Fractional Feynman-Kac equation for weak ergodicity breaking

Area of Science:

• Statistical Physics
• Anomalous Diffusion
• Stochastic Processes

Background:

• Continuous-time random walk (CTRW) models anomalous subdiffusion with power-law waiting times, leading to subdiffusion (x^2 ~ t^α).
• In closed systems, CTRW exhibits weak ergodicity breaking, where time averages diverge from ensemble averages due to long waiting times.
• The Feynman-Kac equation describes Brownian motion functionals, but extensions are needed for CTRW.

Purpose of the Study:

• To derive forward and backward fractional Feynman-Kac equations for continuous-time random walks (CTRWs) in a binding potential.
• To analyze specific time averages, including occupation fraction and time-averaged position in a harmonic field, using these new equations.
• To investigate the dynamics of weak ergodicity breaking and the convergence of fluctuations.

Main Methods:

• Derivation of novel forward and backward fractional Feynman-Kac equations tailored for CTRW dynamics.
• Application of these equations to calculate probability density functions and moments for time averages in specific potentials.
• Analysis of the asymptotic behavior (t → ∞) of these time averages and their convergence properties.

Main Results:

• The derived fractional Feynman-Kac equations successfully describe functionals of CTRW in a binding potential.
• Both the occupation fraction and time-averaged position are shown to be random variables for long times (t → ∞), except for α = 1.
• The study elucidates the convergence dynamics of fluctuations, providing insights into weak ergodicity breaking.

Conclusions:

• Fractional Feynman-Kac equations provide a powerful tool for analyzing CTRW with complex potentials and ergodicity breaking.
• The results confirm the persistent randomness of time averages in subdiffusive systems, highlighting deviations from standard diffusion.
• This work advances the understanding of anomalous transport phenomena and the fundamental nature of ergodicity in physical systems.