1,547 research outputs found
Spectral properties of fractional Fokker-Plank operator for the L\'evy flight in a harmonic potential
We present a detailed analysis of the eigenfunctions of the Fokker-Planck
operator for the L\'evy-Ornstein-Uhlenbeck process, their asymptotic behavior
and recurrence relations, explicit expressions in coordinate space for the
special cases of the Ornstein-Uhlenbeck process with Gaussian and with Cauchy
white noise and for the transformation kernel, which maps the fractional
Fokker-Planck operator of the Cauchy-Ornstein-Uhlenbeck process to the
non-fractional Fokker-Planck operator of the usual Gaussian Ornstein-Uhlenbeck
process. We also describe how non-spectral relaxation can be observed in
bounded random variables of the L\'evy-Ornstein-Uhlenbeck process and their
correlation functions.Comment: 10 pages, 5 figures, submitted to Euro. Phys. J.
Levy targeting and the principle of detailed balance
We investigate confined L\'{e}vy flights under premises of the principle of
detailed balance. The master equation admits a transformation to L\'{e}vy -
Schr\"{o}dinger semigroup dynamics (akin to a mapping of the Fokker-Planck
equation into the generalized diffusion equation). We solve a stochastic
targeting problem for arbitrary stability index of L\'{e}vy drivers:
given an invariant probability density function (pdf), specify the jump - type
dynamics for which this pdf is a long-time asymptotic target. Our
("-targeting") method is exemplified by Cauchy family and Gaussian target
pdfs. We solve the reverse engineering problem for so-called L\'{e}vy
oscillators: given a quadratic semigroup potential, find an asymptotic pdf for
the associated master equation for arbitrary
Long-memory Gaussian processes governed by generalized Fokker-Planck equations
It is well-known that the transition function of the Ornstein-Uhlenbeck
process solves the Fokker-Planck equation. This standard setting has been
recently generalized in different directions, for example, by considering the
so-called -stable driven Ornstein-Uhlenbeck, or by time-changing the
original process with an inverse stable subordinator. In both cases, the
corresponding partial differential equations involve fractional derivatives (of
Riesz and Riemann-Liouville types, respectively) and the solution is not
Gaussian. We consider here a new model, which cannot be expressed by a random
time-change of the original process: we start by a Fokker-Planck equation (in
Fourier space) with the time-derivative replaced by a new fractional
differential operator. The resulting process is Gaussian and, in the stationary
case, exhibits a long-range dependence. Moreover, we consider further
extensions, by means of the so-called convolution-type derivative.Comment: 24, accepted for publicatio
Manifestations of projection-induced memory: General theory and the tilted single file.
Over the years the field of non-Markovian stochastic processes and anomalous diffusion evolved from a specialized topic to mainstream theory, which transgressed the realms of physics to chemistry, biology and ecology. Numerous phenomenological approaches emerged, which can more or less successfully reproduce or account for experimental observations in condensed matter, biological and/or single-particle systems. However, as far as their predictions are concerned these approaches are not unique, often build on conceptually orthogonal ideas, and are typically employed on an ad hoc basis. It therefore seems timely and desirable to establish a systematic, mathematically unifying and clean approach starting from more fine-grained principles. Here we analyze projection-induced ergodic non-Markovian dynamics, both reversible as well as irreversible, using spectral theory. We investigate dynamical correlations between histories of projected and latent observables that give rise to memory in projected dynamics, and rigorously establish conditions under which projected dynamics is Markovian or renewal. A systematic metric is proposed for quantifying the degree of non-Markovianity. As a simple, illustrative but non-trivial example we study single file diffusion in a tilted box, which, for the first time, we solve exactly using the coordinate Bethe ansatz. Our results provide a solid foundation for a deeper and more systematic analysis of projection-induced non-Markovian dynamics and anomalous diffusion
Celebrating Cercignani's conjecture for the Boltzmann equation
Cercignani's conjecture assumes a linear inequality between the entropy and
entropy production functionals for Boltzmann's nonlinear integral operator in
rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities
and spectral gap inequalities, this issue has been at the core of the renewal
of the mathematical theory of convergence to thermodynamical equilibrium for
rarefied gases over the past decade. In this review paper, we survey the
various positive and negative results which were obtained since the conjecture
was proposed in the 1980s.Comment: This paper is dedicated to the memory of the late Carlo Cercignani,
powerful mind and great scientist, one of the founders of the modern theory
of the Boltzmann equation. 24 pages. V2: correction of some typos and one
ref. adde
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