arXiv subjects

Giuseppe Procopio

Publications and source records attributed to Giuseppe Procopio.

14 recordsLinked to original sources

arXiv2026

Brownian motion: non-equilibrium states from equilibrium trajectories -- recovering hydrodynamic regimes from prepared displacement measurements

Owing to the Chapman-Kolmogorov equation for Markovian dynamics,any equilibrium trajectory of a Brownian particle in a solvent fluid can be viewed as the superposition of an uncountable number of non-equilibrium states. This property permits the unraveling of fine details of fluid-particle interactions at microscales defined by its non-equilibrium properties from the analysis of a single Brownian trajectory and to connect them to the hydrodynamics of the solvent fluid, simply considering the lower-order (second) moments of particle position in trapped conditions. In this way, the acceleration due to thermal-hydrodynamic fluctuational forces is isolated from the other factors and the short-time displacement statistics is completely determined by the correlation properties of the fluctuational thermal-hydrodynamic force. This approach not only confirms the $t^{5/2}$-law obtained by Boynewicz et al. (2026), related to fluid inertial effects, but indicates that this scaling may be superseded by a $t^4$-scaling at very short times once the correlated nature of the stochastic forcings is taken into account. The latter result is related to the regularity properties of particle velocity realizations.

cond-mat.stat-mech↗

arXiv2026

The fractal dimension of Brownian dynamics in liquids

The classical Einstein-Langevin theory of Brownian motion assumes a memoryless thermal bath, establishing a universal fractal dimension of $d_v = 3/2$ for the velocity fluctuations of a particle. In this Letter, we demonstrate experimentally and theoretically that fluid-inertial memory effects fundamentally redefine the fractal scaling of these fluctuations. In analyzing highly resolved measurements of Brownian microspheres in liquids, we show that the non-Markovian hydrodynamic thermal noise establishes a distinct velocity fractal dimension of $d_v = 7/4$. Coupled with theoretical analysis of non-equilibrium short-time dynamics and the initial scaling of the velocity autocorrelation function, this result establishes the non-equilibrium universality class of Brownian motion in fluid media possessing a finite non-vanishing density.

cond-mat.stat-mech↗

arXiv2025

Steady inertial flow of a compressible fluid in a spatially periodic channel under large pressure drops: a multiscale semi-analytical approach

Spatially-periodic channels are increasingly attracting attention as an efficient alternative to packed columns for a number of analytical and engineering processes. In incompressible flows, the periodic geometry allows to compute the flow structure by solving the Navier-Stokes (NS) equations in the minimal periodic cell of the structure, however large the pressure gradient. Besides, when gas flow under large pressure drops are dealt with, the velocity field is not periodic because of the density dependence on pressure. In this case, the momentum balance equations must be solved numerically on the entire channel, thereby requiring massive computational effort. Based on the marked separation of scales between the length of the periodic cell and the overall channel length characterizing many applications we develop a general method for predicting both the large-scale pressure and velocity profiles, and the small-scale flow structure. The approach proposed is based on the assumption that the local dimensionless pressure drop as a function of the Reynolds number, say $g({\rm Re})$, can be estimated from the solution of the incompressible NS equations within the minimal periodic cell of the channel. From the knowledge of $g({\rm Re})$, the pressure profile $P(Z)$ vs the large scale axial coordinate $Z$ is derived analytically by quadratures. We show how qualitatively different profiles $P(Z)$ can be obtained depending on the equation of state the gas. The approach is validated by comparing the predicted profiles with the full-scale numerical solution of the compressible NS equations in different axially-symmetric periodic channel geometries.

physics.flu-dyn↗

arXiv2025

Green function and singularities in Stokes flow confined by cylindrical walls

In this article, the Green function for the Stokes flow in the interior, exterior, and annular regions bounded by cylindrical walls is derived as a function of the pole position and expressed invariantly both at the field and pole points. Specifically, the Green function is obtained using a cylindrical harmonic expansion of the Stokes flow within the bitensorial formulation. This formulation allows us to obtain higher-order singularities within the same domains, such as the confined Couplet and Stresslet, by simply differentiating the Green function at its pole. Moreover, the confined Sourcelet and its associated multipoles are derived from the Green function through a new method that enforces the reciprocal properties of the Stokes flow. The resulting singularities are then employed to address hydrodynamic problems involving active and passive colloids interacting with cylindrical walls, such as sedimenting particles in the annular cylindrical region and the attractive or repulsive hydrodynamic forces exerted by the cylindrical boundaries on a microswimmer.

physics.flu-dyn↗

arXiv2025

Long-tailed dissipationless hydromechanics: weak thermalization and ergodicity breaking

We analyze the dynamic properties of dissipationless Generalized Langevin Equations in the presence of fluid inertial kernels possessing power-law tails, $k(t) \sim t^{-\kappa}$. While for $\kappa >1$ the dynamics is manifestly non ergodic, no thermalization occurs, and particle motion is ballistic, new phenomena arise for $0 < \kappa <1$. In this case, a form of weak thermalization appears in the presence of thermal/hydrodynamic fluctuations and attractive potentials. However, the absence of dissipation clearly emerges once an external constant force is applied: an asymptotic settling velocity cannot be achieved as the expected value of the particle velocity diverges.

cond-mat.stat-mech↗

arXiv2024

Fluid-particle interactions and fluctuation-dissipation relations I -- General linear theory and basic fluctuational patterns

The article provides a unitary and complete solution to the fluctuation-dissipation relations for particle hydromechanics in a generic fluid, accounting for the hydrodynamic fluid-particle interactions (including arbitrary memory kernels in the description of dissipative and fluid inertial effects) in linear hydrodynamic regimes, via the concepts of fluctuational patterns. This is achieved by expressing the memory kernels as a linear superposition of exponentially decaying modes. Given the structure of the interaction with the internal degrees of freedom, and assuming the representation of the thermal force as a superposition of modal contributions, the fluctuation-dissipation relation follows simply from the moment analysis of the corresponding Fokker-Planck equation, imposing the condition that at equilibrium all the internal degrees of freedom are uncorrelated with particle velocity. Moreover, the functional structure of the resulting equation of motion corresponds to the principle of complete decoupling amongst the internal degrees of freedom. The theory is extended to the case of confined geometries, by generalizing previous results including the effect of fluid inertia.

cond-mat.stat-mech↗

arXiv2024

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

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.

cond-mat.stat-mech↗

arXiv2024

Fluid-particle interactions and fluctuation-dissipation relations III -- Correlated fluctuations, regularity and added mass

The fluctuation-dissipation theory is grounded on the Langevin condition expressing the local independence between the thermal force and the particle velocity history. Upon hydrodynamic grounds, it is reasonable to relax this condition in order to account for the correlated fluid fluctuations, especially in the case of liquids, consistently with the inclusion of acoustic effects and with the finite speed of propagation of internal shear stresses. We show that the introduction of correlated stochastic processes in the basic fluctuational patterns defined in Giona et al. (2024), preserves the global fluctuation-dissipation relation, connecting diffusivity to the global friction factor, and the resulting velocity fluctuations become almost everywhere smooth functions of time. Moreover, a fluctuational added mass arises as a consequence of correlations. This leads to a fluctuation-inertia relation, connecting the fluctuational added mass at microscale to its occurrence for macroscopic objects.

cond-mat.stat-mech↗

arXiv2024

Dynamic fluctuation-dissipation theory for Generalized Langevin Equations: constructive constraints, stability and realizability

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.

cond-mat.stat-mech↗

arXiv2024

On the ergodicity breaking in well-behaved Generalized Langevin Equations

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.

cond-mat.stat-mech↗

arXiv2023

On the theory of body motion in confined Stokesian fluids

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.

physics.flu-dyn↗

arXiv2023

On the Hinch-Kim dualism between singularity and Fax\'en operators in the hydromechanics of arbitrary bodies in Stokes flows

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.

physics.flu-dyn↗

arXiv2023

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, 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.

cond-mat.stat-mech↗

arXiv2023

Another normality is possible. Distributive transformations and emergent Gaussianity

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.

cond-mat.stat-mech↗