4 An Introduction to Emergence Dynamics in Complex Systems
137
This means that if there exists a sufficient amount of negative entropy flow,
the system can be expected to maintain an ordered configuration. Prigogine and
colleagues thus claimed that “Nonequilibrium may be a source of order”, which
forms the base of the dissipative structure theory [7].
The emergence of dissipative structure depends on the degree of deviation from the
equilibrium state of the system. In the small-deviation regime, the system still keeps
its thermodynamic property, and the minimum entropy principle applies. In the linear
regime, numerous theories such as linear response theory and dissipation-fluctuation
theorem have been proposed. As the system is driven so far from the equilibrium state
that the thermodynamic branch becomes unstable, structural branches may emerge
and replace the thermodynamic branch [7, 8]. This can be mathematically described
in terms of dynamical system theory [9, 10].
Denote the macrostate of a complex system as
u (t) = (u 1 , u 2 , . . . , u n ), the
evolution of the state can be described as
d
u /dt =
f
u , ε
,
(4.4)
where
f = (f 1 , f 2 , . . . , f n ) is the nonlinear function vector, and ε are a group of
control parameters. Equation (4.4) is usually a group of coupled nonlinear equations
and can be extensively discussed by using theories of dynamic systems, and the
stability of possible states and bifurcations have been exhaustively studied in the
past decades. Readers can refer any textbook on nonlinear dynamics and chaos to
gain a detailed understanding [11, 12].
Considering the spatial effect, i.e. u = u(r, t). The simplest spatial effect in
physics is the diffusion process, which is given by Fick’s law as the proportional
relation between the flux and the gradient of matter condensation in space:
J = −D∇u,
(4.5)
where D is the diffusion coefficient. Therefore in this case the governing equation of
motion can be written as
∂u/∂t = f (u, ε) + D∇
2 u,
(4.6)
where D denotes the diffusion coefficient. Equation (4.6) is called the reaction–
diffusion equation. This equation and related mechanism were first proposed by
Alan Turing in 1952 as a possible source of biological organism [13]. The reaction
term f (u, ε) gives the local dynamics, which is the source of spatial inhomogeneity.
The diffusion term tends to erase the spatial differences of the state u, thus it is the
source of spatial homogeneity. Both mechanisms appears at the right hand side of
Eq. (4.6) and compete, resulting in an self-organized state. Equation (4.6) and related
dynamical systems have been extensively explored in the past a few years, and rich
137
This means that if there exists a sufficient amount of negative entropy flow,
the system can be expected to maintain an ordered configuration. Prigogine and
colleagues thus claimed that “Nonequilibrium may be a source of order”, which
forms the base of the dissipative structure theory [7].
The emergence of dissipative structure depends on the degree of deviation from the
equilibrium state of the system. In the small-deviation regime, the system still keeps
its thermodynamic property, and the minimum entropy principle applies. In the linear
regime, numerous theories such as linear response theory and dissipation-fluctuation
theorem have been proposed. As the system is driven so far from the equilibrium state
that the thermodynamic branch becomes unstable, structural branches may emerge
and replace the thermodynamic branch [7, 8]. This can be mathematically described
in terms of dynamical system theory [9, 10].
Denote the macrostate of a complex system as
u (t) = (u 1 , u 2 , . . . , u n ), the
evolution of the state can be described as
d
u /dt =
f
u , ε
,
(4.4)
where
f = (f 1 , f 2 , . . . , f n ) is the nonlinear function vector, and ε are a group of
control parameters. Equation (4.4) is usually a group of coupled nonlinear equations
and can be extensively discussed by using theories of dynamic systems, and the
stability of possible states and bifurcations have been exhaustively studied in the
past decades. Readers can refer any textbook on nonlinear dynamics and chaos to
gain a detailed understanding [11, 12].
Considering the spatial effect, i.e. u = u(r, t). The simplest spatial effect in
physics is the diffusion process, which is given by Fick’s law as the proportional
relation between the flux and the gradient of matter condensation in space:
J = −D∇u,
(4.5)
where D is the diffusion coefficient. Therefore in this case the governing equation of
motion can be written as
∂u/∂t = f (u, ε) + D∇
2 u,
(4.6)
where D denotes the diffusion coefficient. Equation (4.6) is called the reaction–
diffusion equation. This equation and related mechanism were first proposed by
Alan Turing in 1952 as a possible source of biological organism [13]. The reaction
term f (u, ε) gives the local dynamics, which is the source of spatial inhomogeneity.
The diffusion term tends to erase the spatial differences of the state u, thus it is the
source of spatial homogeneity. Both mechanisms appears at the right hand side of
Eq. (4.6) and compete, resulting in an self-organized state. Equation (4.6) and related
dynamical systems have been extensively explored in the past a few years, and rich