Thermodynamic uncertainty relation for energy transport in transient regime -- Model study

TL;DR

This paper examines a transient thermodynamic uncertainty relation (TUR) for energy transport in out-of-equilibrium systems, analyzing exactly solvable models and the influence of particle statistics on TUR bounds.

Contribution

It extends the TUR to transient regimes, explores its validity across different models and statistics, and provides conditions for violations and proofs of bounds in weak-coupling regimes.

Findings

Universal TUR ratio ≥ 2 for all models.

Tighter TUR bound holds for Bose-Einstein statistics.

Violations of the tighter bound occur for Fermi-like and mixed statistics in certain regimes.

Abstract

We investigate transient version of the recently discovered thermodynamic uncertainty relation (TUR) which provides a precision-cost trade-off relation for certain out-of-equilibrium thermodynamic observables in terms of net entropy production. We explore this relation in the context of energy transport in a bipartite setting for three exactly solvable toy model systems (two coupled harmonic oscillators, two coupled qubits and a hybrid coupled oscillator-qubit system) and analyze the role played by the underlying statistics of the transport carriers in TUR. Interestingly, for all these models, depending on the statistics, the TUR ratio can be expressed as a sum or a difference of an universal term which is always greater or equal to 2 and a corresponding entropy production term. We find that the generalized version of the TUR, originating from the universal fluctuation symmetry is always satisfied. However, interestingly, the specialized TUR, a tighter bound, is always satisfied for the coupled harmonic oscillator system obeying Bose-Einstein statistics. Whereas, for both the coupled qubit, obeying Fermi-like statistics and the hybrid qubit-oscillator system with mixed Fermi-Bose statistics, violation of tighter bound is observed in certain parameter regimes. We have provided conditions for such violations. We also provide a rigorous proof following the non-equilibrium Green's function approach that the tighter bound is always satisfied in the weak-coupling regime for generic bipartite systems.