  Recent progress in out-of-equilibrium closed quantum systems has
significantly advanced the understanding of mechanisms behind their evolution
towards thermalization. Notably, the concept of non-thermal fixed points
(NTFPs) - responsible for the emergence of spatio-temporal universal scaling in
far-from-equilibrium systems - has played a crucial role in both theoretical
and experimental investigations. In this work, we introduce a differential
equation that has the universal scaling associated with NTFPs as a solution.
The advantage of working with a differential equation, rather than only with
its solution, is that we can extract several insightful properties not
necessarily present in the solution alone. How the differential equation is
derived allows physical interpretation of the universal exponents in terms of
the time dependence of the amplitude of the distributions and their momentum
scaling. Employing two limiting cases of the equation, we determined the
universal exponents related to the scaling using the distributions near just
two momentum values. We established a solid agreement with previous
investigations by validating this approach with three distinct physical
systems. This consistency highlights the universal nature of scaling due to
NTFPs and emphasizes the predictive capabilities of the proposed differential
equation. Moreover, under specific conditions, the equation predicts a
power-law related to the ratio of the two universal exponents, leading to
implications concerning particle and energy transport. This suggests that the
observed power-laws in far-from-equilibrium turbulent fluids could be related
to the universal scaling due to NTFPs, potentially offering new insights into
the study of turbulence.
