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Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
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Thermodynamic Uncertainty Relation for Arbitrary Initial States.

Kangqiao Liu1, Zongping Gong1, Masahito Ueda1,2,3

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|October 16, 2020
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Summary
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This study introduces a finite-time thermodynamic uncertainty relation (TUR) applicable to any initial state, overcoming previous limitations. The new TUR, derived from the Cramér-Rao inequality, bounds accumulated current variance by final-time current.

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Area of Science:

  • Non-equilibrium thermodynamics
  • Statistical mechanics
  • Information theory

Background:

  • The thermodynamic uncertainty relation (TUR) is a fundamental principle linking nonequilibrium currents and entropy production.
  • Existing TURs have limitations, requiring specific initial states or infinite-time averages, restricting their practical application.

Purpose of the Study:

  • To derive a generalized finite-time thermodynamic uncertainty relation (TUR) valid for arbitrary initial states.
  • To explore the applicability of the derived TUR to feedback-controlled systems and discrete-time Markov chains.

Main Methods:

  • Derivation of a finite-time TUR using the Cramér-Rao inequality.
  • Analysis of accumulated current variance and its lower bound.
  • Application to feedback control scenarios and discrete-time Markov chains.

Main Results:

  • A finite-time TUR is established, bounding accumulated current variance by the instantaneous current at the final time.
  • The derived TUR successfully explains experimental observations in feedback-controlled processes.
  • A novel TUR for discrete-time Markov chains with nonsteady initial states is derived, offering exponentially improved bounds.

Conclusions:

  • The developed finite-time TUR significantly expands the applicability of TURs in nonequilibrium thermodynamics.
  • The findings provide a theoretical framework for understanding feedback control effects on thermodynamic uncertainty.
  • The results offer substantial improvements for analyzing discrete-time systems, particularly those with nonsteady initial states.