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Area of Science:

  • Statistical Physics
  • Non-Markovian Dynamics
  • Stochastic Processes

Background:

  • Understanding fluctuations in dynamical observables of non-Markovian processes is limited.
  • A key challenge is the absence of a theoretical framework for evaluating large deviation functions.

Purpose of the Study:

  • To develop a theoretical framework for analyzing fluctuations in linear diffusions with time delay.
  • To compute generating functions of current-type observables and derive large deviation functions.

Main Methods:

  • Utilized a Markovian embedding procedure to transform the non-Markovian system into an infinite set of coupled differential equations.
  • Solved matrix Riccati differential equations (RDEs) to compute generating functions.
  • Analyzed properties of RDEs and continuous-time algebraic Riccati equations (CAREs).

Main Results:

  • Derived explicit expressions for the scaled cumulant generating function (SCGF), preexponential factors, and the effective process.
  • Identified the generic fixed point for RDE solutions in the long-time limit.
  • Described special behaviors at the limits of the SCGF domain, relating to fluctuation relations for heat and entropy production.

Conclusions:

  • The developed method provides a tractable approach to study fluctuations in non-Markovian processes with time delays.
  • The findings offer new insights into the dynamics of fluctuations and their connection to thermodynamic quantities.
  • This work lays the groundwork for further investigations into large deviation theory in complex systems.