32
2 Complexity Time Bomb
2.16 Appendix 1: How Harmless Behavior Can Become
Critical
In the case of traffic flow, we have seen that a system can get out of control when the
interaction strength (e.g. the density) is too high. Why can a change in the density
make normal and “harmless” behavior become uncontrollable? To understand this
better, Roman Mani, Lucas Böttcher, Hans J. Herrmann, and I studied collisions
in a system of equally sized particles moving in one dimension,
32 which is similar
to Newton’s Cradle.
33 We assumed that the particles tended to oscillate elastically
around equally spaced equilibrium points, while being exposed to random forces
generated by the environment.
The following summarizes the main observations made: If the distance between
the equilibrium points of neighboring particles is large enough, each particle oscillates around its equilibrium point with normally distributed velocities, and all particles have the same small variance in speed. However, when the separation between
the equilibrium points reaches the diameter of the particles, we find a cascade-like
transmission of momentum between particles.
34 Surprisingly, the variance of particle
speeds rapidly increases towards the boundaries—it could even go to infinity with
increasing system size. Due to cascading interactions of particles, this makes their
speeds unpredictable and uncontrollable. While every particle in separation performs
a normal dynamics, which is not excessive at all, their interactions can cause an
extreme behavior of the system.
2.17 Appendix 2: Loss of Synchronization in Hierarchical
Systems
When many socio-economic processes are happening simultaneously while having
feedbacks on each other, a puzzling kind of systemic instability can occur, which
is highly relevant for our complex societies, since many socio-economic processes
happen at an increasing pace.
For the sake of illustration, let us first discuss hierarchically organized systems in
physics. There, elementary particles form atoms, atoms form chemical compounds,
these form solid bodies, and together they may form a planet, which is part of a
planetary system, and a galaxy. Similarly, we know from biology and the social
sciences that cells make up organs, which collectively form a human body. Humans,
in turn, tend to organize themselves in groups, cities, organizations and nations.
Importantly, the stability of such hierarchies is based on two important principles.
First, the forces are strongest at the bottom, and second, the changes are slowest at
32 Mani et al. [18]: https://www.youtube.com/watch?v=RbWSal3aay8.
33 See http://www.youtube.com/watch?v=0LnbyjOyEQ8.
34 See https://www.youtube.com/watch?v=RbWSal3aay8.
2 Complexity Time Bomb
2.16 Appendix 1: How Harmless Behavior Can Become
Critical
In the case of traffic flow, we have seen that a system can get out of control when the
interaction strength (e.g. the density) is too high. Why can a change in the density
make normal and “harmless” behavior become uncontrollable? To understand this
better, Roman Mani, Lucas Böttcher, Hans J. Herrmann, and I studied collisions
in a system of equally sized particles moving in one dimension,
32 which is similar
to Newton’s Cradle.
33 We assumed that the particles tended to oscillate elastically
around equally spaced equilibrium points, while being exposed to random forces
generated by the environment.
The following summarizes the main observations made: If the distance between
the equilibrium points of neighboring particles is large enough, each particle oscillates around its equilibrium point with normally distributed velocities, and all particles have the same small variance in speed. However, when the separation between
the equilibrium points reaches the diameter of the particles, we find a cascade-like
transmission of momentum between particles.
34 Surprisingly, the variance of particle
speeds rapidly increases towards the boundaries—it could even go to infinity with
increasing system size. Due to cascading interactions of particles, this makes their
speeds unpredictable and uncontrollable. While every particle in separation performs
a normal dynamics, which is not excessive at all, their interactions can cause an
extreme behavior of the system.
2.17 Appendix 2: Loss of Synchronization in Hierarchical
Systems
When many socio-economic processes are happening simultaneously while having
feedbacks on each other, a puzzling kind of systemic instability can occur, which
is highly relevant for our complex societies, since many socio-economic processes
happen at an increasing pace.
For the sake of illustration, let us first discuss hierarchically organized systems in
physics. There, elementary particles form atoms, atoms form chemical compounds,
these form solid bodies, and together they may form a planet, which is part of a
planetary system, and a galaxy. Similarly, we know from biology and the social
sciences that cells make up organs, which collectively form a human body. Humans,
in turn, tend to organize themselves in groups, cities, organizations and nations.
Importantly, the stability of such hierarchies is based on two important principles.
First, the forces are strongest at the bottom, and second, the changes are slowest at
32 Mani et al. [18]: https://www.youtube.com/watch?v=RbWSal3aay8.
33 See http://www.youtube.com/watch?v=0LnbyjOyEQ8.
34 See https://www.youtube.com/watch?v=RbWSal3aay8.