Information Geometry of Spatially Periodic Stochastic Systems

Area of Science:

• Statistical Physics
• Nonlinear Dynamics
• Information Geometry

Background:

• Stochastic processes are fundamental in modeling complex systems.
• Understanding the influence of external forces on these processes is crucial.
• Information geometry provides a framework to analyze the distinguishability of states.

Purpose of the Study:

• To investigate the effect of spatially periodic deterministic forces on stochastic process information geometry.
• To analyze the evolution of probability density functions (PDFs) under different force landscapes.
• To determine how information length (L ∞) serves as a diagnostic for underlying forces.

Main Methods:

• Numerical solution of the Fokker-Planck equation.
• Simulation of periodically repeated Gaussian initial conditions.
• Analysis of information length (L ∞) as a function of initial position and force parameters.

Main Results:

• Initial Gaussian peaks maintain shape near equilibrium (x=0) but broaden near unstable points (x=1).
• Information length (L ∞) is smaller for initial positions closer to the unstable point.
• The calculated L ∞ qualitatively mirrors the applied deterministic force, highlighting its diagnostic power.

Conclusions:

• The interplay between initial PDF and deterministic forces significantly shapes the evolution of stochastic processes.
• Information length (L ∞) effectively characterizes the underlying force landscape.
• This study demonstrates the utility of information geometry in understanding complex system dynamics.