Entropic balance with feedback control: information equalities and tight inequalities

We consider overdamped physical systems evolving under a feedback-controlled fluctuating potential and in contact with a thermal bath at temperature $T$. A Markovian description of the dynamics, which keeps only the last value of the control action, is advantageous – both from the theoretical and the practical side – for the entropy balance. Novel second-law equalities and bounds for the extractable work are obtained, the latter being both tighter and easier to evaluate than those in the literature based on the whole chain of controller actions. The Markovian framework also allows us to prove that the bound for the extractable work that incorporates the unavailable information is saturated in a wide class of physical systems, for error-free measurements. These results are illustrated in model systems. For imperfect measurements, there appears an interval of measurement uncertainty, including the point at which work ceases to be extracted, where the new Markovian bound is tighter than the unavailable information bound.

💡 Research Summary

The paper investigates overdamped Brownian particles that are coupled to a thermal bath at temperature T and subjected to a feedback‑controlled, time‑dependent potential V(x,c). The control variable c is updated at regular measurement times based on the particle’s position xₙ via a conditional probability Θ(c|x). While the joint sequence of positions and controls ({x₀,…,xₙ},{c₀,…,cₙ}) is generally non‑Markovian, the authors introduce a Markovian framework that retains only the most recent control value cₙ as part of the system state. Between measurements the control is frozen and the particle evolves according to a Fokker‑Planck equation with the current potential V(x,cₙ). In the long‑time limit the joint distribution P(x,c,t) becomes periodic with period equal to the measurement interval Δtₘ.

Using this description the authors derive an exact entropy balance. The internal energy change during a Fokker‑Planck interval splits into work performed by an external force W_FP and heat Q exchanged with the bath; the control update contributes an additional work W_m, and the total work satisfies ⟨W⟩=⟨W_FP⟩+⟨W_m⟩=−⟨Q⟩. The entropy of the joint process S_{xc}=S_c+S_{x|c} evolves as Δ_FP S_{x|c}=β⟨Q⟩+Σ_{xc}, Δ_m S_{x|c}=−Δ_m I, where Σ_{xc} is the non‑negative entropy production during the Fokker‑Planck evolution and Δ_m I is the change in mutual information between x and c at the measurement instant. From these relations the authors obtain the central inequality β⟨W⟩ = Σ_{xc} − Δ_m I ≥ −Δ_m I, which is tighter than the previously known bound based on the full control history. The latter involves the transfer entropy I_{c} of the entire control chain, a quantity that is difficult to compute because the control sequence is non‑Markovian. The authors prove that I_{c} ≥ Δ_m I, so the Markovian bound (−Δ_m I) is always at least as restrictive as the older bound (−I_{c}). When the system is in equilibrium (no steady currents) the housekeeping entropy Σ_{hk}^{xc}=0 and all dissipation originates from feedback; for non‑equilibrium steady states Σ_{hk}^{xc}>0 adds an extra positive term.

A particularly striking result concerns error‑free measurements (Θ(c|x)=1_c(x)). In this case the authors show that the bound becomes an equality: β⟨W⟩ = I_u − I_{x}, where I_u is the “unavailable information” introduced in earlier work and I_{x} is the transfer entropy of the measured position chain. Thus, for a broad class of systems (arbitrary potential V(x,c) and measurement interval Δtₘ) the previously derived inequality β⟨W⟩ ≥ I_u − I_{x} is actually saturated. For non‑zero external forces the detailed‑balance condition fails, but in the limit of very long measurement intervals (Δtₘ→∞) the propagator approaches the non‑equilibrium steady‑state distribution, and the equality generalises to β⟨W⟩ = Σ_{hk}^{xc} + I_u − I_{x}.

To illustrate the theory, the authors study a concrete information engine where the potential can be flipped, V(x,c)=c U(x) with c=±1. The base potential U(x) is linear within a finite interval and periodic at the boundaries. The feedback rule in the error‑free case inverts the potential whenever the particle’s energy would decrease, thereby extracting work. Imperfect measurements are modelled by a finite resolution Δx, which smooths the conditional probability Θ(c|x). Numerical simulations (dimensionless units) show that the average extracted work ⟨W⟩ becomes positive (no extraction) beyond a critical measurement error Δx₀. The Markovian bound (−Δ_m I) predicts Δx₀ accurately, whereas the bound based on the full control history overestimates the region where work can be extracted. Moreover, for intermediate errors there exists a range where the new Markovian bound is tighter than the unavailable‑information bound, confirming the theoretical predictions.

In summary, the paper makes several important contributions: (i) it introduces a Markovian description that retains only the latest control action, simplifying the entropy‑balance analysis; (ii) it derives a new, tighter work‑extraction bound that is easier to evaluate than previous transfer‑entropy‑based bounds; (iii) it proves that for error‑free measurements the bound involving unavailable information is saturated for a wide class of systems; (iv) it extends the results to non‑equilibrium steady states by incorporating housekeeping entropy; and (v) it validates the theory with a realistic model engine, demonstrating practical relevance for designing and analysing feedback‑controlled nanoscale machines. These findings advance both the theoretical foundations of stochastic thermodynamics with information and the practical assessment of information‑driven engines.