How isolated is enough for an "isolated" system in statistical mechanics?

Abstract: Irreversible processes are frequently adopted to account for the entropy increase in classical thermodynamics. However, the corresponding physical origins are not always clear, e.g. in a free expansion process, a typical model in textbooks. In this letter, we study the entropy change during free expansion for a particle with the thermal de Broglie wavelength ($\lambda_{T}$) in a one-dimensional square trap with size $L$. By solely including quantum dephasing as an irreversible process, we recover classical result of entropy increase in the classical region ($L\gg\lambda_{T}$), while predict prominent discrepancies in the quantum region ($L\ll\lambda_{T}$) because of non-equilibrium feature of trapped atoms after expansion. It is interesting to notice that the dephasing, though absent in classical system, is critical to clarify mysteries in classical thermodynamics.