Fluctuations of dynamical observables in linear diffusions with time delay: A Riccati-based approach

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.