# Order out of Randomness: Self-Organization Processes in Astrophysics

## Abstract

Self-organization is a property of dissipative nonlinear processes that are governed by a global driving force and a local positive feedback mechanism, which creates regular geometric and/or temporal patterns, and decreases the entropy locally, in contrast to random processes. Here we investigate for the first time a comprehensive number of (17) self-organization processes that operate in planetary physics, solar physics, stellar physics, galactic physics, and cosmology. Self-organizing systems create spontaneous “*order out of randomness*”, during the evolution from an initially disordered system to an ordered quasi-stationary system, mostly by quasi-periodic limit-cycle dynamics, but also by harmonic (mechanical or gyromagnetic) resonances. The global driving force can be due to gravity, electromagnetic forces, mechanical forces (e.g., rotation or differential rotation), thermal pressure, or acceleration of nonthermal particles, while the positive feedback mechanism is often an instability, such as the magneto-rotational (Balbus-Hawley) instability, the convective (Rayleigh-Bénard) instability, turbulence, vortex attraction, magnetic reconnection, plasma condensation, or a loss-cone instability. Physical models of astrophysical self-organization processes require hydrodynamic, magneto-hydrodynamic (MHD), plasma, or N-body simulations. Analytical formulations of self-organizing systems generally involve coupled differential equations with limit-cycle solutions of the Lotka-Volterra or Hopf-bifurcation type.

## Keywords

Astrophysics Planetary physics Stellar physics Solar physics Self Organization Limit cycle dynamics Instabilities Lotka Volterra systems Hopf bifurcation## 1 Introduction

*Self-organization is the spontaneous often seemingly purposeful formation of spatial, temporal, spatio-temporal structures or functions in systems composed of few or many components. In physics, chemistry, and biology, self-organization occurs in open systems driven away from thermal equilibrium. The process of self-organization can be found in many other fields also, such as economy, sociology, medicine, technology* (Haken 2008). Self-organization creates “*order out of randomness*” that is opposite to random processes with increasing entropy. Self-organization is a spontaneous process that does not need any control by an external force. It is often initiated by random fluctuations where the local reaction is amplified by a positive feedback mechanism. It can evolve into a stationary cyclic dynamics governed by a (strange) attractor, and develops, as a result of many microscopic interactions, a macroscopic regular geometric spatial pattern (Nicolis and Prigogine 1977; Kaufman 1993, 1996). In this review we compile for the first time a comprehensive set of self-organizing systems observed or inferred in astrophysics. For each astrophysical self-organizing system we discuss a physical model, generally in terms of a system-wide driving force and a positive feedback mechanism, which by mutual interactions evolve into a self-organized quasi-stationary pattern that is different from a random structure. Note that the term “self-organization” should not be confused with the term “self-organized criticality” (Bak et al. 1987; Pruessner 2012; Aschwanden et al. 2016), which is just one (of many) self-organizing complex systems, producing power law-like size distributions of scale-free avalanche events, whereas self-organizing systems usually evolve into a specific quantized (spatial or temporal) scale that is not scale-free.

Self-organization processes in astrophysics: The symbols in the last column indicate the following system characteristics: LC = nonlinear systems with limit cycle(s), I = instabilities, R = resonances, E = entropy, S = regular spatial pattern, T = regular temporal pattern, ? = conjectural. The stellar QPOs include also accretion disks and “coronas” of compact objects and supermassive black holes

Observed phenomenon | Driver mechanism | Feedback mechanism | Characteristics |
---|---|---|---|

Planetary spacing | gravity | harmonic orbit resonances | R, S, T |

Saturn rings and moons | gravity | harmonic orbit resonances | R, S, T |

Protoplanetary disks | rotation | Hall-shear instability | I, S |

Jupiter’s red spot | temperature gradient | inverse MHD turbulent cascade | I, S |

Saturn’s hexagon | circumpolar jet-stream | diocotron instability | I, S |

Planetary entropy | solar radiation | planetary infrared emission | E |

Solar photospheric granulation | temperature gradient | Rayleigh-Bénard instability | I, S |

Solar magnetic fields | solar dynamo, rotation | buoyancy, kink instability | I(?), S |

Solar magnetic Hale cycle | differential rotation | twisted magnetic field relaxation | LC, I(?), S, T |

Solar flare loops | chromospheric evaporation | coronal condensation | I, LC[?], T(?) |

Solar radio pulsations | nonthermal particles | loss-cone instability | LC, I[?], T |

Solar zebra radio bursts | nonthermal particles | double plasma resonance | R, S, IT[?] |

nonthermal particles | Langmuir-whistler coalescence | R, S, IT[?] | |

Star formation | gravity | radiation and recombination | I, S, T |

Stellar quasi-periodic oscillations | rotation | magneto-rotational instability | LC, I(?), T |

Pulsar superfluid unpinning | rotation | Magnus force | I(?), S[?] |

Galaxy formation | gravity, rotation | density waves, reaction-diffusion | S, I(?) |

Cosmology | Big Bang expansion | inflationary ΛCDM model | I(?), LC(?], S[?], T[?] |

Non-astrophysical self-organization processes

Field | Phenomenon | Reference |
---|---|---|

Ionosphere | Stimulation of electromagnetic emission | Leyser (2001) |

Internal gravity waves | Aburjania et al. (2013) | |

Acoustic gravity waves | Kaladze et al. (2008) | |

Magnetosphere | Substorm dynamics | Sharma et al. (2001) |

Substorm current sheet model | Valdivia et al. (2003) | |

Magnetospheric vortex formation | Yoshida et al. (2010) | |

2D MHD transverse Kelvin-Helmholtz instability | Miura (1999) | |

Turbulent relaxation of magnetic fields | ||

Plasma physics | Superconducting ring magnet vortex | Yoshida et al. 2010 |

Magnetic reconnection in laboratory | Yamada et al. (2010) | |

Magnetic reconnection in laboratory | Zweibel and Yamada (2009) | |

Physics | Coupled pendulums | Tanaka et al. (1997) |

Spontaneous magnetization | Boesiger et al. (1978) | |

Laser | Zeiger and Kelley (1991) | |

Superconductivity | Vazifeh and Franz (2013) | |

Bose-Einstein condensation | Nagy et al. (2008) | |

Chemistry | Molecular self-assembly | Lehn (2002) |

Supramolecular soft matter | Müller and Parisi (2015) | |

Reaction-diffusion systems | Kolmogorov et al. (1937) | |

Oscillating reactions | Bray (1921) | |

Oscillating catalytic reaction | Cox et al. (1985) | |

Liquid crystals | Rego et al. (2010) | |

Self-assembled monolayers | Love et al. (2005) | |

Langmuir-Blodgett films | Ritu (2016) | |

Growth of SiGe nanostructures | Aqua et al. (2013) | |

Biology | Biological systems | Camazine et al. (2001) |

Pattern formation in slime molds and bacteria | Camazine et al. (2001) | |

Feeding aggregations of bark beetles | Camazine et al. (2001) | |

Synchronized flashing among fireflies | Camazine et al. (2001) | |

Fish schooling | Camazine et al. (2001) | |

Nectar source selection by honey bees | Camazine et al. (2001) | |

Trail formation in ants | Camazine et al. (2001) | |

Swarm raids of army ants | Camazine et al. (2001) | |

Colony thermoregulation in honey bees | Camazine et al. (2001) | |

Comb patterns in honey bee colonies | Camazine et al. (2001) | |

Wall building by ants | Camazine et al. (2001) | |

Termite mound building | Camazine et al. (2001) | |

Construction algorithms in wasps | Camazine et al. (2001) | |

Dominance hierarchies in paper wasps | Camazine et al. (2001) | |

Social science | Social evolutionary systems | Leydesdorff (1993) |

Learning algorithms | Geach (2012) | |

Coevolution in interdependent networks | Wang et al. (2014) | |

Computer science | Cybernetics | Ashby (1947) |

Cellular automata | Gacs (2000) | |

Random graphs | Brooks (2009) | |

Multi-agent systems | Kernbach (2008) | |

Small-world networks | Watts and Strogatz (1998) | |

Power grid network simulations | Rohden et al. (2012) | |

Self-organizing maps | Kohonen (1989) | |

Cloud computing systems | Zhang et al. (2010) | |

Moore’s Law | Georgiev et al. (2016) |

In this review we discuss 17 different astrophysical processes that exhibit self-organization. For the definition of the term “self-organization” we proceed pragmatically. A nonlinear dissipative process qualifies to be called a “self-organization” process if it fulfills at least one of the following six criteria: (S) a spatially ordered pattern that is significantly different from a random pattern; (T) a temporally ordered (e.g., quasi-periodic) structure that is significantly different from random time intervals; (E) a system with negative entropy change (\(dS < 0\)); (LC) a nonlinear dissipative system with limit-cycle behavior (which by definition produces quasi-periodic temporal oscillations); (R) a nonlinear dissipative system with resonances; and (I) a nonlinear dissipative system that is driven by an external force and counter-acted by a positive feedback force, triggered by an instability or turbulence. We classify the 17 analyzed self-organization processes with these defining criteria (S, T, E, LC, R, I) in Table 1.

The structure of this review is organized by the various subfields in astrophysics, such as planetary physics (Sect. 2), solar physics (Sect. 3), stellar physics (Sect. 4), galactic physics (Sect. 5), and cosmology (Sect. 6). A discussion of randomness, self-organization, and self-organized criticality processes (Sect. 7) and a summary of the conclusions (Sect. 8) is given at the end. Each description of the 17 self-organization processes is annotated with a critical assessment at the end of each Section. The selected 17 cases are all explicitly addressed as self-organization processes by the authors of the cited studies, but we are aware that there are many more phenomena in astrophysics that implicitly qualify as self-organization processes, although they are often not labeled as such in the original literature. As a caveat, our review thus contains some bias towards citations with author-identified self-organization processes.

## 2 Planetary Physics

### 2.1 Planetary Spacing

Our solar system exhibits planet distances \(R_{i}, i=1,\ldots,n\) from the Sun that are not randomly distributed, but rather follow a regular pattern that has been quantified with the Titius-Bode law, known since 250 years. The original Titius-Bode law approximated the planet distance ratios by a factor of two, i.e., \(R_{i+1}/R_{i} \approx2\), while a generalized Titius-Bode law specified the relationship with a logarithmic spacing and a constant geometric progression factor \(Q\), i.e., \(R_{i+1}/R_{i}=Q\) (Blagg 1913). According to Kepler’s third law, a distance ratio \(Q\) corresponds to a period ratio \(q=T_{i+1}/T_{i}=Q ^{(3/2)}\) of the orbital time periods \(T\). However, both the original and the generalized Titius-Bode law represent empirical laws without a physical model.

Planet spacing with low harmonic ratios \(q\) of their orbital time periods \(T\), such as q=(m:n), with \(n=1,2,3\) and \(m \ge n+1\), have been interpreted as harmonic (mechanical) orbit resonances and are expected to occur in a self-organizing system with stable long-lived orbits (Laplace 1829; Peale 1976), especially in systems with resonant chains (e.g., Mills et al. 2016). One of the main heuristic understandings of chaos and instability is that a planet system is generated by overlapping resonances (Wisdom 1980), which explains why not all (or even the majority of) planet systems have exact (2-body) harmonic resonances (Daniel Fabrycky, private communication).

Most recently, the Laplacian 3-body resonances have been studied in great detail in the TRAPPIST-1 exo-planet system (e.g., Luger et al. 2017; Scholkmann 2017), which contains 7 planets and is continuously monitored by the Kepler mission. The 6 planet spacings of TRAPPIST-1 closely match the low-harmonic ratios \(q=(4:3), (3:2), (5:3)\) within an accuracy of \({\lesssim }1\%\) (Luger et al. 2017; Scholkmann 2017; Aschwanden and Scholkmann 2017).

*Solar System Dynamics*by Murray and Dermott (1999) and recently reviewed in Lissauer and Murray (2007) and Musielak and Quarles (2015), building on the work of Isaac Newton, Jean le Rond d’Alembert, Alexis Clairaut, Joseph-Louis Lagrange, Pierre-Simon Laplace, Heinrich Bruns, Henri Poincaré, and Leonard Euler. Some restricted solutions yield stationary orbits in the Lagrangian points L1 to L5. Numerical searches for periodic orbits and resonances based on approximations to harmonic oscillators (similar to the physical model of coupled pendulums) yield the following nominal resonance location \(a_{3}\) for a third body that orbits between the primary and secondary body (internal resonance) (Murray and Dermott 1999),

Observed orbital periods and distances of the planets from the Sun, and the nearest predicted harmonic orbit resonances (H1:H2) or order of resonances \((l,k)\), orbital periods \(T~\mbox{[yr]}\), observed and best-fit semi-major axes \(a_{\mathit{obs}}\) and \(a_{\mathit{harm}}\), and ratios \(a_{\mathit{harm}}/a_{\mathit{obs}}\)

Planet | Number | | | H1 | H2 | Period [yr] | \(a_{\mathit{obs}} \ \mbox{[AU]}\) | \(a_{\mathit{harm}} \ \mbox{[AU]}\) | Ratio \(a_{\mathit{harm}}/a_{\mathit{obs}}\) |
---|---|---|---|---|---|---|---|---|---|

Mercury | 1 | 3 | 2 | 5 | 2 | 0.241 | 0.39 | 0.391 | 1.002 |

Venus | 2 | 2 | 3 | 5 | 3 | 0.615 | 0.72 | 0.711 | 0.988 |

Earth | 3 | 1 | 1 | 2 | 1 | 1.000 | 1.00 | 0.958 | 0.957 |

Mars | 4 | 3 | 2 | 5 | 2 | 1.881 | 1.52 | 1.504 | 0.989 |

Ceres | 5 | 3 | 2 | 5 | 2 | 4.601 | 2.77 | 2.823 | 1.019 |

Jupiter | 6 | 3 | 2 | 5 | 2 | 11.862 | 5.20 | 5.179 | 0.996 |

Saturn | 7 | 2 | 1 | 3 | 1 | 29.457 | 9.54 | 9.225 | 0.967 |

Uranus | 8 | 1 | 1 | 2 | 1 | 84.018 | 19.19 | 18.943 | 0.987 |

Neptune | 9 | 1 | 2 | 3 | 2 | 164.78 | 30.07 | 30.129 | 1.002 |

Pluto | 10 | 284.40 | 39.48 | … | … | ||||

Mean | 0.99 ± 0.02 |

Alternative mechanisms besides gravitational N-body resonance self-organization have been proposed also, such as: (i) Hierarchical self-organization processes based on sequential resonance accretion (starting with the accretion of massive objects first) and 2-body resonance capture of planetesimals in the primordial solar nebula (Patterson 1987); (ii) plasma self-organization driven by the development to minimum energy states of the generic solar plasma during protostar formation (Wells 1989a, 1989b, 1990); (iii) subsequent mass ejections into planetary rings around a central rotating body with magnetic field properties predicted by stochastic electrodynamics (Surdin 1980), (iv) retarded gravitational 2-body resonance, i.e., macroscopic quantization of orbital parameters due to finite gravitational propagation speed (Gine 2007); or (v) quantization of orbital periods in terms of the quantum-mechanical Schrödinger equation (Perinova et al. 2007; De Neto et al. 2007; Scardigli 2007; Chang 2013).

### Critical Assessment

*The spacing of planets*, *moons*, *or exo*-*planets exhibit quantized values that correspond to low*-*harmonic ratios according to some studies*, *in which large period ratios of planet pairs are interpreted as gaps with missing* (*un*-*detected or non*-*existing*) *planets*. *A regular pattern of orbital periods* (*T*), *produced by low*-*harmonic ratios of orbital resonances* (*R*), *causes then also a regular pattern in planetary spacings* (*S*), *via Kepler’s third law*. *Other studies find that harmonic ratios are rare for exo*-*planets with orbital periods of less than a few years*. *The physical model of Lagrangian mean*-*motion resonances predicts exact harmonic ratios* (*in resonant chains*), *but secular disturbances*, *planet migrations*, *and overlapping resonances* (*Wisdom*1980) *may cause slowly*-*varying deviations*. *Nevertheless*, *the fact that harmonic ratios fit the planet orbital periods in the order of a few percents*, *strongly indicates the presence of a self*-*organizing system*, *opposed to randomness* (*Table *1: *qualifiers R*, *S*, *T*).

### 2.2 Planetary Rings and Moons

If we hypothesize that planets and moons are preferentially located at low-harmonic orbits, how do we explain the existence of gaps in a ring system, such as the Cassini division or the Encke gap in Saturn’s ring system? If moons form by accretion of planetesimals that orbit in close proximity to the accreting moon, a gap will result after sweeping over many nearby orbits, with the growing moon sitting in the middle of the gap. Therefore, gaps and moons are essentially cospatial in a long-term stable system. The most prominent “shepherding moon” in Saturn’s ring system is the satellite Mimas, which is responsible for the strongest resonance, i.e., the Cassini Division, a 4700-km gap between Saturn’s A and B rings (Porco and Hamilton 2000; McFadden et al. 1999, 2007). The two smaller moons Janus and Epimetheus cause the sharp outer edge of the A ring. The 320-km Encke gap in the outer A ring is believed to be controlled by the 20-km diameter satellite Pan. At Uranus, Cordelia and Ophelia have the role of “shepherding moons”. The moon Galatea plays a similar role in Neptune’s Adams ring.

The idea of self-organization in planetary rings has already been raised by Gor’kavyi and Fridman (1991). Gravitational forces and collisional deflection represent the drivers, while harmonic orbit resonances produce a feedback mechanism to organize the flat planetary ring plane into discrete rings, mostly because harmonic orbits tend to be more stable statistically. If the phenomenon of harmonic orbit resonances would not exist, randomized collisions only would determine the dynamics of ring particles, leading to a smooth and homogeneous planetary disk (or an asteroid belt or Oort cloud), rather than to quantized rings. The spatial pattern of rings with quantized ratios in their distance from the center of a planet (e.g., Saturn, Jupiter, Uranus, Neptune) thus is a manifestation of a self-organizing “ordered” scheme beyond a random pattern.

### Critical Assessment

*The argument to explain the harmonic structure of planetary rings is identical to the previously discussed case of planetary distances*, *because both are believed to be produced by the same stabilizing effect of orbital resonances with low*-*harmonic ratios*. *The arrangement of rings in quantized distances reveals a regular pattern in space* (*S*) *and time* (*T*) *that is beyond randomness*, *governed by mechanical resonances *(*R*). *These properties* (*R*, *S*, *T*) *argue in favor of a self*-*organization system* (*Table *1: *qualifiers R*, *S*, *T*).

### 2.3 Protoplanetary Disks

The formation of planets can obviously be seen as a self-organizing process, creating “order out of randomness”. The interstellar gas, initially randomly distributed in a molecular cloud, collapses under its own gravity to form a young stellar object. Unless it loses its angular momentum, the gas cannot directly fall onto the newly born star: its angular velocity would increase and matter would be centrifugally expelled at larger radii. In the frame co-rotating with the gas, the effective gravitational potential is minimal in a plane, where dissipative processes allow the protoplanetary (or circum-stellar) disk to form. Dozens of such disks have now been observed over a range of wavelengths (McCaughrean and O’dell 1996), and their link to planet formation casts no doubt, since planets have been observed in older “debris disks” (Kospal et al. 2009). The imaging of dust emission, whether thermal or scattered, has revealed a number of large-scale structures in protoplanetary disks. Such features include spiral arms (Muto et al. 2012; Benisty et al. 2015) or cavities in the innermost regions of the disk (e.g., Andrews et al. 2011). The former are generally attributed to the excitation of density waves by massive planets, while the latter could result from accretion and/or photo-evaporation of the inner disk (Alexander et al. 2006; Koepferl et al. 2013). Horseshoe-shaped dust concentrations have also been identified in several disks (Fukagawa et al. 2013; van der Marel et al. 2013); it is commonly agreed that these could correspond to large-scale anticyclonic vortices in the gas flow (Birnstiel et al. 2013).

### Critical Assessment

*The argument of a self*-*organization process in the evolution of a protoplanetary disk is mostly made in terms of the spatially emerging order* (*S*), *which starts from random*-*like turbulent flows with a complex fine structure and ends up in almost equidistantly ordered rings*. *From the* 3-*D MHD simulations it appears that the Hall*-*shear instability* (*I*) *acts as a feedback mechanism to organize an initially “chaotic” disk into an ordered system of axisymmetric bands*. *These properties* (*I*, *S*) *argue for a self*-*organization process* (*Table *1: *qualifiers I*, *S*).

### 2.4 Jupiter’s Red Spot

*Great Red Spot*since 187 years (or possibly since 350 years), which indicates a high-pressure zone of a persistent anticyclonic storm (Fig. 6). The vortex-like velocity field in Jupiter’s Red Spot has been derived and rendered in Fig. 7 by Simon et al. (2014). The temperature of Jupiter’s atmospheres above the Great Red Spot is measured to be hundreds of degrees warmer than simulations based on solar heating alone can explain (O’Donoghue et al. 2016). The Great Red Spot has a width of \({\approx}16{,}000\mbox{ km}\) and rotates counter-clockwise with a period of \(\approx3\) days. The longitude of the Great Red Spot oscillated with a 90-day period (Link 1975; Reese and Beebe 1977). Why can such an ordered, stable, long-lived structure exist in the (randomly) turbulent atmosphere of a gas giant? Why would it not decay into similar turbulent structures as observed in the surroundings? There exists a similar feature in Neptune’s atmosphere, visible during 1989–1994, called the

*Great Dark Spot*.

Early interpretations associated Jupiter’s Great Spot with a Korteweg-de Vries soliton solution (Maxworthy and Redekopp 1976), a solitary wave solution to the intermediate-geostrophic equations (Nezlin et al. 1996), a Taylor column, a Rossby wave, or a hurricane (Marcus 1993). A geostrophic wind or current results from the balance between pressure gradients and Coriolis forces. One theoretical explanation that was put forward is the self-organization of vorticity in turbulence: The Jovian vortices reflect the behavior of quasi-geostrophic vortices embedded in an east-west wind with bands of uniform potential vorticity (Marcus 1993). Numerical simulations based on the quasi-geostrophic equations for a Boussinesq fluid in a uniformly rotating and stably stratified environment indicated the self-organization of the flow into a large population of coherent vortices (McWilliams et al. 1994). This scenario suggests an evolution from initial turbulent (random) to coherent (ordered) large-scale structures.

The vortex-solution of the Rossby wave equation gives not only a solution resembling of the Great Red Spot but is also similar to a drift soliton in plasma where the Coriolis force (in a rotating atmosphere) is replaced by the Lorenz force (in a magnetized plasma) (Petviashvili 1980). The similarity of these phenomena not only indicates similar self-organizing principle behind them, but also may hint that the Great Dark Spot is the result of a MHD process.

### Critical Assessment

*The argument of Jupiter’s Great Red Spot being a self*-*organizing structure is mostly based on the emergence of a stable ordered large*-*scale structure* (*S*), *which is opposite to random*-*like turbulent small*-*scale structures*. *It is also the longevity of this large*-*scale structure that sets the Great Red Spot apart from short*-*lived small*-*scale turbulent structures*. *The physical process has been modeled with MHD simulations*, *essentially showing an inverse MHD turbulent cascade* (*from small to large scales*), *as it is known in* 2-*D turbulence* (*I*). *Thus*, *self*-*organization is established based on the properties I*, *S* (*Table *1: *qualifiers I*, *S*).

### 2.5 Saturn’s Hexagon

^{∘}N a hexagonal cloud pattern that was first discovered in the 1980s by the Voyager mission (Godfrey 1988), which was later imaged with high resolution by the Cassini Orbiter (Baines et al. 2009). The images obtained by Cassini revealed that the structure consists of two elements: a hexagonal circumpolar jet-stream and a North Polar vortex (NPV), see Fig. 8. Recently, Rostami et al. (2017) showed by computational simulations that the cloud pattern can be described as a coupled dynamical system consisting of the hexagonal circumpolar jet-stream and the NPV, resulting in a self-organized stable hexagonal pattern. The hexagonal shape is formed in a specific region of the turbulent flow between the jet-stream and the NPV that rotate with different speeds; the hexagonal shape is stabilized by the NPV. The concentric ring structure surrounding the vortices at the north and south pole and their peculiar temperature distribution (Fletcher et al. 2008), the occurrence of auroras at the poles (Dyudina et al. 2016), as well as an electrodynamic coupling of Saturn with his moons, e.g. Enceladus (Pontius and Hill 2006; Tokar et al. 2006), indicate that the cloud structures seem on the poles may be also related to plasma-physical and electrical phenomena. Indeed, laboratory studies of plasma discharge showed structures occurring at the diocotron instability (analogous to the Kelvin-Helmholtz instability in fluid mechanics) that resemble structures (discharge and cloud formations) at planetary poles (Parett 2007).

### Critical Assessment

*The argument of Saturn’s hexagon self*-*organizing structure is*, *as in the case of Jupiter’s Great Spot*, *mostly based on the emergence of a stable ordered large*-*scale structure* (*S*), *which is opposite to random*-*like turbulent small*-*scale structures*. *The modeling of the cloud structures based on fluid dynamics or plasma physics shows that instabilities* (*I*) *are involved in the organization process*. *The self*-*organization is thus established by properties I and S*.

### 2.6 Planetary Entropy

*the greenhouse effect, the hydrologic cycle of water, the global circulation of the atmosphere and oceans, are essentially dissipative structures supported by the supply of negentropy and making up the global self-organizing system whose characteristic is the climate on the Earth*(Izakov 1997).

Global energy flux budgets and *Trenberth diagrams* for the climates of terrestrial and gas giant planets are given in Read et al. (2016).

### Critical Assessment

*Since the entropy flux is increasing in random processes*, *we can conclude that processes with decreasing entropy fluxes are non*-*random processes*, *which is one of the definitions of self*-*organization here*. *The entropy flux calculation of the Earth’s atmosphere is made by assuming energy flux balance between the incoming solar radiation and the outgoing infrared emission from Earth*. *Based on this estimate of the entropy flux change* (*E*) *we can conclude that the atmosphere including its weather and climate changes have self*-*organizing capabilities* (*Table *1: *qualifier E*).

## 3 Solar Physics

### 3.1 Photospheric Granulation

Thus, the Lorenz model demonstrates that a temperature gradient (for instance below the photosphere) transforms (a possibly turbulent) random motion into a highly-organized rolling motion (due to the Rayleigh-Bénard instability) and this way organizes the plasma into nearly equi-sized convection rolls that have a specific size (such as \(w \approx1500\mbox{ km}\) for solar granules). The self-organization process thus creates order (of granules with a specific size) out of randomness (of the initial turbulent spectrum).

Since convection is the main energy transport process inside the Sun down to \(0.7 R_{\odot}\), larger convection rolls than the granulation pattern can be expected. Krishan (1991, 1992) argues that the Kolmogorov turbulence spectrum \(N(k) \propto k^{(-5/3)}\) extends to larger scales and possibly can explain the observed hierarchy of structures (granules, mesogranules, supergranules, and giant cells) by the same self-organization process.

*mini-granules*. Mini-granules are very mobile and short-lived. They are predominantly located in places of enhanced turbulence and close to strong magnetic fields in inter-granular lanes. The equivalent size of detected granules was estimated from the circular diameter of the granula’s area. The resulting

*probability density functions (PDF)*for 36 independent snapshots are shown in gray on the left frame of Fig. 12. The average PDF (the red histogram) changes its slope in the scale range of \({\approx} 600\mbox{--}1300~\mbox{km}\). This varying power law PDF is suggestive that the observed ensemble of granules may consist of two populations with distinct properties: regular granules and mini-granules. A decomposition of the observed PDF showed that the best fit is achieved with a combination of a power law function (for mini-granules) and a Gaussian function (for granules). Their sum fits the observational data. Mini-granules do not display any characteristic (“dominant”) scale. This non-Gaussian distribution of sizes implies that a more sophisticated mechanism with more degrees of freedom may be at work, where any small fluctuation in density, pressure, velocity and magnetic field may have significant impact and affect the resulting dynamics. It is worth to note that a recent direct numerical simulation attempt (Van Kooten and Cranmer 2017) produced the PDF of granular size in agreement with the observed one in Fig. 12. The authors concluded that the population of mini-granules is intrinsically related to non-linear turbulent phenomena, whereas Gaussian-distributed regular granules originate from near-surface convection.

### Critical Assessment

*The size distribution of granulation cells in the solar photosphere does not form a power law distribution*, *but clearly shows a preferred spatial scale of*\(\approx1000~\textit{km}\), *which renders a regular spatial pattern* (*S*), *rather than a scale*-*free distribution*. *However*, *a power law distribution has been found for the newly discovered “mini*-*granules” in a size range of* 100*–*600* km*, *which contradicts a self*-*organizing convective process that creates bubbles of equal sizes*. *The physical process of convection that is driven by a temperature gradient and the Rayleigh*-*Bénard instability* (*I*) *is well*-*understood and known as the Lorenz model*. *A caveat is how much the magnetic field plays a role in the solar convection zone*, *requiring a model with magneto*-*convection and hydromagnetic* (*Parker and Kruskal*-*Schwarzschild*) *instabilities*. *Anyway*, *a self*-*organization process is warranted based on the preferred scale of convective rolls* (*Table *1: *qualifiers I*,*S*).

### 3.2 Magnetic Field Self-Organization

How is the solar magnetic field organized and how does the resulting magnetic field self-organize into stable structures? It is said that sunspots and pores represent the basic stable structures that are visible in the photosphere, but their sub-photospheric formation (driven by the solar dynamo) and stability are long-standing problems. In the following we discuss a few papers that explicitly use the term “self-organization” in this context.

Takamaru and Sato (1997) propose a self-organization system that evolves intermittently and undergoes self-adaptively local maxima and minima of energy states. The nonlinear interactions of twisting multiple flux tubes lead to local helical kink instabilities, resulting in the formation of a knotted structure. Intermittent reconnection with neighbored flux tubes in the knotted structure releases energy and restores the original configuration, a process that exhibits self-organization in an open complex nonlinear system where energy is externally and continuously supplied.

Vlahos and Georgoulis (2004) state that non-critical self-organization appears to be essential for the formation and evolution of solar active regions, since it regulates the emergence and evolution of solar active regions, perhaps characterized by a percolation process (Schatten 2007, 2009), while the energy release process is governed by self-organized criticality. Georgoulis (2005, 2012) explores various (scaling and multi-scaling, fractal and multi-fractal) image-processing techniques to measure the expected self-organization of turbulence in solar magnetic fields. However, no difference was found in the turbulence spectrum between flaring and non-flaring active regions.

Chumak (2007) proposes a dynamic self-organization model of the active region evolution in terms of a diffuse aggregation process of magnetic flux tubes in the upper levels of the solar convection zone. The physical model is governed by hydrodynamics, magnetic forces, and additional random forces.

*The simulations reveal two basic steps in the process of spontaneous formation of stable structures that are the key for understanding the magnetic self-organization of the Sun and the formation of pores and sunspots: (1) formation of small-scale filamentary magnetic structures associated with concentrations of vorticity and whirlpool-type motions, and (2) merging of these structures due to the vortex attraction, caused by converging downdrafts around magnetic concentration below the surface*, reaching magnetic field strengths of \(B \approx1500\) G at the surface and \(B \approx6000\) G in the interior. The structure was found to remain stable for at least several hours. Examples of the simulated formation and evolution of magnetic structures are shown in Fig. 13.

As a disclaimer, we have to be aware that these 3-D MHD simulations capture a local box only, rather than being global. A self-consistent generation of magnetic flux, simulated on a global scale that includes the entire spherical convection zone of the Sun, is presented in Miesch et al. (2000), which produces laminar and turbulent states, driven by the differential solar rotation. Related work describes convection and dynamo action in rapidly rotating suns (Brown et al. 2010), or in large-scale dynamos with turbulent convection and shear (Käpylä et al. 2012). In order to understand the basic mechanism of the formation of magnetic flux concentrations, numerical 3-D MHD simulations were performed that study the turbulence contributions to the mean magnetic pressure in a strongly stratified isothermal layer with a large plasma beta (Brandenburg et al. 2012). By applying a weak uniform horizontal mean magnetic field, the *negative effective magnetic pressure instability (NEMPI)* is activated, which reduces the turbulence and thus the turbulent pressure. If this reduction is more than the magnetic pressure, then the weakly magnetized region will have a reduced total pressure, which leads to a collapse of the field into a stronger tube. Since this mechanism generates order from turbulence, it can be considered to be a self-organization process (Robert Cameron, private communication). In the global 3-D MHD simulations of Hotta et al. (2014), an efficient small-scale dynamo generates the magnetic field, which has a feedback on the poleward meridional flows, and thus displays the characteristic feedback feature of a self-organizing process. A simulation of the convective dynamo in the solar convective envelope has been conducted by Fan and Fang (2014), which is driven by the solar radiative diffusive heat flux, exhibiting irregular cyclic behavior with oscillation time scales ranging from about 5 to 15 yr and undergoes irregular polarity reversals, as it is typical for self-organizing limit cycles far off a stationary equilibrium.

### Critical Assessment

*Ideas of applying self*-*organization processes to generate the magnetic field in the solar convection zone or in the solar corona are mentioned only briefly in the reviewed papers* (*or not at all*), *but no quantitative models or measurements are presented that would allow us to discriminate which magnetic structures have a random pattern and which ones exhibit some ordered pattern*. *The magnetic flux on the solar surface was found to have a power law size distribution* (*Parnell et al*. 2009), *which is rather consistent with a self*-*organized criticality process*. *The envisioned feedback mechanisms include the kink instability*, *the NEMPI instability*, *percolation*, *diffuse aggregation*, *and vortex attraction*, *but none of these processes has been characterized in emerging flux simulation in terms of self*-*organization*. *So*, *we can observe spatial patterns of photospheric magnetic flux patches* (*S*), *but are not sure which instability* (*I*) *enacts self*-*organization* (*Table *1: *qualifiers I*(?), *S*).

### 3.3 The Hale Cycle

The global magnetic field of the Sun undergoes a cyclic transition from a global poloidal field to a highly-stressed toroidal field in 11 years, switching the magnetic polarity during this process, so that the original polarity is restored after two cycles, yielding a 22-yr cycle that is called the (magnetic) “Hale cycle”. There exist over 2000 publications about the solar magnetic activity cycle. A recent review can be found in Hathaway (2015).

A chaotically modulated stellar dynamo was modeled also based on bifurcation theory, where modulation of the basic magnetic cycle and chaos occur as a natural consequence of a star that is in transition from a non-magnetic state to one with periodically reversing fields (Tobias et al. 1995).

### Critical Assessment

*The solar cycle is a very periodic phenomenon with little variation in each cycle*, *which is a classic example of a nonlinear dissipative system with limit*-*cycle behavior* (*LC*), *such as the Hopf bifurcation* (*Appendix *C) *or Lotka*-*Volterra equation system* (*Appendix *D). *The limit cycle produces a regular temporal pattern* (*T*), *and the cycle variation modulates the magnetic flux and area on the solar surface like*-*wise* (*S*). *The physics of the solar cycle is also well*-*understood in terms of the Babcock*-*Leighton model*, *where the differential solar rotation is the driver*, *and a twisted magnetic field relaxation mechanism acts as the feedback mechanism*. *The underlying instability still needs to be identified and may depend on both the shallow dynamo or the deep dynamo in the tachocline at the bottom of the convection zone* (*Table *1: *qualifiers LC*, *I*[?], *S*, *T*).

### 3.4 Evaporation-Condensation Cycles

Solar observations show that coronal loops routinely harbor flows that result from the complex physics of the solar transition region (e.g., Peter et al. 2006). Upflows can generally be understood as the result of heated plasma from the chromosphere ascending into coronal loops (chromospheric evaporation), as modeled from EUV, soft X-ray, and hard X-ray observations. These upflows frequently happen during solar flares, but equally occur as a consequence of other coronal heating mechanisms also, in active regions, in Quiet Sun regions (explosive events, EUV brightenings), and even in coronal holes (plumes, jets). At the same time there is numerous evidence for downflows, also called *“coronal rain”* or *“coronal condensation”*, mostly observed in H\(\alpha\) (first reported by Leroy 1972) and UV lines of cooler temperatures (Schrijver 2001; De Groof et al. 2005). The combined pattern of upflows and downflows is also referred to as *“evaporation-condensation cycle”* (Krall and Antiochos 1980), which we consider under the aspect of a self-organization process here.

Numerical 1-D hydrodynamic simulations of the condensation of plasma in loops of wide ranges of lengths and temperatures (\(10~\mbox{Mm} \le \mbox{L} \le 300~\mbox{Mm}\); \(0.2~\mbox{MK} \le \mbox{T} \le 2~\mbox{MK}\)) reproduce the cyclic pattern, starting with chromospheric evaporation, followed by coronal condensation, then motion of the condensation region to either side of the loop, and finally loop reheating with a period of 1 h to 4 days (Müller et al. 2003, 2004, 2005). It is found that the radiatively-driven thermal instability occurs about an order of magnitude faster than the Rayleigh-Taylor instability, which can occur in a loop with a density inversion at its apex also (Müller et al. 2003). Simulations with different heating functions reveal that the process of catastrophic cooling is not initiated by a drastic decrease of the total loop heating rate, but rather results from a loss of equilibrium at the loop apex as a natural consequence of quasi-steady footpoint heating (Müller et al. 2004; Peter et al. 2012). The same effect of a loss of equilibrium can occur in the case of repetitive impulsive heating (e.g., Mendoza-Briceno et al. 2005; Cargill and Bradshaw 2013).

Uzdensky (2007a, 2007b) proposes a similar self-organization process for coronal heating. This self-regulating process keeps the coronal plasma roughly marginally collisionless. The driver of the self-organization process is the magnetic reconnection in the collisional Sweet-Parker regime. The feedback mechanism is the inhibition of magnetic reconnection triggered by density increases due to chromospheric evaporation. After some time, the conductive and radiative cooling lowers the density again below the critical value and fast reconnection sets in again. Thus, the self-organization process is made of repeating cycles of fast reconnection, evaporation, plasma cooling, and re-building of magnetic stress. A similar self-regulation mechanism controlled by marginal collisionality in magnetic reconnection is explored in Cassak et al. (2008) and Imada and Zweibel (2012). The cyclic behavior has been simulated with a 1-D hydrodynamic model that is driven by gravity and the density dependence of the heating function (Imada and Zweibel 2012).

### Critical Assessment

*The evaporation*-*condensation scenario of coronal loops predicts a quasi*-*periodic time pattern*, *but not much is known about the degree of periodicity*, *and whether this corresponds to a quasi*-*periodic self*-*organizing limit cycle*. *The quasi*-*periodic patterns discovered by Froment et al*. (2015, 2017), *which exhibit phase*-*shifted oscillations between the emission measures and temperatures in active regions*, *reveal large fluctuations in the emission measure versus temperature diagram* (*Fig*. 20), *which may indicate strong nonlinearities near the limit cycle or inadequate background subtraction in the differential emission measure analysis*. *Although the physics of the evaporation*-*condensation cycle is well understood*, *to deduce the time evolution of the heating rate*, *electron density*, *and temperature from observational data*, *adequate background subtraction needs to be performed in order to establish whether the observations are well described by a limit*-*cycle system* (*Table *1: *qualifiers I*, *LC*[?], *T*[?]).

### 3.5 Quasi-Periodic Radio Bursts

*remote-sensing*observations with current radio instruments.

Self-organizing systems with limit cycles were initially applied to loss-cone instabilities occurring in the aurora, where two types of waves (electrostatic and upper hybrid waves) exchange energy in a limit cycle, driven by the loss-cone instability (Trakhtengerts 1968), a concept that was then applied to solar radio pulsations also (Zaitsev 1971; Zaitsev and Stepanov 1975; Kuijpers 1978; Bardakov and Stepanov 1979; Aschwanden and Benz 1988). The two-component nonlinear systems of self-organization are controlled either by wave-wave interactions, or by wave-particle interactions (also called a *quasi-linear diffusion* process).

**k**-space, the particle system is described by its density distribution \(f (\mathbf{p}, t)\) in momentum space, \(\varGamma( \mathbf{p}, \mathbf{k}, f )\) is the wave growth rate, \(\gamma( \mathbf{p}, \mathbf{k}, f )\) is the wave damping rate, \(\mathbf {v}_{g}(\mathbf{k})\) is the group velocity of the emitted waves, \(\hat{D}_{ji}(\mathbf{p}, \mathbf{k}, N)\) is the quasi-linear diffusion tensor, and \(S\mbox{--}L\) indicates a source \((S)\) minus a loss term \((L)\). Equation (23) is the wave equation that describes the balance between emission and growth and damping rate, while Eq. (24) describes the evolution of the particle distribution. The interaction between waves and particles is expressed by the quasi-linear diffusion tensor. In addition, there is a source term \(S\) of particles (with large pitch angles), as well as a loss term \(L\) (which quantifies the precipitating particles with small pitch angles out of the loss-cone).

### Critical Assessment

*The quasi*-*periodicity of solar radio bursts as observed in dynamic spectra is the most convincing signature of a self*-*organizing process*, *in contrast to a time series with random time intervals*, *as it would be expected for self*-*organized criticality models*. *The spatial counterpart* (*S*) *of the quasi*-*periodic temporal scales* (*T*) *is generally not observed due to the lack of radio images with high spatial resolution*. *Nevertheless*, *quasi*-*periodic time intervals are consistent with a limit cycle* (*LC*) *of a nonlinear dissipative system*, *but there are many plasma instabilities that can operate as a positive feedback mechanism*, *either in terms of wave*-*particle interactions* (*e*.*g*., *loss*-*cone or beam instabilities*), *or wave*-*wave interactions*. *Thus*, *more data modeling*, *possibly with high*-*resolution imagery and magnetic field modeling is required to identify the relevant instabilities that control a self*-*organizing process in the generation of quasi*-*periodic radio bursts* (*Table *1: *qualifiers T*, *LC*, *I*[?]).

### 3.6 Zebra Radio Bursts

*zebra burst*(Fig. 23), which reveals drifting parallel bands of quasi-stationary radio emission with harmonic frequency ratios. Theoretical interpretations include (i) models with interactions between electrostatic waves and whistlers, and (ii) radio emission at the double-plasma resonance (Kuijpers 1975, 1980; Zheleznyakov and Zlotnik 1975a,b; Mollwo 1983; Winglee and Dulk 1986; Chernov 2006; Chen et al. 2011),

In recent models, the double-plasma resonance mechanism faces a number of difficulties in explaining the dynamics of zebra stripes (i.e., sharp changes of the frequency-drift rate, a large number of stripes, frequency splitting of stripes, super-fine millisecond structure), because the magnetic field and density cannot change as rapidly. Improved models are in progress (Karlicky et al. 2001; LaBelle et al. 2003; Kuznetsov and Tsap 2007; Karlicky and Yasnov 2015). New calculations concern the increments of the upper-hybrid waves under double-plasma resonance conditions, the ring distribution of high-speed electrons with relativistic corrections, different temperatures of the background plasma, and optimum wave numbers (Benacek et al. 2017). It has been shown that the optimum increment for electron velocities is \(v \approx0.1\mbox{ c}\), with a narrow dispersion. If the speed is \(\approx0.2\mbox{ c}\), the increment sharply decreases and the flux maxima are washed out in the continuum for several cyclotron harmonic numbers \(s\). Thus, these calculations show the inefficiency of the double-plasma resonance mechanism. Under such conditions it becomes clear, that the double-plasma resonance mechanism cannot explain the majority of zebra stripes. An additional complication is the simultaneous occurrence of decimetric millisecond spikes.

In the whistler model, all the aforementioned properties of zebra burst stripes have been explained by physical processes that occur during the coalescence of Langmuir waves \((l)\) with whistler waves \((w)\), producing transverse waves, \(l + w \mapsto t\) (Kuijpers 1975; Chernov 1976, 1990, 2006, 2011). Langmuir waves and whistlers can be generated by the same fast particles trapped in magnetic islands (Berney and Benz 1978).

The spatial structure of zebra radio bursts is believed to originate in magnetic islands after coronal mass ejections. Therefore the close connection of zebra bursts with fiber bursts is simply explained by the acceleration of fast particles in magnetic reconnection regions in the lower or upper part of magnetic islands.

A wavelike or saw-tooth frequency drift of stripes was explained by the switching of the whistler instability from the normal Doppler-cyclotron resonance into the anomalous one (Fig. 2b in Chernov 1990). Such switching should lead to a synchronous change of the frequency drift of stripes and spatial drift of the radio source, since whistlers generated at normal and anomalous resonances move in opposite directions. New injections of fast particles cause sharp changes in the frequency drift rate and oscillation pattern of zebra stripes. Low frequency absorption (i.e., black stripes of zebra bursts) are explained by quenching of the plasma wave instability due to diffusion of fast particles by whistler waves.

### Critical Assessment

*The most striking pattern that hints to a self*-*organization process is the periodic appearance of bands in dynamic spectra of some solar radio bursts*, *which is interpreted in terms of gyroharmonic resonances* (*R*). *In principle*, *a periodic pattern in radio frequency can be mapped to a periodic pattern in spatial structures* (*S*), *using the plasma frequency relationship*\(f_{p} \propto\sqrt{n_{e}}\)*and a density model*\(n_{e}(h)\)*as a function of the altitude*\(h\), *while there is no obvious periodic pattern of temporal structures* (*T*) *expected*. *The driver mechanism that produces zebra bursts is likely to be a population of nonthermal particles*, *while the counter*-*acting feedback mechanism has been modeled in terms of electrostatic waves*, *whistler waves*, *or the double*-*plasma resonance*. *The observational verification of any of these wave types is still very challenging with remote*-*sensing techniques* (*Table *1: *qualifiers R*, *S*, *I*[?]). *Note that the spatial pattern of a zebra skin has also been classified as a self*-*organization process in biology* (*e*.*g*., *Camazine et al*. 2001), *where the light and dark pigmentation is created by the diffusive interaction of chemical activation* (*driver*) *and inhibition* (*feedback*) *during the embryonic development*.

## 4 Stellar Physics

### 4.1 Star Formation

Similar low-dimensional models were developed for the hot X-ray emitting gas of elliptical galaxies (Kritsuk 1992, 1993, 1996). The gas is described as an open system with mass and energy sources determined by stellar mass loss, supernova explosions, and radiative cooling. The gas condensation due to thermal instabilities is also accounted for with a mass sink term. A study of the dynamics of this nonlinear closed-box system proved that the steady states in the temperature–density plane are generally unstable. Numerical simulations demonstrated the existence of stable periodic solutions, describing a cyclic process of gas accumulation and heating due to the stellar sources followed by gas cooling and dropping out of the halo. The limit cycle emerges after a sequence of saddle-node and saddle-connection bifurcations (also known as the *fold* bifurcation and the *blue sky catastrophe*) while the system responds to the growth of condensation efficiency with a first-order phase transition (Kritsuk 1996). The limit cycle disappears at a higher condensation efficiency, following a Hopf bifurcation. The bifurcations occur naturally, due to the shape of the radiative cooling function, and this behavior is preserved in a wide range of gas metallicities. Moving beyond the one-zone model required multi-dimensional numerical simulations, which revealed an instability resulting in the emergence of filamentary network of condensation waves propagating in the hot gas (Kritsuk 1994). These condensation waves are similar in nature to the “galactic drips” proposed by Mathews (1997) as an alternative explanation for the presence of young (\(\sim5\mbox{--}10~\mbox{Gyrs}\)) stellar populations observed in many elliptical galaxies.

Subsequent simulations of interstellar turbulence and star formation include isothermal models of molecular clouds and larger-scale multi-phase models to simulate the formulation of molecular clouds. They *show how self-organization in highly compressible magnetized turbulence in the multi-phase interstellar medium can be exploited to generate realistic initial conditions for star formation* (Kritsuk et al. 2011, 2017; Padoan et al. 2016). Multiple states of star-forming clouds have been identified in 3-D MHD simulations: gravity splits the clouds into two populations, one low-density turbulent state, and one high-density collapse state (Collins et al. 2012). However, it would be premature to say that we fully understand the dynamics of self-gravitating turbulent ISM, despite the recent progress achieved (Banerjee and Kritsuk 2017).

### Critical Assessment

*The evidence for self*-*organization in the star formation process is the morphological change from an initial randomized molecular cloud to concentrations in a number of small localized* (*H II*) *zones in galaxies*, *which represent spatially ordered structures* (*S*), *in contrast to the initially uniform randomness of the interstellar gas*. *A size distribution of ordered structures*, *however*, *has not been quantified yet*. *The corresponding ordered time structures* (*T*) *are produced by repetitive violent bursts of star formation*. *The physical process of self*-*organization in star formation is modeled in terms of highly compressible magnetized turbulence*, *which can trigger instabilities ending with high*-*density collapses* (*I*) (*Table *1: *qualifiers I*, *S*, *T*).

### 4.2 Stellar and other Quasi-Periodic Oscillations

The origins of quasi-periodic oscillations (QPO) observed from various stellar sources (pulsars, cataclysmic variable stars, neutron stars, binary stars, active galactic nuclei, etc.) are largely not understood. Interpretations have focused on attributing the overall variability to accretion fluctuations, with the QPO produced by modulation of the accretion, for example by resonance-like interactions between natural rotational and orbital frequencies or—more germane to this article—nonlinear chaotic dynamics perhaps with modulated limit cycles, which point to self-organizing systems.

A number of attempts to detect and characterize deterministic chaos from astronomical time series data have been made. For example, the irregular X-ray variability of the neutron star Her X-1 has been analyzed with the method of Procaccia (1985) and detection of a low-dimensional attractor (\(D \approx 2.3\)) and some higher-dimensional chaos was inferred for the accretion disk (Voges et al. 1987). The light curves of three long-period cataclysmic variable stars have been analyzed with the technique of Grassberger and Procaccia (1983a,b) in the search of an attractor dimension, but the light curves could be modeled with a periodic and a superimposed random component (Cannizzo et al. 1990). Evidence for a low-dimensional attractor with a dimension of \(D \approx 1.5\) was found in the Vela pulsar with a correlation sum technique (Harding et al. 1990).

However, much of this earlier work has proved to be questionable. For example the result found by Voges et al. (1987) was disputed by Norris and Matilsky (1989), who concluded that the insufficient signal-to-noise ratio does not allow to distinguish from an ordinary attractor contaminated with noise. Since the attractor dimension is equivalent to the number of coupled differential equations, we would expect a lowest attractor dimension of \(D = 2\) for the Lotka-Volterra equation system, or \(D = 3\) for the Lorenz model. Based on a careful simulation study of non-chaotic random data Harding et al. (1990) questioned the significance of their own result quoted above. They concluded “It appears that the correlation sum estimator for dimension is unable to distinguish between chaotic and random processes.”

This important cautionary remark is reinforced by at least two key theoretical results. Eckmann and Ruelle (1992) presented an elementary proof that the correlation dimension estimated with the Grassberger-Procaccia algorithm cannot exceed the value \(2\log_{10}(N)\), where \(N\) is the number of points in the time series. (One finds in the astronomical literature a number of dimension estimates approximating this value, suggesting that they are entirely spurious.) Eckmann and Ruelle (1992) disproved several then traditional views in this context, showing that essentially any correlation dimension can be found for entirely random (i.e. lacking any “deterministic chaos”) colored random data simply by choosing the appropriate index for the power-law power spectrum of the data. They concluded “These results have implications on the experimental study of deterministic chaos as they indicate that the sole observation of a finite fractal dimension from the analysis of a time series is not sufficient to infer the presence of a strange attractor in the system dynamics.”

All in all, these theoretical limits and the realities of signal-to-noise and length of available time series cast a pall on the quest for evidence of nonlinear dynamics in astronomical systems that continues to some extent today. However the landscape of time domain astronomy is improving with respect to time coverage, sampling cadence and signal-to-noise, and perhaps prospects for more definitive characterization of underlying dynamics of variable objects will improve accordingly.

On the other side of the theory-observation coin, two simple physical metaphors incorporating self-organization ideas have inspired independent quasi-stochastic models reproducing some features of the observed variability of accretion sources. The dripping handrail (Scargle et al. 1993) evokes an analogy between astrophysical accretion on the one hand and the accumulation, flow, and dripping of moisture on a stairway’s handrail on the other hand. These authors quantified the quasi-periodic oscillations (QPO) of the low-mass X-ray binary star (LMXB) Scorpius X-1 with a wavelet based power spectrum that was found to be consistent with the spectrum computed for a dripping handrail accretion model, a simple dynamical system that exhibits transient chaos (Scargle et al. 1993; Young and Scargle 1996). This highly oversimplified picture nevertheless explains the \(1/f\) and QPO features—typically though to be separate phenomena—as two aspects of a single physical process, notably ascribing the variability as quasi-random due to non-linear dynamics in a constant external accretion flow (and not due to a postulated random accretion).

The sandpile metaphor independently inspired a self organization model (Mineshige et al. 1994a,b; Mineshige and Negoro 1999), similar to the dripping handrail, physically somewhat more realistic in that its 2D geometry allowed treatment of angular momentum transport within the accretion disk. On the other hand these authors postulated randomness for accretion, although quasi-randomness is generated automatically by nonlinearities in their model even with steady accretion, for the same reasons as with the dripping handrail model.

The fluctuation power spectra of accreting black holes, neutron stars, and white dwarfs that are accreting gas from a stellar companion sometimes exhibit peaks at certain frequencies (Remillard and McClintock 2006; van der Klis 2006). These are also seen from some supermassive black holes powering active galactic nuclei (Smith et al. 2017). These peaks are called *“quasi-periodic oscillations” (QPOs)*, since they are usually not very narrow. The frequencies observed in the neutron star and white dwarf sources are time-dependent, usually being positively correlated with the luminosity (proportional to the mass-accretion rate). The black hole sources exhibit two classes of QPOs, separated in frequency by a factor of at least 30. Only the *high-frequency quasi-periodic oscillations (HFQPOs)* have a fixed frequency, which is slightly below that of the innermost stable orbit in the accretion disk (and therefore inversely proportional to the mass of the black hole). However, these HFQPOs have a relatively small duty cycle. A mysterious property of many of them is the 3:2 ratio of the two highest frequencies (Wagoner 2008). There is no complete physical theory that can explain this fact, and no numerical simulations reproduce the observed QPOs. The accretion disks are very turbulent, driven by the conversion of the differential rotational energy via the magneto-rotational instability. In addition, they are subject to viscous and thermal instabilities, on time scales greater than the orbital period at that radius. The hotter “corona” of the neutron star and black hole disks appears to up scatter the cooler thermal photons from the disk into X-rays, without seriously demodulating them.

### Critical Assessment

*A number of oscillatory light curves from various types of stars have been recorded*, *which clearly establish the presence of non*-*random ordered time structures *(*T*). *The spatial counterparts* (*S*), *of course*, *cannot be resolved in stellar distances*. *The time evolution is generally quasi*-*periodic*, *which is typical for nonlinear dissipative systems with limit cycles* (*LC*), *and low*-*dimensional attractors have been identified from those time series*. *The physical mechanism or instability* (*I*) *that is responsible for stellar quasi*-*periodic oscillations is less clear*, *but an accretion model* (*i*.*e*., *the dripping handrail model*) *has been proposed* (*Table *1: *qualifiers T*, *LC*, *I*[?]).

### 4.3 Pulsar Superfluid Unpinning

Pulsar glitches are generally interpreted in terms of the self-organized criticality model, due to the scale-invariant, power law-like distributions of sizes and exponential waiting time distributions (Melatos et al. 2008; Espinoza et al. 2011). In this scenario, superfluid vortices pin metastably in macroscopic domains and unpin collectively via nearest-neighbor avalanches. Recent quantum mechanical simulations, in which the evolution of the pinned, decelerating superfluid is described by the time-dependent Gross-Pitaevskii equation, have identified two knock-on processes responsible for mediating vortex avalanches: local, hydrodynamic, nearest-neighbor repulsion and nonlocal, acoustic-wave unpinning (Warszawski et al. 2012). The simulations also reproduce the size and waiting-time statistics in observational data, albeit over a relatively small dynamic range because computational limitations restrict the simulated system to \(<200\) vortex lines at present (Warszawski and Melatos 2011; Melatos and Warszawski 2015). Alternatively, Melatos and Warszawski (2009) propose a noncritical self-organization process (which they call “coherent noise” according to Sneppen and Newman 1997), where the global Magnus force acts uniformly on vortices trapped in a range of pinning potentials and undergoing thermal creep. In this scenario, Melatos and Warszawski (2009) find that vortices again unpin collectively, without nearest-neighbor avalanches, but still produce a scale-free size distribution as observed. The microscopic self-organization processes of nuclear matter in neutron star crusts has also been simulated in crystalline lattices, *where the system organizes itself into exotic structures* (Sebille et al. 2011; Caplan and Horowitz 2017). When the magnetic field in the superconducting stellar interior is included, the vortex lines and magnetic flux tubes self-organize into a turbulent, reconnecting tangle, sustained by stellar braking (Drummond and Melatos 2017). The flux tubes act as pinning centers as well, widening the scope for vortex line avalanches to occur.

In the pulsar glitch self-organization process, the driver is the electromagnetic braking of the star, while the feedback mechanism is the local interplay between the superfluid Magnus force and the pinning potentials, which regulate semi-coherent unpinning (Cheng et al. 1988). An alternative yet analogous scenario involving elastic stresses (star quakes) has also been proposed (Middleditch et al. 2006). A promising theoretical framework that is applicable to a wide variety of self-organizing systems of this kind is the mean-field model of a state-dependent Poisson process, introduced originally in the context of forest fires (Daly and Porporato 2006), and solar flares (Wheatland 2008), and generalized recently to neutron stars (Fulgenzi et al. 2017) and biological applications (Miles and Keener 2017). The model makes quantitative predictions of size and waiting time distributions and size-waiting time correlations as a function of the driving rate, independent of the detailed microphysics.

### Critical Assessment

*One manifestation of self*-*organization is the lattice grid of nuclei in the neutron superfluid zone of a neutron star*, *which is a highly ordered spatial* (*S*) *structure* (*like a crystal*), *opposed to a random*-*like thermodynamic fluid in normal stars*. *The physical model involves the rapid rotation of a pulsar* (*driver*), *and a feedback mechanism is given by the inhibition of vortex motion in the superfluid unpinning potentials* (*I*). *The feedback mechanism maintains a semi*-*coherent unpinning*, *which represents a spatial ordered structure* (*S*) *also*. *A caveat of this model is that observations of neutron star glitches exhibit scale*-*free power law distributions*, *which are typical for self*-*organized criticality models* (*Table *1: *qualifiers S*[?], *I*[?]).

## 5 Galactic Physics

Observable galaxies arise when gas (“baryons”) flows into concentrations (“halos”) of dark matter and forms stars, which themselves radiate and excite residual gas to radiate in a number of ways. The morphologies of the galaxies we now see is a time-slice of ongoing processes that include: (i) build-up of halos massive enough to retain gas within a framework of intersecting cell-walls and filaments; (ii) continuing gas inflow (only about half of the baryons are currently within star-forming halos); (iii) outflow of gas (and some recycling) driven by winds and jets from bursts of star formation, supernovae, and central supermassive black holes; (iv) gravitationally driven encounters between halos, described as major mergers (when the masses are comparable), producing spiral arms and disks), and minor mergers or captures of little galaxies by large ones (which can make star streams, rings, disks, and central bulges, and in the process initiate driving of spiral arms). The starting point is a random distribution of small (\({\approx}10^{-5}\)) density fluctuations in the distribution of dark matter through the universe, with the spectrum of those fluctuations described by \(N(\delta\rho) \propto(\delta\rho)^{-1}\) (the Harrison-Zeldovich spectrum). A constraint throughout is that the mass of central black holes is close to \({\approx}0.8 \times 10^{-3}\) of the mass of stars through much of the cosmic history.

Over several decades now, many groups have modeled this scenario of a universe with N-body simulations (with N gradually increasing from \(10^{6}\) to \(10^{10}\) and more), and a brief summary of the results is *“Any correct description of our universe must look very much like*\(\varLambda\)*CDM on large scales, seeded by a nearly scale-invariant fluctuations spectrum that is dominated by dark energy”* (Bullock and Boylan-Kolchin 2017). That is, theory and observations agree well for length scales of a megaparsec and more. The situation on small scales is very much less satisfactory (Bullock and Boylan-Kolchin 2017; Naab and Ostriker 2017; Freeman 2017; Concelice 2014). One approach has been the “zoom simulation” (Springel et al. 2005) that switches from large scale considerations to something like the size of a galaxy and includes gas processes, star formation, and dust attenuation within either an adaptive mesh refinement or a smoothed particle hydrodynamic code. Naab and Ostriker (2017) show results (2011–2015) from six groups. Each resembles some real galaxy (e.g., see figures in Concelice 2014), but resemble all to the scenario called “flocculent” (Elmegreen and Elmegreen 1987), rather than the “grand design” (Elmegreen 2011). In addition, it is a general principle that a theoretical process that mimics the real world, does not prove its correctness (an argument often used in discussions of biological evolution).

The next question is what physical model can produce spirals (at least numerically calculated), and which scenario can be described in terms of self-organization? Binney and Merrifield (1998) state that the arms nearly always trail; they are bright because the young stars have formed there, but there is some enhanced density of old stars as well (\(\approx40\%\)). Gas is needed to sustain the process, so galaxies of the Hubble type S0 generally do not show arms, and bars and companions can be drivers. However Binney and Merrifield (1998) consider spiral arms, although appearing as prominent features, not to be very important in the great scheme.

First, it is necessary to understand how some galaxies develop disks. Jeans (1915), starting with methods due to Boltzmann and treating stars as the gas particles, concluded that a non-spherical system with stellar motions describable as gas streams could not be static. Lindblad (1927, 1925, 1926) recognized that the Milky Way, or at least the parts of it he could study, is rotating, that spiral arms seen in the newly-recognized extragalactic nebulae would wind up fairly quickly, and that a pattern of higher density in the arms, rotating more slowly than matter, could be more stable. Lindblad describes his mathematical methods as deriving from work by Poincare, also applied to gases.

Sufficient angular momentum produces disks. Lindblad thought the rotation might arise from galaxy encounters or mergers, though we now associate mergers with destruction of disks and formation of ellipticals. Instabilities tend to form warps and bars (which the Milky Way has both) (Bland-Hawthorn and Gerhard 2016), and it is true that while \(2/3\) of S-type galaxies now have bars, they were rare at \(z=1\), but disks still survive. Ostriker and Peebles (1973) proposed in their highly-cited paper that an extended, dark, spheroidal halo would permit survival. Bland-Hawthorn and Gerhard (2016) show NGC 3 spirals rectified to face-on might be confused with the Milky Way if we could see it face-on from outside. None has two dominant arms of the type of M51 galaxy, but none shows the complexity as the products of “Zoom-in simulations” (Springel et al. 2005). It is perhaps significant that the Milky Way also belongs to the rare “green valley” category of galaxies that are neither blue, vigorous star formers, nor red and dead.

Comments specific to the Milky Way include that it reached its mostly 2-armed state about 9 Gyrs ago (Francis and Anderson 2009), that its present conditions was probably triggered by a first encounter with the Sagittarius dwarf spheroidal galaxy (Purcell et al. 2011). M31 incidentally also has a “driving companion” and both galaxies have their dwarf spheroidal companions largely organized in a planar thin structure that is also not understood.

According to Freeman (2017) true bulges come from mergers, while instabilities in disks provide bars and pseudo bulges as in the Milky Way. Our thick disk (which does not have arms, nor do other thick disks (they are nearly but not quite ubiquitous)) formed 11–12 Gyrs ago, at \(z=2\mbox{--}2.5\) equivalent time, which is very close to the peak of the star formation rate for the local universe as a whole (Madau and Dickinson 2014) and probably also for the Milky Way (though we expect to know more about this topic when the Gaia data are fully in and analyzed), but it is likely that the Milky Way spiral pattern requires more than one mechanism. The oldest spiral reported so far (in a sea of dwarf irregular structures imaged by HST in its deep field) is labeled as Q2343-BX442 (Law et al. 2015) and is of the “companion driven” variety, being seen at a redshift that corresponds to an age near 11 Gyrs, and evolution tends to move galaxies from *flocculent* to *grand design* (Francis and Anderson 2009).

There are also many galaxy-evolution issues that do not obviously interact with spiral structure, for instance the correlation of stellar masses with central *Super Massive Black Hole* masses (Heckman and Best 2014) and deciding whether they radiate enough ultraviolet to re-ionize the universe at \(z \approx6\) (Stark 2016). Locally first, if not temporally first, come perturbations exerted on disks by a companion galaxy or bar. The former, with a swing amplifier, undoubtedly makes things to start as tidal distortions and end up looking like grand design spirals (Toomre 1981). And an ensemble of molecular clouds in the potential of a barred galaxy comes to trace out a spiral type S with their star formation (energy dissipation) (Combes and Gerin 1985). But see Toomre (1977) for how far the purely gravitational processes had come 40 years ago. Woltjer (1965), on the other hand, was certain (and has been since at least Woltjer 1959) that magnetic fields had to be part of the answer, since the energy densities in the Galactic plane in field, cosmic rays, and random gas motions are roughly equal, and both field lines and gas motions seemed to be at least partially along the arms.

This brings us to the density wave theory of Lin and Shu (1964) and Shu (2016). They decided to tackle the problem for the middle, that is, to impose pitched arms on the disk density structure and gravitational potential and see what happened. The short answer was “Not much”; That is the pattern persisted, soliton-like, for several times the rotation period of the model galaxy, and the pitch-angles of real galaxies look like the calculated ones (Pour-Imani et al. 2016). The modern version naturally looks a good deal more complex than the 1964 original, but also has some applicability to the rings of Saturn and hot-Jupiter formation around other stars (Shu 2016).

The chief competing theory for some years, applicable particularly to spiral in the past and to flocculent ones in *Stochastic Self-Propagating Star Formation (SSPSF)*, but forward by Mueller and Arnett (1976) and further developed by Gerola and Seiden (1978). The idea is that a random fluctuation in gas density yields a small burst or star formation; winds from the stars and supernovae move outward, compressing gas, which, in turn, forms stars. Differential rotation in the disk, which will eventually wind up and spoil arms, in the short term (\({\approx}10^{8}\) years), so the galactic rotation period stretches out those regions of propagating star formation into arc-shaped features. Thus SSPSF is a possible “starter” to establish S-shaped perturbations to the Lin and Shu (1964) process started. Auer (1999) combined the processes in roughly this fashion and regarded the result as a good way of looking at the initiation, development, and eventual washing out of spiral structure. Since all processes (Naab and Ostriker gas outflows, Lin and Shu magnetic confinement, SSPSF, and event companion driving) require the presence of gas to form new stars and make the arms visible spiral arms necessarily transient, on time scale from \(10^{8}\) to \(10^{10}\) years.

In summary, the structure of our Milky Way was triggered when the Sagittarius dwarf first passed through the disk about 9 Gyrs ago (Purcell et al. 2011). The arms have been preserved by a density wave (Lin and Shu 1964) with a swing amplifier (Toomre 1981, 1977), but an image reconstructed from HI data and starcluster information suggests with the bar as a likely additional driver (Combes and Gerin 1985), but an image reconstructed from HI and starclusters data (Bland-Hawthorn and Gerhard 2016; Elmegreen 2011) that we also have transient, flocculent spurs and other structures (Elmegreen and Elmegreen 1987). Gas inflow, which continues along the filaments that connect galaxies (Faucher-Giguere and Angles-Alcazar 2017), tends to make the disk larger and less dense and capable of continuing star formation (Naab and Ostriker 2017).

We turn now to the four cases of spiral formation or preservation that appear to be most closely connected with self-organization.

More complex galactic models with chaotic orbits and massive central masses (possibly attributed to a central black hole) were investigated by Kalapotharakos et al. (2004). Small central masses with a ratio of \(m < 0.005\) were found to organize chaotic orbits with Lyapunov exponents too small to develop chaotic diffusion during a Hubble time. Large central masses \((m \gtrsim 0.004)\), produce about the same amount of chaotic orbits, but the Lyapunov exponents are larger, so that the secular evolution evolves into a new equilibrium. The underlying self-organization mechanism converts chaotic orbits into ordered orbits of the *Short Axis Tube* type (Kalapotharakos et al. 2004).

A spatio-temporal self-organization in galaxy formation has been found from a relationship between the number of star formation peaks (per unit time) and the size of the temporal smoothing window function (used to define the peaks), holding over a range of \(\Delta t = 10\mbox{--}1000~\mbox{Myr}\) (Cen 2014). This finding reveals that the superficially chaotic process of galaxy formation is underlined by temporal self-organization up to at least one Gyr (Cen 2014).

The observed hierarchy of galactic structures, from giant cellular voids to enormous superclusters, with galaxies distributed within intricate networks of arcs and cells, clearly indicates some self-organizing process that is not consistent with a random distribution (Krishan 1991, 1992). Two different scenarios are usually considered: (i) the hot dark matter (HDM) scenario with initial large-scale structures that fragment into smaller ones, and (ii) the cold dark matter (CDM) scenario where the smaller structures form first, coalescing then to larger galactic structures. However, besides the self-organization of structures seen in luminous matter, the existence of dark matter (Trimble 1987) may have its own “dark self-organization”. Related self-organization processes may drive gravitational clustering and/or turbulent cascading (Krishan 1991, 1992).

### Critical Assessment

*According to the Hubble galaxy classification*, *galaxies can be formed in different morphologies*, *from ellipticals* (*type E*0-*E*7) *to normal spirals* (*type Sa*-*Sc*) *and barred spirals* (*SBa*-*SBc*), *which all represent a spatial pattern* (*S*) *observed in our present time*-*slice*. *The evolution from an initial random*-*like state to a well*-*ordered spatial structure* (*S*) *with a spiral pattern reveals the action of a self*-*organization process*. *Physical models of galaxy formation include at least three scenarios that are more or less consistent with the*\(\varLambda\)*CDM cosmology*: (*i*) *Interaction with a nearby companion*, (*ii*) *the Lin*-*Shu density wave theory*, *or* (*iii*) *flocculent bursts of star formation that get dragged out by differential rotation*. *Thus*, *a combination of gravity and rotation is a most likely driving force*, *while the feedback force or self*-*organizing instability is still open and is currently investigated with large*-*scale numerical N*-*body simulations*. (*Table *1: *qualifiers S*, *I*[?]).

## 6 Cosmology

Self-organization inherently involves regulated change. That is, it must be dynamical, involving driver forces and positive feedback mechanisms. Each of these features has an associated scale. When looking for the self-organizational aspects of cosmology, relevant scale sizes run from (at least) as small as the Planck scale (\({\approx}10^{-33}\mbox{ cm}\)) to the size of the observable Universe (\({\approx}10^{28}\mbox{ cm}\)), spanning over 60 orders of magnitude. Perhaps even more challenging, modern science has yet to reveal exactly whether and how one may correctly separate our understanding of particles and fields from that of the space-time in which they exist, especially at scales near to and less than the Planck scale. For the purposes of this review, we will simply focus on the organizational aspects of mainstream cosmology.

### 6.1 Einstein-de Sitter and \(\varLambda\)CDM Models

The first goal of cosmology is to understand how space changes with time. Einstein “interpreted gravity as a manifestation of geometry” (Misner et al. 1973). He showed us that a “4-D space-time” formed by merging 3-D space and the time dimension into one continuum (space-time), can respond to any form of energy. Einstein’s equation couples the curvature of space-time, i.e. gravity, to the stress-energy inside it. Therefore, placing energy sources inside a well-chosen unified 4-D space-time geometry is the best way to quantify how they coevolve. To examine how order developed, we recognize that the expansion dynamics of space-time sensitively depends upon the amount(s) of each energy component inside it, and the evolving organization of those energy components depends upon the expansion dynamics. Such coupling or “feedback” can lead to an evolution that “self-organizes”. Thus, when space-time contains matter and/or vacuum energy, interesting processes can emerge.

Einstein’s first solution to his equation assumed that space-time contained uniformly distributed normal matter, but was static; however, quickly realizing that space-time containing only matter would collapse, unless another component was included to resist it, he added a positive “cosmological constant term”. This is the equivalent of a uniform vacuum energy that counter-balances the curvature-producing effect of matter. Soon thereafter de Sitter produced a model envisioning a maximally symmetric space whose “metric” (curvature) is the same *at all times and all places*, which also included both uniformly distributed matter and a cosmological term to balance it. For such a space-time (de Sitter model 1), one may chose a time coordinate and its associated family of space like coordinates (“slicings”) that correspond to specific values of that time coordinate, to thereby represent geometrically flat (Euclidean), positively-curved, or negatively-curved 3-D spaces. However, the de Sitter (model 1) space does not restrict this choice, so it does not select a specific cosmology, *per se*. In other words, maximally symmetric de Sitter (model 1) space may just rest! However, *maximally symmetric space-times are … not reasonable models of the real world* (Carroll 2004). Then, de Sitter found that in a space-time with only a cosmological constant and no matter, test particles would accelerate away from one another! It is this second version (de Sitter model 2) with accelerating expansion that is normally associated with the “de Sitter space”. This was progress.

It was eventually recognized that our actual Universe (i) is geometrically flat \((k=0)\), (ii) has been expanding for almost 14 billion years, (iii) its cosmic fluid has passed through stages where its dominant component was radiation, then matter, and then vacuum energy, and (iv) we have been in a quasi-de Sitter accelerating expansion stage for the last 6 billion years! (Fig. 31). There is compelling evidence that the cosmic fluid is made up of normal matter and radiation, a form of matter that does not emit or absorb electromagnetic radiation (*Cold Dark Matter or “CDM”*), and vacuum energy (*“Dark Energy”*). It is then no surprise that the current cosmological model (“\(\varLambda\)CDM”) is based upon the Friedman Equations, although it is still undecided whether the vacuum energy component has a constant value \((\Lambda)\) or is changing with time (Fig. 31).

### 6.2 Evolution of Matterless Space-Time

The self-organizational concepts have been applied to the creation of de Sitter (model 2) space-time (only vacuum energy) and to the more relevant Friedman-LeMaitre space-time that obeys the Friedman equation. Creation of de Sitter (model 2) space-time from quantum fluctuations, combining causality and gravity with quantum theory, was discussed by Ambjorn et al. (2008). Viewed as a self-organization process, many microscopic constituents exhibit a collective behavior and give rise to a unified, smooth space-time macrostructure in this model. However, one must keep in mind that this pure de Sitter (model 2) space-time, without matter, is maximally symmetric and too broad to reflect the real Universe.

In another approach quantum gravity is described as a network that self-organizes into a discrete 4-D universe, in analogy to the ferro-magnetic Ising model for space-time vertices with an anti-ferromagnetic Ising model for the links. The ground state self-organizes as a new type of low-clustering graph with finite Hausdorff dimension 4 (Trugenberger 2015). Once again, this work does not appear to directly lead to the \(\varLambda\)CDM model.

### 6.3 Evolution of Space-Time with Matter and Radiation

The largest order out of random process in astrophysics today is the production of the observed large scale structure of galaxies and clusters of galaxies throughout the cosmos. The \(\varLambda\)CDM model, together with the theory of cosmic inflation, lay the foundation (i) for generating the initial conditions for structure formation, (ii) for creating matter and radiation, and (iii) for the subsequent hierarchical growth of the structure of matter via gravitational instability.

Because Einstein’s Equation relates space-time curvature (i.e., gravity) to the stress-energy of its contents, fluctuations of energy density will generate fluctuations of curvature. “Inflation theory” envisions a very early burst of quasi-de Sitter (model 2) expansion during near-Planck scale stochastic quantum fluctuations of the inflation-driving scalar energy field (“inflaton”) generates the corresponding space-time curvature fluctuations that expand superluminally to semi-classical scales. As inflation ends, the inflaton transfers its remaining energy into radiation and particles that, during the first few minutes of the Big Bang, evolve through a nucleosynthesis stage into a plasma of “normal matter” (comprised primarily of hydrogen and helium atoms and electrons) and gravitationally-interacting-only Dark Matter. As the continuing expansion further cools the cosmic fluid (plasma) further, the theory goes, it is attracted by the curvature fluctuations (gravitational potentials) originating from the inflaton field, and ultimately collapses into the structure we see today.

In a very recent paper, Ge and Wang (2017) set forth an approach to derive cosmological dynamics starting with the physics of quantum entanglement. Building upon earlier ideas that space-time geometry could be the result of the entanglement of macroscopic quantum states, together with recent work by Jakobson hypothesizing a relationship between Einstein’s equation for gravity and vacuum entanglement of quantum states, Ge and Wang (2017) were able to derive the flat-Universe Friedman equations (34) and (35) above. It will be interesting to see if further work exploiting the apparent deep connection between quantum information theory and the emergence of space-time will successfully be applied to the entire \(\varLambda\)CDM Universe paradigm with its inflaton field, dark matter, and dark energy.

### 6.4 The Cosmic Microwave Background

Measurements of the *Cosmic Microwave Background (CMB)* over the past 25 years strongly support the idea that at about 400,000 years into the Big Bang the temperature of the H/He plasma dropped to around 3000 degrees K, allowing the electrically charged free electrons and nuclei to then combine into neutral atoms. At this point, known as “recombination”, electromagnetic radiation (photons) that had previously enabled the plasma to resist, gravitational collapse was released, carrying the image of the last surface from which it scattered. Continued expansion of the Universe then caused the wavelengths of the released photons to stretch from visible to microwave values.

*Cosmic Background Explorere (COBE)*satellite work (Mather et al. 1991; Boggess et al. 1992), followed by the

*Wilkinson Microwave Anisotropy Probe (WMAP)*(Bennett et al. 2013; Hinshaw et al. 2013), and Planck satellites Planck Collaboration (2016), with increasing sensitivity and resolution, precisely mapped that 13.8-billion-year-old microwave image of the celestial sphere of the cosmos which encoded much of the physics of the early Universe (Fig. 32).

### 6.5 Formation of Large Scale Structure

So how do the MS and IS relate to the theme of this review? They show that the \(\varLambda\)CDM model, including inflation, well describes the initial conditions for the physical evolution of the large-scale structure of the Universe to the present time. The most interesting feature of the underlying cosmological model is that a period of exponential inflation can temporarily remove all disorder within the causal speck of space-time that then grows into our current observable Universe. At the end of inflation, the matter content of the cosmic fluid is created from the remaining inflaton energy and re-heats to a temperature at or above that envisioned by grand unified particle theories. Then, the “Big Bang” ensues. This model ensures that the post-inflation evolution of the Universe is basically isotropic and homogenous, as the Friedman Equation assumes. However, it is most fortunate for us that the large-scale structure is not perfectly isotropic and homogeneous, but exhibits a definite ordering into filaments, cluster nodes and voids on scales smaller than ≈100 Mpc where galaxies and stars, heavier elements, and life came to be.

### 6.6 Self-Organization and Logistic Growth

*carrying capacity*(or maximum amount of resource) in ecological models. The authors argue that the logistic growth model (Verhulst law), which describes nonlinear growth phenomena in a closed system with a limited total resource quantity \((D_{\infty})\), is a concept that agrees with the nonlinear theory of structure formation, and thus indicates a self-organized universe. The self-organizational aspect is the predicted feedback that the growth rate \(dD(r)/dr\) of the fractal dimension \(D\) is decreasing to zero (in the asymptotic limit) when the scale is increased \(r \mapsto\infty\) (in Eq. (37)). Interestingly, this model predicts an almost finite universe, where the mass or energy asymptotically vanishes at large distances (\(r \gtrsim 250~\mbox{Mpc}\)). It also predicts 2-D galactic structures at large distances, and 1-D structures (filaments or curvi-linear threads) at nearby galactic distances of \({r \lesssim 25~\mbox{Mpc}}\).

### 6.7 Self-Organization of Interacting Cosmic Fluid Components

Self-organization of components of the cosmic fluid into stars and galaxies was covered earlier in this review where dark and baryonic matter are assumed to only interact gravitationally. However, non-gravitational interactions of dark matter and dark energy have also been studied in a cosmological model with diffusion (Szydlowski and Stachowski 2016). The state variables of the density parameter for matter (dark and visible) and of the rate of growth of energy transfer between the dark sectors can be coupled using the Lotka-Volterra framework, from which it was demonstrated that the de Sitter solution is a global attractor for all trajectories in the phase space (Szydlowski and Stachowski 2016). In a related approach, called the “Jungle Universe”, the dynamics of homogeneous and isotropic Friedman-Lemaitre universes are considered as a special case of a generalized Lotka-Volterra system, where the competitive species are the barotropic fluids that fill the universe (Perez et al. 2014).

### Critical Assessment

*The large*-*scale structure of the Universe is seen to have resulted from a combination of quantum*-*fluctuation*-*seeded gravitational collapse and more complex particle physics*, *all the way back to the Big Bang*. *Self*-*organization concepts applied to cosmology are extremely scanty in literature* (*amounting to a few sentences in a few cosmology papers*) *and appear not to relate to the*\(\varLambda\)*CDM model*. *Quantitative measurements of spatial* (*S*) *or temporal structures* (*T*) *that discriminate against random patterns*, *identification of nonlinear dissipative systems with driver and positive feedback mechanisms*, *critical instabilities* (*I*), *and possible limit*-*cycle* (*LC*) *equilibria need to be identified*. *In conclusion*, *there is a lot of room for modeling of cosmological models in terms of self*-*organization* (*Table *1: *qualifiers S*[?], *T*[?], [*I*?], *LC*[?]).

## 7 Discussion

In this interdisciplinary review we aim to point out some universal properties of nonlinear systems governed by self-organization. We discussed 6 cases in planetary physics, 6 cases in solar physics, 3 cases in stellar physics, one case in galactic physics, and some tentative ideas in cosmology, amounting to 17 systems in the field of astronomy and astrophysics (Table 1). Self-organizing systems, however, have been found in many more scientific disciplines, such as in ionospheric physics, magnetospheric physics, plasma physics, physics, chemistry, biology, social science, and computer science, as the 51 examples compiled in Table 2 demonstrate.

Characteristics of randomness, self-organization, and self-organized criticality systems or processes

Parameter | Randomness process | Self-organizing process | Self-organized criticality process |
---|---|---|---|

Dynamics: | events | limit cycle | avalanches |

Temporal structure: | intermittent | quasi-periodic | intermittent, scale-free |

Temporal size distribution: | exponential | quantized | power law |

Spatial structure: | random | ordered | fractal, scale-free |

Spatial size distribution: | Gaussian | quantized | power law |

Entropy evolution: | increasing | decreasing | invariant |

Physical condition: | independency | positive feedback | critical threshold |

### 7.1 Characteristics of Random Processes

Random or stochastic processes can be characterized with the statistics of independent events. The mathematical distribution of independent events can be derived from rolling dices, which leads to a binomial distribution and can be approximated by a Gaussian function (also called normal distribution) in the limit of an infinite number of dices, or with a Poisson distribution or exponential distribution in the limit of rare events. A time series of random events consists of irregular, intermittent events and the resulting power spectrum is characterized by white noise (i.e., a flat power spectrum \(f(\nu)=\text{const}\)). We can consider random processes in time or in space. If spatial structures are produced by a random process, their size distribution is theoretically a Gaussian function, with a well-defined mean and standard deviation, where the mean defines a specific preferred spatial scale. From the thermodynamic or information (theory) point of view, the entropy is increasing with time in random processes. Examples of random processes are Brownian motion of gas molecules, diffusion processes, the detected photons from a star, electrical current fluctuations due to thermal noise, or patterns of sand at the beach (Fig. 38a). (For a concise summary of the statistics of random processes see Sect. 4 in Aschwanden 2011).

### 7.2 Characteristics of Self-Organized Criticality

Self-organized criticality systems (Bak et al. 1987; Pruessner 2012; Aschwanden et al. 2016) are completely different from random processes, which is experimentally and observationally demonstrated by the appearance of scale-free power law distributions of spatial and temporal sizes. Avalanches in self-organized criticality systems represent coherent structures in the time domain (1/f-noise), in contrast to incoherent noise in random systems. Quantitatively, a size distribution of avalanches in a self-organized criticality system can be simulated from chain reactions of nearest-neighbor interactions in a lattice grid, where a critical threshold of the gradient (or curvature radius) between next-neighbor interactions has to be exceeded, before an avalanche can start. The avalanches occur intermittently in such a complex system, and the intervening time intervals (waiting times) obey an exponential random distribution function. The spatial structure of the avalanches is fractal (or multi-fractal), which corresponds to a power law distribution also. The reason why such a dissipative nonlinear system is called “self-organizing”, (e.g., the critical slope of a sandpile) is the fact that the system automatically maintains the critical state of avalanching without external control, as long as the energy input into this open system is steady and stochastic. The microscopic structure of a self-organizing system is maintained in the time average, and thus the entropy of the system is invariant when averaged over many avalanches. Examples of self-organizing systems are sand piles (Fig. 38c), earthquakes, solar flares, forest fires, stock market fluctuations, etc. (see a representative list of phenomena in Sect. 1 of Aschwanden 2011).

### 7.3 Characteristics of Self-Organization

Although the two terms “self-organization” and “self-organized criticality” sound confusingly similar, they characterize two completely different types of dissipative nonlinear systems, to which terminology we will adhere for historical reasons. The only commonality between the two systems is that both are open nonlinear dissipative systems with external energy input, and that both maintain some system property in an automated way without external control. A self-organizing system has always a primary driving force, and a secondary counter-acting force that acts as a stabilizing feedback reaction to the driving force, in order to ensure long-term stability of the resulting quasi-equilibrium state. A system can only be called to be a self-organization process, when the feedback mechanism leads to a quasi-stationary stabilization of the combined system. Otherwise, a system with a negative feedback (or none) will evolve away from a stationary state and end in a catastrophic way.

What sets a self-organizing system apart from a random system is the ability to create *“order out of chaos”*, or better “order out of randomness”, since the term “chaos” is already used in nonlinear physics to characterize a particular type of nonlinear system behavior that is non-deterministic. Therefore, the property of self-organization is also called spontaneous order, a process where some form of overall order arises from local interactions between parts of an initially disordered system. In principle, every ordered structure that is significantly different from randomized distributions requires an ordering mechanism, or a self-organizing process. We have seen in this review that every self-organizing mechanism observed in astrophysics can be modeled by a system of coupled differential equations, among which the type of a Lotka-Volterra system is most prominent. These differential equations describe the interaction between a driving force and a secondary feedback force, which generally have a limit-cycle solution, and therefore can sustain a quasi-stationary oscillation near the limit cycle, which is also called attractor (or strange attractor if it has a fractal structure). The quasi-stationary, quasi-periodic system dynamics near a limit cycle is the essential characteristic of self-organization processes.

What are the size distributions of a self-organizing system? Since the limit cycle represents a fixed value of a time period (which often corresponds also to a fixed value of a spatial structure), the size distribution is expected to be peaked or quantized at this particular value. For instance, the solar granulation exhibits a fixed spatial scale of \(w \approx1500\mbox{ km}\) for convection cells (granules), and a temporal scale (or life time) of \(\tau\approx8\mbox{--}10~\mbox{min}\). In the case of planetary systems, each planet has its own attractor and limit-cycle dynamics, which are moreover weakly coupled by harmonic ratios in a N-body system. Since the limit-cycle solution often contains some random noise, the size distribution is generally not sharply quantized like a delta-function, but rather broadened to a single or multiple Gaussian functions, see Gaussian distribution of granules in Fig. 12, right panel.

Since self-organizing systems create “order out of randomness”, the entropy is decreasing during the evolution from an initially disordered system to a self-organized limit-cycle behavior. It is also said that self-organizing systems evolve into a dynamics far away from thermal equilibrium. A limit cycle is defined by a critical point (in phase space) around which the quasi-stationary oscillation dynamics occurs, where the amplitude of the oscillation is a measure how far off the system evolves from a system equilibrium solution. The thermal equilibrium therefore corresponds to the asymptotic limit of a vanishing limit-cycle amplitude, where every dynamic system variable becomes constant.

In Fig. 38b we show the pattern of sand dunes, which self-organized in an interplay between gravity and wind, forming ripple patterns with a fixed ripple separation scale, which is distinctly different from sand piles generated by self-organized criticality avalanching (Fig. 38c), or from a sand beach shaped by random processes.

### 7.4 The Physics of Self-Organization Systems

It should be clear by know that the term “self-organization” merely expresses a category of a nonlinear dissipative system behavior, which is a general property of complex system behavior, but does not define a specific physical model for an observed phenomenon. The choice of a particular physical model that is applied to an observed phenomenon is a matter of interpretation. We can study the dynamic system behavior with purely mathematical models (of coupled differential equation systems) without specifying a physical application (e.g., see examples in textbook by Strogatz (1994, Chap. 7)). However, since we are most interested in obtaining physical insights from the observed phenomena, we identified the underlying driving forces and positive feedback mechanisms for each of the 17 studied astrophysical phenomena (Table 1).

A summary of observed astrophysical phenomena and the self-organizing driver forces and feedback mechanisms is given in Table 1, based on the concepts offered in the reviewed publications. Drivers can be the gravitational force, the centrifugal force (from rotation), differential rotation, solar radiation, temperature gradients, convection, magnetic stressing, plasma evaporation, acceleration of nonthermal particles, or the cosmic expansion. Feedback mechanisms involve mostly instabilities (i.e., the magneto-rotational or Balbus-Hawley instability, the Rayleigh-Bénard instability, turbulence, vortex attraction, magnetic reconnection, plasma condensation, loss-cone instability), but also resonances (mechanical orbit resonance, double plasma resonance). While instabilities mostly evolve into limit-cycle behavior, which constrains one specific time scale, resonances can produce ordered structures at multiple quantized values (such as harmonic orbit resonances of planets, or magnetic harmonics in upper hybrid waves of solar radio bursts).

We described the underlying physical models with systems of coupled differential equations in this review. Very few equation systems can be analytically solved, if at all. For instance, even the basic Lotka-Volterra equation system has a transcendental (implicit) solution (Appendix D). Consequently, the more complex cases that involve a hydrodynamic approach (photospheric granulation, chromospheric evaporation, star formation, galaxy formation), an MHD approach (protoplanetary disks, solar magnetic fields, the Hale cycle), or N-body problems (planetary spacing, planetary rings and moons), have to be studied by numerical simulations. Numerical solutions of coupled differential equation systems can now easily be obtained with numerical minimization algorithms (in form of time profiles \(X(t), Y(t)\), or phase diagrams \(Y(X)\), see Fig. 18).

## 8 Conclusions

- 1.
Self-organization is a very multi-disciplinary subject that has been applied in planetary physics, solar physics, stellar physics, galactic physics, cosmology, ionospheric physics, magnetospheric physics, laboratory plasma physics, condensed matter physics, chemistry, biology, social science, and computer science.

- 2.
Self-organizing systems in astrophysics create spontaneous

*order out of randomness*, during the evolution from an initially disordered system to an ordered and more regular quasi-stationary system, via: (i) quasi-periodic limit-cycle dynamics, and/or (ii) resonances (i.e., harmonic mechanical resonances, or harmonics of the gyrofrequency). - 3.
Self-organizing processes are not controlled from outside, but are driven by global forces (inside an open dissipative system), such as gravity, rotation, thermal pressure, or acceleration of nonthermal particles, in the case of astrophysical applications.

- 4.
The limit-cycle behavior of astrophysical self-organization processes occurs due to a positive feedback mechanism that couples with the primary driver. This feedback mechanism is often an instability, such as the magneto-rotational instability, the Rayleigh-Bénard convection instability, turbulence, vortex attraction, magnetic reconnection, plasma condensation, or a loss-cone instability.

- 5.
Physical models of an astrophysical self-organization process require a hydrodynamic approach (photospheric granulation, chromospheric evaporation, star formation, galaxy formation), an MHD approach (protoplanetary disks, solar magnetic field, Hale cycle), or N-body simulations (planetary spacing, planetary rings and moons).

- 6.
The entropy in self-organization processes is decreasing during the evolution from an initially disordered system to a self-organized limit-cycle behavior, in contrast to random processes where the entropy increases, or to self-organized criticality systems where the entropy remains invariant in the long-term time average.

- 7.
The Lotka-Volterra equation system represents a useful tool to study the dynamical behavior of nonlinear dissipative systems, which are likely to evolve into a limit-cycle behavior for long-lived quasi-stationary phenomena.

While the modeling of systems with self-organization was severely hampered in the past, due to the mathematical difficulty of finding analytical solutions for coupled integro-differential equation systems, it is expected that the use of numerical computer simulations will be capable to produce realistic models in the future, for hydrodynamic, MHD, and N-body problems.

## Notes

### Acknowledgements

The author acknowledges the hospitality and partial support for two workshops on “Self-Organized Criticality and Turbulence” at the *International Space Science Institute (ISSI)* at Bern, Switzerland, during October 15–19, 2012, and September 16–20, 2013, as well as constructive and stimulating discussions (in alphabetical order) with Robert Cameron, Sandra Chapman, Paul Charbonneau, Daniel Fabrycky, Clara Froment, Hermann Haken, Henrik Jeldtoft Jensen, Lucy McFadden, and Nick Watkins. This work was partially supported by NASA contract NNX11A099G “Self-organized criticality in solar physics”.

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