# gst: increase transport with disorder

<< Overcoming the detrimental effect of disorder at the nanoscale is very hard since disorder induces localization and an exponential suppression of transport efficiency. (AA) unveil novel and robust quantum transport regimes achievable in nanosystems by exploiting long-range hopping. (They) demonstrate that in a 1D disordered nanostructure in the presence of long-range hopping, transport efficiency, after decreasing exponentially with disorder at first, is then enhanced by disorder  [disorder-enhanced transport regime] until, counterintuitively, it reaches a disorder-independent transport regime, persisting over several orders of disorder magnitude in realistic systems. >>️

Nahum C. Chávez, Francesco Mattiotti, et al. Disorder-Enhanced and Disorder-Independent Transport with Long-Range Hopping: Application to Molecular Chains in Optical Cavities. Phys. Rev. Lett. 126, 153201. Apr 12,  2021.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.153201

Horacio M. Pastawski. Driving Transport with High Disorder. Physics 14, 57. Apr 12, 2021. 

https://physics.aps.org/articles/v14/57

Also

keyword 'disorder' in FonT

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

keyword 'disordine' in Notes (quasi-stochastic poetry)

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

# gst: influence of disorder on the spreading and entanglement properties of coined quantum walks.

AA << investigate the influence of disorder on the spreading and entanglement properties of coined quantum walks. Specifically, (AA) consider quantum walks on the line and explore the effects of quenched disorder in the coin operations. (They) find that coin disorder alters the usual ballistic transport properties of coined quantum walks considerably and yields an extremely slow dynamics with strong evidence for localization behavior. (They) investigate this slow dynamics by comparing different properties of the walker occupation probability with the standard Hadamard walk. (They) find that the walker distribution, and a number of properties associated with it, are significantly altered by the coin disorder. Special focus is given to the influence of coin disorder on entanglement properties. (AA) observe that generically, coin disorder decreases the coin-walker entanglement. The behavior of the entanglement properties further supports the premise that coin disorder induces localization in coined quantum walks. >>

Louie Hong Yao, Sascha Wald. Coined Quantum Walks on the Line: Disorder, Entanglement and Localization. arXiv: 2303.15978v1 [quant-ph]. doi: 10.48550/ arXiv.2303.15978. 28 Mar 28, 2023.

https://arxiv.org/abs/2303.15978

Also

Voli a casaccio. Notes. Oct 01, 2006. 

(quasi-stochastic poetry)

https://inkpi.blogspot.com/2006/10/2066-voli-casaccio.html

keyword 'disorder' in FonT

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

keyword 'disordine' in Notes 

(quasi-stochastic poetry)

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

keyword 'walk' | 'walking' in FonT

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

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

keyword 'passo lieve' | 'walk' | 'walking' in Notes

(quasi-stochastic poetry)

https://inkpi.blogspot.com/search?q=passo+lieve

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

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

Keywords: gst, disorder, quantum physics, walk, walking, coined quantum walks

# gst: chaos controlled and disorder driven phase transitions by breaking permutation symmetry

<< Introducing disorder in a system typically breaks symmetries and can introduce dramatic changes in its properties such as localization. At the same time, the clean system can have distinct many-body features depending on how chaotic it is. >>

<< In this work the effect of permutation symmetry breaking by disorder is studied in a system which has a controllable and deterministic regular to chaotic transition. >>

<< Results indicate a continuous phase transition from an area-law to a volume-law entangled phase irrespective of whether there is chaos or not, as the strength of the disorder is increased. The critical disorder strength obtained by finite size scaling, indicate a strong dependence on whether the clean system is regular or chaotic to begin with. >>

<< Additionally, (AA) find that a relatively small disorder is seen to be sufficient to delocalize a chaotic system. >>

Manju C, Arul Lakshminarayan, Uma Divakaran. Chaos controlled and disorder driven phase transitions by breaking permutation symmetry. arXiv: 2406.00521v1 [quant-ph]. Jun 1, 2024. 

https://arxiv.org/abs/2406.00521

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

Keywords: gst, transition, chaos, disorder 

FonT: who is Manju C?

# gst: when homogeneous systems meet dissipation and disorder.

AA << investigate the localization and topological properties of the non-equilibrium steady state (NESS) in a one-dimensional homogeneous system. (Their) results demonstrate that, despite the absence of disorder in the initial system, the NESS can exhibit localization under bond dissipation. >>

<< Conversely, for an initially flat-band system with compactly localized wave functions, the NESS is delocalized. >>

<< Furthermore, (They) find that the initial localization characteristics of the system significantly influence the localization properties of the NESS. Drawing upon the concept of Bose-Einstein condensate broadening in cold-atom experiments, along with experimental data, (AA) systematically characterize the impact of disorder on the localization and topological properties of the NESS. >>

<< The phase diagram reveals that the NESS can be topologically non-trivial even when the initial state is topologically trivial, and that the topological triviality of the initial state weakens the topological non-triviality of the NESS. >>

Xixi Feng, Ao Zhou, et al. When Homogeneous Systems Meet Dissipation and Disorder. arXiv: 2502.11383v1 [cond-mat.dis-nn]. Feb 17, 2025. 

https://arxiv.org/abs/2502.11383

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

Keywords: gst, order, disorder, disorder & fluctuations, transitions

# gst: evolving disorder and chaos induces acceleration of elastic waves.

<< Static or frozen disorder, characterised by spatial heterogeneities, influences diverse complex systems, encompassing many-body systems, equilibrium and nonequilibrium states of matter, intricate network topologies, biological systems, and wave-matter interactions. >>

AA << investigate elastic wave propagation in a one-dimensional heterogeneous medium with diagonal disorder. (They) examine two types of complex elastic materials: one with static disorder, where mass density randomly varies in space, and the other with evolving disorder, featuring random variations in both space and time. (AA) results indicate that evolving disorder enhances the propagation speed of Gaussian pulses compared to static disorder. Additionally, (They) demonstrate that the acceleration effect also occurs when the medium evolves chaotically rather than randomly over time. The latter establishes that evolving randomness is not a unique prerequisite for observing wavefront acceleration, introducing the concept of chaotic acceleration in complex media. >>️

M. Ahumada, L. Trujillo, J. F. Marín. Evolving disorder and chaos induces acceleration of elastic waves. arXiv: 2403.02113v1 [cond-mat.dis-nn]. Mar 4, 2024. 

https://arxiv.org/abs/2403.02113

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

Keywords: gst, waves, elastic, chaos, transition

# gst: disorder can induce and modulate topological structures

 << One of the boons of topological insulators is that their conducting edge states are protected from disorder in the material’s structure. Turn up the disorder too much, however, and these topological features are usually lost. In 2018, experiments demonstrated the opposite behavior: in so-called topological Anderson insulators (TAIs), the addition of disorder can generate protected edge states, turning a trivial insulator into a topological one. Now, a team (..) reports a new type of TAI realized in a photonic crystal. Their experiments show that the crystal exhibits a wealth of topological phenomena that hadn’t been seen previously. >> 

Matteo Rini. Let Disorder Dictate Topology. Physics 13, s123. Sep 24, 2020. 

https://physics.aps.org/articles/v13/s123    

 AA << directly observe the disorder-induced topological phase transition from a trivial insulator to a TAI with robust chiral edge states. (They) also demonstrate topological heterostructures that host edge states at interfaces between domains with different disorder parameters. >> 

Gui-Geng Liu, Yihao Yang, et al. Topological Anderson Insulator in Disordered Photonic Crystals. Phys. Rev. Lett. 125, 133603. Sep 24, 2020.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.125.133603

# gst: strange attractors in complex networks

<< Disorder and noise in physical systems often disrupt spatial and temporal regularity, yet chaotic systems reveal how order can emerge from unpredictable behavior. Complex networks, spatial analogs of chaos, exhibit disordered, non-Euclidean architectures with hidden symmetries, hinting at spontaneous order. Finding low-dimensional embeddings that reveal network patterns and link them to dimensionality that governs universal behavior remains a fundamental open challenge, as it needs to bridge the gap between microscopic disorder and macroscopic regularities. >>

<< Here, the minimal space revealing key network properties is introduced, showing that non-integer dimensions produce chaotic-like attractors. >>

Pablo Villegas. Strange attractors in complex networks. Phys. Rev. E 111, L042301. Apr 15, 2025. 

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.111.L042301.  

arXiv: 2504.08629v1 [cond-mat.stat-mech] . Apr 11, 2025.

https://arxiv.org/abs/2504.08629

Also: disorder, disorder & fluctuations, noise, network, attractor, chaos, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, disorder, disorder & fluctuations, noise, networks, attractors, self-similarity, chaos 

# gst: when a phase transition could be controlled by a fixed point of infinite disorder

AA << apply a real-space block renormalization group approach to study the critical properties of the random transverse-field Ising spin chain with multispin interactions. First (They) recover the known properties of the traditional model with two-spin interactions by applying the renormalization approach for arbitrary size of the block. For the model with three-spin couplings (They) calculate the critical point and demonstrate that the phase transition is controlled by an infinite disorder fixed point. (AA) have determined the typical correlation-length critical exponent, which seems to be different from that of the random transverse Ising chain with nearest-neighbor couplings. Thus this model represents a new infinite disorder universality class. >>️

Ferenc Iglói, Yu-Cheng Lin. Random quantum Ising model with three-spin couplings. arXiv:2503.18690v1 [cond-mat.dis-nn]. Mar 24, 2025.

https://arxiv.org/abs/2503.18690

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

Keywords: gst, disorder, order, transition

# gst: a case of stabilization by weak noise

<< An elusive state of matter - quantum spin liquid - may actually be enhanced rather than suppressed by disorder as seen in a compound that contains praseodymium, zirconium, and oxygen (Pr2Zr2O7) >>

<< Disorder is generally thought to be detrimental to creating materials with unusual magnetism or other quantum phenomena. However, a team found that weak disorder surprisingly stabilizes a rare quantum state called a quantum spin liquid >>

Stirring up a quantum spin-liquid with disorder. Dec 13, 2017.

https://m.phys.org/news/2017-12-quantum-spin-liquid-disorder.html

J.-J. Wen, S. M. Koohpayeh, et al. Disordered Route to the Coulomb Quantum Spin Liquid: Random Transverse Fields on Spin Ice in Pr2Zr2O7. Phys. Rev. Lett. 2017; 118 (107206). doi: 10.1103/PhysRevLett.118.107206. Mar 8, 2017.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.107206

# gst: conventional and anomalous mobility edges in a quasiperiodic chain.

<< Mobility edges (MEs) constitute the energies separating the localized states from the extended ones in disordered systems. Going beyond this conventional definition, recent proposal suggests for an ME which separates the localized and multifractal states in certain quasiperiodic systems - dubbed as the anomalous mobility edges (AMEs). >>

<< In this study, (AA) propose an exactly solvable quasiperiodic system that hosts both the conventional and anomalous mobility edges under proper conditions. (They) show that with increase in quasiperiodic disorder strength, the system first undergoes a delocalization to localization transition through an ME of conventional type. >>

<< Surprisingly, with further increase in disorder, (They) obtain that a major fraction of the localized states at the middle of the spectrum turn multifractal in nature. Such unconventional behavior in the spectrum results in two AMEs, which continue to exist even for stronger quasiperiodic disorder. >>

AA << numerically obtain the signatures of the coexisting MEs complement it through analytical derivation using Avila's global theory. In the end (They) provide important signatures from the wavepacket dynamics. >>️

Sanchayan Banerjee, Soumya Ranjan Padhi, Tapan Mishra. Emergence of distinct exact mobility edges in a quasiperiodic chain. arXiv: 2503.19834v1 [cond-mat.quant-gas]. Mar 25, 2025.️

https://arxiv.org/abs/2503.19834

Also: edge, order, disorder, waves, transition,  in https://www.inkgmr.net/kwrds.html 

Keywords: gst, edge, order, disorder, waves, transition 

# gst: stagnation points controlling the onset and strength of chaotic fluctuations (in viscoelastic porous media flows)

<< Viscoelastic porous media flows become chaotic beyond critical flow conditions, impacting processes including enhanced oil recovery and targeted drug delivery. Understanding how geometric details of the porous medium affect the onset and strength of the chaotic flows can lead to fundamental insights and potential optimization of such processes. Recently, it has been argued that geometric disorder in the medium suppresses chaotic fluctuations. In contrast, (AA) demonstrate that disorder can also significantly enhance fluctuations given a different originally ordered configuration. (AA) show that the occurrence of stagnation points in the flow field is the vital factor controlling the onset and strength of fluctuation, providing a general and intuitive understanding of how pore geometry affects this important class of complex viscoelastic flows. >>

Simon J. Haward, Cameron C. Hopkins, Amy Q. Shen. Stagnation points control chaotic fluctuations in viscoelastic porous media flow. PNAS. 118 (38)  e2111651118. doi: 10.1073/ pnas.2111651118. Sep 21, 2021. 

https://www.pnas.org/content/118/38/e2111651118

https://twitter.com/ShenU_OIST/status/1438306244998033408

Also

keyword 'elastic' | 'turbulence' | 'disorder' in FonT

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

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

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

keyword 'elastico' | 'turbolento' | 'disordine' in Notes

(quasi-stochastic poetry)

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

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

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

keywords: viscoelastic flows, porous media, stagnation, elastic turbulence, chaos, chaotic fluctuations, geometric disorder.

# gst: intertwining order, disorder, and hierarchy

<< Nature hosts a wealth of materials showcasing intricate structures intertwining order, disorder, and hierarchy, delivering resilient multifunctionality surpassing perfect crystals or simplistic disordered materials. The engineering of such materials through nanoparticle assembly represents a burgeoning field, poised with potential to yield sustainable material systems rivaling or exceeding biological functionalities. >>

AA << review delineates the fundamental concept of complexity in the context of nanoscale materials. >>

Xiaoming Mao, Nicholas Kotov. Complexity, Disorder, and Functionality of Nanoscale Materials. arXiv: 2401.09567v1 [cond-mat.soft]. Jan 17, 2024. 

https://arxiv.org/abs/2401.09567

Also: nano, particle, order, disorder, in  https://www.inkgmr.net/kwrds.html

Keywords: nano, particle, order, disorder 

# gst: intrinsic disorder bars intruders (from biomolecular condensates)

<< It was recently discovered that eukaryotic cells are host to a multiplicity of biomolecular condensates. These condensates typically contain protein and/or RNA components with intrinsically disordered regions (IDRs). While IDRs have been proposed and demonstrated to play many roles in condensate biology, (AA) suggest here an additional crucial role of IDRs, which is to exclude unwanted “intruders” from condensates. This exclusion effect arises from the large conformational entropy of IDRs; i.e., there is a high free-energy cost to occupying space that would otherwise be available to the IDRs. By combining polymer theory with sticker-spacer simulations, (They) show that the relevant insertion free energy increases with the concentration of IDRs in the condensate as well as with intruder size, attaining a linear scaling with surface area for large intruders. >>

AA << find that at realistic IDR concentrations, particles as small as the size of an average protein (4 nm in diameter) can be 97% excluded from condensates. To overcome this entropic barrier, molecules must interact favorably with condensate components to be recruited as clients. Application of the developed size-exclusion theory to biological condensates suggests that condensate IDRs may play a generic exclusionary role across organisms and types of condensates. >>️

Vladimir Grigorev, Ned S. Wingreen, Yaojun Zhang. Conformational Entropy of Intrinsically Disordered Proteins Bars Intruders from Biomolecular Condensates. PRX Life 3, 013011. Feb 14, 2025.

https://journals.aps.org/prxlife/abstract/10.1103/PRXLife.3.013011

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

Keywords: gst, order, disorder, disorder & fluctuations, transitions

gst: actually and counterintuitively a coherent jump could generate disorder.

AA << consider a quantized version of a model for “random walk in random environment.” (..) For a ring geometry (a chain with periodic boundary condition) it features a delocalization-transition as the bias in increased beyond a critical value, indicating that the relaxation becomes underdamped. Counterintuitively, the effective disorder is enhanced due to coherent hopping. >>

Ben Avnit, Doron Cohen. Quantum walk in stochastic environment. Phys. Rev. E 108, 054111. Nov 7, 2023. 

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.108.054111

Also:  Voli a casaccio. Notes (quasi-stochastic poetry). Oct 01, 2006.

https://inkpi.blogspot.com/2006/10/2066-voli-casaccio.html

Also: 'random', 'transition', 'disorder', 'chaos', in https://www.inkgmr.net/kwrds.html

Keywords: gst, walk, random walk, quantum walk, qu-walk, jump, transition, disorder, chaos

# s-gst-music: Hexacorda mollia inside acyclical order and disorder

<< A new composition for string quartet takes listeners on a journey into the weird world of soft matter >>

<< Hexacorda mollia revolves around the theme of order and disorder in soft matter—materials like liquid crystals and biological organisms that can be easily deformed by thermal fluctuations or an external stress, such as an electric or magnetic field. For many of these systems, this form of disorder is required to create ordered structures that can perform some function >>

<< Music is always going through various states of order to disorder and back again, in ways that [scientists] might recognize as phase transitions >>

Katherine Wright. Arts & Culture: A Quartet for Soft Matter. Physics 9, 92. Aug 5, 2016.

http://physics.aps.org/articles/v9/92

# gst: apropos of subdiffusive behaviors, the boomerang effect in classical stochastic models.

<< The phenomenon of Anderson localization, occurring in a disordered medium, significantly influences the dynamics of quantum particles. A fascinating manifestation of this is the “quantum boomerang effect” (QBE), observed when a quantum particle, propelled with a finite initial velocity, reverses its average trajectory, eventually halting at its starting point. >>

AA << uncover evidence of a similar effect in classical systems, characterized by the absence of typical diffusion processes. (Their) investigation encompasses both simplified probabilistic models and more complex phenomenological models that link classical with quantum mechanics. The results indicate that the boomerang effect is not confined to the quantum realm and may also be present in diverse systems exhibiting subdiffusive behavior. >>️

Santiago Zamora, Lisan M. M. Durao, et al. Boomerang effect in classical stochastic models. Phys. Rev. E 111, 024104. Feb 3, 2025.

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.111.024104 

Also: behav, particle, disorder, disorder & fluctuations, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, behavior, particles, disorder, disorder & fluctuations, stocasticity, boomerang effect

# gst: order and chaos in systems of coaxial vortex pairs

Fig. B.12: Ex. with 4 interact. vortex pairs

AA << have analyzed interactions between two and three coaxial vortex pairs, classifying their dynamics as either ordered or chaotic based on strengths, initial sizes, and initial horizontal separations.  >>️

They << found that periodic cases are scattered among chaotic ones across different initial configurations. Quasi-periodic leapfrogging typically occurs when the initial distances between the vortex pairs are small and cannot coexist with vortex-pair overtake. When the initial configuration splits into two interacting vortex pairs and a single propagating vortex pair, the two interacting pairs consistently exhibit periodic leapfrogging. For the smallest initial horizontal separations, the system predominantly exhibits chaotic or quasi-periodic motions rather than periodic leapfrogging with a single frequency. This behavior is due to the strong coupling between all three vortex pairs. When the pairs are in close proximity, more complex and chaotic dynamics emerge instead of periodic motion. >>

Their << findings indicate that quasi-periodic leapfrogging and chaotic interactions generally occur when the three vortex pairs have similar strengths and initial sizes. Conversely, discrepancies in these parameters cause the system to disintegrate into two subsystems: a single propagating vortex pair and two periodically leapfrogging pairs. >>️

️

Christiana Mavroyiakoumou, Wenzheng Shi. Order and Chaos in Systems of Coaxial Vortex Pairs. arXiv: 2502.07002v1 [physics.flu-dyn]. Feb 10, 2025. ️

https://arxiv.org/abs/2502.07002

Also: chaos, vortex, order, disorder, disorder & fluctuations, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, chaos, vortex, order, disorder, disorder & fluctuations

# gst: an analogy between geometrical frustrations and nonreciprocal systems

<< A system is geometrically frustrated when its members cannot find a configuration that simultaneously minimizes all their interaction energies, (..). A nonreciprocal system is one whose members have conflicting, asymmetric goals, as exemplified by an ecosystem of predators and prey. >>️

<< Ryo Hanai (..) has identified a powerful mathematical analogy between those two types of dynamical systems. >>️

Peter Littlewood. Nonreciprocal Frustration Meets Geometrical Frustration. Physics 17, 32. Feb 26, 2024. 

https://physics.aps.org/articles/v17/32

Ryo Hanai. Nonreciprocal Frustration: Time Crystalline Order-by-Disorder Phenomenon and a Spin-Glass-like State. Phys. Rev. X 14, 011029. Feb 26, 2024. 

https://journals.aps.org/prx/abstract/10.1103/PhysRevX.14.011029

Also: order, disorder, disorder & fluctuations, noise, analogy, in https://www.inkgmr.net/kwrds.html

Keywords: gst, analogy, order, disorder, noise

# gst: apropos of disorder & fluctuations

a 'synthetic disorder & fluctuations' from these two blogs ... 

keyword 'disorder' in FonT 

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

keyword 'disordine' in Notes (quasi-stochastic poetry) 

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

keyword 'error' | 'fuzzy' | 'noise'  in FonT

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

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

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

keywords 'errore' | 'errori' in Notes (quasi-stochastic poetry)

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

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

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

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

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

keyword 'waves' in FonT 

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

keyword 'onda' in Notes 

(quasi-stochastic poetry) 

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

keyword 'gst' (General Sistem Theory) in FonT 

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

# gst: criticality and multistability in quasi-2D turbulence

       Fig. 1(a) Helmholtz resonators

<< Two-dimensional (2D) turbulence, despite being an idealization of real flows, is of fundamental interest as a model of the spontaneous emergence of order from chaotic flows. The emergence of order often displays critical behavior, whose study is hindered by the long spatial and temporal scales involved. >>

Here AA << experimentally study turbulence in periodically driven nanofluidic channels with a high aspect ratio using superfluid helium. (They) find a multistable transition behavior resulting from cascading bifurcations of large-scale vorticity and critical behavior at the transition to quasi-2D turbulence consistent with phase transitions in periodically driven many-body systems. >>

AA << demonstrate that quasi-2D turbulent systems can undergo an abrupt change in response to a small change in a control parameter, consistent with predictions for large-scale atmospheric or oceanic flows. >>️

Filip Novotny, Marek Talir, et al. Critical behavior and multistability in quasi-two-dimensional turbulence. arXiv: 2406.08566v1 [physics.flu-dyn]. Jun 12, 2024.

https://arxiv.org/abs/2406.08566

Also: order, disorder, disorder & fluctuations, turbulence, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, order, disorder, disorder & fluctuations, criticality, turbulence, transition