Rattling vs energy

Rattling plays the same role in non-equilibrium steady-state distributions (for typical systems) as energy does in the Boltzmann distribution. This suggests comparing properties of Rattling $\mathcal{R}$ with those of energy, to see if $\mathcal{R}$ can be as powerful as energy in various contexts. Below we present a more intuitive (physics) and more formal (math) perspective on this comparison:

Physics perspective#

First notice that there is no fundamental sense in which state energy should be any more “natural” or “simple” to deal with than Rattling. Energy is more familiar to us, but ratting is in fact easier to access experimentally using only local measurements - by just checking the state exit rate. In contrast, there is actually no simple local way to measure the energy of a given state (partly because only relative energies are physically meaningful - so there is no absolute quantity to measure). The main reason why energy $U(x)$ is so useful in equilibrium systems is because it is conserved (which leads to the Boltzmann distribution), and composable (which means that the energy of any complex system state is just the sum of all constituent subsystem energies and interaction energies). This composability of energy often allows us to analytically calculate the energy of a state, even if we can’t measure it directly. One core question here is whether Rattling of a state is as easy to calculate as state energy. At first this seems challenging, but some progress can, surprisingly, be made – see discussion on this here for some initial steps.

See the page on rattling thermodynamics for research directions that this opens.

More formally#

Jacob…