Thermodynamics of Superdiffusion Generated by Lévy-Wiener Fluctuating Forces

Area of Science:

• Statistical Physics
• Stochastic Processes
• Non-linear Dynamics

Background:

• Scale-free Lévy motion generalizes Wiener processes, modeling natural signals via overdamped Langevin equations.
• These processes are crucial for understanding non-Gaussian noise in complex systems.
• The stationary state of such systems is typically determined by system parameters.

Purpose of the Study:

• To analyze the dynamics of a Brownian-like particle driven by external forces and uncorrelated Lévy fluctuations.
• To investigate the detailed balance condition in Markov processes with stability index α < 2.
• To explore the consequences of non-Gibbsian stationary states on fluctuation-dissipation theorems.

Main Methods:

• Modeling particle dynamics using an overdamped Langevin stochastic differential equation with Lévy noise.
• Analysis of the Markov process properties, including stationary probability density and current.
• Derivation of the fluctuation-dissipation theorem under weak external forcing.

Main Results:

• The analyzed Markov process with stability index α < 2 violates detailed balance.
• A non-vanishing current exists in the stationary state, indicating a non-equilibrium condition.
• The non-Gibbsian nature of the stationary state impacts the fluctuation-dissipation theorem.

Conclusions:

• Systems driven by scale-free Lévy motion can exhibit non-equilibrium stationary states.
• Violation of detailed balance is a key characteristic of these non-Gibbsian states.
• Understanding these states is crucial for accurately describing complex physical phenomena and deriving generalized physical laws.