# gst: Lévy flights and Lévy walks under stochastic resetting.

<< Stochastic resetting is a protocol of starting anew, which can be used to facilitate the escape kinetics. (AA)  demonstrate that restarting can accelerate the escape kinetics from a finite interval restricted by two absorbing boundaries also in the presence of heavy-tailed, Lévy-type, α

-stable noise. However, the width of the domain where resetting is beneficial depends on the value of the stability index α determining the power-law decay of the jump length distribution. For heavier (smaller α) distributions, the domain becomes narrower in comparison to lighter tails. >>

<< Additionally, (AA) explore connections between Lévy flights (LFs) and Lévy walks (LWs) in the presence of stochastic resetting. First of all, (They) show that for Lévy walks, the stochastic resetting can also be beneficial in the domain where the coefficient of variation is smaller than 1. Moreover, (They) demonstrate that in the domain where LWs are characterized by a finite mean jump duration (length), with the increasing width of the interval, the LWs start to share similarities with LFs under stochastic resetting. >>️

Bartosz Żbik, Bartłomiej Dybiec. Lévy flights and Lévy walks under stochastic resetting. Phys. Rev. E 109, 044147. April 22, 2024.

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.109.044147

Also: keyword Lévy in FonT

https://flashontrack.blogspot.com/search?q=Lévy&m=1

Also: escape, noise, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, escape, noise, stochastic resetting, Lévy

# s-gst: Lévy flight hypothesis, not only for predation ...

<< When sharks and other ocean predators can’t find food, they abandon Brownian motion, the random motion seen in swirling gas molecules, for Lévy flight — a mix of long trajectories and short, random movements found in turbulent fluids [1] >>

<<  Lévy flights interspersed with Brownian motion can describe the animals' hunting patterns. Birds and other animals (including humans) [2] follow paths that have been modeled using Lévy flight (e.g. when searching for food).[3] >>

<< (..) a growing body of research on generative mechanisms suggests that Lévy walks can arise freely as by-products of otherwise innocuous behaviours; consequently their advantageous properties are purely coincidental. This suggests that the Lévy flight foraging hypothesis should be amended, or even replaced, by a simpler and more general hypothesis. This new hypothesis would state that 'Lévy walks emerge spontaneously and naturally from innate behaviours and innocuous responses to the environment but, if advantageous, then there could be selection against losing them'.  [4] >>

[1] - https://en.m.wikipedia.org/wiki/L%C3%A9vy_flight

[2] - Reynolds, Gretchen (January 1, 2014). "Navigating Our World Like Birds and some authors have claimed that the motion of bees". The New York Times.

http://well.blogs.nytimes.com/2014/01/01/navigating-our-world-like-birds-and-bees/?_r=0

[3] - Sims, David W., Reynolds, Andrew M. et al.  "Hierarchical random walks in trace fossils and the origin of optimal search behavior". Proceedings of the National Academy of Sciences. doi:10.1073/pnas.1405966111. Retrieved 16 July 2014.

http://m.pnas.org/content/111/30/11073.abstract

[4] - Reynolds A. Liberating Lévy walk research from the shackles of optimal foraging. Review article. Phys Life Rev. 2015.

http://www.ncbi.nlm.nih.gov/m/pubmed/25835600/?i=5&from=/26205677/related

# gst: near a critical point, switching between exploitation and exploration, to approach life with Lévy's (chaotic) walk

<< Lévy walks are common biological movements. However, the functional advantages of Lévy walks emerging near a critical point are poorly understood. >>

AA << showed that there could be functional advantages associated with Lévy walks emerging near a critical point, including a large dynamic range to stimuli and highly flexible switching between exploitation and exploration. >>

Masato S. Abe. Functional advantages of Lévy walks emerging near a critical point. PNAS.  doi: 10.1073/ pnas.2001548117.  Sep 14, 2020.

https://www.pnas.org/content/early/2020/09/11/2001548117

Chaotic 'Lévy walks' are a good strategy for animals. Riken. Sep 17, 2020.

https://phys.org/news/2020-09-chaotic-lvy-good-strategy-animals.html   

Also

Lévy flight hypothesis, not only for predation ...  Nov 22, 2015.

https://flashontrack.blogspot.com/2015/11/s-gst-levy-flight-hypothesis-not-only.html

keyword 'Lévy' in FonT

https://flashontrack.blogspot.com/search?q=L%C3%A9vy&m=1

# s-gst: controlling light-harvesting in molecular ensembles

AA << propose that the trap is a Frenkel exciton state formed much below the main exciton band edge due to an environmentally induced heavy-tailed Lévy disorder. This points to disorder engineering as a new avenue in controlling light-harvesting in molecular ensembles >>

Merdasa A, Jiménez Á, et al. Single
Lévy states-disorder induced energy funnels in molecular aggregates. Nano Lett. 2014 Dec 10;14(12):6774-81. doi: 10.1021/nl5021188. Epub 2014 Nov 7. PubMed PMID:
25349900.

http://pubs.acs.org/doi/abs/10.1021/nl5021188

also

"Frenkel exciton state" in:

https://journals.aps.org/search/

https://arxiv.org/find/all/1/all:+AND+state+AND+Frenkel+exciton/0/1/0/all/0/1

# s-gst-music: basic structural patterns

<< Lévy motion model captures basic structural patterns in classical as well as in folk music >>

http://m.phys.org/news/2016-01-musical-melodies-laws-foraging-animals.html

Gunnar A. Niklasson and Maria H. Niklasson. Non-Gaussian distributions of melodic intervals in music: The Lévy-stable approximation. EPLA, 2015, EPL (Europhysics Letters), Volume 112, Number 4. dx.doi.org/10.1209/0295-5075/112/40003

http://iopscience.iop.org/article/10.1209/0295-5075/112/40003

# gst: turbulence spreading and anomalous diffusion on combs.

<< This (AA) paper presents a simple model for such processes as chaos spreading or turbulence spillover into stable regions. In this simple model the essential transport occurs via inelastic resonant interactions of waves on a lattice. The process is shown to result universally in a subdiffusive spreading of the wave field. The dispersion of this spreading process is found to depend exclusively on the type of the interaction process (three- or four-wave), but not on a particular underlying instability. The asymptotic transport equations for field spreading are derived with the aid of a specific geometric construction in the form of a comb. >>

<< The results can be summarized by stating that the asymptotic spreading proceeds as a continuous-time random walk (CTRW) and corresponds to a kinetic description in terms of fractional-derivative equations. The fractional indexes pertaining to these equations are obtained exactly using the comb model. >>

<< A special case of the above theory is a situation in which two waves with oppositely directed wave vectors couple together to form a bound state with zero momentum. This situation is considered separately and associated with the self-organization of wave-like turbulence into banded flows or staircases. >>

<< Overall, (AA) find that turbulence spreading and staircasing could be described based on the same mathematical formalism, using the Hamiltonian of inelastic wave-wave interactions and a mapping procedure into the comb space. Theoretically, the comb approach is regarded as a substitute for a more common description based on quasilinear theory. Some implications of the present theory for the fusion plasma studies are discussed and a comparison with the available observational and numerical evidence is given. >>

Alexander V. Milovanov, Alexander Iomin, Jens Juul Rasmussen. Turbulence spreading and anomalous diffusion on combs. Phys. Rev. E 111, 064217 – Published 24 June, 2025

https://journals.aps.org/pre/abstract/10.1103/cmf5-sf8x#d93038b1-6671-419d-be1e-219e5bf78523

Also: waves, turbulence, walk, self-assembly, instability, chaos, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, waves, turbulence, walk, self-assembly, instability, chaos, comb model, inelastic resonant interactions, inelastic wave-wave interactions, continuous-time random walk, self-organization of wave-like turbulence, Lévy flights, Lévy walks

# gst: when generalized diffusion could result from stochastic processes.

<< Despite the success of fractional Brownian motion (fBm) in modeling systems that exhibit anomalous diffusion due to temporal correlations, recent experimental and theoretical studies highlight the necessity for a more comprehensive approach of a generalization that incorporates heterogeneities in either the tracers or the environment. >>

AA present << a modification of Lévy's representation of fBm for the case in which the generalized diffusion coefficient is a stochastic process. (They) derive analytical expressions for the autocovariance function and both ensemble- and time-averaged mean squared displacements. Further, (AA)  validate the efficacy of the developed framework in two-state systems, comparing analytical asymptotic expressions with numerical simulations. >>️

Adrian Pacheco-Pozo, Diego Krapf. Fractional Brownian motion with fluctuating diffusivities. Phys. Rev. E 110, 014105. Jul 1, 2024.

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.110.014105

Also: disorder, fluctuations, network, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, fractional Brownian motion, fBm, Lévy, disorder, fluctuations, anomalous, network, transition