I'm trying to differentiate a thermal system in local equilibrium (and slow time varying) v/s a non-equilibrium system.
For a thermal system which is slowly time varying, how does one define local thermalization/local equilibrium/local temperature?
-> One thought is take a snapshot of the (energy) distribution of the particles (bosons or fermions) in a small volume of the system. Average it over a time $\Delta t$, where $\Delta t$ > particle mean free path. Try to fit a bose einstein distribution (or fermi dirac) which minimizes the rms error with the actual distribution. This would then give the local temperature.
-> Local thermalization: One can say that the small volume has reached local thermalization if the rms error is < some epsilon.
-> But the problem is that the value of epsilon is subjective (not a well defined number). So, the concept of local thermalization/temperature seems not an extremely well-defined concept.
Is there a more formal and quantitative definition of a thermal system being in "local equilibrium" instead of a "non-equilibrium" system?