Stochastic Hydrodynamic Velocity Field and the Representation of Langevin Equations

The fluctuation-dissipation theorem, in the Kubo original formulation, is based on the decomposition of the thermal agitation forces into a dissipative contribution and a stochastically fluctuating term. This decomposition can be avoided by introducing a stochastic velocity field, with correlation properties deriving from linear response theory. Here, this field is adopted as the comprehensive hydrodynamic/fluctuational driver of the kinematic equations of motion in the absence of any external forcing. With this description, it is shown that the Langevin equations for a Brownian particle interacting with a solvent fluid become particularly simple as it is no longer necessary to integrate the momentum equation, and can be applied even in those cases in which the classical approach, based on the concept of a stochastic thermal force, displays intrinsic difficulties e.g., in the presence of the Basset force. This formulation provides in the overdamped approximation a more regular formulation consistent with the statistical properties of particle velocity. This approach is also applied in the broader context of the dynamic theory of Generalized Langevin Equations in order to analyze the stochastic realizability of the dynamics. A condition based on the spectral properties of the Fredholm operator associated with the Kubo correlation function is derived to assess the stochastic realizability in the broad sense.