Descending from infinity: convergence of tailed distributions

Area of Science:

• Probability theory
• Stochastic processes
• Statistical mechanics

Background:

• Long-tailed distributions commonly appear in natural phenomena.
• Understanding their behavior under dynamic processes is crucial.
• Stochastic dynamics often simplify distributions over time.

Purpose of the Study:

• To analyze the relaxation of long-tailed distributions under specific stochastic dynamics.
• To characterize the effects of different relaxation rates on tail behavior.
• To investigate the spreading dynamics of initial delta-function distributions.

Main Methods:

• Mathematical modeling of stochastic processes.
• Analysis of probability distribution functions.
• Derivation of analytical solutions for relaxation dynamics.

Main Results:

• Linear relaxation exponentially suppresses, but does not eliminate, long tails.
• Stronger-than-linear relaxation immediately suppresses long tails.
• Stronger decay can induce transient peaks and dual diffusive spreading regimes from delta-function initial states.

Conclusions:

• The rate of stochastic relaxation critically determines the fate of long-tailed distributions.
• Non-linear relaxation offers a mechanism for rapid tail suppression but introduces complex transient behaviors.
• These findings have implications for modeling systems with heavy tails and non-equilibrium dynamics.