The world around us is highly complex. Still, there are simple patterns and laws that have been discovered that are useful and practical in describing this complex world. We can also build simple and useful tools and machines. Indeed, a detailed understanding of complexity has not been necessary to develop a complex technological society. The reason it is possible to describe simply many properties of the world is because of the separation of scales. When the separation of scales does not work, concepts from the modern study of complex systems become essential.

The idea of the separation of scales is that there are three types of processes: slow, dynamic and fast. We treat each of these processes using a different approach.

The slow processes we consider to be static. All of the parameters describing these slow processes are fixed (frozen). We specify them once and for all, and we consider them to be the constraints on the system as far as the dynamic and fast processes.

The dynamic processes are the ones we treat using Newtonian laws of physics, logical laws of computers, or other dynamic laws that are particularly suited to the system we are studying. These are the mechanical laws of the system. It can be tricky to write down the equations that describe these dynamic processes, but the idea is that once this is done we can predict the dynamics precisely and directly.

The fast processes are averaged over. The idea is that these processes are so fast that we only see the changes they make as a blur. This blur can, however, be important in determining the dynamic processes. This means that if we are trying to figure out the dynamic laws, we have to do the averaging over the fast processes. But once they are averaged over (once and for all), we can ignore them except for the average properties they generate. For example, pressure and temperature are average (collective) properties we can describe and measure and relate to the microscopic motion of particles.

The problem with the idea of separation of scales in complex systems is that the different scales of behavior become coupled. This is captured in a simple way in the study of chaos and fractals. Chaos describes the coupling of different scales through the influence over time in the dynamics of the system (amplification or dissipation of differences). Fractals describe the causal/logical relationship between behaviors of the system at different scales. Since the behavior of the system at different scales are related, our descriptions should include these relationships.

In summary, the use of separation of scales in the real world is to average the fast degrees of freedom (thermodynamics) and discuss their influence on the dynamic degrees of freedom (Newtonian Mechanics) while keeping fixed the slow degrees of freedom.

**Related concepts:** thermodynamics, statistical mechanics, ensemble

Copyright © 2011 Yaneer Bar-Yam All rights reserved.