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Symmetry of deterministic ratchets.

V M Rozenbaum1, I V Shapochkina2, Y Teranishi3

  • 1Chuiko Institute of Surface Chemistry, National Academy of Sciences of Ukraine, Generala Naumova Street 17, Kiev 03164, Ukraine.

Physical Review. E
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This study reveals hidden symmetries in ratchets, explaining conditions for directional motion absence or presence. It details how force symmetry and potential energy structure dictate particle velocity in pulsating and forced ratchets.

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Area of Science:

  • Statistical Mechanics
  • Non-equilibrium Thermodynamics
  • Soft Matter Physics

Background:

  • Brownian motion describes particle movement influenced by random thermal forces.
  • Ratchet mechanisms utilize asymmetric potentials or forces to rectify random motion into directed transport.
  • Understanding particle dynamics in biased and unbiased force fields is crucial for nanoscale transport applications.

Purpose of the Study:

  • To analytically derive the average particle velocity for overdamped Brownian motion in unbiased, periodic force fields.
  • To uncover and prove hidden symmetries governing ratchet behavior.
  • To identify conditions for the absence or presence of the ratchet effect and analyze velocity dependencies on force symmetries.

Main Methods:

  • Derivation of an analytical expression for average particle velocity as a series expansion in the inverse friction coefficient.
  • Analysis of force harmonics' summation indices to reveal ratchet symmetries.
  • Classification of conditions based on spatial and temporal force structures (multiplicative, additive) and potential symmetries (shift, symmetric, antisymmetric, universal).

Main Results:

  • A compact analytical representation for average particle velocity was obtained.
  • Hidden symmetries of ratchets were proven, linked to the symmetry of applied force harmonics.
  • Conditions for absent ratchet effect and velocity being an even/odd functional of the force were established.
  • Fundamental differences in velocity dependence for pulsating and forced ratchets were identified based on potential asymmetry.

Conclusions:

  • The study provides a comprehensive understanding of ratchet dynamics and symmetries in unbiased periodic systems.
  • The findings clarify how different force and potential structures influence directed particle transport.
  • Specific conditions were identified where pulsating ratchets with multiplicative potential energy exhibit restricted directional motion, with exceptions in the inertial regime.