Multidimensional thermodynamic uncertainty relations

Abstract

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