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My main interests are biological systems, but the question is general.

I was trained in computational biology, and virtually all quantitative models of biological processes I've encountered in my undergraduate studies were of probabilistic nature - Hidden Markov Models, Bayesian Networks etc. This makes sense considering the complexity of these processes. These are systems with many degrees of freedom interacting in often non-linear ways, and so of course there is no hope of describing them with a manageable set of equations the way one would, say, the motion of planets in the solar system, and of course one has to resort to using stochastic models.

Nevertheless, I've seen (usually in the context of biophysics) many cases of complex biological processes modeled as low dimensional (usually non-linear) dynamical systems, with impressive predictive and explanatory power.

I've been reading about center manifold theory, which deals with high dimensional systems in which many of the dimensions are superfluous, in the sense that perturbations in these directions rapidly die off and so an effective low-dimensional description of the system is possible. It's a beautiful theory, but does not really answer the question, which becomes “Why do dynamical systems describing real-world processes have so many negative eigenvalues in their Jacobian near equilibria?”

Questions:

  1. What property of complex systems (or certain observables of such systems) makes them modelable as low dimensional systems?

  2. Why do so many real-world systems seem to have this property?

  3. In particular, am I right that this happens more often than one would expect in biological systems?

Technical or non-technical answers are both welcome.

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  • $\begingroup$ I don't know much about the subject, but I'd be very interested to look at some examples, if you could add them to your question. $\endgroup$
    – Philip
    Commented Aug 23, 2020 at 17:56
  • $\begingroup$ Reminds me of quantum effects averaging out to zero at the macro-level. $\endgroup$
    – DKNguyen
    Commented Aug 23, 2020 at 19:45
  • $\begingroup$ Have you heard of Sethna's sloppy models? If not I'd recommend (lassp.cornell.edu/sethna/Sloppy/index.html) as an interesting perspective on your questions $\endgroup$
    – Dwagg
    Commented Aug 23, 2020 at 20:17

3 Answers 3

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Here are a couple of factors that come into play:

  • If we take a given real-life system described by, say, 2000 variables, chances are that:
    • a lot of them aren't fully independent or nonequivalent, and can be disregarded - such as in a crystal, whose symmetries greatly simply its description;
    • a large number of the remaining ones can be aggregated with little loss of information - like the populations of different species playing the same role in an ecological model;
    • there is some sort of loss in the system, of energy for instance, which simplifies, limits its asymptotic behavior - restricting it to a region of its state space considerably smaller (and of fewer dimensions) than the whole state space;
    • the system is subject to noise which, no matter how small, typically sets to zero the chance of finding the system on one of its (likely many) unstable solutions: a further restriction on the region of state space we have to consider.
  • And if this system has, for instance, 100 parameters, likewise:
    • it's very unlikely that all of them are equally important - taking a number of the top influential ones allows one to explain a large part of the system's variability;
    • some parameters will probably not be fully independent, or will be fixed by intrinsic or external factors, being effectively constants, rather than parameters.

When the noise mentioned above doesn't cancel out in average (and have a vanishingly small influence on the system), then statistical descriptions are probably unavoidable.

That's for questions 1 and 2; as for question 3, it's perhaps a matter of personal opinion: after all, how much "one would expect"? Given the factors listed above, one might argue that it's to be expected that more systems should be describable in lower dimensions. At any rate, we should not forget that we give more attention to such systems, regardless of them being typical or not, simply because those are the ones we can better understand - which might lead to the impression that they are more common than they actually are.

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The equilibria with positive eigenvalues would tend to blow up if they were perturbed even slightly, right? Which means that, in settings in which those kinds of perturbations are likely, we would not tend to find systems in those equilibria. They would already have "blown up" locally, and moved out of that region of the manifold. So maybe it's just that, over time, the world has evolved so that systems are found in the equilibria which are stable given their context, which means that for all the kinds of perturbations they will experience in their environment, the eigenvalues are negative.

As for why a complex system can often be modeled with a low number of parameters. Well, generally those few parameters turn out to be averages or aggregates of the more numerous degrees of freedom that are visible on a smaller scale. So the question is, how come we can apply the laws of physics only to those aggregate properties instead of accounting for all the true degrees of freedom?

You could write the dynamical law in question for each particle in the system. But if the system is linear, then you can add up all those equations, and you end up with the same law, only applied to the object as a whole, with its aggregate properties. By virtue of conservation, the interactions between individual particles of the system will cancel out, and the law will refer only to influences coming from the outside world.

And that's why it's important that, to the extent that the system is non-linear, those non-linear perturbations decay quickly. Because until they do, you do have to consider the parts of the system separately, so you're forced to use more parameters to describe it.

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These questions are answered by renormalization group (RG) theory. Briefly, when you take a system with many degrees of freedom (DOF) and repeatedly coarse-grain it, the coarse-grained description describes a flow in the space of probability distributions. This flow has the property that many microscopic details become irrelevant, which is elevated to a technical term. In condensed-matter and particle physics, we normally discuss this flow in terms of the Hamiltonian, but the concept applies to any family of probability distributions and thus applies universally.

To understand intuitively why there are so many irrelevant directions, the key idea is that most DOF only interact locally, so that a large system is approximately constructed out of approximately non-interacting subsystems. Then, under coarse-graining, the so-called renormalized description has weaker interactions than the original description. Irrelevance in RG is thus an extension of the irrelevance of higher moments in the central limit theorem.

To make these ideas precise, the language of field theory is needed. The modern view crystallized in the 1970's with the work of Ken Wilson, although this was based on the related, but distinct, concept of renormalization as used in particle physics since the 1950's.

There are two important caveats with respect to biological systems: first, the usual RG is developed in terms of spatial locality. If you have a system that is not organized spatially, it can be difficult to apply the orthodox methods. This is a currently active area of research. Second, the RG approach works well when there is a large separation of scales between the microscale and the macroscale. This criterion may not always be met in biological systems.

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