Multidimensional thermodynamic uncertainty relations

TL;DR

This paper generalizes thermodynamic uncertainty relations to multidimensional observables, enabling analysis of high-dimensional systems and providing tighter bounds on heat engine performance and system mobility.

Contribution

It introduces a multidimensional thermodynamic uncertainty relation based on the generalized Cr{á}mer-Rao inequality, extending previous scalar relations.

Findings

Provides a tighter constraint on heat engine performance.

Establishes a connection between uncertainty relations and differential mobility.

Shows that equilibrium fluctuation-dissipation relation is necessary for equality.

Abstract

We extend a class of recently derived thermodynamic uncertainty relations to vector-valued observables. In contrast to the scalar-valued observables examined previously, this multidimensional thermodynamic uncertainty relation provides a natural way to study currents in high-dimensional systems and to obtain relations between different observables. Our proof is based on the generalized Cr{\'a}mer-Rao inequality, which we interpret as a relation between physical observables and the Fisher information. This allows us to develop high-dimensional versions of both the original, steady state uncertainty relation and the more recently obtained generalized uncertainty relation for time-periodic systems. We apply the multidimensional uncertainty relation to obtain a new constraint on the performance of steady-state heat engines, which is tighter than previous bounds and reveals the role of heat-work correlations. As a second application, we show that the uncertainty relation is connected to a bound on the differential mobility. As a result of this connection, we find that a necessary condition for equality in the uncertainty relation is that the system obeys the equilibrium fluctuation-dissipation relation.