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Area of Science:

  • Statistical Physics
  • Mathematical Physics
  • Stochastic Processes

Background:

  • Run-and-tumble models are crucial for describing particle movement with intermittent stops.
  • Existing kinetic equations for these models have limitations in capturing complex dynamics.
  • Fractional calculus offers a powerful framework for modeling anomalous transport phenomena.

Purpose of the Study:

  • To introduce and analyze a new generalization of kinetic equations for run-and-tumble models.
  • To derive a class of generalized fractional kinetic (GFK) and telegraph-type equations.
  • To explore the connection between these generalized equations and underlying stochastic processes.

Main Methods:

  • Derivation of generalized kinetic equations from run-and-tumble dynamics.
  • Analysis of the resulting GFK and telegraph-type equations.
  • Explicit solution in the Laplace domain.
  • Interpretation of solutions as probability density functions of transformed stochastic processes.

Main Results:

  • A new class of generalized fractional kinetic (GFK) and telegraph-type equations with two or three parameters was established.
  • An explicit expression for the solution in the Laplace domain was derived.
  • The fundamental solution of the GFK equation was linked to the probability density function of a specific stochastic process.
  • Special cases including generalized telegraph models and fractional diffusion equations were discussed.

Conclusions:

  • The introduced generalization provides a flexible framework for modeling complex kinetic phenomena.
  • The GFK equation offers a unified approach to various fractional diffusion and telegraph processes.
  • The connection to stochastic processes deepens the understanding of anomalous transport in physical systems.