Collective ordering behaviors are typical macroscopic manifestations embedded in complex systems and can be ubiquitously observed across various physical backgrounds. Elements in complex systems may self-organize via mutual or external couplings to achieve diverse spatiotemporal coordinations. The order parameter, as a powerful quantity in describing the transition to collective states, may emerge spontaneously from large numbers of degrees of freedom through competitions. In this minireview, we extensively discussed the collective dynamics of complex systems from the viewpoint of order-parameter dynamics. A synergetic theory is adopted as the foundation of order-parameter dynamics, and it focuses on the self-organization and collective behaviors of complex systems. At the onset of macroscopic transitions, slow modes are distinguished from fast modes and act as order parameters, whose evolution can be established in terms of the slaving principle. We explore order-parameter dynamics in both model-based and data-based scenarios. For situations where microscopic dynamics modeling is available, as prototype examples, synchronization of coupled phase oscillators, chimera states, and neuron network dynamics are analytically studied, and the order-parameter dynamics is constructed in terms of reduction procedures such as the Ott-Antonsen ansatz, the Lorentz ansatz, and so on. For complicated systems highly challenging to be well modeled, we proposed the eigen-microstate approach (EMP) to reconstruct the macroscopic order-parameter dynamics, where the spatiotemporal evolution brought by big data can be well decomposed into eigenmodes, and the macroscopic collective behavior can be traced by Bose-Einstein condensation-like transitions and the emergence of dominant eigenmodes. The EMP is successfully applied to some typical examples, such as phase transitions in the Ising model, climate dynamics in earth systems, fluctuation patterns in stock markets, and collective motion in living systems. Topics Nonlinear systems, Coupled oscillators, Kuramoto models, Chaos synchronization, Phase transitions, Chimeras, Neurodynamics, Statistical physics An order parameter is an important quantity measuring the symmetry breaking and the emergence of new orderings in complex systems. Traditionally, the order parameter is empirically introduced based on the understanding of specific physical backgrounds and the dynamics of complex systems. As a matter of fact, an order parameter should be a quantity embedded in state space, representing a part of degrees of freedom. Synergetic theory proposed the slaving principle with the physical language of the competitions between slow and fast degrees of freedom and the emergence of slow modes. This can be understood as a physics version of the central-manifold theorem and the adiabatic elimination method. Technically, the slaving principle provides a diagonalization procedure for identifying the slow mode. At the onset of a macroscopic transition, the slow mode becomes unstable and evolves to be the order parameter signifying the new collective ordering in complex systems. This idea has been extensively applied to various collective behaviors of complex systems. Recent years witnessed the storming explorations of large-scale data that are collected from various complex systems. One is faced with the challenge of disclosing intrinsic orderness and self-organization embedded complex data. The recently proposed eigen-microstate approach (EMA) has revealed these eigenmodes and transitions among these different macroscopic states. This approach actually offers a numerical way of identifying various modes and picking up the key order parameters at the onset of transitions. I. INTRODUCTION Understanding the intrinsic mechanism of collective behaviors of coupled units has become a focus for a variety of fields, such as biological neurons, circadian rhythm, chemically reacting cells, and even social systems.1 Various collective dynamics, such as collective sustained oscillation in excitable networks, synchronization of coupled oscillators, swarming and flocking in active matters, and spatiotemporal pattern formations in nonlinear media, have been extensively observed. These behaviors manifest in different ways, but they share some common intrinsic mechanisms. Some properties of collective behaviors depend on the complexity of the system, while the other properties, such as phase transitions, may be the collaboration-induced emergence and can be described by low-dimensional dynamics with macroscopic variables.2 Discovering the method to simplify the system is just as important and fascinating as the discovery of its complexity. An order parameter is a key quantity in measuring the symmetry breaking and the emergence of new order in complex systems, which was first introduced to study phase transitions in thermodynamic systems with the aid of statistical physics.3 This concept was thereafter extended to studies of non-equilibrium behaviors in complex systems, and the dynamical behaviors of order parameters found their more comprehensive applications in spatiotemporal behaviors in various systems ranging from physics and chemistry to biology and even social and economic systems.4 The emergence of ordered dynamics is the process of dimension reduction of the dynamical space of a complex system. Traditionally, one empirically “defines” or “constructs” an order parameter based on observations, experiences, or intuitions. It should be stressed that the existence of an order parameter is a natural and intrinsic feature for the emerging properties of complex systems. According to synergetic theory,4 order parameters are intrinsically embedded in the state variables of complex systems as slow modes. The emergence of order parameters is the consequence of competitions between slow and fast modes. At the onset of critical points, the slow degrees of freedom embedded in the huge number of stable variables may become unstable and grow to emerge, which are determined by the center manifold theorem or the slaving principle. Like most cases in physics, the simplification and low-dimensional reduction are associated with the symmetry property of the system. As the identity of gas particles is the foundation of statistical mechanics and collective variables as temperature and pressure, the identity of the coupled units in a complex system is also related to some order parameters.
Collective ordering behaviors are typical macroscopic manifestations embedded in complex systems and can be ubiquitously observed across various physical backgrounds. Elements in complex systems may self-organize via mutual or external couplings to achieve diverse spatiotemporal coordinations. The order parameter, as a powerful quantity in describing the transition to collective states, may emerge spontaneously from large numbers of degrees of freedom through competitions. In this minireview, we extensively discussed the collective dynamics of complex systems from the viewpoint of order-parameter dynamics. A synergetic theory is adopted as the foundation of order-parameter dynamics, and it focuses on the self-organization and collective behaviors of complex systems. At the onset of macroscopic transitions, slow modes are distinguished from fast modes and act as order parameters, whose evolution can be established in terms of the slaving principle. We explore order-parameter dynamics in both model-based and data-based scenarios. For situations where microscopic dynamics modeling is available, as prototype examples, synchronization of coupled phase oscillators, chimera states, and neuron network dynamics are analytically studied, and the order-parameter dynamics is constructed in terms of reduction procedures such as the Ott-Antonsen ansatz, the Lorentz ansatz, and so on. For complicated systems highly challenging to be well modeled, we proposed the eigen-microstate approach (EMP) to reconstruct the macroscopic order-parameter dynamics, where the spatiotemporal evolution brought by big data can be well decomposed into eigenmodes, and the macroscopic collective behavior can be traced by Bose-Einstein condensation-like transitions and the emergence of dominant eigenmodes. The EMP is successfully applied to some typical examples, such as phase transitions in the Ising model, climate dynamics in earth systems, fluctuation patterns in stock markets, and collective motion in living systems.
Topics
Nonlinear systems, Coupled oscillators, Kuramoto models, Chaos synchronization, Phase transitions, Chimeras, Neurodynamics, Statistical physics
An order parameter is an important quantity measuring the symmetry breaking and the emergence of new orderings in complex systems. Traditionally, the order parameter is empirically introduced based on the understanding of specific physical backgrounds and the dynamics of complex systems. As a matter of fact, an order parameter should be a quantity embedded in state space, representing a part of degrees of freedom. Synergetic theory proposed the slaving principle with the physical language of the competitions between slow and fast degrees of freedom and the emergence of slow modes. This can be understood as a physics version of the central-manifold theorem and the adiabatic elimination method. Technically, the slaving principle provides a diagonalization procedure for identifying the slow mode. At the onset of a macroscopic transition, the slow mode becomes unstable and evolves to be the order parameter signifying the new collective ordering in complex systems. This idea has been extensively applied to various collective behaviors of complex systems. Recent years witnessed the storming explorations of large-scale data that are collected from various complex systems. One is faced with the challenge of disclosing intrinsic orderness and self-organization embedded complex data. The recently proposed eigen-microstate approach (EMA) has revealed these eigenmodes and transitions among these different macroscopic states. This approach actually offers a numerical way of identifying various modes and picking up the key order parameters at the onset of transitions.
I. INTRODUCTION
Understanding the intrinsic mechanism of collective behaviors of coupled units has become a focus for a variety of fields, such as biological neurons, circadian rhythm, chemically reacting cells, and even social systems.1 Various collective dynamics, such as collective sustained oscillation in excitable networks, synchronization of coupled oscillators, swarming and flocking in active matters, and spatiotemporal pattern formations in nonlinear media, have been extensively observed. These behaviors manifest in different ways, but they share some common intrinsic mechanisms. Some properties of collective behaviors depend on the complexity of the system, while the other properties, such as phase transitions, may be the collaboration-induced emergence and can be described by low-dimensional dynamics with macroscopic variables.2 Discovering the method to simplify the system is just as important and fascinating as the discovery of its complexity.
An order parameter is a key quantity in measuring the symmetry breaking and the emergence of new order in complex systems, which was first introduced to study phase transitions in thermodynamic systems with the aid of statistical physics.3 This concept was thereafter extended to studies of non-equilibrium behaviors in complex systems, and the dynamical behaviors of order parameters found their more comprehensive applications in spatiotemporal behaviors in various systems ranging from physics and chemistry to biology and even social and economic systems.4
The emergence of ordered dynamics is the process of dimension reduction of the dynamical space of a complex system. Traditionally, one empirically “defines” or “constructs” an order parameter based on observations, experiences, or intuitions. It should be stressed that the existence of an order parameter is a natural and intrinsic feature for the emerging properties of complex systems. According to synergetic theory,4 order parameters are intrinsically embedded in the state variables of complex systems as slow modes. The emergence of order parameters is the consequence of competitions between slow and fast modes. At the onset of critical points, the slow degrees of freedom embedded in the huge number of stable variables may become unstable and grow to emerge, which are determined by the center manifold theorem or the slaving principle.
Like most cases in physics, the simplification and low-dimensional reduction are associated with the symmetry property of the system. As the identity of gas particles is the foundation of statistical mechanics and collective variables as temperature and pressure, the identity of the coupled units in a complex system is also related to some order parameters.