Multidimensional thermodynamic uncertainty relations

By A. Dechant. Journal of Physics A: Mathematical and Theoretical, 2018

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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 Crámer–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.

Subfields: Applied Mathematics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/69d7d763b0367b50574dd58e9566d223dd23b019

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