  We proved when random-variable fluctuations obey the central limit theorem
the equality of the uncertainty relation corresponds to the thermodynamic
equilibrium state. The inequality corresponds to the thermodynamic
non-equilibrium state. The uncertainty relation is a quantum-mechanics
expression of the second law of thermodynamics originated in wave-particle
duality. Formulas of mean square-deviations changes adjusted by random
fluctuations under the minimal uncertainty relation are obtained. Finally, an
assumption is made which is waiting for examination. We except phase
transitions in our discussion.
