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In this work, we proposed a smooth transition wave equation from a quantum toclassical regime in the framework of von Neumann formalism for ensembles andthen obtained an equivalent scaled equation. This led us to develop a scaledstatistical theory following the well-known Wigner-Moyal approach of quantummechanics. This scaled nonequilibrium statistical mechanics has in it all theingredients of the classical and quantum theory described in terms of acontinuous parameter displaying all the dynamical regimes in between the twoextreme cases. Finally, a simple application of our scaled formalism consistingof reflection from a mirror by computing various quantities, includingprobability density plots, scaled trajectories, and arrival times, wasanalyzed.

Quantum Classical Transition for Mixed States: The Scaled Von Neumann Equation