Boltzmann-Langevin Equation and Hydrodynamic Fluctuations

The one-particle distribution function which satisfies the Boltzmann equation is interpreted as the average of the phase-space function $\Sigma {i=1\mathrm{}}^N\delta (\overrightarrow{\mathrm{r}}-{\overrightarrow{\mathrm{r}}}_i)\delta (\overrightarrow{\mathrm{v}}-{\overrightarrow{\mathrm{v}}}_i)$. The equation of motion for this function is a generalized Langevin equation. This equation is the linear Boltzmann equation to which a fluctuating force term is added. An expression for the second moment of this force, in terms of the Boltzmann kernel and the equilibrium second moment of the distribution function, is derived in analogy with the known procedure involving the Langevin equation. The second moments of the fluctuating pressure tensor and the heat-flow vector are evaluated by using the first Chapman-Enskog approximation. They are equal to the expressions derived by Landau and Lifshitz, using thermodynamic fluctuation theory in relation to the linearized hydrodynamic equations.