# Characteristic length, characteristic time and complexity of the process

Different physical processed (starting from elementary particles or even below to the universe itself) have different length scales $L$ and different characteristic times $T$. Larger processed tend to live at slower pace (very roughly, for electron orbitals it is $(10^{-10}\text{m},10^{-15}\text{s})$ while for stars - $(10^{11}\text{m},10^{12}\text{s})$).

But going into details, objects of the same size have different characteristic times. For example, a human time scale (from 'blink of an eye' (100ms) to the lifespan (70ys)) is much shorter that for a rock of the same size.

The question is if such relation (i.e. $T$ vs. $L$) is related to complexity of a process (either in terms of precision/robustness of it, long range correlations, or structure of interactions between its components)?

Any 2D plot of "time scale" vs "length scale" of different processed would be appreciated.

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I'm not sure this is physics. However, I also think that it doesn't make sense to say that an object has a time or length scale. Rather it is processes or interactions which can be said to have a time or length scale associated with them. Humans have a shorter lifetime than rocks because the chemical reactions of life are very different than the chemical activity in a rock. – Colin K Sep 22 '11 at 16:00
@ColinK: Physics of complex systems - very much. I used 'object' (in this context) as a synonym of 'process'. The question is not about a particular example 'rock vs human', but about general properties. The thing with reaction is only a manifestation of a broader phenomenon (number of components, and of possible interactions between them). – Piotr Migdal Sep 22 '11 at 16:13
There is a bit in the introduction to Callen's text on thermodynamics where he says the subject is about the study of situations where the fast things have finished and the slow things haven't gotten started yet. I found it very enlightening and would suggest it to anyone who has not yet seen it. – dmckee Sep 22 '11 at 16:39

There is absolutely no relation between the length/time scale relation and the complexity of the phenomenon. The graph you are looking for has a log-axis for L and for T, and a black region for $L>T/c$ which is the speed-of-light bound on the allowed time scales for change in a system of size L. You can make complexity happen in our universe in the biological region, which will be around the heartrate/mass power-law for animals. T

Humans live far away from this boundary, but unstable elemetary particles are mostly right on it. But there is no relation between these natural scales and the complexity of a process. A supernova core collapse is a very simple phenomenon in terms of biology, but it happens quickly for a huge object, and the time scale for cellular division can be slowed down by freezing and thawing, without changing the inherent complexity of the process.

The rule of thumb is that you can make complexity happen in our universe in the biological region, which will be around the heartrate/mass power-law for animals.

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I don't mean only complexity of biological objects, and when it comes to the relation I would rather expect something statistically than holding all the time. Speed of light is of course a limit, but most of processes have their time scale much longer then that of the sound wave propagation speed. – Piotr Migdal Sep 29 '11 at 12:14
@Piotr: I focused only on biology, because complexity of biological systems dwarfs everything else. – Ron Maimon Sep 29 '11 at 14:09

It might be useful to think about how fundamental particles combine step by step into more complex patterns (for lack of a better word), what constraints there are on these with regards to time (stability) and how much energy is involved (total conversion vs chemistry, for example).

We have a number of fundamental particles, most of which are too unstable to give rise to complex phenomena (the top quark doesn't even form hadrons) and some of which don't interact much (neutrinos, possibly WIMP dark matter). What we do have to work with is up and down quarks that form hadrons and electrons that with them form electrically neutral atoms.

Not all atoms are stable for long times. Those that are can interact chemically, forming larger, more complex structures. Chemical processes are slower, less energetic and happen over larger distances than interactions between fundamental particles.

The next step in complexity isn't necessarily obvious, but I'd suggest self-replicators (life) as the logical one. We don't know what the smallest or least complex possible self-replicator would be, but the smallest known bacteria are far larger than single molecules.

I don't know if this is much of an answer, but it might be a sketch for how to think about an answer.

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