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From classical to active particles: mathematical tools for social dynamics and behavioural economics

Marina Dolfin, Leone Leonida

TL;DR

The work interrogates how to mathematically describe collective social and economic dynamics arising from large populations of living agents. It foregrounds the kinetic theory of active particles (KTAP) as a unifying differential-structure framework that extends classical Boltzmann theory to systems with an activity variable and multiple functional subsystems, capturing learning and decision-making processes. By contrasting classical and active particle formalisms, it exposes the nonlinear, nonlocal, and multi-physics nature of interactions, including proliferation and destruction, and develops spatially inhomogeneous and spatially homogeneous models with explicit operators and domains such as $\Omega[f]$. The paper then charts research perspectives toward a mathematical theory of socio-economic dynamics, emphasizing nonlinear interactions, evolving interaction rules, and multiscale validation, with potential applications to brain-guided learning, behavioral economics, and scientific machine learning. Its synthesis offers a principled path to translate microscopic agent rules into emergent collective behavior, informing both theory and real-world analyses of social dynamics.

Abstract

This essay provides a critical overview of the mathematical kinetic theory of active particles, which is used to model and study collective systems consisting of interacting living entities, such as those involved in behavior and evolution. The main objective is to study the interactions of large systems of living entities mathematically. More specifically, the study relates to the complex features of living systems and the mathematical tools inspired by statistical physics. The focus is on the mathematical description of these interactions and their role in deriving differential systems that describe the aforementioned dynamics. The paper demonstrates that studying these interactions naturally yields new mathematical insights into systems in the natural sciences and behavioral economics.

From classical to active particles: mathematical tools for social dynamics and behavioural economics

TL;DR

The work interrogates how to mathematically describe collective social and economic dynamics arising from large populations of living agents. It foregrounds the kinetic theory of active particles (KTAP) as a unifying differential-structure framework that extends classical Boltzmann theory to systems with an activity variable and multiple functional subsystems, capturing learning and decision-making processes. By contrasting classical and active particle formalisms, it exposes the nonlinear, nonlocal, and multi-physics nature of interactions, including proliferation and destruction, and develops spatially inhomogeneous and spatially homogeneous models with explicit operators and domains such as . The paper then charts research perspectives toward a mathematical theory of socio-economic dynamics, emphasizing nonlinear interactions, evolving interaction rules, and multiscale validation, with potential applications to brain-guided learning, behavioral economics, and scientific machine learning. Its synthesis offers a principled path to translate microscopic agent rules into emergent collective behavior, informing both theory and real-world analyses of social dynamics.

Abstract

This essay provides a critical overview of the mathematical kinetic theory of active particles, which is used to model and study collective systems consisting of interacting living entities, such as those involved in behavior and evolution. The main objective is to study the interactions of large systems of living entities mathematically. More specifically, the study relates to the complex features of living systems and the mathematical tools inspired by statistical physics. The focus is on the mathematical description of these interactions and their role in deriving differential systems that describe the aforementioned dynamics. The paper demonstrates that studying these interactions naturally yields new mathematical insights into systems in the natural sciences and behavioral economics.
Paper Structure (13 sections, 26 equations, 3 figures)

This paper contains 13 sections, 26 equations, 3 figures.

Table of Contents

  1. Motivations and Plan of the Paper
  2. Theories of the kinetic theory of classical and active particles
  3. Differential structures for the dynamics of classical particles
  4. Differential structures for the dynamics of active particles
  5. Critical analysis and selection of a differential structure
  6. Complexity of the collective dynamics of living systems
  7. Interactions of active particles
  8. Interactions of classical particles
  9. Interactions of active particles
  10. Interactions in spatial homogeneity
  11. Towards research perspectives
  12. On the dynamics of interaction's rules
  13. On a quest to a mathematical theory

Figures (3)

  • Figure 1: Sequential steps of the modelling approach
  • Figure 2: Blue color direct arrows: Gain of the low level a-particles and loss of the high level a-particles. Red color crossing arrows: Loss of the low level a-particles an d gain of the high level a-particles.
  • Figure 3: Brain-guided learning and decision-making