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Visualizzazione dei post in ordine di data per la query weak. Ordina per pertinenza Mostra tutti i post
Visualizzazione dei post in ordine di data per la query weak. Ordina per pertinenza Mostra tutti i post

venerdì 28 marzo 2025

# gst: noise as endogenous control

<< Systems biology faces the monumental task of synthesizing vast amounts of molecular facts into a cohesive whole. While it is recognized that stochasticity is ubiquitous in the cell, with noise playing a role in gene expression, differentiation, and switching (..), it is still commonly argued that the cell functions in spite of the noisy cellular environment. >>️

<< Stochastic systems have a control-theoretic interpretation in which noise plays the role of endogenous control. In the weak-noise limit, relevant at low temperatures or in large populations, control is optimal and an exact mathematical mapping from noise to control is described, where the maximizing the probability of a state becomes the control objective. >>

<< In Langevin dynamics noise is identified directly with control, while in general Markov jump processes, which include chemical reaction networks and electronic circuits, (AA) use the Doi-Zel'dovich-Grassberger-Goldenfeld-Peliti path integral to identify the `response' or `tilt' field π as control, which is proportional to the noise in the semiclassical limit. This solves the longstanding problem of interpreting π. >>

AA << illustrate the mapping on multistable chemical reaction networks and systems with unstable fixed points. The noise-control mapping builds intuition for otherwise puzzling phenomena of stochastic systems: why the probability is generically a non-smooth function of state out of thermal equilibrium; why biological mechanisms can work better in the presence of noise; and how agentic behavior emerges naturally without recourse to mysticism. >>

Eric De Giuli. Noise equals endogenous control. arXiv: 2503.15670v2 [q-bio.MN]. Mar 24, 2025.

Also: noise, disorder & fluctuations, transition, dance, jazz, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, noise, disorder, transition, ethno, crosstalk, gossip, misinformation, fake news, dance, jazz

mercoledì 5 marzo 2025

# gst: amplification of turbulence through multiple planar shocks.


AA << study the amplification of isotropic, incompressible turbulence through multiple planar, collisional shocks, using analytical linear theory. There are two limiting cases (They) explore. The first assumes shocks occur rapidly in time such that the turbulence does not evolve between shocks. Whereas the second case allows enough time for turbulence to isotropize between each shock. For the latter case, through a quasi-equation-of-state, we show that the weak multi-shock limit is agnostic to the distinction between thermal and vortical turbulent pressures, like an isotropic volumetric compression. >>

<< When turbulence does not return to isotropy between shocks, the generated anisotropy -- itself a function of shock strength -- can feedback on amplification by further shocks, altering choices for maximal or minimal amplification. >>

<< In addition for this case, (AA) find that amplification is sensitive to the shock ordering. (They) map how choices of shock strength can impact these amplification differences due to ordering, finding, for example, shock pairs which lead to identical mean post-shock fields (density, temperature, pressure) but maximally distinct turbulent amplification. >>️

Michael F. Zhang, Seth Davidovits, Nathaniel J. Fisch. Amplification of turbulence through multiple planar shocks. arXiv: 2502.18708v1 [astro-ph.GA]. Feb 25, 2025. 

Also: waves, turbulence, vortex, crack, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, waves, turbulence, vortex, shocks 


venerdì 3 gennaio 2025

# gst: soliton dynamics over a disordered topography

AA << report on the dynamics of a soliton propagating on the surface of a fluid in a 4-m-long canal with a random or periodic bottom topography. Using a full space-and-time resolved wave field measurement, (They) evidence, for the first time experimentally, how the soliton is affected by the disorder, in the context of Anderson localization, and how localization depends on nonlinearity. >>

<< For weak soliton amplitudes, the localization length is found in quantitative agreement with a linear shallow-water theory. For higher amplitudes, this spatial attenuation of the soliton amplitude is found to be enhanced. >>

<< Behind the leading soliton slowed down by the topography, different experimentally unreported dynamics occur: fission into backward and forward nondispersive pulses for the periodic case, and scattering into dispersive waves for the random case. >>

Guillaume Ricard, Eric Falcon. Soliton Dynamics over a Disordered Topography. Phys. Rev. Lett. 133, 264002. Dec 27, 2024.

Also: soliton, waves, disorder, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, soliton, waves, disorder


martedì 31 dicembre 2024

# gst: stochastic adaptation, stochastic resonance.

<< Stochastic resonance is a phenomenon possessed by some nonlinear oscillators, in which a weak signal is boosted by noise. Both the monostable and bistable Duffing oscillators can exhibit this property. However, stochastic resonance has a strong frequency-dependence, as only a band of frequencies may be boosted by noise. >>️

<< Adaptive oscillators are a subset of nonlinear oscillators that can learn the features of an external force. (In this AA work), an adaptive state is added to a Duffing oscillator. This adaptive state enables the Duffing adaptive oscillator to outperform stochastic resonance over a wide range of frequencies by learning a resonance condition. >>️

Edmon Perkins. Comparison of stochastic adaptation and stochastic resonance. Phys. Rev. E 110, 064225. Dec 27, 2024.

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

Keywords: gst, fluctuations, noise, transition, stochasticity, stochastic adaptation, stochastic resonance


venerdì 6 dicembre 2024

# gst: anomalous oscillation modes of (superfluid) pendant droplets; horizontal translation on a flat surface, bouncing off at the corner and vertical oscillations at the edge.

<< Droplets should exhibit various dynamical phenomena when adhered to a surface; not all of them are realized in classical fluids. Visualization of superfluid  4^He (helium-4) pendant droplets revealed that the droplets were horizontally translated on a flat surface, bouncing off at the corner, known as the Noether mode that reflects the translation symmetry. >>️

<< The droplets exhibited another mode in vertical oscillations with high amplitude that included oscillation of the droplet edge. The oscillation period remained constant even as the droplets grew, exhibiting an anomalously weak size dependence. The high mobility of the droplet edges owing to the superfluidity was a crucial factor for the appearance of these anomalous modes. >>️

Keita Onodera, Ryuma Nagatomo, et al. Anomalous Oscillation Modes of Superfluid Pendant Droplets. Phys. Rev. Lett. 133, 216001. Nov 19, 2024.

Also: drop, droplet, droploid, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, drop, droplet, droploid, transition


mercoledì 4 settembre 2024

# gst: attractive-repulsive interaction in coupled qu-oscillators, when a symmetry-breaking state could be a prominent state.

AA << have explored the role of attractive repulsive coupling in shaping the collective behavior of coupled oscillators in the quantum domain. >>

<< A direct simulation (..) showed a symmetry breaking transition from quantum limit cycle to quantum oscillation death state with increasing coupling strength. >>️

<< This is in contrast to the quantum symmetry-breaking transitions reported earlier where inhomogeneity emerges from the homogeneous steady state. >>️

<< The phenomenon is general as it occurs at both weak and deep quantum regime. Specially, in the deep quantum regime where quantum noise is strong, yet it can not wash out the inhomogeneous state indicating that the symmetry-breaking state is indeed a prominent state. >>️️

Bulti Paul, Biswabibek Bandyopadhyay, Tanmoy Banerjee. Attractive-repulsive interaction in coupled quantum oscillators. arXiv: 2408.12972v1 [quant-ph]. Aug 23, 2024.

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

Keywords: gst, transition, symmetry-breaking transition, noise, attractive-repulsive interaction, coupled qu-oscillator


martedì 30 luglio 2024

# gst: collapse of a toroidal bubble inducing shock waves

<< When bubbles collapse near a wall, they typically experience an asymmetric deformation. This collapse leads to the creation of a jet that strikes the bubble interface, causing the formation of a toroidal bubble and the subsequent release of a water-hammer shock. >>️

AA << findings demonstrate that shock waves emitted from the toroidal bubble consistently propagate toward the central axis of the torus, resulting in significant pressure shocks along the axis, similar to the water-hammer shock formed during the collapse of a spherical bubble. >>️

<< In contrast, weak pressure waves are generated in the transverse directions, leading to relatively weaker pressure peaks. Furthermore, the wall-pressure peak induced by the toroidal bubble is approximately three times higher than that induced by the spherical bubble. >>️

Cheng Liu, Xiaobin Yang, et al. Investigations on the shock wave induced by collapse of a toroidal bubble. Phys. Rev. E 110, 015103. Jul 16, 2024. 

Also: bubble, drop, waves, collapse, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, bubble, drop, waves, collapse


venerdì 26 luglio 2024

# gst: Resonancelike emergence of chaos in complex networks of damped-driven nonlinear systems.

AA << solve a critical outstanding problem in this multidisciplinary research field: the emergence and persistence of spatiotemporal chaos in complex networks of damped-driven nonlinear oscillators in the significant weak-coupling regime, while they exhibit regular behavior when uncoupled. >>

They << uncover and characterize the basic physical mechanisms concerning both heterogeneity-induced and impulse-induced emergence, enhancement, and suppression of chaos in starlike and scale-free networks of periodically driven, dissipative nonlinear oscillators. >>️

Ricardo Chacon, Pedro J. Martínez. Resonancelike emergence of chaos in complex networks of damped-driven nonlinear systems. Phys. Rev. E 110, 014209. Jul 19, 2024. 

Also: network, chaos, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, network, resonance, chaos


giovedì 9 maggio 2024

# gst: bubble phases sliding over periodically modulated substrates

AA << analyze a bubble-forming system composed of particles with competing long-range repulsive and short-range attractive interactions driven over a quasi-one-dimensional periodic substrate. >>️

They << find various pinned and sliding phases as a function of substrate strength and drive amplitude. When the substrate is weak, a pinned bubble phase appears that depins elastically into a sliding bubble lattice. For stronger substrates, (AA) find anisotropic bubbles, disordered bubbles, and stripe phases. Plastic depinning occurs via the hopping of individual particles from one bubble to the next in a pinned bubble lattice, and as the drive increases, there is a transition to a state where all of the bubbles are moving but are continuously shedding and absorbing individual particles. This is followed at high drives by a moving bubble lattice in which the particles can no longer escape their individual bubbles. >>️

<< When the bubbles shrink due to an increase in the attractive interaction term, they fit better inside the pinning troughs and become more strongly pinned, leading to a reentrant pinning phase. For weaker attractive terms, the size of the bubbles becomes greater than the width of the pinning troughs and the depinning becomes elastic with a reduced depinning threshold. >>
C. Reichhardt, C.J.O. Reichhardt. Sliding dynamics for bubble phases on periodic modulated substrates. Phys. Rev. Research 6, 023116. May 2, 2024.

Also: bubble, elastic, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, bubble, pinned bubble, elastic, transition




lunedì 6 maggio 2024

# gst: random walk model for dual cascades in wave turbulence.

<< Dual cascades in turbulent systems with two conserved quadratic quantities famously arise in both two-dimensional hydrodynamic turbulence and also in wave turbulence based on four-wave interactions. >>

<< in wave turbulence the systematic spectral fluxes observed in a dual cascade do not require an irreversible dynamical mechanism, rather, they arise as the inevitable outcome of blind chance. >>️️

Oliver Bühler. Random walk model for dual cascades in wave turbulence. Phys. Rev. E 109, 055102. May 1, 2024. 

Also: waves, turbulence, random, weak, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, waves, turbulence, weak turbulence, random, random walks


sabato 4 maggio 2024

# gst: sudden noise-induced expansions beyond the onset of a steady symmetry-breaking bifurcation.

AA << consider fluid flows (..) subject to a steady symmetry-breaking bifurcation and forced by a weak noise acting on a slow timescale. >>

The validity of their proposed model << is tested on the flow past a sudden expansion for a given Reynolds number and different noise amplitudes. At a very low numerical cost, the statistics obtained from the amplitude equation accurately reproduce those of long-time direct numerical simulations. >>

Yves-Marie Ducimetière, Edouard Boujo, François Gallaire. Noise-induced transitions past the onset of a steady symmetry-breaking bifurcation: The case of the sudden expansion. Phys. Rev. Fluids 9, 053905. May 3, 2024. 

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

Keywords: gst, transition, noise



giovedì 18 gennaio 2024

# gst: pseudo epileptic seizures in self-organized bistability

<< Self-organized bistability (SOB) stands as a critical behavior for the systems delicately adjusting themselves to the brink of bistability, characterized by a first-order transition. >>️

(AA) << embark on a theoretical exploration that extends the boundaries of the SOB concept on a higher-order network (implicitly embedded microscopically within a simplicial complex) while considering the limitations imposed by coupling constraints. >>️

AA << use continuous synchronization diagrams and statistical data from spontaneous synchronized events to demonstrate the crucial role SOB plays in initiating and terminating temporary synchronized events. (They) show that under weak coupling consumption, these spontaneous occurrences closely resemble the statistical traits of the epileptic brain functioning. >>
Md Sayeed Anwar, Nikita Frolov, Alexander E. Hramov, Dibakar Ghosh. Self-organized bistability on globally coupled higher-order networks. arXiv: 2401.02825v1 [nlin.AO]. Jan 5, 2024.

Also: transition, self-assembly, brain, in: https://www.inkgmr.net/kwrds.html

Keywords: gst, transition, self-assembly, bistability, self-organized bistability, brain, epileptic seizure

sabato 13 maggio 2023

# gst: the robustness of a edge spiking.


<< Excitable media, ranging from bioelectric tissues and chemical oscillators to forest fires and competing populations, are nonlinear, spatially extended systems capable of spiking. Most investigations of excitable media consider situations where the amplifying and suppressing forces necessary for spiking coexist at every point in space. In this case, spiking requires a fine-tuned ratio between local amplification and suppression strengths. But, in Nature and engineered systems, these forces can be segregated in space, forming structures like interfaces and boundaries. Here, (AA) show how boundaries can generate and protect spiking if the reacting components can spread out: even arbitrarily weak diffusion can cause spiking at the edge between two non-excitable media. This edge spiking is a robust phenomenon that can occur even if the ratio between amplification and suppression does not allow spiking when the two sides are homogeneously mixed. >> ️

Colin Scheibner, Hillel Ori, et al. Spiking at the edge. arXiv:2304.06940v1 [cond-mat.soft]. Apr 14, 2023. 


Keywords: gst, excitable media, weak, spike, edge spiking  




sabato 22 aprile 2023

# gst: when droplets are capable of self-propulsion as if they were surfing on a self-generated wave.


<< active droplets can move autonomously or oscillate between confining walls (..). Those behaviors could provide a clue about how life emerged from inanimate material. >>️

<< in the past decades, it has become clear that weak physical interactions among biomolecules are a crucial part of the answer. Such interactions allow some molecules to stay together transiently while avoiding others, which can lead to the spontaneous formation of droplets whose composition differs from their surroundings. Although biochemist Alexander Oparin suggested such ideas a century ago (..), experimental corroboration arrived only recently >>️

<< The key contribution of Demarchi and his collaborators is to demonstrate that droplet drift can enhance the heterogeneity of substrate and product. The resulting positive feedback allows droplets to move continuously as if they were surfing on a self-generated wave. >>️

David Zwicker. Droplets Come to Life. Physics 16, 45. Mar 20, 2023. 

AA << find that condensates move toward the center of a confining domain when this feedback is weak. Above a feedback threshold, they exhibit self-propulsion, leading to oscillatory dynamics. Moreover, catalysis-driven enzyme fluxes can lead to interrupted coarsening, resulting in equidistant condensate positioning, and to condensate division. >>
Leonardo Demarchi, Andriy Goychuk, et al. Enzyme-Enriched Condensates Show Self-Propulsion, Positioning, and Coexistence. Phys. Rev. Lett. 130, 128401. Mar 20, 2023.

Also

'drop', 'droplet', 'droploid', 'transition' in 

Keywords: gst, drop, droplet, waves, transition, liquid-liquid phase transition, nonequilibrium systems




lunedì 3 gennaio 2022

# gst: weird but not so weird dynamics, basins with tentacles could be common in high-dimensional systems.


<< Basins of attraction are fundamental to the analysis of dynamical systems (..). Over the years, many remarkable properties of basins have been discovered (..), most notably that their geometry can be wild, as exemplified by Wada basins (..), fractal basin boundaries (..), and riddled or intermingled basins (..). Yet despite these foundational studies, much remains to be learned about basins, especially in systems with many degrees of freedom. >>

AA show that for locally-coupled Kuramoto oscillators << high-dimensional basins tend to have convoluted geometries and cannot be approximated by simple shapes such as hypercubes. Although they are impossible to visualize precisely (because of their high dimensionality), (they) present evidence that these basins have long tentacles that reach far and wide and become tangled with each other. Yet sufficiently close to its own attractor, each basin becomes rounder and more simply structured, somewhat like the head of an octopus. >>

<< In terms of (AA) metaphor, almost all of a basin’s volume is in its tentacles, not its head. This finding is not limited to Kuramoto oscillators. (AA) provide a simple geometrical argument showing that, as long as the number of attractors in a system grows subexponentially with system size, the basins are expected to be octopus-like. As further evidence of their genericity, basins of this type were previously found in simulations of jammed sphere packings (..) where they were described as “branched” and “threadlike” away from a central core (..) and accurate methods were developed for computing their volumes (.,). There is also enticing evidence of octopus-like basins in neuronal networks (..), power grids (..), and photonic couplers (..). >>

<< Figure 4 is a further attempt to visualize the structure of high-dimensional basins, now by examining randomly oriented two-dimensional (2D) slices of state space, either far from a twisted state or close to one. (..) Despite the fact that each basin is connected (..)  the basins look fragmented in this 2D slice. >>

 Fig. 4(a): << Perhaps another metaphor than tentacles—a ball of tangled yarn—better captures the essence of the basin structure in this regime, far from any attractor, in which differently colored threads (representing different basins) are interwoven together in an irregular fashion. >>

Fig. 4(b): << The basin structure near an attractor is strikingly different. (..) the basins near an attractor are organized like an onion. >>

Yuanzhao Zhang, Steven H. Strogatz. Basins with tentacles. arXiv: 2106.05709v3 [nlin.AO]. Nov 2, 2021. 



Also

Reshaping Kuramoto model, when a collective dynamics becomes chaotic, with a surprisingly weak coupling. Dec 27, 2021.


Keywords: gst, dynamical systems, high-dimensional systems, Kuramoto oscillator, attractors, basin of attraction 



lunedì 27 dicembre 2021

# gst: reshaping Kuramoto model, when a collective dynamics becomes chaotic, with a surprisingly weak coupling.

<< The emergence of collective synchrony from an incoherent state is a phenomenon essentially described by the Kuramoto model (..) Collective synchronization is a phenomenon in which an ensemble of heterogeneous, self-sustained oscillatory units (commonly known as oscillators) spontaneously entrain their rhythms. This is a pervasive phenomenon observed in natural systems and man-made devices, covering a wide range of spatio-temporal scales, from cell aggregates to swarms of fireflies >>

<< However, this is only partly true, (..) Kuramoto’s perturbative phase-reduction approach is valid for weak coupling. Specifically, oscillator heterogeneity and interactions appear at zeroth and linear orders in the coupling constant, respectively. >> 

AA << have introduced the ‘enlarged Kuramoto model’; a population of phase oscillators in which three-body interactions enter in a perturbative way. Remarkably, this makes a world of difference, drastically reshaping the traditional Kuramoto scenario. The ‘enlarged Kuramoto model’ exhibits a variety of unsteady states, including collective chaos and hyperchaos. >>

Ivan Leon, Diego Pazo. Enlarged Kuramoto Model: Secondary Instability and Transition to Collective Chaos. arXiv: 2112.00176v1 [nlin.AO]. Nov 30, 2021.


Also

More on the three-body problem (695 families of collisionless orbits). FonT. Oct 16, 2017. 


Keywords: gst, behav, instability, Kuramoto model, three-body interactions, chaos, collective chaos, hyperchaos.

giovedì 8 luglio 2021

# gst: apropos of 'disordered interactions', localization and dissociation of bound states and mapping to chaotic billiards concerning two particles on a chain

AA << consider two particles hopping on a chain with a contact interaction between them. At strong interaction, there is a molecular bound state separated by a direct gap from a continuous band of atomic states. Introducing weak disorder in the interaction, the molecular state becomes Anderson localized (exponential localization of all energy eigenstates,). At stronger disorder, part of the molecular band delocalizes and dissociates due to its hybridization to the atomic band. (AA) characterize these different regimes by computing the density of states, the inverse participation ratio, the level-spacing statistics and the survival probability of an initially localized state.  >>️

<< The atomic band is best described as that of a rough billiard for a single particle on a square lattice that shows signatures of quantum chaos. In addition to typical ``chaotic states'', (AA) find states that are localized along only one direction. These ``separatrix states'' are more localized than chaotic states, and similar in this respect to scarred states, but their existence is due to the separatrix iso-energy line in the interaction-free dispersion relation, rather than to unstable periodic orbits. >> 

Hugo Perrin, Janos K. Asboth, et al.  Two particles on a chain with disordered interaction: Localization and dissociation of bound states and mapping to chaotic billiards. arXiv: 2106.09603v1. Jun 17, 2021. 



mercoledì 17 marzo 2021

# phys: trace the birth and evolution of a quasiparticle entity

<< The idea of quasiparticles was first introduced in the 1930s by physicist Lev Landau, who was trying to gain a better understanding of complex quantum systems. >>

<< Over the past decades, physicists worldwide have been trying to gain a better understanding of non-equilibrium dynamics in quantum many-body systems. Some studies investigated what are known as quasiparticles, disturbances or entities in physical systems that exhibit behavior similar to that of particles. >>

In this study AA << observed three distinct regimes of impurity evolution marked by dynamic transitions. These regimes then link initial few-body and later many-body physical dynamics. >>

Ingrid Fadelli. Observing the birth of a quasiparticle. Mar 11, 2021.


<<  These results offer a systematic picture of polaron formation from weak to strong impurity interactions. They reveal three distinct regimes of evolution with dynamical transitions that provide a link between few-body processes and many-body dynamics. Our measurements reveal universal dynamical behaviour in interacting many-body systems and demonstrate new pathways to study non-equilibrium quantum phenomena. >>

Magnus G. Skou, Thomas G. Skov, et al. Non-equilibrium quantum dynamics and formation of the Bose polaron. 
Nat. Phys. (2021). doi: 10.1038/ s41567-021-01184-5. Feb 25, 2021.


"three distinct regimes"  




lunedì 4 gennaio 2021

# gst: apropos of waves perturbed by weak turbulences

<< Wave phenomena are ubiquitous at all scales throughout the Universe, and where there are waves, there’s the potential for wave turbulence. Wave turbulence is the physical state that arises in a continuous medium when a large number of wave modes interact with each other randomly. >>

<< A specific case of wave turbulence in which weakly nonlinear waves propagate in an unbounded space is described by weak turbulence theory. Capturing the dynamics of this regime will be helpful to accurately model large fluid systems. >>

<< In a rotating fluid, the predicted wave spectrum for interacting weakly nonlinear inertial waves is perturbed by the formation of geostrophic modes—columnar vortices aligned with the rotation axis (..) (AA) successfully suppress geostrophic modes with the addition of two honeycomb-patterned plates, allowing them to confirm the predictions of weak turbulence theory. >>

Sergey Nazarenko. Verifying Weak Turbulence Theory.  Physics 13, 194. Dec 14, 2020. 


Eduardo Monsalve, Maxime Brunet, et al. Quantitative Experimental Observation of Weak Inertial-Wave Turbulence. Phys. Rev. Lett. 125, 254502. Dec 14, 2020.





sabato 2 gennaio 2021

# gst: the strong impact on nanosheets by the weak van der Waals force

<< Van der Waals is a weak force that allows neutral molecules to attract one another through randomly fluctuating dipoles, depending on distance. Though small, its effects can be seen in the macro world, like when geckos walk up walls. >> 

<< Van der Waals forces are everywhere and, essentially, at the nanoscale everything is sticky, (..) When you put a large, flat particle on a large, flat surface, there's a lot of contact, and it's enough to permanently deform a particle that's really thin and flexible. >> 
Matt Jones. 

<< the ubiquitous, "weak" van der Waals force was sufficient to indent a rigid silver nanosheet. The phenomenon suggests possible applications in nanoscale optics or catalytic systems. >>

<< In further experiments, (..) nanospheres could be used to control the shape of the deformation, from single ridges when two spheres are close, to saddle shapes or isolated bumps when the spheres are farther apart. >> 

Mike Williams. Weak force has strong impact on nanosheets. Rice University. Dec 15, 2020. 


Sarah M. Rehn, Theodor M. Gerrard-Anderson, et al. Mechanical Reshaping of Inorganic Nanostructures with Weak Nanoscale Forces. Nano Lett. doi: 10.1021/ acs.nanolett.0c03383. Dec 10, 2020. 


Also

keyword 'van der Waals' in FonT