Infinite invariant densities due to intermittency in a nonlinear oscillator

Meyer, Philipp G.; Kantz, Holger

doi:10.1103/physreve.96.022217

Cited by 21 publications

(18 citation statements)

References 27 publications

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“…Several AR(2) processes with different periods therefore lead to a successive increase of the background noise for longer time. This emergent scaling is an alternative interpretation to dynamical long range correlations, the origin of which is still not fully understood [5,7,24].…”

Section: Decomposition Of One Time Seriesmentioning

confidence: 99%

A simple decomposition of European temperature variability capturing the variance from days to a decade

Meyer

Kantz

2019

Clim Dyn

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We analyze European temperature variability from station data with the method of detrended fluctuation analysis. This method is known to give a scaling exponent indicating long range correlations in time for temperature anomalies. However, by a more careful look at the fluctuation function we are able to explain the emergent scaling behaviour by short time relaxation, the yearly cycle and one additional process. It turns out that for many stations this interannual variability is an oscillatory mode with a period length of approximately 7-8 years, which is consistent with results of other methods. We discuss the spatial patterns in all parameters and validate the finding of the 7-8 year period by comparing stations with and without this mode.

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Section: Decomposition Of One Time Seriesmentioning

confidence: 99%

A simple decomposition of European temperature variability capturing the variance from days to a decade

Meyer

Kantz

2019

Clim Dyn

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“…The goal of this manuscript is to consider the case where S is increasing with time, as opposed to saturating to a limit, but still all this structure remains intact when the appropriate modifications are made, namely we must use the tool of non-normalizable Boltzmann-Gibbs statistics [2,4,15]. This idea, which is discussed at length below, harnesses the tools of infinite-ergodic theory, which has been well established as a mathematical theory for several decades (see e.g., [16][17][18][19][20][21][22][23]), and recently also in other physical systems, such as [10,[24][25][26].…”

Section: Preliminaries a A Recap Of Statistical Mechanicsmentioning

confidence: 99%

Infinite ergodic theory meets Boltzmann statistics

Aghion,

Kessler,

Barkai

2019

Preprint

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We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat external potential field, in contact with a thermal bath. For an infinite system size, the particles may escape the force field and diffuse freely at large length scales. The partition function diverges and hence the standard canonical ensemble fails. This is replaced with tools stemming from infinite ergodic theory. The Boltzmann factor exp(−V (x)/kBT ), even though not normalized, still describes integrable observables, like energy and occupation times. This infinite density is derived heuristically using an entropy maximization principle, as well as via a first-principles calculation using an eigenfunction expansion in the continuum of low-energy states. It provides a statistical law for regions in space where the force field is important, for example particles in the vicinity of a wall or in an optical trap. The ergodic properties of integrable observables are given by the Aaronson-Darlin-Kac theorem. A generalised virial theorem is derived, showing how the virial coefficient describes the delay in the diffusive spreading of the particles, found at large distances.

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“…Writing that lim t→∞ Z * P (x, t) = exp[−βU (x)], means that the Boltzmann factor is reached at sufficiently long times and plays the role of an infinite invariant density [9][10][11][12][13]. This paves the way to formulating a non-equilibrium statistical framework which is based on concepts from the infinite ergodic theory relating ensemble and time averages of non-normalizable densities.…”

Section: Introductionmentioning

confidence: 99%

Thermodynamics of a Brownian particle in a non-confining potential

Farago

2021

Preprint

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We consider the overdamped Brownian dynamics of a particle starting inside a square potential well which, upon exiting the well, experiences a flat potential where it is free to diffuse. We calculate the particle's probability distribution function (PDF) at coordinate x and time t, P (x, t), by solving the corresponding Smoluchowski equation. The solution is expressed by a multipole expansion, with each term decaying t 1/2 faster than the previous one. At asymptotically large times, the PDF outside the well converges to the Gaussian PDF of a free Brownian particle. The average energy, which is proportional to the probability of finding the particle inside the well, diminishes as E ∼ 1/t 1/2 . Interestingly, we find that the free energy of the particle, F , approaches the free energy of a freely diffusing particle, F0, as δF = F − F0 ∼ 1/t, i.e., at a rate faster than E. We provide analytical and computational evidences that this scaling behavior of δF is a general feature of Brownian dynamics in non-confining potential fields. Furthermore, we argue that δF represents a diminishing entropic component which is localized in the region of the potential, and which diffuses away with the spreading particle without being transferred to the heat bath.

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Infinite invariant densities due to intermittency in a nonlinear oscillator

Cited by 21 publications

References 27 publications

A simple decomposition of European temperature variability capturing the variance from days to a decade

A simple decomposition of European temperature variability capturing the variance from days to a decade

Infinite ergodic theory meets Boltzmann statistics

Thermodynamics of a Brownian particle in a non-confining potential

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