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Kramers' theory for diffusion on a periodic potential.

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Summary

Kramers' turnover theory is extended to particle motion on periodic potentials. The new model accurately predicts hopping rates and diffusion coefficients, including finite barrier corrections, validated by simulations.

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Area of Science:

  • Chemical Physics
  • Condensed Matter Physics
  • Statistical Mechanics

Background:

  • Kramers' turnover theory is a foundational model for chemical reaction rates in condensed phases.
  • The theory traditionally simplifies the system's dynamics, particularly the bath's influence.
  • Extending the theory to more complex potentials and bath interactions is crucial for accurate predictions.

Purpose of the Study:

  • To extend Kramers' turnover theory (also known as PGH theory) to particle motion on a periodic potential coupled to a dissipative harmonic bath.
  • To incorporate the dynamics of the collective unstable normal mode more accurately.
  • To derive analytical expressions for hopping rates, diffusion coefficients, and finite barrier corrections.

Main Methods:

  • Extension of Kramers' turnover theory by defining a small parameter related to the deviation of the collective bath mode.
  • Utilizing second-order classical perturbation theory.
  • Analytical derivation of hopping rates, diffusion coefficient, and finite barrier corrections.
  • Numerical simulations for validation using a cosine potential, ohmic friction, and varying barrier heights.

Main Results:

  • The effective potential along the unstable normal mode remains periodic with renormalized mass or lattice length.
  • Analytical expressions for hopping rates and diffusion coefficients were derived.
  • Finite barrier corrections to Kramers' turnover theory were successfully obtained.
  • The analytical results show good agreement with numerical simulation data.

Conclusions:

  • The extended theory provides a more accurate description of particle dynamics on periodic potentials with dissipative baths.
  • The method successfully incorporates collective bath mode dynamics and yields corrections for finite barriers.
  • This work offers a refined theoretical framework for understanding transport phenomena in condensed matter systems.