Typicality

We claim that the Rattling hypothesis holds (i.e., predicts steady-state probability) for “typical” dynamical systems. On the one hand, “typical” here expresses an uncontrolled approximation – for a given system, it may be hard to determine a-priori how well Rattling will hold. On the other hand, it points to the intuition that Rattling should hold most of the time in real-world scenarios.

So what does “typical” mean exactly?#

Here we are referring to the concept of typicality – a related set of ideas and results that emerged over the last 50 years. Pankaj Mehta described it nicely in his 2024 paper “Theory is dead. Long live theory! For a 21st century statistical physics of life”:

Wigner showed that the statistical structure of nuclear resonances in the Uranium atom could be reproduced by modeling the Hamiltonian as a large symmetric random matrix.¹ Winger’s conjectured that when a system becomes complex enough, many of its statistical properties will be “typical” and indistinguishable from a large random system subject to the same physical constraints as the object of study (i.e. is drawn from the same ensemble). In the case of Uranium, the relevant physical constraint Wigner identified was unitarity, which is why he considered symmetric random matrices to model the Hamiltonian of Uranium. More recently, this line of thinking has found great success in condensed matter theory in the context of Eigenstate Thermalization Hypothesis, which uses typicality to explain why even a single eigenstate of a sufficiently large manybody quantum system can be accurately described by statistical mechanics.²

As Freeman Dyson explained, this is a fundamentally “new kind of statistical mechanics, in which we renounce exact knowledge not of the state of a system, but the nature of the system itself.”³ What is nice about this approach is that it suggests a natural framework for thinking about living systems. Because it is agnostic to the origin of the constraints, it does not matter whether the relevant constraints originate from physical conservation laws, dynamics, geometry, energetics, evolutionary history, or even from specialized biological functions. For this reason, these ideas offer a promising avenue for thinking about how to model complex biological systems.

In the last few years this program has been successfully applied to understand many properties of complex microbial communities, suggesting that it can be operationalized to gain biological insights⁴. Moreover, modern generative models of proteins also naturally fall into this rubric. When stripped of all their bells and whistles, generative models of proteins are probabilistic models for sequence generation that respect the evolutionary and functional constrains embodied in amino acid conservation patterns.⁵

This is also the essence of “maximum entropy (MaxEnt) modeling” – where in the absence of having detailed measurements of the system parameters, we assume these to take their most likely values, constrained by some macroscopic or global parameters that we can measure. This technique has been applied in many systems, and even works to model neuron firing statistics and even neuron connectivity. In fact, MaxEnt is one way to derive the Boltzmann distribution, as well as our Rattling hypothesis (see Sec.2.2 in Supplement of our paper).

Adversarial fine-tuning#

The main intuition behind “typicality” is that the system is large, complex, and did not suffer any adversarial fine-tuning. This is reminiscent of how a Neural Network can perfectly classify all “typical” cat images, unless someone adjusts the pixels slightly just right, so as to make it think it’s a dog (see here). In our case, such adversarial fine-tuning need not come from a malevolent agent, but from some pre-existing selection pressures. E.g., a biological organism is not a “typical” dynamical system as it has been fine-tuned by evolution. A pendulum is not a “typical” dynamical system because it’s too simple. A crystal is not because it’s too ordered (not complex enough).

So with this, the question arises: how typical exactly are dynamical systems where Rattling hypothesis holds? Can we construct a random ensemble representing all real-world dynamical systems, and then see what’s the probability of Rattling-friendly systems?