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Piotr Garbaczewski

Publications and source records attributed to Piotr Garbaczewski.

At least 37 records · Page 2Linked to original sources
arXiv2011

Dynamics of confined Levy flights in terms of (Levy) semigroups

The master equation for a probability density function (pdf) driven by Lévy noise, if conditioned to conform with the principle of detailed balance, admits a transformation to a contractive strongly continuous semigroup dynamics. Given a priori a functional form of the semigroup potential, we address the ground-state reconstruction problem for generic Lévy-stable semigroups, for {\em all} values of the stability index $μ\in (0,2)$. That is known to resolve an invariant pdf for confined Lévy flights (e.g. the former jump-type process). Jeopardies of the procedure are discussed, with a focus on: (i) when an invariant pdf actually is an asymptotic one, (ii) subtleties of the pdf $μ$-dependence in the vicinity and sharply {\em at} the boundaries 0 and 2 of the stability interval, where jump-type scenarios cease to be valid.

cond-mat.stat-mech↗
arXiv2011

Levy targeting and the principle of detailed balance

We investigate confined Lévy flights under premises of the principle of detailed balance. The master equation admits a transformation to Lévy - Schrödinger 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 $0<μ<2$ of Lévy drivers: given an invariant probability density function (pdf), specify the jump - type dynamics for which this pdf is a long-time asymptotic target. Our ("$μ$-targeting") method is exemplified by Cauchy family and Gaussian target pdfs. We solve the reverse engineering problem for so-called Lévy oscillators: given a quadratic semigroup potential, find an asymptotic pdf for the associated master equation for arbitrary $μ$.

cond-mat.stat-mech↗
arXiv2010

Heavy-tailed targets and (ab)normal asymptotics in diffusive motion

We investigate temporal behavior of probability density functions (pdfs) of paradigmatic jump-type and continuous processes that, under confining regimes, share common heavy-tailed asymptotic (target) pdfs. Namely, we have shown that under suitable confinement conditions, the ordinary Fokker-Planck equation may generate non-Gaussian heavy-tailed pdfs (like e.g. Cauchy or more general Lévy stable distribution) in its long time asymptotics. For diffusion-type processes, our main focus is on their transient regimes and specifically the crossover features, when initially infinite number of the pdf moments drops down to a few or none at all. The time-dependence of the variance (if in existence), $\sim t^γ$ with $0<γ<2$, in principle may be interpreted as a signature of sub-, normal or super-diffusive behavior under confining conditions; the exponent $γ$ is generically well defined in substantial periods of time. However, there is no indication of any universal time rate hierarchy, due to a proper choice of the driver and/or external potential.

cond-mat.stat-mech↗
arXiv2010

Thermalization of random motion in weakly confining potentials

We show that in weakly confining conservative force fields, a subclass of diffusion-type (Smoluchowski) processes, admits a family of "heavy-tailed" non-Gaussian equilibrium probability density functions (pdfs), with none or a finite number of moments. These pdfs, in the standard Gibbs-Boltzmann form, can be also inferred directly from an extremum principle, set for Shannon entropy under a constraint that the mean value of the force potential has been a priori prescribed. That enforces the corresponding Lagrange multiplier to play the role of inverse temperature. Weak confining properties of the potentials are manifested in a thermodynamical peculiarity that thermal equilibria can be approached \it only \rm in a bounded temperature interval $0\leq T < T_{max} =2ε_0/k_B$, where $ε_0$ sets an energy scale. For $T \geq T_{max}$ no equilibrium pdf exists.

cond-mat.stat-mech↗
arXiv2009

Cauchy flights in confining potentials

We analyze confining mechanisms for Lévy flights evolving under an influence of external potentials. Given a stationary probability density function (pdf), we address the reverse engineering problem: design a jump-type stochastic process whose target pdf (eventually asymptotic) equals the preselected one. To this end, dynamically distinct jump-type processes can be employed. We demonstrate that one "targeted stochasticity" scenario involves Langevin systems with a symmetric stable noise. Another derives from the Lévy-Schrödinger semigroup dynamics (closely linked with topologically induced super-diffusions), which has no standard Langevin representation. For computational and visualization purposes, the Cauchy driver is employed to exemplify our considerations.

math-ph↗
arXiv2009

Lévy flights in inhomogeneous environments

We study the long time asymptotics of probability density functions (pdfs) of Lévy flights in different confining potentials. For that we use two models: Langevin - driven and (Lévy - Schrödinger) semigroup - driven dynamics. It turns out that the semigroup modeling provides much stronger confining properties than the standard Langevin one. Since contractive semigroups set a link between Lévy flights and fractional (pseudo-differential) Hamiltonian systems, we can use the latter to control the long - time asymptotics of the pertinent pdfs. To do so, we need to impose suitable restrictions upon the Hamiltonian and its potential. That provides verifiable criteria for an invariant pdf to be actually an asymptotic pdf of the semigroup-driven jump-type process. For computational and visualization purposes our observations are exemplified for the Cauchy driver and its response to external polynomial potentials (referring to Lévy oscillators), with respect to both dynamical mechanisms.

cond-mat.stat-mech↗
arXiv2009

Levy flights and Levy -Schroedinger semigroups

We analyze two different confining mechanisms for Lévy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Levy-Schroedinger semigroups which induce so-called topological Levy processes (Levy flights with locally modified jump rates in the master equation). Given a stationary probability function (pdf) associated with the Langevin-based fractional Fokker-Planck equation, we demonstrate that generically there exists a topological Lévy process with the very same invariant pdf and in the reverse.

cond-mat.stat-mech↗
arXiv2009

Levy flights in confining potentials

We analyze confining mechanisms for Lévy flights. When they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport. Two classes of stochastic jump - type processes are considered: those driven by Langevin equation with Lévy noise and those, named by us topological Lévy processes (occurring in systems with topological complexity like folded polymers or complex networks and generically in inhomogeneous media), whose Langevin representation is unknown and possibly nonexistent. Our major finding is that both above classes of processes stay in affinity and may share common stationary (eventually asymptotic) probability density, even if their detailed dynamical behavior look different. That generalizes and offers new solutions to a reverse engineering (e.g. targeted stochasticity) problem due to I. Eliazar and J. Klafter [J. Stat. Phys. 111, 739, (2003)]: design a Lévy process whose target pdf equals a priori preselected one. Our observations extend to a broad class of Lévy noise driven processes, like e.g. superdiffusion on folded polymers, geophysical flows and even climatic changes.

cond-mat.stat-mech↗
arXiv2009

Levy flights, dynamical duality and fractional quantum mechanics

We discuss dual time evolution scenarios which, albeit running according to the same real time clock, in each considered case may be mapped among each other by means of an analytic continuation in time. This dynamical duality is a generic feature of diffusion-type processes. Technically that involves a familiar transformation from a non-Hermitian Fokker-Planck operator to the Hermitian operator (e.g. Schroedinger Hamiltonian), whose negative is known to generate a dynamical semigroup. Under suitable restrictions upon the generator, the semigroup admits an analytic continuation in time and ultimately yields dual motions. We analyze an extension of the duality concept to Levy flights, free and with an external forcing, while presuming that the corresponding evolution rule (fractional dynamical semigroup) is a dual counterpart of the quantum motion (fractional unitary dynamics).

cond-mat.stat-mech↗
arXiv2008

Modular Schrödinger equation and dynamical duality

We discuss quite surprising properties of the one-parameter family of modular (Auberson and Sabatier (1994)) nonlinear Schrödinger equations. We develop a unified theoretical framework for this family. Special attention is paid to the emergent \it dual \rm time evolution scenarios which, albeit running in the \it real time \rm parameter of the pertinent nonlinear equation, in each considered case, may be mapped among each other by means of an "imaginary time" transformation (more seriously, an analytic continuation in time procedure).

quant-ph↗
arXiv2008

Entropy and time: Thermodynamics of diffusion processes

We give meaning to the first and second laws of thermodynamics in case of mesoscopic out-of-equilibrium systems which are driven by diffusion processes. The notion of the entropy production is analyzed. The role of the Helmholtz extremum principle is contrasted to that of the more familiar entropy extremum principles.

cond-mat.stat-mech↗
arXiv2007

Information dynamics: Temporal behavior of uncertainty measures

We carry out a systematic study of uncertainty measures that are generic to dynamical processes of varied origins, provided they induce suitable continuous probability distributions. The major technical tool are the information theory methods and inequalities satisfied by Fisher and Shannon information measures. We focus on a compatibility of these inequalities with the prescribed (deterministic, random or quantum) temporal behavior of pertinent probability densities.

cond-mat.stat-mech↗
arXiv2007

Information functionals and the notion of (un)certainty: RMT - inspired case

Information functionals allow to quantify the degree of randomness of a given probability distribution, either absolutely (through min/max entropy principles) or relative to a prescribed reference one. Our primary aim is to analyze the "minimum information" assumption, which is a classic concept (R. Balian, 1968) in the random matrix theory. We put special emphasis on generic level (eigenvalue) spacing distributions and the degree of their randomness, or alternatively - information/organization deficit.

quant-ph↗
arXiv2007

Indeterminacy relations in random dynamics

We analyze various uncertainty measures for spatial diffusion processes. In this manifestly non-quantum setting, we focus on the existence issue of complementary pairs whose joint dispersion measure has strictly positive lower bound.

cond-mat.stat-mech↗
arXiv2006

Information dynamics in quantum theory

Shannon entropy and Fisher information functionals are known to quantify certain information-theoretic properties of continuous probability distributions of various origins. We carry out a systematic study of these functionals, while assuming that the pertinent probability density has a quantum mechanical appearance $ρ\doteq |ψ|^2$, with $ψ\in L^2(R)$. Their behavior in time, due to the quantum Schrödinger picture evolution-induced dynamics of $ρ(x,t)$ is investigated as well, with an emphasis on thermodynamical features of quantum motion.

quant-ph↗
arXiv2005

Entropy methods in random motion

We analyze a contrasting dynamical behavior of Gibbs-Shannon and conditional Kullback-Leibler entropies, induced by time-evolution of continuous probability distributions. The question of predominantly purpose-dependent entropy definition for non-equilibrium model systems is addressed. The conditional Kullback-Leibler entropy is often believed to properly capture physical features of an asymptotic approach towards equilibrium. We give arguments in favor of the usefulness of the standard Gibbs-type entropy and indicate that its dynamics gives an insight into physically relevant, but generally ignored in the literature, non-equilibrium phenomena. The role of physical units in the Gibbs-Shannon entropy definition is iscussed.

cond-mat.stat-mech↗
arXiv2005

Differential entropy and time

We give a detailed analysis of the Gibbs-type entropy notion and its dynamical behavior in case of time-dependent continuous probability distributions of varied origins: related to classical and quantum systems. The purpose-dependent usage of conditional Kullback-Leibler and Gibbs (Shannon) entropies is explained in case of non-equilibrium Smoluchowski processes. A very different temporal behavior of Gibbs and Kullback entropies is confronted. A specific conceptual niche is addressed, where quantum von Neumann, classical Kullback-Leibler and Gibbs entropies can be consistently introduced as information measures for the same physical system. If the dynamics of probability densities is driven by the Schrödinger picture wave-packet evolution, Gibbs-type and related Fisher information functionals appear to quantify nontrivial power transfer processes in the mean. This observation is found to extend to classical dissipative processes and supports the view that the Shannon entropy dynamics provides an insight into physically relevant non-equilibrium phenomena, which are inaccessible in terms of the Kullback-Leibler entropy and typically ignored in the literature.

quant-ph↗