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The thermodynamic uncertainty relation, a trade-off between system cost and precision, does not hold for underdamped Brownian motion. A novel counterexample demonstrates this violation, challenging previous conjectures.

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Area of Science:

  • Statistical Mechanics
  • Non-equilibrium Thermodynamics
  • Physical Systems

Background:

  • The thermodynamic uncertainty relation establishes a fundamental trade-off between the energetic cost of driving a system and the precision of its output.
  • Previous proofs were limited to discrete systems and overdamped Brownian motion, leaving the validity for underdamped systems (where inertia is significant) an open question.
  • Conjectures based on numerical evidence suggested the relation holds even for underdamped systems.

Purpose of the Study:

  • To investigate the validity of the thermodynamic uncertainty relation for underdamped Brownian motion.
  • To construct a counterexample that disproves the conjecture regarding the universality of the thermodynamic uncertainty relation in underdamped systems.

Main Methods:

  • Development of a discrete, thermodynamically consistent model inspired by a pendulum clock's escapement mechanism.
  • Analysis of an underdamped harmonic oscillator in thermal equilibrium to demonstrate the violation.
  • Simulations of a continuous underdamped system with a potential landscape mimicking an escapement mechanism.

Main Results:

  • The thermodynamic uncertainty relation is disproven for underdamped Brownian motion.
  • Oscillations of an underdamped harmonic oscillator in thermal equilibrium were shown to break the relation.
  • Simulations confirmed the violation in a continuous underdamped system resembling an escapement.

Conclusions:

  • The thermodynamic uncertainty relation is not universally applicable to all autonomous systems, specifically failing in the presence of inertia (underdamped motion).
  • The proposed escapement-inspired model serves as a concrete counterexample, highlighting the limitations of the previously conjectured universality.
  • This finding necessitates a re-evaluation of the thermodynamic uncertainty relation's scope and implications for systems with inertial effects.