216 research outputs found
Relativistic analysis of stochastic kinematics
The relativistic analysis of stochastic kinematics is developed in order to
determine the transformation of the effective diffusivity tensor in inertial
frames. Poisson-Kac stochastic processes are initially considered. For
one-dimensional spatial models, the effective diffusion coefficient
measured in a frame moving with velocity with respect to the rest
frame of the stochastic process can be expressed as .
Subsequently, higher dimensional processes are analyzed, and it is shown that
the diffusivity tensor in a moving frame becomes non-isotropic with
, and ,
where and are the diffusivities parallel and orthogonal
to the velocity of the moving frame. The analysis of discrete Space-Time
Diffusion processes permits to obtain a general transformation theory of the
tensor diffusivity, confirmed by several different simulation experiments.
Several implications of the theory are also addressed and discussed
On the influence of reflective boundary conditions on the statistics of Poisson-Kac diffusion processes
We analyze the influence of reflective boundary conditions on the statistics
of Poisson-Kac diffusion processes, and specifically how they modify the
Poissonian switching-time statistics. After addressing simple cases such as
diffusion in a channel, and the switching statistics in the presence of a
polarization potential, we thoroughly study Poisson-Kac diffusion in fractal
domains. Diffusion in fractal spaces highlights neatly how the modification in
the switching-time statistics associated with reflections against a complex and
fractal boundary induces new emergent features of Poisson-Kac diffusion leading
to a transition from a regular behavior at shorter timescales to emerging
anomalous diffusion properties controlled by walk dimensionality of the fractal
set
Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part II Irreversibility, Norms and Entropies
In this second part, we analyze the dissipation properties of Generalized
Poisson-Kac (GPK) processes, considering the decay of suitable -norms and
the definition of entropy functions. In both cases, consistent energy
dissipation and entropy functions depend on the whole system of primitive
statistical variables, the partial probability density functions , while the corresponding energy
dissipation and entropy functions based on the overall probability density
do not satisfy monotonicity requirements as a function of time.
Examples from chaotic advection (standard map coupled to stochastic GPK
processes) illustrate this phenomenon. Some complementary physical issues are
also addressed: the ergodicity breaking in the presence of attractive
potentials, and the use of GPK perturbations to mollify stochastic field
equations
Markovian nature, completeness, regularity and correlation properties of Generalized Poisson-Kac processes
We analyze some basic issues associated with Generalized Poisson-Kac (GPK)
stochastic processes, starting from the extended notion of the Markovian
condition. The extended Markovian nature of GPK processes is established, and
the implications of this property derived: the associated adjoint formalism for
GPK processes is developed essentially in an analogous way as for the
Fokker-Planck operator associated with Langevin equations driven by Wiener
processes. Subsequently, the regularity of trajectories is addressed: the
occurrence of fractality in the realizations of GPK is a long-term emergent
property, and its implication in thermodynamics is discussed. The concept of
completeness in the stochastic description of GPK is also introduced. Finally,
some observations on the role of correlation properties of noise sources and
their influence on the dynamic properties of transport phenomena are addressed,
using a Wiener model for comparison
Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part I Basic theory
This article introduces the notion of Generalized Poisson-Kac (GPK) processes
which generalize the class of "telegrapher's noise dynamics" introduced by Marc
Kac in 1974, usingPoissonian stochastic perturbations. In GPK processes the
stochastic perturbation acts as a switching amongst a set of stochastic
velocity vectors controlled by a Markov-chain dynamics. GPK processes possess
trajectory regularity (almost everywhere) and asymptotic Kac limit, namely the
convergence towards Brownian motion (and to stochastic dynamics driven by
Wiener perturbations), which characterizes also the long-term/long-distance
properties of these processes. In this article we introduce the structural
properties of GPK processes, leaving all the physical implications to part II
and part III
Modal representation of inertial effects in fluid–particle interactions and the regularity of the memory kernels
This article develops a modal expansion (in terms of functions exponentially decaying with time) of the force acting on a micrometric particle and stemming from fluid inertial effects (usually referred to as the Basset force) deriving from the application of the time-dependent Stokes equation to model fluid–particle interactions. One of the main results is that viscoelastic effects induce the regularization of the inertial memory kernels at t=0, eliminating the 1/√t-singularity characterizing Newtonian fluids. The physical origin of this regularization stems from the finite propagation velocity of the internal shear stresses characterizing viscoelastic constitutive equations. The analytical expression for the fluid inertial kernel is derived for a Maxwell fluid, and a general method is proposed to obtain accurate approximations of it for generic complex viscoelastic fluids, characterized by a spectrum of relaxation times
Multiphase Partitions of Lattice Random Walks
Considering the dynamics of non-interacting particles randomly moving on a
lattice, the occurrence of a discontinuous transition in the values of the
lattice parameters (lattice spacing and hopping times) determines the uprisal
of two lattice phases. In this Letter we show that the hyperbolic hydrodynamic
model obtained by enforcing the boundedness of lattice velocities derived by
Giona (2018) correctly describes the dynamics of the system and permits to
derive easily the boundary condition at the interface, which, contrarily to the
common belief, involves the lattice velocities in the two phases and not the
phase diffusivities. The dispersion properties of independent particles moving
on an infinite lattice composed by the periodic repetition of a multiphase unit
cell are investigated. It is shown that the hyperbolic transport theory
correctly predicts the effective diffusion coefficient over all the range of
parameter values, while the corresponding continuous parabolic models deriving
from Langevin equations for particle motion fail. The failure of parabolic
transport models is shown via a simple numerical experiment
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 -th order ambient flow and the -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 -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
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