Thermodynamic uncertainty relation for generalized time-reversal observables

Abstract

Time-reversal symmetry plays an essential role in the thermodynamic uncertainty relations, which bound the fluctuations of observables in terms of the associated dissipation. In fact, thermodynamic uncertainty relations are typically derived under the assumption that the observable of interest is antisymmetric under time reversal. This also suggests that existing thermodynamic uncertainty relations are restricted to a limited class of observables. In this paper, we mitigate this restriction by introducing a new class of observables that do not exhibit the exact antisymmetry but change the sign under time reversal. We call it generalized time reversal and derive a broadly applicable thermodynamic uncertainty relation for observables with this condition. The generalization is achieved by direct statistical arguments on the observable distributions and holds for both deterministic and stochastic dynamics. We demonstrate the derived thermodynamic uncertainty relation with observables subjected to rectifications, showing that the precision of generalized observables remains expressible in terms of the dissipative cost. The result extends the scope of thermodynamic uncertainty relations beyond the reach of previous frameworks relying on the exact antisymmetry.