There's a nice short Azimuth post on Jarzynski's equality by [Eric Downes](, and a longer post on somewhat related issues by [Matteo Smerlak](

> So it basically says: if you want to find the most interesting observed quantities, don’t start by looking for commutations. That’s the value of the result in my opinion, to show us where we shouldn’t look and therefore to guide us to where we should look.

If you want to look for conserved quantities that don't commute with the Hamiltonian, you could start by looking at the simple example that Brend**A**n and I describe in [our paper]( Other examples will work in a similar way: each state $i$ where the observable $O$ takes value $O_i$ has probabilities $p_{j i}$ of evolving to other states $j$, with the property that

$$ O_i = \sum_j O_j p_{j i} $$

Here I'm describing the discrete-time case, where time evolution is given by a stochastic operator. Our paper discusses a continuous-time example, where time evolution is given by an infinitesimal stochastic Hamiltonian. They're similar but different.

I guess the fun part would be to find concrete, natural examples where this happens.