# What is the definition of “Complexity” in physics? Is it quantifiable?

I don't know much about the discipline of "Complex systems studies" but I know in the field of "Statistical mechanics" there is much talk about the "Complexity of the system". Like "...the state of this system is more complex..." or "...as we see the complexity of the system is arising..." and so on.

My question is:

1. How "Complexity" is actually defined in physics?
2. Is "Complexity" a quantifiable property of the system? I mean can we define a quantity like $\mathfrak{C}$ representing the measure of complexity of the system?
• @sammygerbil Most of my google searches ends in pages discussing the mathematical discipline of complex systems and doesn't refer to its physical meaning. Though the physical papers I reached just talk about complexity in general and don't give a precise definition of the term. – Hamed Begloo Dec 1 '16 at 19:41
• What research have you done to find answers? When I googled "complexity physics" I got the wikipedia article on Complex Systems as #1 and a Nature article about the emerging role of computational complexity in theoretical physics as #2. At #3 is an earlier issue of Nature Physics Insight devoted to the topic of complexity. – sammy gerbil Dec 1 '16 at 19:46
• The Insights commentary points out that the subject is difficult to define. This is a common problem with all emerging subjects : it takes some time to develop a consensus about the scope of the subject. The term mostly described a collection of topics such as chaos theory. There is as yet no definition of a physical or mathematical property called "complexity" - although it is easy to imagine that one could be devised. – sammy gerbil Dec 1 '16 at 19:53
• @sammygerbil So it seems the notion of complexity is not yet formalized (at least) in physics. But like the sentences I provided it seems people are treating it as a physical quantity. Therefore as you said, it is likely that they're going to provide a rigorous definition of the term. – Hamed Begloo Dec 1 '16 at 20:00
• That's right. Complexity studies is an emerging science; it takes time for the scientific community to reach a consensus about its scope and its defining concepts or principles. At the same time complexity refers to an undefined property of systems. It is not certain that it will develop into a new science, it might be superseded by some other way of looking at the same problems. – sammy gerbil Dec 1 '16 at 20:31

Being somewhere in the wide and diffuse field of complex-systems science, I am not aware of any generally accepted definition of complexity, let alone one that yields a measurable quantity. It’s more an “I know it when I see it” thing, and in my experience, there is common agreement that the term cannot and does not need to be defined rigorously (for reasons that I elaborate later).

I have many times witnessed that scientists dealing with more complex systems expressing a tongue-in-cheek superiority over those dealing with less complex but still complex systems, saying that they were not really complex. In another example, the 90-page review paper The Structure and Function of Complex Networks applies the adjective complex to networks only a handful of times and not at all in a way that could serve as a definition.

If I were to define the field, I would say probably say something along the lines of:

Complex-systems science investigates phenomena that emerge from the (complex) interplay of perfectly or at least well understood components.

Thus, if you so wish, you can define a complex system as one that is principially capable of exhibiting such emergent phenomena. Of course, these definitions inevitably inherit vagueness from terms such as emergence, well understood, or pricipially capable. Moreover, once we understand a complex system, it becomes a well-understood component itself.

However, once we go down to specific research, these intricacies of the definition do not matter anymore: As long as your research gains insights on phenomena that are not yet understood, it yields new knowledge – whether the system exhibiting the phenomenon is complex or not does not really matter in this respect.

• I agree that just dealing with the intricacies existing between a complex system's constituents doesn't matter itself. But I think the existence of complexity as a property of the system as a whole is really important. For example we know the study of the details of a "Chaotic system" is not useful since they doesn't provide any useful information or pattern for us. But as we proceed it turns out that in long term the system shows some recognizable (and mostly informative) patterns and the patterns depend exactly on the very complex nature of the system..... – Hamed Begloo Dec 2 '16 at 21:23
• ..... so it means knowing the measure of complexity of a system is important. By the way I was recently reading a popular science book named "The Big Picture". In chapter 4th of the book which discusses about complexity it says when a system evolves from one state of equilibrium to another state of equilibrium the complexity initially arises and then goes down which seemed an interesting quantitative thought about "Complexity" to me but since it was a popular science book, it didn't give a formal description of the idea. – Hamed Begloo Dec 2 '16 at 21:24
• the patterns depend exactly on the very complex nature of the system – What exactly do you mean by this? 1) Chaos does not need complexity. 2) What do you mean by depend exactly? 3) Chaos usually depends on the system parameters, but they usually do not have any relation with complexity. – Wrzlprmft Dec 2 '16 at 22:29
• Sorry. You're right. So far I wrongly thought that being stochastic necessarily means being complex too but now I know I was wrong. By "...depends exactly..." I meant stochastic systems doesn't seem to show any patterns in short term but in long enough temporal intervals some patterns began to be shown and the properties of these patterns is actually related to the random(stochastic) nature of the system. Though I think it's not important either. Now at least I know that complexity doesn't have anything to do with "Determinism" and "Randomness". – Hamed Begloo Dec 4 '16 at 20:45
• @HamedBegloo: Be careful. Chaos is crucially different from stochasticity. One of its key features is that it occurs in completely deterministic systems. Stochasticity/randomness however can indeed be linked to complexity: (Besides quantum randomness) we use stochastic models, if we have influencing factors in a system that are too complex to be modeled precisely, e.g., the effect of air turbulence on a pendulum. Also, sufficiently complex (chaotic) systems are impossible to distinguish from stochastic ones. E.g., pseudo-random number generators are nothing but complex chaotic maps. – Wrzlprmft Dec 4 '16 at 21:02

As others have said, complexity is still a notion that does not have one and only one definition. However, you did ask for a quantitative definition, so one example of complexity of a quantum field (or even a quantum lattice system) I've seen in the context of quantum gravity is:

"the minimum number of quantum gates needed to take you from the ground state of the system (or really any distinguished state) to the state in question."

This definition is due to Suskind (pg 7) to the best of my knowledge. It's not the only way to do it, but generally, quantum information theory is interested in complexity of a quantum system and its relationship to entanglement.

• Thank you for your answer. Do you think it's possible to build a definition of complexity in the context of "Statistical mechanics"? Like relating it to system's microstate, the system's distance from equilibrium, the ensembles, etc. Or in the context of "Analytical mechanics"? Like relating it to configuration of the system or the system's evolution path. You know I want the definition to be in the context of a formalism rather than being in the context of a theory( like quantum mechanics, classical mechanics, quantum gravity, etc). – Hamed Begloo Dec 2 '16 at 21:21
• I think it's unfair to not put QM and analytical mechanics on the same footing as a "formalism." They are both based off a lagrangian and the fiber bundle to which it is associated. The difference only comes with the definition of states (a point on the manifold and its cotangent space for analytical mechanics vs. a linear combination of points on the cotangent space or the manifold for quantum mechanics). Then there is some difference in their update rules and how you get physical predictions (i.e. there is a Born rule for QM but not classical mechanics) <Answer continued...> – Bobak Hashemi Dec 3 '16 at 10:12
• But inherently I don't see why you couldn't make the definition you want. Distinguish an 'equilibrium distribution' in phase space. Then call the complexity of an ensemble the distance between it and the distinguished distribution. I'm sure you can be more creative than "minimum number" of some group of transformation as well. Since you have continuous distributions on phase space, probably the phase space integral of some function of the distributions makes more sense. The issue here, is I'm not sure what usefulness there will be to such an exercise... finding the "distance from equilibrium" – Bobak Hashemi Dec 3 '16 at 10:19
• That makes sense too. But just for sake of clarification I meant in the scope of "Theoretical mechanics" whether being in any of its formulations, some theories bring their own axioms into the formulation and make it a unique description of nature under that formulation. I would rather name what you described as "Analytical mechanics" actually "Classical analytical mechanics"(analytical mechanics as a formalism manipulated by the axioms of classical mechanics) and the latter "Quantum analytical mechanics". However these are just differences in terminology and aren't important. – Hamed Begloo Dec 4 '16 at 20:47
• On the other hand your second comment gave good insights to me. But about the usefulness of the subject: I think it's important to define a "Magnitude of complexity" of the system for achieving "How much data do you need" to describe Time-evolution of a system. For example in "Non-equilibrium thermodynamics" it may help to determine states of the system at any moment of a non-quasistatic process or at least it may show the possibility or impossibility of determining these states. – Hamed Begloo Dec 4 '16 at 20:47

I agree with the previous statements, that no formally and broadly well accepted definition of complexity exists. Still, an intuitive idea of what is complexity is being developed, somehow. The problem is that the formalization of such idea changes, considerably, from field to field (one example being that of Suskind, above, which is rather vage since it relies on the also vage concept of "any distinguished state"---the case of the "ground state" as a distinguished state may be misleading since there are systems whose ground state is considered to be complex on its own and, therefore, the definition cannot be applied without getting into a contradiction). A vage description of the idea is as follows:

A complex system is one that cannot be modeled using a model (both, deterministic or stochastic) with a small number of adjustable parameters. In other words, the amount of information required to provide the "minimal" or more "succint" description of the system is "large".

To get a better idea about this, you can read the book of Murray Gell-Man (The Quark and the Jaguar), the works of James P. Crutchfield and the work of Jorma Rissanen about the Minimum Description Length (MDL) principle. Gell-Man elaborates more on the intuitive idea. Crutchfield and Rissanen elaborates more on how to formalize the idea, although with quite different approaches; the former more on "complex processes" and the later more on "complex structures/states". All of them give an idea on how to quantify how much complex a system is.

• I kindly appreciate your book recommendations. Can you also provide a resource for me which talks about the concept of complexity intertwined with the field of statistical mechanics? I also will be glad if you consider reading my comments under other answers and give an opinion. – Hamed Begloo Dec 2 '16 at 21:39
• Hi, apologies for the later reply. Regarding books, I have in mind a few I consider interesting but, not necessarily involving statistical mechanics; at least not in its traditional sense. The problem is that, as already mentioned, there is no a formal definition of complexity. Usually, the idea is to combine approaches from statistical mechanics with approaches coming from computer science and statistical modeling. I highly recommend you to read the PhD. thesis of Cosma Rohila Shalizi (link). – Juan I. Perotti Dec 8 '16 at 14:09