Position distribution in a generalized run-and-tumble process.

Area of Science:

• Statistical Mechanics
• Stochastic Processes
• Active Matter Physics

Background:

• The standard run-and-tumble model describes active particle motion in one dimension.
• Active particles exhibit complex dynamics influenced by noise and their environment.
• Generalizing these models is crucial for understanding diverse physical phenomena.

Purpose of the Study:

• To investigate a generalized class of stochastic processes described by dⁿx/dtⁿ = v₀σ(t).
• To analytically determine the behavior of mean-squared displacement and position distribution for all n > 0.
• To identify and characterize a localization transition within this generalized framework.

Main Methods:

• Exact analytical computation of mean-squared displacement and position distribution.
• Extension of the process to non-integer values of n.
• Numerical computation using importance sampling methods for the rate function.
• Calculation of the cumulant-generating function for specific values of n.

Main Results:

• A localization transition occurs at n = 1/2, with distinct behaviors for n > 1/2 (superdiffusion), n < 1/2 (localization), and n = 1/2 (logarithmic growth).
• The position distribution exhibits time-dependent behavior for n ≥ 1/2 and approaches a stationary state for n < 1/2.
• The late-time tails of the position distribution follow a large deviation form with a computable rate function Φn(z).

Conclusions:

• The generalized stochastic process exhibits a tunable localization transition controlled by the parameter n.
• The analytical and numerical results provide a comprehensive understanding of the system's dynamics across different regimes.
• This work offers a versatile framework for studying active matter and related stochastic phenomena.