Speaker
Massimiliano Giona
(University of Rome La Sapienza DICMA)
Description
Langevin equations driven by vector-valued Wiener noise
represent the prototypical model
of evolution equations for a physical system driven
by a deterministic velocity field in the presence of
superimposed stochastic fluctuations. The statistical
nature of a Wiener process
can be
regarded as the natural legacy of a large number ansatz, in which
the effects of many unknown and uncorrelated perturbations justifies
the Gaussian nature for the increments of the stochastic forcing.
Analogously, in dealing with stochastic field equations
(stochastic partial differential equations),
${\partial \phi({\bf x},t)}/{\partial t} = {\mathcal N}[\phi({\bf x},t)]
+ a(\phi({\bf x},t)) \, f_s({\bf x},t)
$,
the most
common assumption for the stochastic spatio-temporal
forcing $f_s({\bf x},t)$ is its delta-correlated
nature
in space and time ("derivative of a Wiener process").
Notwithstanding the analytical advantages, the assumption
of stochastic perturbation of Wiener nature entails some
intrinsic shortcomings. The most striking one is
the unbounded speed of propagation of stochastic
perturbations that, at a microscopic level, is one-to-one
with the fractal nature (almost nowhere differentiability)
of the graph of a generic realization of a Wiener process.
The resolution of the infinite propagation velocity problem
has been proposed by C. Cattaneo in the form of a
hyperbolic diffusion equation, now bearing his name.
In 1974 M. Kac
provided
a simple stochastic model, for which the associated probability
density function is a solution of the Cattaneo equation.
In point of fact, it is well known that the
Cattaneo model in spatial dimension higher than one
does not admit any stochastic interpretation and that
the solutions of the Cattaneo model do not preserve positivity.
In order to overcome this problem and to provide
a stochastic background to the extended thermodynamic
theories of irreversible phenomena, the original
Kac model has been recently extended and generalized in any spatial dimension
via the concept of Generalized Poisson-Kac (GPK) processes.
In this presentation, after a brief review of GPK theory
we discuss some new results and applications
in statistical physics.
Specifically:
(i) Motivated by the title of the present conference
"On the Uniformity of Laws of Nature", it is addressed
how Poisson-Kac and GPK processes permit to resolve the
"singularities" in the solutions of classical
parabolic transport equations. This is not only related to the
resolution of the paradox of infinite propagation velocity,
but involves also the description of boundary-layer dynamics and the
group properties of the associated Markov operator.
(ii) The latter issue is closely related to the
intrinsic "spinorial" statistical description
of GPK processes, that naturally emerges from the relativistic
description of stochastic kinematics.
(iii) It is addressed how the application
of GPK fluctuations in stochastic partial differential
equation ensures the preservation of positivity of the
field variable (if required by physical principles, for instance
whenever $\phi({\bf x},t)$ represents a concentration)
and avoids the occurrence of
diverging correlation function, problem that arises
even in the
simplest (linear) stochastic partial differential equations
in the presence of delta-correlated noise fields.
The most striking example is the Edwards-Wilkinson model in
spatial dimensions higher than one.
(iv) Finally, the application of GPK is addressed in connection
with the modeling of systems of interacting particles.
Primary author
Massimiliano Giona
(University of Rome La Sapienza DICMA)
