  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 $U(\textbf{R}_{\rm a})<U(\textbf{R}_{\rm
b})\Rightarrow U(\lambda\textbf{R}_{\rm a})<U(\lambda\textbf{R}_{\rm b})$.
"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.
