Periodic-orbit determination of dynamical correlations in stochastic processes.

Area of Science:

• Statistical mechanics
• Dynamical systems theory
• Probability theory

Background:

• Markov processes are fundamental in modeling systems with discrete states and transitions.
• Large-deviation theory provides tools to analyze rare events in probabilistic systems.
• Kalman maps offer a related framework for analyzing system dynamics.

Purpose of the Study:

• To establish the equivalence between large-deviation statistics of discrete-time Markov processes and their corresponding Kalman maps.
• To demonstrate the utility of Kalman maps in describing time correlation functions of Markov processes.

Main Methods:

• Mathematical analysis of discrete-time, finite-state Markov processes.
• Comparison of large-deviation statistical quantities.
• Utilizing unstable periodic orbits within Kalman maps.

Main Results:

• Demonstrated complete coincidence between large-deviation statistical quantities of Markov processes and their Kalman map equivalents.
• Showcased that time correlation functions in Markov processes can be accurately described by unstable periodic orbits in Kalman maps.

Conclusions:

• The Kalman map provides a powerful and equivalent framework for understanding the large-deviation statistics of discrete-time Markov processes.
• Unstable periodic orbits within Kalman maps offer a valuable method for characterizing time correlation functions in these systems.