Chaotic properties of systems with Markov dynamics

Area of Science:

• Statistical Physics
• Dynamical Systems Theory
• Information Theory

Background:

• Continuous-time Markov processes are fundamental in modeling complex systems.
• Calculating dynamic partition functions and associated entropies is crucial for understanding system dynamics.
• Existing methods often struggle with complex interacting systems.

Purpose of the Study:

• To introduce a general computational approach for the dynamic partition function of continuous-time Markov processes.
• To establish a connection between Ruelle topological pressure and large deviation functions.
• To derive and compute the Kolmogorov-Sinai entropy for specific stochastic systems.

Main Methods:

• Developing a general framework for dynamic partition function computation.
• Identifying the Ruelle topological pressure with the large deviation function.
• Constructing the finite Kolmogorov-Sinai entropy for continuous-time Markov processes.
• Applying the method to a symmetric exclusion process and an infinite-range Ising model.

Main Results:

• A general method for computing dynamic partition functions is presented.
• The Ruelle topological pressure is shown to be equivalent to the large deviation function.
• The first finite Kolmogorov-Sinai entropy for these processes is constructed.
• Exact calculations for the topological pressure and entropies of an infinite-range Ising model are provided.

Conclusions:

• The presented approach offers a unified framework for analyzing the statistical properties of continuous-time Markov processes.
• This work provides novel tools for quantifying complexity and information flow in stochastic systems.
• The findings pave the way for deeper insights into the behavior of interacting many-body systems.