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Large deviation function for a driven underdamped particle in a periodic potential.

Lukas P Fischer1, Patrick Pietzonka1, Udo Seifert1

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Summary
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We investigate current fluctuations in underdamped Brownian motion using large deviation theory. Our findings provide new bounds and calculation methods for particle current, differing from overdamped dynamics.

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Area of Science:

  • Statistical Mechanics
  • Non-equilibrium Physics
  • Complex Systems

Background:

  • Brownian motion describes random particle movement.
  • Underdamped Brownian motion includes inertial effects, unlike simpler overdamped models.
  • Large deviation theory analyzes rare events in stochastic processes.

Purpose of the Study:

  • To explore current fluctuations in underdamped Brownian motion.
  • To develop and compare methods for calculating the large deviation function of particle current.
  • To establish bounds on the large deviation function and assess their tightness.

Main Methods:

  • Application of large deviation theory to underdamped Brownian motion in a 1D periodic potential.
  • Derivation of a large deviation functional for empirical phase space density.
  • Calculation of the large deviation function using the cumulant generating function.

Main Results:

  • An explicit expression for the large deviation functional of empirical phase space density was derived.
  • Several bounds on the large deviation function of particle current were obtained.
  • The tightness of these bounds was assessed numerically for a cosine potential.

Conclusions:

  • The study presents novel approaches to analyzing current fluctuations in underdamped systems.
  • The derived methods and bounds offer insights into non-equilibrium statistical mechanics.
  • Comparison with overdamped dynamics highlights differences due to inertial effects.