Non-Markovian Brownian dynamics and nonergodicity

Area of Science:

• Physics
• Statistical Mechanics
• Non-equilibrium Systems

Background:

• Brownian motion typically exhibits ergodic behavior, meaning time averages equal ensemble averages.
• Non-Markovian dynamics and generalized Langevin equations (GLEs) introduce memory effects, potentially altering standard physical assumptions.
• Understanding ergodicity breaking is crucial for describing complex systems far from equilibrium.

Purpose of the Study:

• To investigate the breaking of ergodicity in generalized Brownian motion.
• To identify the physical conditions and parameters governing this phenomenon.
• To introduce a quantitative measure for the strength of ergodicity breaking.

Main Methods:

• Modeling non-Markovian dynamics using a generalized Langevin equation (GLE).
• Analyzing the effective friction and its role in ergodicity.
• Introducing and defining a novel parameter 'b' to quantify ergodicity breaking.
• Examining the stationary probability density of the embedded Markovian dynamics.

Main Results:

• Ergodicity breaking was observed in a class of generalized Brownian motion.
• A vanishing effective friction was identified as the origin of this phenomenon.
• A new parameter 'b', derived from the memory friction kernel, effectively measures ergodicity breaking strength.
• Ergodicity breaking correlated with a non-unique stationary probability density.

Conclusions:

• The study demonstrates ergodicity breaking in non-Markovian Brownian motion driven by GLEs.
• The introduced parameter 'b' provides a quantitative tool for assessing ergodicity breaking.
• The findings have implications for understanding diverse physical systems, including free, periodic, and confined potentials.