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This study analyzes the telegrapher's equation using a master equation on a lattice. It reveals how energy loss and jumping timescales govern information transport and diffusion dynamics in lattices.

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Area of Science:

  • Statistical Mechanics
  • Lattice Models
  • Information Theory

Background:

  • The telegrapher's equation describes wave propagation and diffusion.
  • Master equations model the time evolution of quantum or stochastic systems.
  • Understanding transport dynamics in lattices is crucial for various physical systems.

Purpose of the Study:

  • To present the telegrapher's equation within a non-local-in-time master equation framework on a lattice.
  • To analyze the influence of characteristic timescales on transport phenomena.
  • To investigate the role of energy loss in finite-velocity diffusion.

Main Methods:

  • Exact solution of the transport equation for different hopping models.
  • Analysis of the second moment in an infinite lattice.
  • Study of probability time evolution on a ring.
  • Investigating timescales in the memory kernel of the finite-velocity approach.

Main Results:

  • Demonstrated how energy loss and jumping timescales constrain positive solutions and affect entropy.
  • Characterized the approach to disordered stationary states on a ring.
  • Provided an analytic treatment of the lattice model.
  • Highlighted the functional role of energy loss in diffusion dynamics.

Conclusions:

  • The characteristic timescales are key to understanding constraints and dynamics in lattice transport.
  • The model offers insights into Shannon entropy, information transport, and wave propagation.
  • Energy loss plays a significant role in finite-velocity diffusion processes.