6,506 research outputs found
The fractional Fisher information and the central limit theorem for stable laws
A new information-theoretic approach to the central limit theorem for stable
laws is presented. The main novelty is the concept of relative fractional
Fisher information, which shares most of the properties of the classical one,
included Blachman-Stam type inequalities. These inequalities relate the
fractional Fisher information of the sum of independent random variables to
the information contained in sums over subsets containing of the random
variables. As a consequence, a simple proof of the monotonicity of the relative
fractional Fisher information in central limit theorems for stable law is
obtained, together with an explicit decay rate
Stochastic processes via the pathway model
After collecting data from observations or experiments, the next step is to
build an appropriate mathematical or stochastic model to describe the data so
that further studies can be done with the help of the models. In this article,
the input-output type mechanism is considered first, where reaction, diffusion,
reaction-diffusion, and production-destruction type physical situations can fit
in. Then techniques are described to produce thicker or thinner tails (power
law behavior) in stochastic models. Then the pathway idea is described where
one can switch to different functional forms of the probability density
function) through a parameter called the pathway parameter.Comment: 15 pages, 7 figures, LaTe
Heat and work distributions for mixed Gauss-Cauchy process
We analyze energetics of a non-Gaussian process described by a stochastic
differential equation of the Langevin type. The process represents a
paradigmatic model of a nonequilibrium system subject to thermal fluctuations
and additional external noise, with both sources of perturbations considered as
additive and statistically independent forcings. We define thermodynamic
quantities for trajectories of the process and analyze contributions to
mechanical work and heat. As a working example we consider a particle subjected
to a drag force and two independent Levy white noises with stability indices
and . The fluctuations of dissipated energy (heat) and
distribution of work performed by the force acting on the system are addressed
by examining contributions of Cauchy fluctuations to either bath or external
force acting on the system
Mesoscopic non-equilibrium thermodynamics approach to non-Debye dielectric relaxation
Mesoscopic non-equilibrium thermodynamics is used to formulate a model
describing non-homogeneous and non-Debye dielectric relaxation. The model is
presented in terms of a Fokker-Planck equation for the probability distribution
of non-interacting polar molecules in contact with a heat bath and in the
presence of an external time-dependent electric field. Memory effects are
introduced in the Fokker-Planck description through integral relations
containing memory kernels, which in turn are used to establish a connection
with fractional Fokker-Planck descriptions. The model is developed in terms of
the evolution equations for the first two moments of the distribution function.
These equations are solved by following a perturbative method from which the
expressions for the complex susceptibilities are obtained as a functions of the
frequency and the wave number. Different memory kernels are considered and used
to compare with experiments of dielectric relaxation in glassy systems. For the
case of Cole-Cole relaxation, we infer the distribution of relaxation times and
its relation with an effective distribution of dipolar moments that can be
attributed to different segmental motions of the polymer chains in a melt.Comment: 33 pages, 6 figure
Exponential Convergence Towards Stationary States for the 1D Porous Medium Equation with Fractional Pressure
We analyse the asymptotic behaviour of solutions to the one dimensional
fractional version of the porous medium equation introduced by Caffarelli and
V\'azquez, where the pressure is obtained as a Riesz potential associated to
the density. We take advantage of the displacement convexity of the Riesz
potential in one dimension to show a functional inequality involving the
entropy, entropy dissipation, and the Euclidean transport distance. An argument
by approximation shows that this functional inequality is enough to deduce the
exponential convergence of solutions in self-similar variables to the unique
steady states
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