Variational, Nonequilibrium, and Optimization Principles of the Coevolution of Structure and Dynamics in Complex Systems

Complex systems fascinate because of the way dynamic microscopic interactions give rise to striking, often unexpected macroscopic structures: convection cells in fluids, patterns in ecosystems, networks in societies, and organization in biology. What unites these diverse examples is the deep link between how the agents in systems move and what structure emerges. While diverse approaches have been proposed, in addition, a unifying language may lie in variational principles and optimal control in stochastic and dissipative regimes which can offer a powerful language for understanding this interplay.

Action principles are among the most unifying ideas in science: from Lagrangian mechanics to quantum field theory, they describe how nature selects pathways. The stochastic-dissipative extensions of the principle of least action in the form of path integrals, such as by Onsager-Machlup and more recent versions provide a natural framework for describing how agents and processes, obeying fundamental physical laws, select the most probable and efficient pathways under constraints. These pathways not only govern system dynamics but also generate—and are constrained by—emergent structures. Feedback between dynamics and structure thus shapes evolution, with frozen accidents and historical contingencies balanced against tendencies toward action-efficient configurations. If dynamics select the most probable, efficient pathways, then structure itself may be seen as the lasting imprint of such pathways. Can such principles also help explain the emergence of complexity?

This Collection aims to gather theoretical, computational, and empirical contributions that advance the use of variational principles to explain and predict structure–dynamics interplay in complex systems. By doing so, we hope to move toward general non-equilibrium thermodynamics capable of grounding complexity science in physics while connecting to diverse domains of application. Contributions are welcome across disciplines, from mathematics and physics to biology, engineering, and social sciences. Themes may include, but are not limited to:

• Stochastic and dissipative formulations of variational principles.
• Path integrals and optimal control.
• Structure formation in non-equilibrium thermodynamics.
• Agent-based simulations and computational models.
• Empirical case studies from physical, chemical, biological, or social systems.
• Comparative perspectives with non-variational approaches.

The aim is to advance a physics-grounded framework for understanding how complex structures emerge and persist under dynamic constraints. The objective of this Collection is to foster dialogue among researchers working on different manifestations of the same fundamental questions: How do dynamics give rise to structure, how structure determines dynamics, and how can variational principles provide the key to understanding this process across scales and systems? Can variational pathways explain the emergence of complex structures from dynamics across nature and society?