On the correlation properties of thermal noise in fluids

The properties of the thermal force driving micron particles in incompressible fluids are studied within the hydrodynamic theory of the Brownian motion. It is shown that the assumption used for the hydrodynamic Langevin equation in its usual form, according to which the random force at a time t and the velocity of the particle at the initial time equal to zero are uncorrelated, leads to super-diffusion of the particle. To obtain the correct Einstein diffusion at long times, the mentioned hypothesis must be abandoned, which however does not contradict causality. The corresponding correlations are explicitly evaluated. We consider also the "color" of thermal noise, recently measured experimentally (Th. Franosch et al., Nature 478, 85 (2011)), and correct the interpretation of these experiments. The time correlation functions for the thermal random force are obtained using the exact solution of the Langevin equation, and on the basis of the theorem that in the linear response theory connects the mobility of the particle and its velocity autocorrelation function.