Thermodynamic uncertainty relation for feedback cooling

TL;DR

This work derives a thermodynamic uncertainty relation (TUR) for measurement-based feedback cooling in classical underdamped Langevin systems, linking cooling efficiency $\epsilon_{\rm I}$ to the entropy-reduction rate and kinetic temperature $T_{\rm kin}$. A novel, unique orthogonal decomposition of the irreversible local mean velocity and the partial entropy production is introduced, enabling a TUR that bounds a generalized current by the entropy production and revealing how maximal cooling with finite power can be achieved. The authors show that attaining $\epsilon_{\rm I}=1$ with finite entropy reduction rate is possible in the asymptotic limit by employing a Kalman-filter-based estimator with large gain, where the fluctuation of the reversible current diverges. They further extend the framework to a particle in a harmonic potential, obtaining a two-direction TUR with kinetic temperatures $T_u$ and $T_v$, and discuss implications for quantum extensions and non-Gaussian dynamics as future directions.

Abstract

Feedback cooling enables a system to achieve low temperatures through measurement-based control. Determining the thermodynamic cost required to achieve the ideal cooling efficiency within a finite time remains an important problem. In this work, we establish a thermodynamic uncertainty relation (TUR) for feedback cooling in classical underdamped Langevin systems, thereby deriving a trade-off between the cooling efficiency and the entropy reduction rate. The obtained TUR implies that simultaneous achievement of the ideal cooling efficiency and finite entropy reduction rate is asymptotically possible by letting the fluctuation of the reversible local mean velocity diverge. This is shown to be feasible by using a feedback control based on the Kalman filter. Our results clarify the thermodynamic costs of achieving the fundamental cooling limit of feedback control from the perspective of the TUR.

Figures (4)

• Figure 1: Schematic of the TUR for feedback cooling, where $\epsilon_{\rm I}$ is the cooling efficiency and $\dot{W}_{\mathrm{ext}}/T_{\mathrm{kin}}$ is the entropy reduction rate. The region above the parabola is prohibited by the obtained TUR \ref{['eq: tradeoff from TUR for feedback cooling']}, where each parabola corresponds to the bound set by TUR for different parameter values (e.g., the feedback gain $G$). In the limit of large feedback gain $G\rightarrow \infty$, the entropy reduction rate maintains a finite value while approaching the maximum cooling efficiency $\epsilon_{\rm I}\rightarrow 1$ (see also Fig. \ref{['fig: achieving max efficiency']}).
• Figure 2: Schematic of the orthogonal decomposition used in this study. The irreversible local mean velocity $\nu_v^{\mathrm{irr}}$ is decomposed into two orthogonal components, which are a marginalized part $\tilde{\nu}_v^{\mathrm{irr}}$ and a residual part $\delta \nu_v^{\mathrm{irr}}$. The irreversible component of a generalized current with a general observable $w(v,y)$ is given by $\mathcal{J}^{w}_{\nu^{\rm irr}} \coloneqq (\gamma T/m^2) \ev{w, \nu_v^{\mathrm{irr}}}_p$. Using the decomposition $\ev{w, \nu_v^{\mathrm{irr}}}_p = \ev{w, \delta \nu_v^{\mathrm{irr}}}_p + \ev{w, \tilde{\nu}_v^{\mathrm{irr}}}_p$, the current $\mathcal{J}^{w}_{\nu^{\rm irr}}$ is decomposed into each component. Each term leads to distinct TURs \ref{['eq: short time TUR for partial entropy production']}-\ref{['eq: short time TUR for marginal entropy production']}.
• Figure 3: Behavior of the entropy reduction rate and the cooling efficiency as functions of the feedback gain $G$. The main plot shows the entropy reduction rate $\dot{W}_{\mathrm{ext}}/T_{\mathrm{kin}}$ (left axis) and the cooling efficiency $\varepsilon_{\mathrm{I}}$ (right axis), where the two plots overlap with each other because the information flow $\dot{I}_{v}=-K/2m$ does not depend on $G$. As $G$ increases, $\varepsilon_{\mathrm{I}}$ asymptotically approaches unity, while the fluctuation associated with the entropy reduction rate (shown in the inset) diverges linearly. This divergence clarifies how asymptotically achieving finite entropy reduction rate at maximum efficiency is possible from the viewpoint of the obtained TUR \ref{['eq: tradeoff from TUR for feedback cooling']}. The parameters are: $m=1$, $\gamma=1$, $T=5$, and $\sigma^2=0.5$.
• Figure 4: Behavior of the cooling efficiency and entropy reduction rate as functions of the feedback gain $G$. The main plot shows the entropy reduction rate $\dot{W}_{\mathrm{ext}, u}/T_u + \dot{W}_{\mathrm{ext}, v}/T_v$ (left axis) and the cooling efficiency $\varepsilon_{\mathrm{I}}$ (right axis), where the two curves overlap with each other because the information flow $\dot{I}_{s}$ does not depend on $G$ in this setup. As $G$ increases, $\varepsilon_{\mathrm{I}}$ asymptotically approaches unity, while the fluctuation associated with the entropy reduction rate (shown in the inset) diverges linearly. This divergence clarifies how asymptotically achieving finite entropy reduction rate at maximum efficiency is possible from the viewpoint of the trade-off relation Eq. \ref{['eq: trade-off with harmonic potential']}. The parameters are: $\omega=1$, $\gamma=10$, $T=1$, and $\sigma=0.2$.