evaluating means, variances, and correlations of hydrodynamic variables. The unbiased estimates are formulated; some alternative, incorrect approaches are presented as cautionary warnings. The expressions are verified by numerical simulations, both at thermodynamic equilibrium and at a nonequilibrium steady state.
Alejandro Garcia. "Estimating Hydrodynamic Quantities in the Presence of Microscopic Fluctuations" Communications in Applied Mathematics and Computational Science (2006): 53-78.