How to use rattling

Q: So what exactly can Rattling theory tell me for my specific system, experiment or question? How do I use it?

First you need to consider if you think the Rattling hypothesis should hold in your system. Generally, it should hold for “typical” systems (see these guidelines). If this applies, then there are several things you can do.

Predict relative likelihoods#

To get the steady-state probability of any system configuration $x$, you need to know its Rattling $\mathcal{R}(x)$ and the normalization $Z$ (just as with Boltzmann distribution):

Rattling is powerful because it is easy to measure $\mathcal{R}(x)$—one just needs to initialize the system in state $x$ and see how fast it leaves that state on average. The generalization for systems with continuous configuration space is $\mathcal{R}(x) = \frac{1}{2} \log \det D(x)$, where $D(x)$ is the effective diffusion tensor at $x$, which can be similarly approximated from how fast it leaves $x$ (from taking a few short system trajectories starting at $x$, and calculating the covariance matrix (see details in our Science paper Materials and Methods, sec 3.2). Note that, empirically, it is easier to measure Rattling of a state than it is to measure energy of a state, since energy cannot be locally defined in terms of transition rates among the states.

Without knowing $Z$, this allows us to predict relative likelihoods:

The coefficient $\gamma$ here is a system-specific constant, which will often be close to 1, but may deviate somewhat if the system doesn’t quite meet the mentioned guidelines (e.g., its configuration space is 6-dimensional rather than 100-dimensional). See here for more details on what $\gamma$ is. Either way, it is easy to estimate empirically from a few state-measurements and then make predictions for the rest.

Predict absolute likelihoods?#

But to predict whether of not a system will self-organize, relative likelihoods are not enough, and so we need to find $Z$. $Z$, however, is not so easy to estimate empirically, since it depends globally on all configurations $x$. Unless we have empirical access to all system states, which is usually impractical, we need some theoretical calculation to find $Z$, and so some theoretical understanding of the global system properties. This is similar to finding the partition function in statistical mechanics, except that calculating energy of a state is well-known, while calculating its rattling is generally an open question.

We have some examples where we were able to successfully do this and predict the phase transition to self-organization (e.g., Vicsek model), but it’s not clear for which systems this is or isn’t possible. In other cases, it may be possible to combine theoretical and empirical methods to estimate $Z$—by sampling a representative set of configurations and measuring their rattling (here you need a sufficient understanding of your system to know how to choose that “representative set”).

Destabilize by adding noise#

Predict self-organized configurations#

Control self-organized configurations#

(our paper examples - prop ratios, control, predict, etc…