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We propose here a formal approach to study collective behaviors intended as

Understanding what “mechanism” allows a system to maintain its coherence and its global robustness is fundamental to study collective behaviors. The general approach usually adopt models already well consolidated in physics, assuming

Ideal models are based on general principles and studied analytically in an infinite volume independent on initial conditions and boundaries. Non-ideal models are based on opposite assumptions, considering

Within the mesoscopic description level of complex processes,

In these

By the classification of forms of emergence introduced by Crutchfield [

Actually, the selection of observables is performed here by the observer and is intended as

Here, to reaffirm these concepts, we consider collective behaviors modeled by variations of geometrical coordinates and spatial properties, e.g., speed, direction, distance and topology of interacting generic agents of Collective Systems represented. Also, by variations in shapes, density, volume and behavior of the collective entity, e.g., swarms, flocks, and protein chains emerging in 3D space. We refer to such collective behaviors as Spatial Collective Behaviors (SCBs) adopted by Collective Systems. The extension of the approach to non-spatial collective behaviors requires adoption of different mesoscopic variables, whereas meta-structural properties should, in any case, represent the coherence deriving from processes of emergence.

The novel ideas in our approach consist of considering SCBs as being established through coherent sequences of spatial configurations, adopted by interacting agents through various corresponding structures over time. The corresponding sequences of new structures are intended here as sequences of phase transitions and their coherence is modeled by using properties of values taken by suitable mesoscopic variables, rather than micro or macroscopic ones as in more usual approaches, where the term configuration denotes the spatial arrangement of agents. We introduce approaches considering emergence as mesoscopic coherence.

It is well known that systems may be studied by using a variety of assumptions. Our research focuses upon so-called

Consider the following cases:

Examples of collective systems, given by the collective motion of living systems,

Examples of collective systems, given by the collective motion of living systems, provided with no cognitive systems, include amoeba, bacterial colonies, cells, and macromolecules. Examples of collective systems, given by the collective motion of

Examples of collective artificial systems, given by

Several cases and distinctions should be considered:

The homogeneous and non-homogeneous cases where generic agents have or do not have the

The number of generic agents may be

The way of interacting—called here the

We recall that a Multiple System is a coherent set of simultaneous or successive systems, modeled by the observer and established by the same elements interacting in different ways,

In other words Multiple Systems are systems with components belonging to more than one system. Collective Beings are particular cases of Multiple Systems where the components are autonomous agents,

The study of Multiple Systems also considers

As introduced above, we will consider the multiplicity of rules, which can be adopted by generic agents in any combination and for various periods of time. The problem in representing such multiplicity is almost intractable at a microscopic level where each rule should be, for instance, formalized in the same way if not limited to different values of a parameter. At a macroscopic level, we lose

The

On the other hand, mesoscopic clustering, allowing identification of mesoscopic variables, may be given without assuming

Such a mesoscopic level of description is suitable to

As stated above, the dynamics of such kinds of complex systems are known only

The coherence of sequences of configurations in collective behavior is represented by meta-structural properties. In this view:

Mesoscopic variables

Properties of sets of such values represent the coherence of sequences of configurations,

Multiple structures are considered to be suitably represented by the values and properties adopted by mesoscopic variables, thereby specifying effective applications of rules of interaction, such as, at a suitable threshold. In the case of SCBs, for example:

_{i}

_{i}

_{i}

_{i}

_{i}

_{i}

_{i}

Macroscopic variables such as measures of _{i}_{i}

We specify that various _{1}_{2}_{n}

Specific generic agents establish Mesoscopic variables. Sets of generic agents establishing mesoscopic variables may even be considered for other properties than their mesoscopic belonging. For instance,

We may consider a number _{k}_{j:1-M}_{i:1-T}

It is possible to introduce the mesoscopic general vector
_{m,j}_{i}_{1}_{i}_{2}_{i}_{M}_{i}_{j}_{i}_{j}_{i}_{i}_{j}_{i}_{i}

It is also possible to consider a matrix _{i}_{k}_{k}_{,m}(_{i}_{k}_{i}_{k}_{i}

This matrix—mesoscopic general vector—fixes the properties valid for single generic agents over time. The values of the vector in the matrix considered above over time are termed meta-elements.

Mathematical properties of sets of meta-elements over time are termed here as meta-structural properties,

Examples of meta-structural properties are:

Properties of the values acquired by mesoscopic variables, single or crossed, such as any regularities including periodicity, quasi-periodicity, chaotic regularities possibly with attractors which characterize specific collective behaviors;

Possible statistical properties of sets of meta-elements detected by suitable techniques like Principal Components (PCs), Recurrence Quantification Analysis (RQA), Multivariate Data Analysis (MDA), Cluster Analysis, Principal Component Analysis (PCA), Time-Series Analysis, Pearson Product Moment Correlation Coefficient (PPMCC);

Properties, e.g., geometrical and statistical, of sets of generic agents constituting mesoscopic variables;

Properties related to the usage of degrees of freedom as introduced above;

Relationships between properties of sets of clustered generic agents and, macroscopic properties such as density, distribution, scale-freeness, numerical properties such as percentages;

Properties of the thresholds adopted for specifying the mesoscopic general vector;

Levels of ergodicity or quasi-ergodicity;

Properties of values of the vector and the matrix considered above over time;

Possible topological properties of network representations such as power laws and scale-freeness.

By the Matrix (2) it is possible to consider some

More generally,

When focusing on generic agents we may consider various cases of different complexities:

All of the generic agents simultaneously possess all the same single mesoscopic property which is constant over time;

All of the generic agents simultaneously possess the same subset, constant over time, of the mesoscopic properties available;

All of the generic agents simultaneously possess a subset, variable over time, of the mesoscopic properties available;

A significant percentage of the generic agents simultaneously possess all the same single mesoscopic property which is constant over time;

A significant percentage of the generic agents simultaneously possess the same subset, constant over time, of the mesoscopic properties available;

A significant percentage of the generic agents simultaneously possess a subset, variable over time, of the mesoscopic properties available;

Any combinations of the previous cases may occur regarding different or the same generic agents.

Since the seven cases above are very general, it is necessary to

It would be interesting to study, for instance, the possible correspondence between such cases and the topological roles of generic agents, e.g., at the center, on the boundary, in the front of or at the bottom, in the case of collective motion; or diffused.

A general view is presented in

Mesoscopic dynamics.

Mesoscopic Dynamics | |
---|---|

Structural cases | Meta-structural properties |

(1) All of the generic agents simultaneously possess all the same single mesoscopic property constant over time; | Trivial meta-structural properties |

(2) All of the generic agents simultaneously possess the same subset, constant over time, of the mesoscopic properties available; | Trivial meta-structural properties |

(3) All of the generic agents simultaneously possess a subset, variable over time, of the mesoscopic properties available; | Significant meta-structural properties |

(4) A significant percentage of the generic agents simultaneously all possess the same single mesoscopic property constant over time; | Non-trivial meta-structural properties |

(5) A significant percentage of the generic agents simultaneously possess the same subset, constant over time, of the mesoscopic properties available; | Non-trivial meta-structural properties |

(6) A significant percentage of the generic agents simultaneously possess a subset, variable over time, of the mesoscopic properties available; | Non-trivial meta-structural properties |

(7) Any combinations of the previous cases may occur regarding different or the same generic agents. | Complex multiple meta-structural properties |

Summary of key concepts used in the meta-structures approach.

Mesoscopic variables | Mesoscopic state variables are invented by the observer in a constructivist manner and represent clusters of agents taking on the same values _{i} |

Meta-elements | Meta-elements are time-ordered sets of values in a discrete temporal representation adopted by mesoscopic variables over time and |

Meta-structural properties | Meta-Structural properties are given by the mathematical properties possessed by ordered sets of values establishing Meta-elements, e.g., statistical, periodic, or correlative. |

Meta-Structure | The term |

Dynamical Systems |

We can consider meta-elements and meta-structural properties useful for characterizing collective behaviors as represented by their mesoscopic changing.

The metastructural approach

As we said, the main aim of MS approach is not predicting but being an intervention on systems:

Induce coherence within sets of elements interacting collectively giving rise to processes of emergence;

Change properties of collective behaviors, allow merging processes;

Maintain or restore the coherence of a collective behavior when possible changes or a loss of coherence occur for any internal and/or external reason;

Destroy or prevent the coherence and related processes of emergence which should be avoided under any circumstances.

An example consists in

For the introduction of the Perturbative Collective Behavior (PCB), we may consider aspects such as:

Component generic agents of the PCB, may be the same generic agents of the collective behavior,

The distribution of PCB agents may be fixed or variable with some regularities;

The percentages of agents acquiring mutations may be fixed or variable and have a fixed duration or be properly distributed over time;

It is possible to obtain various simultaneous or subsequent PCBs.

The possibility of considering

A typical methodology, for example, consists in inserting—according to a proper space-time grid—the muted agents in CB following a doping strategy inspired by

The nature of the possible general approach discussed above recalls effects, which can be considered in populations of cells, when some of them mutate, acquiring different properties, external agents are

The use of such approaches to induce the emergent system to behave by respecting some specific meta-structural properties, as well as the

Conceptually, we can consider at any instant the

The next step should be to use

The above mentioned approaches constitute the research framework where we may consider the implementations of tools suitable for recognizing emergence, even in the absence of the recognition of acquired properties, and to induce, restore, reproduce or even avoid coherence.

The theory of Phase Transitions (PT) using the Spontaneous Symmetry Breaking (SSB) mechanism in Quantum Field Theory (QFT) is well known in the literature. Consider, for example, a marble standing in an unstable equilibrium at the top of a “Mexican Hat” potential well. When a slight variation occurs, thus breaking the equilibrium, the marble will roll down the gradient to some position in the circular valley at the bottom.

The global structure of the dynamic situation obeys general symmetry principles (the Mexican Hat does not change its shape), but the final state is highly asymmetrical. As a classical case, it is a banal situation, but if the marble considered is an infinite state quantum system and the Mexican Hat is the potential, defining its dynamic-evolutionary possibilities with infinite degrees of freedom, the question becomes quite interesting. The rolling down is really a pertinent image for radical emergence phenomena in QFT. When one of the parameters is linked to the available energy changes, the system will distribute itself in one of the many possible ground states, with a consequent energy redistribution characterizing its macroscopic properties. Each “marble position” expresses a different energy arrangement of the system, and in contrast with the classical case, there is no possibility of forecasting any details of the final state; because the renowned quantum dice, about which Einstein expressed concern, are not informationally closed with respect to the observer, whereas the statistics of quantum objects—Fermi-Dirac for fermions and Bose-Einstein for bosons—are radically different from classical statistics and provide a rich phenomenology of organized states.

To be more precise, the key idea is that with an infinite number of states, quantum systems are different and non-unitarily equivalent representations of the same system that are possible, and consequently, phase transitions can structurally modify the system as well. This occurs by means of the SSB, that is, a process that does not allow all the states to be compatible with a given invariant energy value [

What usually happens is, when a given parameter varies, the system will settle into one of its possible fundamental states, thus breaking the symmetry. This leads to balancing by the emergence of long-range correlations associated with Higgs-Goldstone bosons, which act to make the new configuration stable.

The boson-condensed states can be fully considered as forms of macroscopic coherence of the system, and they are peculiar to the quantum statistics, which formally depend upon the indistinguishability of states with respect to the observer. The new system’s phase requires a new description level for its behaviors, so that we can speak of radical emergence.

Many behaviors of great interest in Physics on different scales are included within SSB processes, such as phonons in a crystal, Cooper pairs in superconductivity phenomena, the Higgs mechanism, multiple vacuum states in elementary particle physics, and inflation and formation of the “cosmic landscape” in Quantum Cosmology. It is reasonable to suppose that the fundamental processes for the formation of structures depend, essentially and critically, upon SSB and the QFT “syntax” makes it possible to grasp them.

A question of great interest arises when comparing the “ideal model” of emergence, proposed by the language of dense quantum systems, with the more classical, traditional, and “semi-classical” ones of Prigogine dissipative systems and the self-organization processes at the boundary between order and disorder. This problem is strongly correlated to the emergence of the classical world from the quantum one, and a promising approach is to consider the traditional—classical or semi-classical and critically depending on opportune boundary conditions-self—organization theories as emergent residual “traces” of SSB processes. Most of the complex systems we deal with have finite dimensions and a very high, but not infinite, number of degrees of freedom. One answer could be that these systems are the outcome of a “freezing” of the degrees of freedom typical of the SSB system and all the classical self-organization phenomena are the consequence of quantum processes of symmetry breaking. According to this idea, phenomenological emergence manifestations are a particular case of quantum radical emergence [

How can SSB radical emergence be compared with the phenomenological detection of patterns and how can the radically quantum features be distinguished? In SSB processes, the phase transition is likewise led toward a globally predictable state by an order parameter, that is, we know that there exists a critical value beyond which the system will find a new state and exhibit macroscopic correlations, and here too, a relevant role is played by boundary conditions (all in all, a phonon is the dynamic emergence occurring within a crystal lattice and it does not make any sense out of it). Moreover, in SSB, there exists an “adjustment” transient phase whose description is mostly classical. Where the analogy fails and we can actually speak of an irreducibly nonclassical feature, is the bosonic condensation, which is a non-local phenomenon. In a classical dissipative system, we can, in principle, obtain information on the “fine details” of bifurcation and know where the marble will fall, whereas in SSB processes, this is not possible because of the very nature of the quantum roulette! This point suggests that QFT is the conceptually “ideal” framework to describe emergence, but much less practical in many classical cases, quite common in the “middle way” [

In Haken’s Synergetics [

Finally, we observe that generally, in classical and quantum phase transitions, the stochastic and constructive role of fluctuations is well understood, and “domesticated” by a rich tradition of mathematical methods. As we have already mentioned, the mesoscopic scale is defined as the area of “negotiation” between micro and macro dynamics, and this area is fully defined by the global boundary conditions and microscopic interactions. These aspects allow their treatment, using ideal models. In the case of the collective systems studied by meta-structures, such as groups, swarms and flocks, the mesoscopic, and therefore, fluctuations, have an irreducible centrality still not fully understood. For example, the changes in an observable can have influence on certain aspects of the system and not on others. In particular, this depends on the fact that the inter-relations between agents do not follow simple laws of interaction, and often the effectiveness of a fluctuation depends critically on the existence of an

The fundamental lesson of complex systems is that they cannot be zipped in a single formal model. Emergence processes require the dynamic and integrated use of many different models. In this work, we introduced the general lines of an approach useful when in handling radical emergence, when we cannot make use of the traditional models of Theoretical Physics. They are collective behaviors—unpredictable on the basis of microscopic information (even when it is available)—and systems, which can be neither modeled by a “state equation” with few macroscopic variables. Actually, we cannot speak exactly of “dynamics”, but of “change” in a more general sense. Typical cases are: traffic, animal collective motion, social systems and markets. Even in these intractable cases we do not renounce to the possibility of a scientific investigation. It can be interesting, for example, to investigate the manifestation of specific and significant configurations,

The MS are useful to design collective behaviors by

The authors declare no conflict of interest.