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Operationally accessible uncertainty relations for thermodynamically consistent semi-Markov processes.

Benjamin Ertel1, Jann van der Meer1, Udo Seifert1

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Summary
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Semi-Markov processes, which include temporal memory, are analyzed using a new thermodynamic uncertainty relation. This relation helps distinguish between true semi-Markov processes and those arising from coarse-graining Markov processes.

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Area of Science:

  • Statistical Mechanics
  • Non-equilibrium Thermodynamics
  • Stochastic Processes

Background:

  • Markov processes are fundamental but lack temporal memory.
  • Semi-Markov processes extend Markov processes by incorporating memory effects via a semi-Markov kernel.
  • Thermodynamic consistency in equilibrium requires direction-time independence.

Purpose of the Study:

  • To derive and present an alternative proof for the direction-time independence condition in semi-Markov processes.
  • To establish a thermodynamic uncertainty relation for semi-Markov processes.
  • To develop an inference tool based on a second thermodynamic uncertainty relation for coarse-grained Markov processes.

Main Methods:

  • Recalling path weight definitions for semi-Markov trajectories.
  • Deriving the direction-time independence condition.
  • Proving thermodynamic uncertainty relations for semi-Markov processes, including those emerging from coarse-graining.

Main Results:

  • An alternative derivation of the direction-time independence condition for thermodynamic consistency.
  • A novel thermodynamic uncertainty relation connecting entropy production to steady-state currents (mean and variance).
  • A second thermodynamic uncertainty relation applicable to coarse-grained Markov processes, violation of which indicates non-Markovian origin.

Conclusions:

  • The study provides a robust framework for analyzing memory effects in stochastic processes using thermodynamic principles.
  • The derived uncertainty relations offer quantitative tools for characterizing and distinguishing different types of stochastic processes.
  • The violation of the coarse-graining uncertainty relation serves as a powerful inference method for identifying underlying Markovian dynamics.