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The phenomenon of ergodicity breaking of stochastic dynamics governed by Generalized Langevin Equations (GLE) in the presence of well-behaved exponentially decaying dissipative memory kernels, recently investigated by many authors (Phys. Rev. E {\bf 83} 062102 2011; Phys. Rev. E {\bf 98} 062140 2018; Eur. Phys. J. B {\bf 93} 184 2020), finds, in the dynamic theory of GLE, its simple and natural explanation, related to the concept of dissipative stability. It is shown that the occurrence of ergodicity breakdown for well-behaved dissipative kernels falls, in general, ouside the region of stochastic realizability, and therefore it cannot be observed in physical systems.

Using the initial-value formulation, a dynamic theory for systems evolving according to a Generalized Langevin Equation is developed, providing more restrictive conditions on the existence of equilibrium behavior and its fluctuation-dissipation implications. For systems fulfilling the property of local realizability, that for all the practical purposes corresponds to the postulate of the existence of a Markovian embedding, physical constraints, expressed in the form of dissipative stability and stochastic realizability are derived. If these two properties are met, Kubo theory is constructively recovered, while if one of these conditions is violated a thermodynamic equilibrium behavior does not exist (and this occurs also for `` well-behaved dissipative systems'' according to the classical Kubo theory), with significant implications in the linear response theory.

We propose a theoretical method to decompose the solution of a Stokes flow past a body immersed in a confined fluid in two simpler problems, related separately to the two geometrical elements of these systems: (i) the body immersed in the unbounded fluid (represented by its Fax\'en operators), and (ii) the domain of the confinement (represented by its Stokesian multipoles). Specifically, by using a reflection method, and assuming linear and reciprocal boundary conditions \citep{procopio-giona_pof}, we provide the expression for the velocity field, the forces, torques and higher-order moments acting on the body in terms of: (i) the volume moments of the body in the unbounded ambient flow; (ii) the multipoles in the domain of the confinement; (iii) the collection of all the volumetric moments on the body immersed in all the regular parts of the multipoles considered as ambient flows. A detailed convergence analysis of the reflection method is developed. In the light of practical applications, we estimate the truncation error committed by considering only the lower order moments (thus truncating the matrices) and the errors associated with the approximated expressions available in the literature for force and torques. We apply the theoretical results to the archetypal hydrodynamic system of a sphere with Navier-slip boundary conditions near a plane wall with no-slip boundary conditions, to determine forces and torques on a translating and rotating sphere as a function of the slip length and of the distance of the sphere from the plane. The hydromechanics of a spheroid is also addressed.

We generalize the multipole expansion and the structure of the Fax\'en operator in Stokes flows obtained for bodies with no-slip to generic boundary conditions, addressing the assumptions under which this generalization is conceivable. We show that a disturbance field generated by a body immersed in an ambient flow can be expressed as a multipole expansion the coefficients of which are the moments of the volume forces, independently on the boundary conditions. We find that the dualism between the operator giving the disturbance field of an $n$-th order ambient flow and the $n$-th order Fax\'en operator, referred to as the Hinch-Kim dualism, holds only if the boundary conditions satisfy a property that we call Boundary-Condition reciprocity (BC-reciprocity). If this property is fulfilled, the Fax\'en operators can be expressed in terms of the $(m,n)$-th order geometrical moments of the volume forces (defined in the article). In addition, it is shown that in these cases, the hydromechanics of the fluid-body system is completely determined by the entire set of the Fax\'en operators. Finally, classical boundary conditions of hydrodynamic practice are investigated in the light of this property: boundary conditions for rigid bodies, Newtonian drops at the mechanical equilibrium, porous bodies modeled by the Brinkman equations are BC-reciprocal, while deforming linear elastic bodies, deforming Newtonian drops, non-Newtonian drops and porous bodies modeled by the Darcy equations do not have this property. For Navier-slip boundary conditions on a rigid body, we find the analytical expression for low order Fax\'en operators.

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, we adopt this field as the comprehensive hydrodynamic/fluctuational driver of the kinematic equations of motion. With this description, we show that the Langevin equations for a Brownian particle interacting with a solvent fluid become particularly simple 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. We show that a convenient way for describing hydrodynamic/thermal fluctuations is by expressing them in the form of Extended Poisson-Kac Processes possessing prescribed correlation properties and a continuous velocity density function. We further highlight the importance of higher-order correlation functions in the description of the stochastic hydrodynamic velocity field with special reference to short-time properties of Brownian motion. We conclude by outlining some practical implications in connection with the statistical description of particle motion in confined geometries., Comment: 36 pages, 7 figures

A distributional route to Gaussianity, associated with the concept of Conservative Mixing Transformations in ensembles of random vector-valued variables, is proposed. This route is completely different from the additive mechanism characterizing the application of Central Limit Theorem, as it is based on the iteration of a random transformation preserving the ensemble variance. Gaussianity emerges as a ``supergeneric'' property of ensemble statistics, in the case the energy constraint is quadratic in the norm of the variables. This result puts in a different light the occurrence of equilibrium Gaussian distributions in kinetic variables (velocity, momentum), as it shows mathematically that, in the absence of any other dynamic mechanisms, almost Gaussian distributions stems from the low-velocity approximations of the physical conservation principles. Whenever, the energy constraint is not expressed in terms of quadratic functions (as in the relativistic case), the Juttner distribution is recovered from CMT.