Fluid-particle interactions and fluctuation-dissipation relations II -- Gaussianity and Gaussianity breaking

Abstract

The analysis of fluctuation-dissipation relations developed in Giona et al. (2024) for particle hydromechanics is extended to stochastic forcings alternative to Wiener processes, with the aim of addressing the occurrence of Gaussian equilibrium densities or alternatively the breaking of the Gaussian paradigm at equilibrium. Preliminarly, it is discussed how the determination of the fluctuational patterns starting from the Gaussian approach to Markov processes is practically unfeasible, and the moment analysis provides the simplest way to achieve it. We show the existence of an uncountable family of white-noise processes, different from the distributional derivatives of Wiener processes, and satisfying the requirement of fluctuational independence, i.e. the basic assumption on thermal fluctuations in the Kubo theory based on the Langevin condition. The importance of this extension is that it may provide a transition from mesoscopic to microscopic (event-based) stochastic modeling. In this framework, the derivatives of Wiener processes constitute a very peculiar, albeit continuous, element of this class. The fluctuational patterns driven by non-Wiener stochastic forcings display in general non-Gaussian velocity fluctuations at equilibrium, and the Gaussian case is recovered in the limit of small perturbations. Finally, a fully hydromechanic approach to anomalous diffusion is provided, both in the subdiffusive and in the superdiffusive cases.