I guess my question could be be asked this way:
Is there a thermodynamic quantity that describes how close or far from thermal equilibrium a system is? And if so, how would this quantity relate to the standard deviation of the energy of the systems constituents?
If thermodynamic temperature can be interpreted as the average kinetic energy in a system, is there a quantity defined as the standard deviation?
For example, let's say you poured some hot water into a cup of cold water. The instant you poured it the standard deviation of the system would be...