The work continues a line of research on thermodynamic inference, that is, extracting a system's properties from limited observations, a field that has been developing actively in recent years. Close to it lie studies of rare events in the quantum world, of extracting the laws of nature from noise, and of estimating chaos in systems with unknown rules. All of them share one idea: even incomplete data about random processes carry information about fundamental constraints, such as the rate of energy dissipation. This paper stands out because it uses correlation functions of arbitrary coarse-grained observables rather than specially chosen quantities, and it delivers bounds that hold for an arbitrary time lag in the stationary case.
The authors consider Markov networks with discrete states that may be partially observable, meaning that several microstates are merged into a single mesostate. They analyze the asymmetry of the two-time correlation function between two observables A and B, defined as the difference between ⟨A(t)B(t+τ)⟩ and ⟨B(t)A(t+τ)⟩. Using the Cauchy–Schwarz inequality and the log-sum inequality, they relate this asymmetry to entropy production. For stationary nonequilibrium states, the estimate holds at any lag τ, while for time-dependent processes it requires the vanishing-lag limit. The paper also discusses optimizing the estimate by shifting the observables by constants, which yields tighter lower bounds.
Two main results are obtained. First, for a nonequilibrium stationary state, the lower bound on the entropy production rate σ is expressed through the correlation asymmetry χ(τ) and a quantity D(τ) computed from the correlation functions, and it holds for any τ ≥ 0. Second, for arbitrary time-dependent processes, including relaxation and periodic driving, a similar bound emerges in the limit τ → 0 and is expressed through the derivatives of the correlation functions at zero lag. Numerical examples for a four-state network with coarse graining show that the estimate can capture up to 60% of the total entropy production at small affinities and lags. For a particle on a ring with partially hidden regions, the quality of the estimate reaches about 0.5 when half of the ring is visible.
Unlike neighboring studies, which often rely on special observables or require knowledge of trajectories, this paper uses arbitrary coarse-grained observables and only their correlation functions. That broadens the applicability of thermodynamic inference to experimental systems where only time series of a few macroscopic quantities are available. Moreover, the possibility of optimizing the estimate by shifting the observables by constants offers a practical recipe for improving accuracy without additional measurements.
A natural next step is to study more general transformations of observables, such as linear or nonlinear combinations, in order to further improve the bounds. It is also important to test the robustness of the estimates under limited measurement statistics and in the presence of noise in the identification of mesostates. For time-dependent processes, the question remains open whether the bound can be extended to finite time lags, which matters for experimental setups with limited temporal resolution.
The proposed estimates may be useful in biophysics for analyzing molecular motors and ion channels, where only individual states are observed, as well as in experiments with optical traps and colloidal particles. They also apply to financial time series, where correlation functions are a standard analysis tool.
The next step will be to apply the estimates to real experimental data, for instance to trajectories of molecular machines or to fluorescence correlation spectroscopy. It is also important to study the effect of erroneous coarse graining, when mesostates are defined with noise, and to develop methods for compensating for such errors.
In stochastic thermodynamics, the problem of accurately inferring entropy production from partial observations without knowledge of the hidden dynamics remains open. This work contributes by proposing lower bounds that require neither the full trajectory nor a model of hidden states, but only the correlation functions of the observed quantities. That brings us closer to solving the problem of minimal dissipation in systems with incomplete information.
For a particle on a ring with an asymmetric potential and a constant force, the estimate can reach 100% of the entropy production if the observables are chosen as the sine and cosine of a single spatial frequency and the potential is absent.