In an old factory, an invisible mechanism spins behind the wall. We cannot see the gears, but we hear how differently the floor and the ceiling respond to its work: when the floor hums, the ceiling answers with a delay, and when the ceiling hums, the floor answers differently. This asymmetry of response gives away that something inside is irreversibly rubbing and heating up.

That is roughly the setup of a problem from stochastic thermodynamics. We have a small system, say a molecular motor or an ion channel, in which thermal noise prevents us from measuring every detail precisely. The instrument registers only coarse states: instead of the rotor's exact position, we see "spinning" or "stopped." From such coarse observations it is hard to tell how much energy is dissipated as heat. The authors of the paper show that a lower bound on this dissipation can be extracted from the asymmetry of two-time correlations between two observable quantities.

Formally, the paper considers Markov networks with discrete states, where several microstates are lumped into a single mesostate. For a nonequilibrium steady state, when the flows are constant, the lower bound on the rate of entropy production is expressed through the correlation asymmetry and a quantity computed from correlation functions. This bound holds for any time lag. For time-dependent processes, such as relaxation or periodic driving, a similar bound emerges in the limit of vanishing lag, that is, when the two moments in time nearly coincide.

Numerical examples show that the estimate can capture up to 60% of the total entropy production in a four-state network with coarse graining. For a particle on a ring, where only half of the trajectory is visible, the quality of the estimate reaches about half of the true value. That is no small feat, given that we do not know the hidden dynamics.

This approach applies to experimental data where only time series of a few macroscopic quantities are available: in the biophysics of molecular motors, in optical traps, in financial series. We still cannot see the mechanism, but now we can estimate how much heat it dissipates, simply by listening to how its parts respond to one another.