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Uncertainty relations in stochastic processes: An information inequality approach.
Yoshihiko Hasegawa1, Tan Van Vu1
1Department of Information and Communication Engineering, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan.
Physical Review. E
|July 24, 2019
Summary
Information inequalities provide bounds for statistical inference, similar to thermodynamic uncertainty relations. This study applies information inequalities to Langevin systems, deriving bounds for thermodynamic fluctuations and revealing connections to the Cramér-Rao inequality.
Area of Science:
- Statistical physics
- Information theory
- Non-equilibrium thermodynamics
Background:
- The thermodynamic uncertainty relation sets a precision limit based on entropy production.
- Information inequalities in statistical inference define bounds for estimator accuracy.
- A conceptual link exists between these two types of inequalities.
Purpose of the Study:
- To apply information inequalities to systems described by Langevin equations.
- To derive bounds for thermodynamic quantity fluctuations.
- To explore the relationship between thermodynamic uncertainty and information inequalities.
Main Methods:
- Application of information inequalities, specifically the Cramér-Rao and Chapman-Robbins inequalities.
- Analysis of systems governed by Langevin equations.
- Derivation of fluctuation bounds and equality conditions.
Main Results:
- Derived bounds for thermodynamic fluctuations in Langevin systems.
- Demonstrated that the thermodynamic uncertainty relation is a specific case of the Cramér-Rao inequality, with Fisher information equaling total entropy production.
- Identified stochastic total entropy production as the sole quantity achieving equality in the thermodynamic uncertainty relation.
- Obtained a lower bound for the variance-to-sensitivity ratio using the Chapman-Robbins inequality.
Conclusions:
- Information inequalities offer a powerful framework for understanding thermodynamic fluctuations.
- The Cramér-Rao inequality unifies thermodynamic uncertainty and fluctuation-response relations.
- Stochastic total entropy production plays a critical role in achieving precision limits in thermodynamic systems.
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