Thermodynamic entropic uncertainty relation (2502.06174v3)

Abstract: Thermodynamic uncertainty relations reveal a fundamental trade-off between the precision of a trajectory observable and entropy production, where the uncertainty of the observable is quantified by its variance. In information theory, Shannon entropy is a common measure of uncertainty. However, a clear quantitative relationship between the Shannon entropy of an observable and the entropy production in stochastic thermodynamics remains to be established. In this Letter, we show that an uncertainty relation can be formulated in terms of the Shannon entropy of an observable and the entropy production. We introduce symmetry entropy, an entropy measure that quantifies the symmetry of the observable distribution, and demonstrate that a greater asymmetry in the observable distribution requires higher entropy production. Specifically, we establish that the sum of the entropy production and the symmetry entropy cannot be less than $\ln 2$. As a corollary, we also prove that the sum of the entropy production and the Shannon entropy of the observable is no less than $\ln 2$. As an application, we demonstrate our relation in the diffusion decision model, revealing a fundamental trade-off between decision accuracy and entropy production in stochastic decision-making processes.

• The paper introduces an uncertainty relation that combines observable Shannon entropy with entropy production using the novel concept of symmetry entropy.
• It demonstrates that the sum of entropy production and symmetry entropy is bounded below by ln 2, revealing a fundamental trade-off in thermodynamic systems.
• The framework offers actionable insights for optimizing energy-precision trade-offs in nanosystems and suggests potential extensions to quantum thermodynamics.

Overview of the Thermodynamic Entropic Uncertainty Relation

The paper "Thermodynamic Entropic Uncertainty Relation," authored by Yoshihiko Hasegawa, proposes an innovative framework for understanding uncertainty in stochastic thermodynamics through a thermodynamic entropic uncertainty relation. Unlike traditional approaches that often measure uncertainty via variance, this paper explores uncertainty in terms of Shannon entropy, bringing novel insights into thermodynamic processes.

The entropic uncertainty relation, well-established in quantum mechanics, links position and momentum uncertainty, often utilizing Shannon entropy for quantifying such uncertainty. The novelty of Hasegawa's work lies in translating these entropic insights into the domain of stochastic thermodynamics, which previously lacked a formalized entropic description similar to those in quantum systems. By introducing the concept of symmetry entropy, the paper bridges this gap by linking the asymmetry in the probability distribution of observables to the required entropy production.

Key Claims and Numerical Results

A central claim of the paper is encapsulated in the formulation of an uncertainty relation that combines observable Shannon entropy and entropy production. Hasegawa introduces symmetry entropy, defined as the difference between the Shannon entropy of an observable and the entropy of its absolute value distribution. This measure quantifies the asymmetry of the observable distribution and is shown to be pivotal in determining the entropy production in thermodynamic systems.

The paper posits that the sum of entropy production and symmetry entropy cannot be less than $\ln 2$. Formally, it is demonstrated that:

$\Sigma + \Lambda[P(\Phi)] \geq \ln 2,$

where $\Sigma$ is the entropy production, and $\Lambda[P(\Phi)]$ is the symmetry entropy. This inequality reflects a fundamental trade-off, reminiscent of traditional uncertainty relations where increased precision (or reduced uncertainty) entails higher cost—in this context, with respect to entropy production.

Implications and Future Directions

The implications of establishing such an entropic uncertainty relation are profound, as it provides an additional layer of understanding concerning the interplay between stochastic observables and thermodynamics. Practically, this framework could influence the way we analyze and optimize stochastic processes in engines and nanosystems, where managing entropy production is crucial.

From a theoretical standpoint, these results invite further exploration into whether similar principles could be applied or expanded to quantum thermodynamics, particularly given the ongoing research in quantum speed limits and the role of coherence in quantum thermodynamic systems.

Future developments might explore applications in quantum systems, perhaps offering unified or analogous uncertainty relations that account for coherence and entanglement. Additionally, experimental validations in complex biochemical processes could solidify the practical relevance of these theoretical insights.

Conclusion

Hasegawa's contribution presents a thoughtful extension of uncertainty principles from the field of quantum mechanics to stochastic thermodynamics. By integrating Shannon entropy into this domain, the paper lays foundational grounds for new uncertainty relations that could redefine energy-precision trade-offs in thermodynamic systems. These insights promise to foster both theoretical advancements and practical innovations across multiple disciplines within the physical sciences.