Cramer-Rao Inequality Generalizes the Equilibrium Energy Fluctuation-Response Relation to Nonequilibrium Steady States

Abstract

Equilibrium steady states are fully characterized by the Boltzmann distribution, which depends only on the energy and the temperature. In contrast, nonequilibrium steady states, which describe a wide range of physical and biological phenomena, are generically much more complex, and their stationary distributions can depend on a zoo of kinetic parameters. A central challenge of nonequilibrium statistical mechanics is the identification of universal relations that hold for nonequilibrium steady states despite this complexity. We show here that a form of the Cramer-Rao inequality applies to arbitrary nonequilibrium steady-state distributions and contains no explicit dependence on kinetic parameters. A main original feature in our observation is that this inequality becomes a well-known fluctuation-response relation in equilibrium. The inequality involves three well-defined quantities: the fluctuations of the energy, the response given by the derivative of the average energy with respect to the inverse temperature, and the Fisher information with respect to the inverse temperature. We illustrate this inequality using three representative models: an active particle in a harmonic potential, a simple scheme for kinetic proofreading, and a one-dimensional model for heat conduction between two baths.