190
Z. Zheng
It can be seen from the above analysis of the two-dimensional order-parameter
dynamics that it is much easier than the original (N + 1)-dimensional dynamics
given by (4.134). Moreover, the dynamics of the order parameter can be completely
solvable. Readers may refer [108–111] for a more detailed discussion.
4.5 Remarks
Collective behavior of a complex system implies the emergence of a macroscopic
order from the organizations of populations of units with mutual interactions. Usually
the number of degrees of freedom at the microscopic level is so large that an exact
description of the system at this level is impossible and also unnecessary. Therefore
a macroscopic study of the complex system is significant. In the description of thermodynamics of gas, one only needs to use several variables, e.g. the temperature, the
pressure and the volume to depict the property of a large number of particles moving
in a box. Generally, the macroscopic order of a complex system is originated from
the reduction of degrees of freedom by transiting from microscopic to macroscopic
levels, i.e. and the macroscopic description of the order needs only a few variables.
These macroscopic variables are called the order parameters.
On the other hand, the emergence of order parameters in a system is spontaneous rather than manually selected. This means the ordered state is the result of
self-organization and competitions of units. In this Chapter, we applied the slaving
principle to discuss the competitions among various state variables with different time
scales. We showed that at the critical point, only a few slow variables will conquer
the large number of fast variables and dominate the evolution of the system, and these
slow variables grow up to become the order parameters. In this case the dynamics
can be described by the evolution of these small number of order parameters, which
is a great dimension reduction.
It should be stressed that the idea of the reduction scheme can be extensively
applied to various transitions, such as the emergence of spatiotemporal patterns, the
transitions from partial to global synchronization in coupled oscillators, the transitions from non-integrability to integrability in Hamiltonian systems, thermodynamic
phase transitions in condensed matters, and so on. We think the slaving principle and
more generally the principles in synergetic theory and dissipative-structure theory
are exhibiting their great privileges in exploring the order emergence of complex
systems in recent years.
An effective reduction depends strongly on the motivation, and it can be a projection to a low-dimensional subspace, or a dimension reduction of high-dimensional
systems, or a topology simplification, et al. The reducibility of complex systems
should obey, at least approximately obey the following physical properties:
(1) Symmetry
This is a well-known solvable case in physics, which applies also in the process
of reduction. For example, symmetries in a Hamiltonian system lead to various
Z. Zheng
It can be seen from the above analysis of the two-dimensional order-parameter
dynamics that it is much easier than the original (N + 1)-dimensional dynamics
given by (4.134). Moreover, the dynamics of the order parameter can be completely
solvable. Readers may refer [108–111] for a more detailed discussion.
4.5 Remarks
Collective behavior of a complex system implies the emergence of a macroscopic
order from the organizations of populations of units with mutual interactions. Usually
the number of degrees of freedom at the microscopic level is so large that an exact
description of the system at this level is impossible and also unnecessary. Therefore
a macroscopic study of the complex system is significant. In the description of thermodynamics of gas, one only needs to use several variables, e.g. the temperature, the
pressure and the volume to depict the property of a large number of particles moving
in a box. Generally, the macroscopic order of a complex system is originated from
the reduction of degrees of freedom by transiting from microscopic to macroscopic
levels, i.e. and the macroscopic description of the order needs only a few variables.
These macroscopic variables are called the order parameters.
On the other hand, the emergence of order parameters in a system is spontaneous rather than manually selected. This means the ordered state is the result of
self-organization and competitions of units. In this Chapter, we applied the slaving
principle to discuss the competitions among various state variables with different time
scales. We showed that at the critical point, only a few slow variables will conquer
the large number of fast variables and dominate the evolution of the system, and these
slow variables grow up to become the order parameters. In this case the dynamics
can be described by the evolution of these small number of order parameters, which
is a great dimension reduction.
It should be stressed that the idea of the reduction scheme can be extensively
applied to various transitions, such as the emergence of spatiotemporal patterns, the
transitions from partial to global synchronization in coupled oscillators, the transitions from non-integrability to integrability in Hamiltonian systems, thermodynamic
phase transitions in condensed matters, and so on. We think the slaving principle and
more generally the principles in synergetic theory and dissipative-structure theory
are exhibiting their great privileges in exploring the order emergence of complex
systems in recent years.
An effective reduction depends strongly on the motivation, and it can be a projection to a low-dimensional subspace, or a dimension reduction of high-dimensional
systems, or a topology simplification, et al. The reducibility of complex systems
should obey, at least approximately obey the following physical properties:
(1) Symmetry
This is a well-known solvable case in physics, which applies also in the process
of reduction. For example, symmetries in a Hamiltonian system lead to various