Bridging classical and quantum dynamics with the Wigner–Moyal equation

We present a numerical framework for solving the Wigner–Moyal equation. While Moyal’s form is renowned for its similarity to classical dynamics, it has remained unusable for several decades due to severe numerical instability. This instability arises from the Moyal bracket not being constrained by the uncertainty principle, resulting in unbounded nonlocality. We demonstrate that excessive nonlocality can be suppressed by expanding the observation window to the uncertainty limit, rendering the problem well-posed. Our approach naturally reduces to the Boltzmann equation in regions where quantum effects are negligible; opening a new device simulation methodology that bridges classical and quantum dynamics.