# Complexity of a physical system

Are there any accepted definitions quantifying the complexity of:

a) macroscopic, classical mechanical systems (e.g., a bicycle)

b) microscopic systems (ensembles of atoms)?

References are more than welcome.

• The number of degrees of freedom en.m.wikipedia.org/wiki/… But note that for e.g. a gas, the number of degrees of freedom can be $10^{23}$ or more, but statistical methods can give a good description with far fewer variables than that. So number of degrees of freedom isn't the whole story. – Robin Ekman Oct 14 '14 at 23:58
• @RobinEkman - would changing the size of gears in a bicycle or rearranging them without resizing result in changes in your measure of the bike's complexity? – Deer Hunter Oct 15 '14 at 0:04
• No. The number of degrees of freedom is essentially the number of independent moving parts, counted once for each dimension the part can move in. If you don't change how your parts are connected to each other, this number stays the same. – Robin Ekman Oct 15 '14 at 0:21
• @RobinEkman - okay. I can add as many useless parts to my bike as I like, up until their number reaches that of a plane. Will it mean that my bike has the same complexity as a 787? – Deer Hunter Oct 15 '14 at 0:26

I understand that you're not talking about entropy (which is not the same as complexity or disorder), but $\text{complexity}$ is a rather subjective term. I would assume that the relative complexity of a system (which is broad in and of itself, by the way) would have to do with statistical analysis of its parts, the energy required to achieve a certain state, the amount of ways that this state could be acheived, and the apparent function and resilliancy of the state, though there is no known way to accurately quantify something like $\text{complexity}$.

Complexity seems like an arbitrary valuation. There is fundamentally no distinction between microscopic and macroscopic. However, computational complexity is intensely studied and many research groups attempt to bridge the gaps between when classical molecular dynamics and ab initio or ring polymer quantum mechanical simulations are most appropriate. Here's a sample of coarse graining in biological systems from a collaboration of 4 UChicago profs. http://pubs.acs.org/doi/abs/10.1021/ct4000444

Hope this helps

• There may be some better reviews out there, but it's a bit outside my field and Voth is who I'm most familiar with. I think he also has some youtube videos talking about the concept too. – Nick Nov 25 '14 at 16:58

This is a collection of resources related to complexity.

1. Many physical systems can be represented with graphs:

2. Complexity of a graph or a weighted graph is a notion established by algebraic geometry.

• http://arxiv.org/abs/0705.2284 (The weighted complexity and the determinant functions of graphs)
• Audrey Terras. Zeta functions of graphs: A stroll through the garden. CUP, 2010.
3. Organic chemistry has Wiener index.

4. Software development has Cyclomatic complexity.