# gst: pensive billiards

AA << define a new class of plane billiards - the `pensive billiard' - in which the billiard ball travels along the boundary for some distance depending on the incidence angle before reflecting, while preserving the billiard rule of equality of the angles of incidence and reflection. This generalizes so called `puck billiards' (..), as well as a `vortex billiard', i.e. the motion of a point vortex dipole in 2D hydrodynamics on domains with boundary. (AA) prove the variational origin and invariance of a symplectic structure for pensive billiards, as well as study their properties including conditions for a twist map, the existence of periodic orbits, etc. (AA) also demonstrate the appearance of both the golden and silver ratios in the corresponding hydrodynamical vortex setting. Finally, (AA) introduce and describe basic properties of pensive outer billiards. >>

Theodore D. Drivas, Daniil Glukhovskiy, Boris Khesin. Pensive billiards, point vortices, and pucks. arXiv: 2408.03279v1 [math.DS]. Aug 6, 2024.

https://arxiv.org/abs/2408.03279

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

FonT: 'pensive billiard' evokes images in me that could inspire a series of quasi-stochastic short poems ( https://inkpi.blogspot.com ), but (for now) I will abstain.

Keywords: gst, billiards, pensive billiard, puck billiard, vortex billiard

# gst: asymptotic scaling in a one-dimensional billiard

<< The emergence of power laws that govern the large-time dynamics of a one-dimensional billiard of N point particles is analysed. In the initial state, the resting particles are placed in the positive half-line x≥0 at equal distances. Their masses alternate between two distinct values. The dynamics is initialized by giving the leftmost particle a positive velocity. >>

<< Due to elastic inter-particle collisions the whole system gradually comes into motion, filling both right and left half-lines. As shown by S. Chakraborti, et al. (2022), an inherent feature of such a billiard is the emergence of two different modes: the shock wave that propagates in x≥0 and the splash region in x<0. >>

<< Moreover, the behaviour of the relevant observables is characterized by universal asymptotic power-law dependencies. In view of the finite size of the system and of finite observation times, these dependencies only start to acquire a universal character. To analyse them, (AA) set up molecular dynamics simulations using the concept of effective scaling exponents, familiar in the theory of continuous phase transitions. (They) present results for the effective exponents that govern the large-time behaviour of the shock-wave front, the number of collisions, the energies and momentum of different modes and analyse their tendency to approach corresponding universal values. >>️

<< A characteristic feature of the billiard problem (AA) have considered (..) is the lack of a priori randomness, neither in the distribution of masses nor in the inter-particle distances. Therefore, the emergence of the hydrodynamic power-law asymptotics– pointing to the stochastic background of the underlying process– may be interpreted as a kind of self-averaging in the system.  >>️

Taras Holovatch, Yuri Kozitsky, et al. Effective and asymptotic scaling in a one-dimensional billiard problem. arXiv: 2503.20476v1 [cond-mat.stat-mech]. Mar 26, 2025.

https://arxiv.org/abs/2503.20476

Also: billiard, random, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, billiard, randomness 

# gst: chaotic billiards inside mixed curvatures

<< The boundary of a billiard system dictates its dynamics, which can be integrable, mixed, or fully chaotic. >>️

This AA study << introduces two such billiards: a bean-shaped billiard and a peanut-shaped billiard, the latter being a variant of Cassini ovals. Unlike traditional chaotic billiards, these systems incorporate both focusing and defocusing regions along their boundaries, with no neutral segments. >>

AA << examine both classical and quantum dynamics of these billiards and observe a strong alignment between the two perspectives. For classical analysis, the billiard flow diagram and billiard map reveal sensitivity to initial conditions, a hallmark of classical chaos. In the quantum domain, (AA) use nearest-neighbour spacing distribution and spectral complexity as statistical measures to characterise chaotic behaviour. >>

<< Both classical and quantum mechanical analysis are in firm agreement with each other. One of the most striking quantum phenomena (They) observe is the eigenfunction scarring (both scars and super-scars). Scarring phenomena serve as a rich visual manifestation of quantum and classical correspondence, and highlight quantum suppression chaos at a local level. >>

Pranaya Pratik Das, Tanmayee Patra, Biplab Ganguli. Manifestations of chaos in billiards: the role of mixed curvature. arXiv: 2501.08839v1 [nlin.CD]. Jan 15, 2025.

https://arxiv.org/abs/2501.08839

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

Keywords: gst, billiard, chaos

# gst: even tight-binding billiards could exhibit chaotic behaviors

<< Recent works have established universal entanglement properties and demonstrated validity of single-particle eigenstate thermalization in quantum-chaotic quadratic Hamiltonians. However, a common property of all quantum-chaotic quadratic Hamiltonians studied in this context so far is the presence of random terms that act as a source of disorder. >>

AA << introduce tight-binding billiards in two dimensions, which are described by non-interacting spinless fermions on a disorder-free square lattice subject to curved open boundaries. >>

They <<  show that many properties of tight-binding billiards match those of quantum-chaotic quadratic Hamiltonians (..) these properties indeed appear to be consistent with the emergence of quantum chaos in tight-binding billiards. This statement nevertheless needs to be taken with some care since there exist a sub-extensive (in lattice volume) set of single-particle eigenstates that are degenerate in the middle of the spectrum at zero energy (i.e., zero modes), for which the agreement with RMT (random matrix theory) predictions may not be established. >>

Iris Ulcakar, Lev Vidmar. Tight-binding billiards. arXiv:2206.07078v1 [cond-mat.stat-mech]. Jun 14, 2022. 

https://arxiv.org/abs/2206.07078

Also

keyword 'billiard' in FonT

https://flashontrack.blogspot.com/search?q=billiard

keyword 'chaos' | 'chaotic' in Font

https://flashontrack.blogspot.com/search?q=chaos

https://flashontrack.blogspot.com/search?q=chaotic

keyword 'caos' | 'caotico' in Notes (quasi-stochastic poetry)

https://inkpi.blogspot.com/search?q=caos

https://inkpi.blogspot.com/search?q=caotico

keywords: gst, billiard, chaos, chaotic behavior

# s-gst: tracing nonlocal surreal behaviors ...

<< (..) particles at the quantum level can in fact be seen as behaving something like billiard balls rolling along a table, and not merely as the probabilistic smears that the standard interpretation of quantum mechanics suggests. But there’s a catch – the tracks the particles follow do not always behave as one would expect from “realistic” trajectories, but often in a fashion that has been termed “surrealistic” >>

http://www.cifar.ca/assets/researchers-demonstrate-quantum-surrealism/

Dylan H. Mahler, Lee Rozema, et al. Experimental nonlocal and surreal Bohmian trajectories. Science Advances  19 Feb 2016:
Vol. 2, no. 2, e1501466. DOI: 10.1126/science.1501466

http://dx.doi.org/10.1126/science.1501466

http://advances.sciencemag.org/content/2/2/e1501466.full-text.pdf+html

# 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. 

https://arxiv.org/abs/2106.09603

# gst: solitary wave billiards

<<  In the present work (AA) introduce the concept of solitary wave billiards. I.e., instead of a point particle, (they) consider a solitary wave in an enclosed region and explore its collision with the boundaries and the resulting trajectories in cases which for particle billiards are known to be integrable and for cases that are known to be chaotic. A principal conclusion is that solitary wave billiards are generically found to be chaotic even in cases where the classical particle billiards are integrable. However, the degree of resulting chaoticity depends on the particle speed and on the properties of the potential. >>

J. Cuevas-Maraver, P.G. Kevrekidis, H. Zhang. Solitary wave billiards. arXiv: 2203.09489v1 [nlin.PS]. Mar 17, 2022. 

https://arxiv.org/abs/2203.09489

Also 

keyword 'chaos' | 'chaotic' in Font

https://flashontrack.blogspot.com/search?q=chaos

https://flashontrack.blogspot.com/search?q=chaotic

keyword 'caos' | 'caotico' in Notes (quasi-stochastic poetry)

https://inkpi.blogspot.com/search?q=caos

https://inkpi.blogspot.com/search?q=caotico

keyword | 'soliton' in FonT

https://flashontrack.blogspot.com/search?q=soliton

keywords: gst, waves, solitons, billiard, chaos