Validity conditions for an exponential escape law in throughflow-driven systems

Abstract

An escape law of the form p_b = exp(−1/χ_b), with χ_b the ratio of the free energy delivered to a system within one relaxation time to the barrier separating it from an adjacent configuration, has been proposed for transitions in systems held away from equilibrium by sustained throughflow. Written that way the delivered energy density occupies the position of kT in the Arrhenius factor, and the law appears to assert that a throughflow ratio can serve as an effective temperature. This note asks what the law actually requires. It requires less than it appears to: the exponential form descends from weak-noise asymptotics rather than from equilibrium rate theory, needs neither a bath nor detailed balance nor a fluctuation–dissipation relation, and holds for non-gradient dynamics and non-Gaussian forcing alike. Under that reading the delivered energy is a noise scale rather than a temperature, the formula is unchanged, and a contested commitment is discharged at no cost. What the reinterpretation relocates rather than removes is the identification of the exponent with the barrier, and four regimes are set out, drawn from the literature on activated escape in driven systems, in which that identification fails. The law is further shown to be incompletely specified: its regime boundaries are set by the correlation time of the drive, which no combination of its present variables recovers, and a second dimensionless group is proposed. No new results about escape rates are presented.