Thermodynamic inference from noisy single-molecule time series

Single-molecule or single-particle tracking measurements inherently yield noisy microscopic trajectories, often significantly constrained by the diffraction limit and by the finite rate at which photons are emitted and counted. Here, we study systematically the resulting effects of finite spatial and temporal resolution on one's ability to discern and quantify the arrow of time in microscopic trajectories. Given an experimental time series Y(t) degraded by noise, we consider the problem of estimating the entropy production associated with the corresponding microscopic variable X(t) using two strategies. The first attempts to infer the statistical properties of X(t) from those of Y(t) before estimating the entropy production. The second uses the experimental observable as a proxy for the true microscopic observable, with the entropy production estimator applied directly to Y(t). We prove that both strategies result in lower bounds on the true entropy production. Importantly, noise-degraded observables Y(t) undergo non-Markovian dynamics even when X(t) are Markovian, and non-Markovian entropy production estimators are advantageous. We further note a nontrivial interplay between spatial and temporal resolution: in the presence of detection noise, improving the temporal resolution alone may lead to poorer rather than better entropy production estimates.