Isomorph theory beyond thermal equilibrium

This paper generalizes isomorph theory to systems that are not in thermal equilibrium. The systems are assumed to be R-simple, i.e., have a potential energy that as a function of all particle coordinates $textbf{R}$ obeys the hidden-scale-invariance condition $U(textbf{R}_{rm a})<U(textbf{R}_{rm b})Rightarrow U(lambdatextbf{R}_{rm a})<U(lambdatextbf{R}_{rm b})$. Systemic isomorphs are introduced as lines of constant excess entropy in the phase diagram defined by density and systemic temperature, which is the temperature of the equilibrium state point with average potential energy equal to $U(textbf{R})$. The dynamics is invariant along a systemic isomorph if there is a constant ratio between the systemic and the bath temperature. In thermal equilibrium, the systemic temperature is equal to the bath temperature and the original isomorph formalism is recovered. The new approach rationalizes within a consistent framework previously published observations of isomorph invariance in simulations involving nonlinear steady-state shear flows, zero-temperature plastic flows, and glass-state isomorphs. The paper relates briefly to granular media, physical aging, and active matter. Finally, we discuss the possibility that the energy unit defining reduced quantities should be based on the systemic rather than the bath temperature.