13.2 Growth in a Population and Chaos Theory
563
13.2.3 Connections with Critical Phenomena
In the previous paragraphs, we described how, for non-linear systems, as a parameter
associated with the dynamics of that system changes, the system can reach a ‘critical
point’ where the system undergoes a transition increasing its disorder. We also saw
that as one increases a parameter, say r, the dynamics of certain non-linear systems
show chaos when r reaches a critical value, say r c .
Phase changes, such as water turning from a liquid to a gas, can have disorder
changing drastically upward when there is only a small change in temperature.
Order in this case means some regularity in the material structure. One sign of
approaching chaos is the development of long-range correlations in space as the
system becomes more ordered. Subsystems are formed with dimensions much larger
than the typical separation distances between the fundamental entities.
A sudden increase in disorder means a jump in the entropy while passing
through the phase-change temperature. The process is easily described: At a given
temperature, a solid has more order than a liquid (and liquid more than a gas) for a
given number of atoms. When a solid melts, or a liquid vaporizes, each atom must
take up energy to release it from local attractive forces, including ‘long-range’ forces
between neighboring atoms. Environmental thermal energy is one source for this
melting or vaporizing energy, called latent heat. Latent heat is released or absorbed
during a so-called ‘discontinuous phase transition’. The first derivatives of the Gibbs
free energy appear discontinuous as the temperature or pressure passes through the
phase change point.
It is also possible to have a phase change without any exchange of heat. Such
‘continuous phase transitions’ occur at the so-called critical point. For materials, the
critical point is that state where the liquid and gas phases are in equilibrium and have
the same specific density and entropy, each measured per particle. A characteristic
of continuous phase transitions is the apparent singularities in the susceptibilities,
i.e. the second derivatives of the Gibbs free energy. During a continuous phase
transition, the correlation in the behavior of distant clusters of particles approaches
infinity according to a power law. This can be seen as the onset of a milky
appearance as a clear liquid approaches its critical point. The effect is called ‘critical
obsolescence’, and is caused by large-scale variations in clustering of particles.
Measurable quantities such as the heat capacity, when measured near the critical
point with temperature T c , behave close to the power law: C ∝ |T c − T | −α . Just as
in the case of the logistic map, the exponent α ≈ 0.59 applies to a diverse set of
systems, and so is thought to be a universal constant called the ‘critical exponent’
for that set of systems.
563
13.2.3 Connections with Critical Phenomena
In the previous paragraphs, we described how, for non-linear systems, as a parameter
associated with the dynamics of that system changes, the system can reach a ‘critical
point’ where the system undergoes a transition increasing its disorder. We also saw
that as one increases a parameter, say r, the dynamics of certain non-linear systems
show chaos when r reaches a critical value, say r c .
Phase changes, such as water turning from a liquid to a gas, can have disorder
changing drastically upward when there is only a small change in temperature.
Order in this case means some regularity in the material structure. One sign of
approaching chaos is the development of long-range correlations in space as the
system becomes more ordered. Subsystems are formed with dimensions much larger
than the typical separation distances between the fundamental entities.
A sudden increase in disorder means a jump in the entropy while passing
through the phase-change temperature. The process is easily described: At a given
temperature, a solid has more order than a liquid (and liquid more than a gas) for a
given number of atoms. When a solid melts, or a liquid vaporizes, each atom must
take up energy to release it from local attractive forces, including ‘long-range’ forces
between neighboring atoms. Environmental thermal energy is one source for this
melting or vaporizing energy, called latent heat. Latent heat is released or absorbed
during a so-called ‘discontinuous phase transition’. The first derivatives of the Gibbs
free energy appear discontinuous as the temperature or pressure passes through the
phase change point.
It is also possible to have a phase change without any exchange of heat. Such
‘continuous phase transitions’ occur at the so-called critical point. For materials, the
critical point is that state where the liquid and gas phases are in equilibrium and have
the same specific density and entropy, each measured per particle. A characteristic
of continuous phase transitions is the apparent singularities in the susceptibilities,
i.e. the second derivatives of the Gibbs free energy. During a continuous phase
transition, the correlation in the behavior of distant clusters of particles approaches
infinity according to a power law. This can be seen as the onset of a milky
appearance as a clear liquid approaches its critical point. The effect is called ‘critical
obsolescence’, and is caused by large-scale variations in clustering of particles.
Measurable quantities such as the heat capacity, when measured near the critical
point with temperature T c , behave close to the power law: C ∝ |T c − T | −α . Just as
in the case of the logistic map, the exponent α ≈ 0.59 applies to a diverse set of
systems, and so is thought to be a universal constant called the ‘critical exponent’
for that set of systems.