A Quest Towards a Mathematical Theory of Living Systems

Bellomo, Nicola; Bellouquid, Abdelghani; Gibelli, Livio; Outada, Nisrine

doi:10.1007/978-3-319-57436-3

Cited by 65 publications

(75 citation statements)

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“…We specifically refer to [3] (Chapter 3) for systems where the interacting entities-viewed as active particles-are partitioned into n functional subsystems labeled by the subscript i = 1, . .…”

Section: On the Dynamics Of Active Particlesmentioning

confidence: 99%

“…Here, the first step is the derivation of a mathematical structure consistent with the complexity features of living systems. A recently published book has been devoted to this topic [3], while a specific structure has been presented in our paper. Actually, we do believe that a further step would be particularizing the selection of the entropy to each specific structure corresponding to different classes of systems.…”

Section: Research Perspectivesmentioning

confidence: 99%

“…The interested reader is referred to the book [3] for theoretical aspects and for the rationale to develop specific applications which have been produced to model, for instance, the dynamics of collective learning [4], the immune competition [5], vehicular traffic [6][7][8], the dynamics of crowds [9][10][11], and the collective motion of swarms [12].…”

Section: Introductionmentioning

confidence: 99%

“…The contents are presented as follows: Section 2 defines the mathematical description of a multi-particle system and provides the conceptual reference of the developments proposed in our paper; it is a very concise description extracted from [3] (Chapter 3). Section 3 contains the main results obtained by deriving a new class of models with discrete states of the internal variable and analyzes the related entropy dynamics; this general framework includes further developments, with respect to Section 2, to include the aforementioned Darwinist dynamics and selection.…”

Section: Introductionmentioning

confidence: 99%

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On Entropy Dynamics for Active “Living” Particles

Ełaiw

Alghamdi

Bellomo

2017

Entropy

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This paper presents a modeling approach, followed by entropy calculations of the dynamics of large systems of interacting active particles viewed as living-hence, complex-systems. Active particles are partitioned into functional subsystems, while their state is modeled by a discrete scalar variable, while the state of the overall system is defined by a probability distribution function over the state of the particles. The aim of this paper consists of contributing to a further development of the mathematical kinetic theory of active particles.

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Section: On the Dynamics Of Active Particlesmentioning

confidence: 99%

Section: Research Perspectivesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Entropy Dynamics for Active “Living” Particles

Ełaiw

Alghamdi

Bellomo

2017

Entropy

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“…Unfortunately, there is no multi-dimensional counterpart for this complete cluster predictability. The clustering dynamics of the C-S model under the attractive-repulsive couplings was also discussed in [21,22,32] Swarms theory often interacts with the kinetic theory approaches which might be based on Fokker-Plank methods [35] or stochastic evolutive games within the framework of the so-called kinetic theory of active particles [10]. An example of a kinetic theory approach to swarm modeling is given in [15].…”

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confidence: 99%

A quest toward a mathematical theory of the dynamics of swarms

Bellomo

2017

Math. Models Methods Appl. Sci.

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101

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This paper presents a unified mathematical theory of swarms where the dynamics of social behaviors interacts with the mechanical dynamics of self-propelled particles. The term behavioral swarms is introduced to characterize the specific object of the theory which is subsequently followed by applications. As concrete examples for our unified approach, we show that several Cucker-Smale type models with internal variables fall down to our framework. Subsequently the modeling goes beyond the Cucker-Smale approach and looks ahead to research perspectives. Plan of the paperThe celebrated paper by Cucker and Smale [20] has boosted a huge literature on the modeling, qualitative analysis, and computational applications of the mathematical theory of swarms. Namely, models which describe, within a pseudo-Newtonian framework, the mechanics of many interacting self-propelled particles. Interactions are generally nonlocal and nonlinearly additive. These produce accelerations, where inertia is hidden in the interaction parameters rather than being explicitly taken into account. The mathematical literature in the field has been reviewed and critically analyzed in [3], specifically treated in Sections 5 and 6 which are devoted to the theory of swarms. This topic has been related to the dynamics of crowds as well as to computational models to treat kinetic and multi-agent systems.After Cucker-Smale's (C-S)'s seminal work, the (C-S) model has been extensively studied in literature, for example, we refer to [18] and references therein for mathematical aspects of the model. Among them, we briefly discuss some recent works incorporating social behaviors together with mechanical aspect of the model. Original C-S model describes the temporal evolution of the mechanical variables (positions and momentum of C-S particles), whereas in the modeling for the collective dynamics of biological and social complex systems, one needs to take into account of internal variables such as temperature, spin and excitation [24,27,29] etc, see Section 3.Recently, in [26], the authors found that the one-dimensional C-S model can be converted into the first-order nonlinear consensus model with monotone coupling function on the real line. Then, in 1 arXiv:2006.12932v1 [nlin.AO]

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On the Complex Interaction Between Mathematics and Urban Morphology

Bellomo

Terna²

2019

The Mathematics of Urban Morphology

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A Quest Towards a Mathematical Theory of Living Systems

Cited by 65 publications

References 0 publications

On Entropy Dynamics for Active “Living” Particles

On Entropy Dynamics for Active “Living” Particles

A quest toward a mathematical theory of the dynamics of swarms

On the Complex Interaction Between Mathematics and Urban Morphology

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