Free diffusion bounds the precision of currents in underdamped dynamics

Area of Science:

• Physics
• Statistical Mechanics
• Non-equilibrium Thermodynamics

Background:

• The thermodynamic uncertainty relation (TUR) provides a fundamental link between fluctuations and dissipation in thermodynamic processes.
• Generalizing the TUR to underdamped systems, particularly those with inertia, remains a significant challenge in non-equilibrium statistical mechanics.
• Existing bounds for underdamped dynamics lack transparency and do not consistently reduce to the well-established TUR in the overdamped limit.

Purpose of the Study:

• To investigate the validity and generalization of the thermodynamic uncertainty relation (TUR) for underdamped stochastic dynamics.
• To derive a new, transparent bound for one-dimensional driven diffusion in the underdamped regime.
• To examine the applicability of this bound to more complex systems, including higher dimensions and various observables.

Main Methods:

• Analysis of driven free diffusion in the underdamped regime to identify violations of the overdamped TUR.
• Development of a conjectured bound based on numerical evidence for one-dimensional driven diffusion in a potential.
• Investigation of the behavior of the conjectured bound in the overdamped limit and for observables with higher velocity powers.

Main Results:

• Driven free diffusion in the underdamped regime inherently violates the standard TUR for finite times.
• A new conjectured bound for one-dimensional driven diffusion in a potential is proposed, which converges to the overdamped TUR.
• The conjectured bound is shown to be valid for odd velocity-power observables and its applicability to higher dimensions is discussed.

Conclusions:

• The standard thermodynamic uncertainty relation (TUR) requires careful consideration and potential modification for underdamped systems.
• The proposed bound offers a promising generalization of the TUR for one-dimensional underdamped driven diffusion.
• Further research is needed to fully establish the scope and limitations of this new bound in various physical systems.